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After insertion of these three formulas into (22)–(2.3),
the derivatives of Rnm≅xmFsuperscriptsubscript𝑅𝑛𝑚superscript𝑥𝑚𝐹R_{n}^{m}\cong x^{m}Fitalic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ≅ italic_x start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ita... | Rnm′′/Rnm′superscriptsuperscriptsubscript𝑅𝑛𝑚′′superscriptsuperscriptsubscript𝑅𝑛𝑚′{R_{n}^{m}}^{\prime\prime}/{R_{n}^{m}}^{\prime}italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT / italic_R start_POSTSUBSCRIPT it... | Rnm′′(x)superscriptsuperscriptsubscript𝑅𝑛𝑚′′𝑥\displaystyle{R_{n}^{m}}^{\prime\prime}(x)italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x )
≅\displaystyle\cong≅ | Rnm′′′(x)superscriptsuperscriptsubscript𝑅𝑛𝑚′′′𝑥\displaystyle{R_{n}^{m}}^{\prime\prime\prime}(x)italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT ( italic_x )
≅\displaystyle\cong≅ | Rnm′(x)superscriptsuperscriptsubscript𝑅𝑛𝑚′𝑥\displaystyle{R_{n}^{m}}^{\prime}(x)italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x )
≅\displaystyle\cong≅ | D |
This is achieved by using specific upper and lower triangular transvections to avoid using a discrete logarithm oracle. Building on Lemma 3.2 we construct transvections which are upper triangular matrices.
Here, as per Section 3.1, ω𝜔\omegaitalic_ω denotes a primitive element of 𝔽qsubscript𝔽𝑞\mathbb{F}_{q}blackboar... | Using the row operations, one can reduce g𝑔gitalic_g to a matrix with exactly one nonzero entry in its d𝑑ditalic_dth column, say in row r𝑟ritalic_r.
Then the elementary column operations can be used to reduce the other entries in row r𝑟ritalic_r to zero. | Let i∈{1,…,d−1}𝑖1…𝑑1i\in\{1,\dotsc,d-1\}italic_i ∈ { 1 , … , italic_d - 1 }. Getting the diagonal entry of hℎhitalic_h at position (i,i)𝑖𝑖(i,i)( italic_i , italic_i ) to 1111 requires the following number of operations. We start by adding the column i+1𝑖1i+1italic_i + 1 to column i𝑖iitalic_i as in Line 5. We alre... |
The idea is to eliminate all other entries in the c𝑐citalic_cth column, namely to apply elementary row operations to make the entries in rows i=r+1,…,d𝑖𝑟1…𝑑i=r+1,\ldots,ditalic_i = italic_r + 1 , … , italic_d of column c𝑐citalic_c equal to zero. Specifically, g𝑔gitalic_g is multiplied on the left by the transvec... |
The key idea is to transform the diagonal matrix with the help of row and column operations into the identity matrix in a way similar to an algorithm to compute the elementary divisors of an integer matrix, as described for example in [23, Chapter 7, Section 3]. Note that row and column operations are effected by left... | D |
where Ω⊂ℝdΩsuperscriptℝ𝑑\Omega\subset\mathbb{R}^{d}roman_Ω ⊂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT with d=2𝑑2d=2italic_d = 2 or 3333 for simplicity, and is an open bounded domain with polyhedral boundary ∂ΩΩ\partial\Omega∂ roman_Ω, the symmetric tensor 𝒜∈[L∞(Ω)]symd×d𝒜superscriptsubscrip... | One difficulty that hinders the development of efficient methods is the presence of high-contrast coefficients [MR3800035, MR2684351, MR2753343, MR3704855, MR3225627, MR2861254]. When LOD or VMS methods are considered, high-contrast coefficients might slow down the exponential decay of the solutions, making the method ... |
As in many multiscale methods previously considered, our starting point is the decomposition of the solution space into fine and coarse spaces that are adapted to the problem of interest. The exact definition of some basis functions requires solving global problems, but, based on decaying properties, only local comput... | It is hard to approximate such problem in its full generality using numerical methods, in particular because of the low regularity of the solution and its multiscale behavior. Most convergent proofs either assume extra regularity or special properties of the coefficients [AHPV, MR3050916, MR2306414, MR1286212, babuos85... | In [MR2718268] is shown that the number of eigenvalues that are very large is related to the number of connected sub-regions on τ¯∪τ¯′¯𝜏superscript¯𝜏′\bar{\tau}\cup{\bar{\tau}}^{\prime}over¯ start_ARG italic_τ end_ARG ∪ over¯ start_ARG italic_τ end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT with large coefficien... | C |
Similarly, from a P𝑃Pitalic_P-stable triangle A′B′C′superscript𝐴′superscript𝐵′superscript𝐶′A^{\prime}B^{\prime}C^{\prime}italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, we can also construct △ABC△𝐴�... | Our algorithm given in section 4 (denoted by Alg-One) is different from Alg-DS.
First, step 1 of Alg-One sets the initial value of (r,s,t)𝑟𝑠𝑡(r,s,t)( italic_r , italic_s , italic_t ) differently from the initial value (1,2,3)123(1,2,3)( 1 , 2 , 3 ) used by Alg-DS. |
Our experiment shows that the running time of Alg-A is roughly one eighth of the running time of Alg-K, or one tenth of the running time of Alg-CM. (Moreover, the number of iterations required by Alg-CM and Alg-K is roughly 4.67 times that of Alg-A.) | Comparing the description of the main part of Alg-A (the 7 lines in Algorithm 1) with that of Alg-CM (pages 9–10 of [8]),
Alg-A is conceptually simpler. Alg-CM is claimed “involved” by its authors as it contains complicated subroutines for handling many subcases. | Alg-A computes at most n𝑛nitalic_n candidate triangles (proof is trivial and omitted) whereas Alg-CM computes at most 5n5𝑛5n5 italic_n triangles (proved in [8]) and so as Alg-K.
(by experiment, Alg-CM and Alg-K have to compute roughly 4.66n4.66𝑛4.66n4.66 italic_n candidate triangles.) | B |
Most relevant for our work is the work presented in [20], where a time series model to capture the time-based variation of social-content features is used. We build upon the idea of their Series-Time Structure, when building our approach for early rumor detection with our extended dataset, and we provide a deep analys... | In the lower part of the pipeline, we extract features from tweets and combine them with the creditscore to construct the feature vector in a time series structure called Dynamic Series Time Model. These feature vectors are used to train the classifier for rumor vs. (non-rumor) news classification.
|
We observe that at certain points in time, the volume of rumor-related tweets (for sub-events) in the event stream surges. This can lead to false positives for techniques that model events as the aggregation of all tweet contents; that is undesired at critical moments. We trade-off this by debunking at single tweet le... | . As shown in Table 5, CreditScore is the best feature in overall. In Figure 4 we show the result of models learned with the full feature set with and without CreditScore. Overall, adding CreditScore improves the performance, especially for the first 8-10 hours. The performance of all-but-CreditScore jiggles a bit afte... | The processing pipeline of our classification approach is shown in Figure 2. In the first step, relevant tweets for an event are gathered. Subsequently, in the upper part of the pipeline,
we predict tweet credibilty with our pre-trained credibility model and aggregate the prediction probabilities on single tweets (Cred... | D |
The convergence of the direction of gradient descent updates to the maximum L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT margin solution, however is very slow compared to the convergence of training loss, which explains why it is worthwhile
continuing to optimize long after we have zero training ... | decreasing loss, as well as for multi-class classification with cross-entropy loss. Notably, even though the logistic loss and the exp-loss behave very different on non-separable problems, they exhibit the same behaviour for separable problems. This implies that the non-tail
part does not affect the bias. The bias is a... | We should not rely on plateauing of the training loss or on the loss (logistic or exp or cross-entropy) evaluated on a validation data, as measures to decide when to stop. Instead, we should look at the 00–1111 error on the validation dataset. We might improve the validation and test errors even when when the decrease ... | The follow-up paper (Gunasekar et al., 2018) studied this same problem with exponential loss instead of squared loss. Under additional assumptions on the asymptotic convergence of update directions and gradient directions, they were able to relate the direction of gradient descent iterates on the factorized parameteriz... | Let ℓℓ\ellroman_ℓ be the logistic loss, and 𝒱𝒱\mathcal{V}caligraphic_V be an independent validation set, for which ∃𝐱∈𝒱𝐱𝒱\exists\mathbf{x}\in\mathcal{V}∃ bold_x ∈ caligraphic_V such that 𝐱⊤𝐰^<0superscript𝐱top^𝐰0\mathbf{x}^{\top}\hat{\mathbf{w}}<0bold_x start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over^ start_... | B |
Text Features are derived from a tweet’s text content. We consider 16 text features including lengthOftweet and smile (contain :−>,:−),;−>,;−)..):->,:-),;->,;-)..): - > , : - ) , ; - > , ; - ) . . ), sad, exclamation, I-you-heshe (contain first, second, third pronouns). In addition, we use the natural language Toolkit ... | The performance of user features is similar with the Twitter features, they are both quite stable from the first hour to the last hour. As shown in Table 9, the best feature over 48 hours of the user feature group is UserTweetsPerDays and it is the best feature overall in the first 4 hours, but its rank decreases with ... | For analysing the employed features, we rank them by importances using RF (see 4). The best feature is related to sentiment polarity scores. There is a big bias between the sentiment associated to rumors and the sentiment associated to real events in relevant tweets. In specific, the average polarity score of news even... | In this section, we compare the performance our model with the human rumor debunking websites: snopes.com and urbanlegend.com. Snopes has their own Twitter account141414https://twitter.com/snopes. They regularly post tweets via this account about rumors which they collected and verified. We consider the creation time o... | Twitter Features refer to basic Twitter features, such as hashtags, mentions, retweets. In addition, we derive three more URL-based features. The first is the WOT–trustworthy-based– score which is crawled from the APIs of WOT.com555https://www.mywot.com/en/api. The second is domain categories which we have collected fr... | D |
\mathcal{C}_{k}|a,t)\sum\limits_{l=1}^{m}P(\mathcal{T}_{l}|a,t,\mathcal{C}_{k}%
)\hat{y_{a}},y_{a})sansserif_f start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = roman_arg roman_min start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT ∀ italic_a end_POSTSUBSCRIPT caligraphic_L ( ∑ start_POSTSUBSCRIPT italic_... | We further investigate the identification of event time, that is learned on top of the event-type classification. For the gold labels, we gather from the studied times with regards to the event times that is previously mentioned. We compare the result of the cascaded model with non-cascaded logistic regression. The res... | Learning a single model for ranking event entity aspects is not effective due to the dynamic nature of a real-world event driven by a great variety of multiple factors. We address two major factors that are assumed to have the most influence on the dynamics of events at aspect-level, i.e., time and event type. Thus, we... | Multi-Criteria Learning. Our task is to minimize the global relevance loss function, which evaluates the overall training error, instead of assuming the independent loss function, that does not consider the correlation and overlap between models. We adapted the L2R RankSVM [12]. The goal of RankSVM is learning a linear... | For this part, we first focus on evaluating the performance of single L2R models that are learned from the pre-selected time (before, during and after) and types (Breaking and Anticipate) set of entity-bearing queries. This allows us to evaluate the feature performance i.e., salience and timeliness, with time and type ... | C |
\right)\;.\\
\end{cases}where { start_ROW start_CELL roman_Θ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a end_POSTSUBSCRIPT = [ italic_θ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_a end_POSTSUBSCRIPT... | More broadly, one can establish uniform-in-time convergence for path functionals that depend only on recent states,
as the Monte Carlo error of pM(θt−τ:t|ℋ1:t)subscript𝑝𝑀conditionalsubscript𝜃:𝑡𝜏𝑡subscriptℋ:1𝑡p_{M}(\theta_{t-\tau:t}|\mathcal{H}_{1:t})italic_p start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( ital... | The use of SMC in the context of bandit problems was previously considered for probit [Cherkassky and Bornn, 2013] and softmax [Urteaga and Wiggins, 2018c] reward models,
and to update latent feature posteriors in a probabilistic matrix factorization model [Kawale et al., 2015]. | the combination of Bayesian neural networks with approximate inference has also been investigated.
Variational methods, stochastic mini-batches, and Monte Carlo techniques have been studied for uncertainty estimation of reward posteriors of these models [Blundell et al., 2015; Kingma et al., 2015; Osband et al., 2016; ... | —i.e., the dependence on past samples decays exponentially, and is negligible after a certain lag—
one can establish uniform-in-time convergence of SMC methods for functions that depend only on recent states, see [Kantas et al., 2015] and references therein. | D |
These are also the patients who log glucose most often, 5 to 7 times per day on average compared to 2-4 times for the other patients.
For patients with 3-4 measurements per day (patients 8, 10, 11, 14, and 17) at least a part of the glucose measuremtents after the meals is within this range, while patient 12 has only t... | Table 2 gives an overview of the number of different measurements that are available for each patient.111For patient 9, no data is available.
The study duration varies among the patients, ranging from 18 days, for patient 8, to 33 days, for patient 14. | In order to have a broad overview of different patients’ patterns over the one month period, we first show the figures illustrating measurements aggregated by days-in-week.
For consistency, we only consider the data recorded from 01/03/17 to 31/03/17 where the observations are most stable. | For time delays between carb entries and the next glucose measurements we distinguish cases where glucose was measured at most 30 minutes before logging the meal, to account for cases where multiple measurements are made for one meal – in such cases it might not make sense to predict the glucose directly after the meal... | The insulin intakes tend to be more in the evening, when basal insulin is used by most of the patients. The only difference happens to patient 10 and 12 whose intakes are earlier at day.
Further, patient 12 takse approx. 3 times the average insulin dose of others in the morning. | B |
In this work, we laid out three convolutional layers with kernel sizes of 3×3333\times 33 × 3 and dilation rates of 4, 8, and 12 in parallel, together with a 1×1111\times 11 × 1 convolutional layer that could not learn new spatial dependencies but nonlinearly combined existing feature maps. Image-level context was rep... | To restore the original image resolution, extracted features were processed by a series of convolutional and upsampling layers. Previous work on saliency prediction has commonly utilized bilinear interpolation for that task Cornia et al. (2018); Liu and Han (2018), but we argue that a carefully chosen decoder architect... | To quantify the contribution of multi-scale contextual information to the overall performance, we conducted a model ablation analysis. A baseline architecture without the ASPP module was constructed by replacing the five parallel convolutional layers with a single 3×3333\times 33 × 3 convolutional operation that result... | Image-to-image learning problems require the preservation of spatial features throughout the whole processing stream. As a consequence, our network does not include any fully-connected layers and reduces the number of downsampling operations inherent to classification models. We adapted the popular VGG16 architecture S... |
In this work, we laid out three convolutional layers with kernel sizes of 3×3333\times 33 × 3 and dilation rates of 4, 8, and 12 in parallel, together with a 1×1111\times 11 × 1 convolutional layer that could not learn new spatial dependencies but nonlinearly combined existing feature maps. Image-level context was rep... | A |
loc(Zi)=|Zi|+14=2i−2locsubscript𝑍𝑖subscript𝑍𝑖14superscript2𝑖2\operatorname{\textsf{loc}}(Z_{i})=\frac{|Z_{i}|+1}{4}=2^{i-2}loc ( italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = divide start_ARG | italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | + 1 end_ARG start_ARG 4 end_ARG = 2 start_POSTS... | It is easy to see that 1111-locality implies some sort of palindromic structure of a word. For example, palindromes like the English words radar, refer and rotator are obviously 1111-local, while the palindrome 𝚊𝚋𝚊𝚋𝚊𝚋𝚊𝚊𝚋𝚊𝚋𝚊𝚋𝚊\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}typewriter_... |
Notice that both Zimin words and 1111-local words have an obvious palindromic structure. However, in the Zimin words, the letters occur multiple times, but not in large blocks, while in 1111-local words there are at most 2222 blocks of each letter. With respect to palindromes, we can show the following general result ... | Regarding the locality of Zisubscript𝑍𝑖Z_{i}italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, note that marking x2subscript𝑥2x_{2}italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT leads to 2i−2superscript2𝑖22^{i-2}2 start_POSTSUPERSCRIPT italic_i - 2 end_POSTSUPERSCRIPT marked blocks; further, marking x1subsc... |
Observation 2.1 justifies that in the following, we are only concerned with condensed words (and therefore words with at most 2loc(α)2loc𝛼2\operatorname{\textsf{loc}}(\alpha)2 loc ( italic_α ) occurrences per symbol and total length of at most |X|2loc(α)𝑋2loc𝛼|X|2\operatorname{\textsf{loc}}(\alpha)| italic_X |... | B |
It consists of a three layer CNN that extracts cardiac representations, then two parallel LSTM-based RNNs for modeling the temporal dynamics of cardiac sequences.
Finally, there is a Bayesian framework capable of learning multitask relationships and a softmax classifier for classification. | Zubair et al.[75] detected the R-peak using a non-linear transformation and formed a beat segment around it.
Then, they used the segments to train a three layer 1D CNN with variable learning rate depending on the mean square error and achieved better results than previous state-of-the-art. | In[148] the authors adopt a 3D multi-scale CNN to identify pixels that belong to the RV.
The network has two convolutional pathways and their inputs are centered at the same image location, but the second segment is extracted from a down-sampled version of the image. | They used multi-scale discrete WT to facilitate the extraction of MI features at specific frequency resolutions and softmax regression to build a multi-class classifier based on the learned features.
Their validation experiments show that their method performed better than previous methods in terms of sensitivity and s... | Extensive comparisons with the state-of-the-art show the effectiveness of this method in terms of mean absolute error.
In[161] the authors created an unsupervised cardiac image representation learning method using multi-scale convolutional RBM and a direct bi-ventricular volume estimation using RF. | D |
In search for an effective world model we experimented with various architectures, both new and modified versions of existing ones. This search resulted in a novel stochastic video prediction model (visualized in Figure 2) which achieved superior results compared to other previously proposed models. In this section, we... | In our experiments, we varied details of the architecture above. In most cases, we use a stack of four convolutional layers with 64646464 filters followed by three dense layers (the first two have 1024102410241024 neurons). The dense layers are concatenated with 64646464 dimensional vector with a learnable action embed... | Our basic architecture, presented as part of Figure 2, resembles the convolutional feedforward network from Oh et al. (2015). The input X𝑋Xitalic_X consists of four consecutive game frames and an action a𝑎aitalic_a. Stacked convolution layers process the visual input. The actions are one-hot-encoded and embedded in a... | A stochastic model can be used to deal with limited horizon of past observed frames as well as sprites occlusion and flickering which results to higher quality predictions. Inspired by Babaeizadeh et al. (2017a), we tried a variational autoencoder (Kingma & Welling, 2014) to model the stochasticity of the environment. ... | Figure 2: Architecture of the proposed stochastic model with discrete latent. The input to the model is four stacked frames (as well as the action selected by the agent) while the output is the next predicted frame and expected reward. Input pixels and action are embedded using fully connected layers, and there is per-... | B |
One common approach that previous studies have used for classifying EEG signals was feature extraction from the frequency and time-frequency domains utilizing the theory behind EEG band frequencies [8]: delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–20 Hz) and gamma (20–64 Hz).
Truong et al. [9] used Short... | For the CNN modules with one and two layers, xisubscript𝑥𝑖x_{i}italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is converted to an image using learnable parameters instead of some static procedure.
The one layer module consists of one 1D convolutional layer (kernel sizes of 3333 with 8888 channels). | Zhang et al. [11] trained an ensemble of CNNs containing two to ten layers using STFT features extracted from EEG band frequencies for mental workload classification.
Giri et al. [12] extracted statistical and information measures from frequency domain to train an 1D CNN with two layers to identify ischemic stroke. | One common approach that previous studies have used for classifying EEG signals was feature extraction from the frequency and time-frequency domains utilizing the theory behind EEG band frequencies [8]: delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–20 Hz) and gamma (20–64 Hz).
Truong et al. [9] used Short... | Architectures of all bdsubscript𝑏𝑑b_{d}italic_b start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT remained the same, except for the number of the output nodes of the last linear layer which was set to five to correspond to the number of classes of D𝐷Ditalic_D.
An example of the respective outputs of some of the m𝑚mita... | B |
In this equation, T𝑇Titalic_T is the time taken to negotiate, Eisubscript𝐸𝑖E_{i}italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the energy used by the actuated joint i𝑖iitalic_i, n𝑛nitalic_n denotes the actuated joint number, and dt𝑑𝑡dtitalic_d italic_t signifies the time step.
| In this section, we explore the autonomous locomotion mode transition of the Cricket robot. We present our hierarchical control design, which is simulated in a hybrid environment comprising MATLAB and CoppeliaSim. This design facilitates the decision-making process when transitioning between the robot’s rolling and wal... | Figure 11: The Cricket robot tackles a step of height 2h, beginning in rolling locomotion mode and transitioning to walking locomotion mode using the rear body climbing gait. The red line in the plot shows that the robot tackled the step in rolling locomotion mode until the online accumulated energy consumption of the ... |
Fig. 7 illustrates the hierarchical control design for the autonomous locomotion mode transition. The decision-making process for this transition is accomplished in MATLAB, whereas the control of each separate locomotion mode is enacted in CoppeliaSim. The connection between MATLAB and the physical robot model in Copp... | Similarly, when the robot encountered a step with a height of 3h (as shown in Fig. 12), the mode transition was activated when the energy consumption of the rear track negotiation in rolling mode surpassed the threshold value derived from the previously assessed energy results of the rear body climbing gait. The result... | A |
Under the current models, the advice bits can encode any information about the input sequence; indeed, defining the “right” information to be conveyed to the algorithm plays an important role in obtaining better online algorithms. Clearly, the performance of the online algorithm can only improve with larger number of ... | We introduced a new model in the study of online algorithms with advice, in which the online algorithm can leverage information about the request sequence that is not necessarily foolproof. Motivated by advances in learning-online algorithms, we studied tradeoffs between the trusted and untrusted competitive ratio, as ... |
All the above results pertain to deterministic online algorithms. In Section 6, we study the power of randomization in online computation with untrusted advice. First, we show that the randomized algorithm of Purohit et al. [29] for the ski rental problem Pareto-dominates any deterministic algorithm, even when the lat... | The above observations were recently made in the context of online algorithms with machine-learned predictions.
Lykouris and Vassilvitskii [24] and Purohit et al. [29] show how to use predictors to design and analyze algorithms with two properties: (i) if the predictor is good, then the online algorithm should perform ... | As argued in detail in [9], there are compelling reasons to study the advice complexity of online computation.
Lower bounds establish strict limitations on the power of any online algorithm; there are strong connections between randomized online algorithms and online algorithms with advice (see, e.g., [27]); online alg... | D |
It is important to note that, as it is described in Section 2.2 of [Losada & Crestani, 2016], to construct the depression group, authors first collected users by doing specific searches on Reddit (e.g. “I was diagnosed with depression”) to obtain self-expressions of depression diagnoses, and then they manually reviewed... | In this pilot task, classifiers must decide, as early as possible, whether each user is depressed or not based on his/her writings.
In order to accomplish this, during the test stage and in accordance with the pilot task definition, the subject’s writings were divided into 10 chunks —thus each chunk contained 10% of th... | As said earlier, each chunk contained 10% of the subject’s writing history, a value that for some subjects could be just a single post while for others hundreds or even thousands of them. Furthermore, the use of chunks assumes we know in advance all subject’s posts, which is not the case in real life scenarios, in whic... | Another important aspect of this incremental approach is that since this confidence vector is a value that “summarizes the past history”, keeping track of how this vector changes over time should allow us to derive simple and clear rules to decide when the system should make an early classification. As an example of th... | In the first one, we performed experiments in accordance with the original eRisk pilot task definition, using the described chunks.
However, since this definition assumes, by using chunks, that the total number of user’s writings is known in advance191919Which is not true when working with a dynamic environment, such a... | A |
Considering the computation overhead of the top-s𝑠sitalic_s selection, we use an approximate way to implement it (Lin et al., 2018): given a vector 𝐚∈ℝd𝐚superscriptℝ𝑑{\bf a}\in{\mathbb{R}}^{d}bold_a ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, we first randomly choose a subset S⊂[d]𝑆delimited... |
In this paper, we propose a novel method, called global momentum compression (GMC), for sparse communication in distributed learning. To the best of our knowledge, this is the first work that introduces global momentum for sparse communication in DMSGD. Furthermore, to enhance the convergence performance when using mo... | We use the CIFAR10 and CIFAR100 datasets under both IID and non-IID data distribution. For the IID scenario, the training data is randomly assigned to each worker. For the non-IID scenario, we use Dirichlet distribution with parameter 0.1 to partition the training data as in (Hsu et al., 2019; Lin et al., 2021).
We ado... |
Table 1 shows the empirical results of different methods under IID data distribution. Figure 3 shows the training curves under IID data distribution. We can observe that each method achieves comparable RCC. As for test accuracy, GMC and DGC (w/ mfm) exhibit comparable performance and outperform the other three methods... | Table 2 and Figure 4 show the performance under non-IID data distribution. We can find that GMC can achieve much better test accuracy and faster convergence speed compared to other methods. Furthermore, we can find that the momentum factor masking trick will severely impair the performance of DGC under non-IID data dis... | C |
Although ReLU creates exact zeros (unlike its predecessors sigmoid𝑠𝑖𝑔𝑚𝑜𝑖𝑑sigmoiditalic_s italic_i italic_g italic_m italic_o italic_i italic_d and tanh\tanhroman_tanh), its activation map consists of sparsely separated but still dense areas (Fig. 1LABEL:sub@subfig:relu) instead of sparse spikes.
The same a... | Recently, in k𝑘kitalic_k-Sparse Autoencoders [21] the authors used an activation function that applies thresholding until the k𝑘kitalic_k most active activations remain, however this non-linearity covers a limited area of the activation map by creating sparsely disconnected dense areas (Fig. 1LABEL:sub@subfig:topkabs... | Although ReLU creates exact zeros (unlike its predecessors sigmoid𝑠𝑖𝑔𝑚𝑜𝑖𝑑sigmoiditalic_s italic_i italic_g italic_m italic_o italic_i italic_d and tanh\tanhroman_tanh), its activation map consists of sparsely separated but still dense areas (Fig. 1LABEL:sub@subfig:relu) instead of sparse spikes.
The same a... |
The three separate clusters which are depicted in Fig. 3 and the aggregated density plot in Fig. 4LABEL:sub@subfig:crrl_density_plot between the Identity activation function, the ReLU and the rest show the effect of a sparser activation function on the representation. | ϕ=ReLU(s)italic-ϕ𝑅𝑒𝐿𝑈𝑠\phi=ReLU(s)italic_ϕ = italic_R italic_e italic_L italic_U ( italic_s ).
The ReLU activation function produces sparsely disconnected but internally dense areas as shown in Fig. 1LABEL:sub@subfig:relu instead of sparse spikes. | A |
A new algorithm which can learn from previous experiences is required, and the algorithm with faster learning speed is more desirable. Existing algorithms’ learning method is learning by prediction. It means that UAV knows current strategies with corresponding payoff and it can randomly select another strategy and calc... |
Compared with other algorithms, novel algorithm SPBLLA has more advantages in learning rate. Various algorithms have been employed in the UAV networks in search of the optimal channel selection [31][29], such as stochastic learning algorithm [30]. The most widely seen algorithm–LLA is an ideal method for NE approachin... | In the literatures, most works search PSNE by using the Binary Log-linear Learning Algorithm (BLLA). However, there are limitations of this algorithm. In BLLA, each UAV can calculate and predict its utility for any si∈Sisubscript𝑠𝑖subscript𝑆𝑖s_{i}\in S_{i}italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ it... |
The learning rate of the extant algorithm is also not desirable [13]. Recently, a new fast algorithm called binary log-linear learning algorithm (BLLA) has been proposed by [14]. However, in this algorithm, only one UAV is allowed to change strategy in one iteration based on current game state, and then another UAV ch... | A new algorithm which can learn from previous experiences is required, and the algorithm with faster learning speed is more desirable. Existing algorithms’ learning method is learning by prediction. It means that UAV knows current strategies with corresponding payoff and it can randomly select another strategy and calc... | C |
μ𝜇\muitalic_μ, in the viscous terms in the momentum equation (see equation
3.18), as μnum=min0νnumsubscript𝜇𝑛𝑢𝑚subscript𝑚𝑖subscript𝑛0subscript𝜈𝑛𝑢𝑚\mu_{num}=m_{i}n_{0}\,\nu_{num}italic_μ start_POSTSUBSCRIPT italic_n italic_u italic_m end_POSTSUBSCRIPT = italic_m start_POSTSUBSCRIPT italic_i end_POSTSUB... | μ𝜇\muitalic_μ, in the viscous terms in the momentum equation (see equation
3.18), as μnum=min0νnumsubscript𝜇𝑛𝑢𝑚subscript𝑚𝑖subscript𝑛0subscript𝜈𝑛𝑢𝑚\mu_{num}=m_{i}n_{0}\,\nu_{num}italic_μ start_POSTSUBSCRIPT italic_n italic_u italic_m end_POSTSUBSCRIPT = italic_m start_POSTSUBSCRIPT italic_i end_POSTSUB... | a reduced value μphys=min0νphyssubscript𝜇𝑝ℎ𝑦𝑠subscript𝑚𝑖subscript𝑛0subscript𝜈𝑝ℎ𝑦𝑠\mu_{phys}=m_{i}n_{0}\,\nu_{phys}italic_μ start_POSTSUBSCRIPT italic_p italic_h italic_y italic_s end_POSTSUBSCRIPT = italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRI... | for νnumsubscript𝜈𝑛𝑢𝑚\nu_{num}italic_ν start_POSTSUBSCRIPT italic_n italic_u italic_m end_POSTSUBSCRIPT and νphyssubscript𝜈𝑝ℎ𝑦𝑠\nu_{phys}italic_ν start_POSTSUBSCRIPT italic_p italic_h italic_y italic_s end_POSTSUBSCRIPT would be avoided because it breaks
the conservation of system energy pertaining to the ... | νnum=700subscript𝜈𝑛𝑢𝑚700\nu_{num}=700italic_ν start_POSTSUBSCRIPT italic_n italic_u italic_m end_POSTSUBSCRIPT = 700m2/s and νphys=410subscript𝜈𝑝ℎ𝑦𝑠410\nu_{phys}=410italic_ν start_POSTSUBSCRIPT italic_p italic_h italic_y italic_s end_POSTSUBSCRIPT = 410m2/s: νnumsubscript𝜈𝑛𝑢𝑚\nu_{num}italic_ν start_P... | B |
Let r𝑟ritalic_r be the relation on 𝒞Rsubscript𝒞𝑅\mathcal{C}_{R}caligraphic_C start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT given to the left of Figure 12.
Its abstract lattice ℒrsubscriptℒ𝑟\mathcal{L}_{r}caligraphic_L start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT is represented to the right. | If no confusion is possible, the subscript R𝑅Ritalic_R will be omitted, i.e., we will use
≤,∧,∨\leq,\operatorname{\land},\operatorname{\lor}≤ , ∧ , ∨ instead of ≤R,∧R,∨Rsubscript𝑅subscript𝑅subscript𝑅\leq_{R},\operatorname{\land}_{R},\operatorname{\lor}_{R}≤ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT , ∧ start_P... | The tuples t1subscript𝑡1t_{1}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, t4subscript𝑡4t_{4}italic_t start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT represent a counter-example to BC→A𝐵𝐶→𝐴BC\operatorname{\rightarrow}Aitalic_B italic_C → italic_A for g1subscript𝑔1g_{1}italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRI... | For convenience we give in Table 7 the list of all possible realities
along with the abstract tuples which will be interpreted as counter-examples to A→B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B or B→A𝐵→𝐴B\operatorname{\rightarrow}Aitalic_B → italic_A. | First, remark that both A→B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B and B→A𝐵→𝐴B\operatorname{\rightarrow}Aitalic_B → italic_A are possible.
Indeed, if we set g=⟨b,a⟩𝑔𝑏𝑎g=\langle b,a\rangleitalic_g = ⟨ italic_b , italic_a ⟩ or g=⟨a,1⟩𝑔𝑎1g=\langle a,1\rangleitalic_g = ⟨ italic_a , 1 ⟩, then r⊧gA→... | C |
Q∗(s,a)=maxπQπ(s,a)superscript𝑄𝑠𝑎𝑚𝑎subscript𝑥𝜋superscript𝑄𝜋𝑠𝑎Q^{*}(s,a)=max_{\pi}Q^{\pi}(s,a)italic_Q start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_s , italic_a ) = italic_m italic_a italic_x start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_π end_POSTSUPERSCRI... |
To evaluate the Dropout-DQN, we employ the standard reinforcement learning (RL) methodology, where the performance of the agent is assessed at the conclusion of the training epochs. Thus we ran ten consecutive learning trails and averaged them. We have evaluated Dropout-DQN algorithm on CARTPOLE problem from the Class... |
Reinforcement Learning (RL) is a learning paradigm that solves the problem of learning through interaction with environments, this is a totally different approach from the other learning paradigms that have been studied in the field of Machine Learning namely the supervised learning and the unsupervised learning. Rein... | Q-learning is among the most widely used reinforcement learning (RL) algorithms[4]. It’s based on an incremental dynamic programming technique because of the step by step look-up table representation in which it determines the optimal policy[22]. The Q-learning algorithm employs a table to estimate the optimal action v... | The Gridworld problem (Figure 4) is a common RL benchmark. Its relatively small state space permits the Experience Replay (ER) buffer to store all possible state-action pairs. Moreover, this setup allows for the precise computation of the optimal action value function.
| C |
ADE20K: The ADE20K dataset (Zhou et al., 2017) contains 25,210 images from other existing datasets, e.g, the LabelMe (Russell et al., 2008), the SUN (Xiao et al., 2010), and the Places (Zhou et al., 2014) datasets. The images are annotated with labels belonging to 150 classes for “scenes, objects, parts of objects, and... | ADE20K: The ADE20K dataset (Zhou et al., 2017) contains 25,210 images from other existing datasets, e.g, the LabelMe (Russell et al., 2008), the SUN (Xiao et al., 2010), and the Places (Zhou et al., 2014) datasets. The images are annotated with labels belonging to 150 classes for “scenes, objects, parts of objects, and... | CamVid: The Cambridge-driving Labeled Video Database (CamVid) (Brostow et al., 2008, 2009) contains 10 minutes of video captured at 30 frames per second from a driving automobile’s perspective, along with pixel-wise semantic segmentation annotations for 701 frames and 32 object classes.
| Cityscapes: The Cityscapes dataset (Cordts et al., 2016) contains annotated images of urban street scenes. The data was collected during daytime from 50 cities and exhibits variance in the season of the year and traffic conditions. Semantic, instance wise, and dense pixel-wise annotations are provided, with ‘fine’ anno... |
Collecting large-scale accurate pixel-level annotation is time-consuming and financially expensive. However, unlabeled and weakly-labeled images can be collected in large amounts in a relatively fast and cheap manner. As shown in Figure 2, varying levels of supervision are possible when training deep segmentation mode... | B |
The red line indicates the number of edges that remain in 𝐀¯¯𝐀\bar{{\mathbf{A}}}over¯ start_ARG bold_A end_ARG after sparsification.
It is possible to see that for small increments of ϵitalic-ϵ\epsilonitalic_ϵ the spectral distance increases linearly, while the number of edges in the graph drops exponentially. | The reason can be once again attributed to the low information content of the individual node features and in the sparsity of the graph signal (most node features are 0), which makes it difficult for the feature-based pooling methods to infer global properties of the graph by looking at local sub-structures.
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The proposed spectral algorithm is not designed to handle very dense graphs; an intuitive explanation is that 𝐯maxssubscriptsuperscript𝐯𝑠max{\mathbf{v}}^{s}_{\text{max}}bold_v start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT max end_POSTSUBSCRIPT can be interpreted as the graph signal with the... | We notice that the coarsened graphs are pre-computed before training the GNN.
Therefore, the computational time of graph coarsening is much lower compared to training the GNN for several epochs, since each MP operation in the GNN has a cost 𝒪(N2)𝒪superscript𝑁2\mathcal{O}(N^{2})caligraphic_O ( italic_N start_POSTSUP... | The GNN is then trained to fit its node representations to these pre-determined structures.
Pre-computing graph coarsening not only makes the training much faster by avoiding to perform graph reduction at every forward pass, but it also provides a strong inductive bias that prevents degenerate solutions, such as entire... | D |
Neural random forest imitation enables an implicit transformation of random forests into neural networks. Usually, data samples are propagated through the individual decision trees and the split decisions are evaluated during inference.
We propose a method for generating input-target pairs by reversing this process and... | Random forests and neural networks share some similar characteristics, such as the ability to learn arbitrary decision boundaries; however, both methods have different advantages.
Random forests are based on decision trees. Various tree models have been presented – the most well-known are C4.5 (Quinlan, 1993) and CART ... | The generalization performance has been widely studied. Zhang et al. (2017) demonstrate that deep neural networks are capable of fitting random labels and memorizing the training data. Bornschein et al. (2020) analyze the performance across different dataset sizes.
Olson et al. (2018) evaluate the performance of modern... | In contrast to neural networks, random forests are very robust to overfitting due to their ensemble of multiple decision trees. Each decision tree is trained on randomly selected features and samples.
Random forests have demonstrated remarkable performance in many domains (Fernández-Delgado et al., 2014). |
We now compare the proposed method to state-of-the-art methods for mapping random forests into neural networks and classical machine learning classifiers such as random forests and support vector machines with a radial basis function kernel that have shown to be the best two classifiers across all UCI datasets (Fernán... | A |
In the latter two settings with unknown transition dynamics, all the existing algorithms (Neu et al., 2012; Rosenberg and Mansour, 2019a, b) follow the gradient direction with respect to the visitation measure, and thus, differ from most practical policy optimization algorithms. In comparison, OPPO is not restricted to... | Our work is based on the aforementioned line of recent work (Fazel et al., 2018; Yang et al., 2019a; Abbasi-Yadkori et al., 2019a, b; Bhandari and Russo, 2019; Liu et al., 2019; Agarwal et al., 2019; Wang et al., 2019) on the computational efficiency of policy optimization, which covers PG, NPG, TRPO, PPO, and AC. In p... |
Broadly speaking, our work is related to a vast body of work on value-based reinforcement learning in tabular (Jaksch et al., 2010; Osband et al., 2014; Osband and Van Roy, 2016; Azar et al., 2017; Dann et al., 2017; Strehl et al., 2006; Jin et al., 2018) and linear settings (Yang and Wang, 2019b, a; Jin et al., 2019;... |
A line of recent work (Fazel et al., 2018; Yang et al., 2019a; Abbasi-Yadkori et al., 2019a, b; Bhandari and Russo, 2019; Liu et al., 2019; Agarwal et al., 2019; Wang et al., 2019) answers the computational question affirmatively by proving that a wide variety of policy optimization algorithms, such as policy gradient... | for any function f:𝒮→ℝ:𝑓→𝒮ℝf:{\mathcal{S}}\rightarrow\mathbb{R}italic_f : caligraphic_S → blackboard_R. By allowing the reward function to be adversarially chosen in each episode, our setting generalizes the stationary setting commonly adopted by the existing work on value-based reinforcement learning (Jaksch et al.... | B |
\mathbf{W};\mathcal{D})]blackboard_E start_POSTSUBSCRIPT italic_q ( bold_W | italic_ν ) end_POSTSUBSCRIPT [ caligraphic_L ( bold_W ; caligraphic_D ) ] for the variational parameters ν𝜈\mathbf{\nu}italic_ν with gradient-based optimization using the local reparameterization trick (Kingma et al., 2015).
After training ha... | Since their approach is limited to the ReLU activation function, Peters and Welling (2018) extended their work to the signsign\operatorname*{sign}roman_sign activation function.
This involves several non-trivial changes since the sign activation, due to its zero derivative, requires that the local reparameterization tr... | where f~(w)~𝑓𝑤\tilde{f}(w)over~ start_ARG italic_f end_ARG ( italic_w ) is an arbitrary differentiable function with a similar functional shape as f(w)𝑓𝑤f(w)italic_f ( italic_w ).
For instance, in case of the sign activation function f(w)=sign(w)𝑓𝑤sign𝑤f(w)=\operatorname*{sign}(w)italic_f ( italic_w ) = roman... |
While most activation binarization methods use the sign function which can be seen as an approximation of the tanh\tanhroman_tanh function, Cai et al. (2017) proposed a half-wave Gaussian quantization that more closely resembles the predominant ReLU activation function. | In the forward pass, the solid red line is followed which passes the two piecewise constant functions Q𝑄Qitalic_Q and signsign\operatorname*{sign}roman_sign whose gradient is zero almost everywhere (red boxes).
During backpropagation, the dashed green line is followed which avoids these piecewise constant functions an... | A |
We thank Prof. Henry Adams and Dr. Johnathan Bush for very useful feedback about a previous version of this article. We also thank Prof. Mikhail Katz and Prof. Michael Lesnick for explaining to us some aspects of their work. We thank Dr. Qingsong Wang for bringing to our attention the paper [76] which was critical for ... | In [80, Theorem 8.10], Z. Virk provided a proof of the Corollary below which takes place at the simplicial level. The proof we give below exploits the hyperconvexity properties of L∞(X)superscript𝐿𝑋L^{\infty}(X)italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_X ) and also our isomophism theorem, Theorem... | Albeit for the notation sFillRadksubscriptsFillRad𝑘\mathrm{sFillRad}_{k}roman_sFillRad start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, the above stability result should be well known to readers familiar with applied algebraic topology concepts – we state and prove it here however to provide some background for those r... | In Section 3, we construct a category of metric pairs. This category will be the natural setting for our extrinsic persistent homology. Although being functorial is trivial in the case of Vietoris-Rips persistence, the type of functoriality which one is supposed to expect in the case of metric embeddings is a priori no... | In this section we cover the background needed for proving our main results. We alert readers that, in this paper, the same notation can mean either a simplicial complex itself or its geometric realization, interchangeably. The precise meaning will be made clear in each context.
| D |
One initial observation is that the overall Completion Time for both groups was remarkably similar. With the exception of Tasks 1 and 5, where t-viSNE users performed faster than GEP, in general the results have not shown any statistically significant difference. To answer RQ1, we detected no statistically significant... |
Figure 9: Results of the comparative study: the top charts show completion time and tool supportiveness (as judged by participants) for all the tasks of the study, and the bottom row includes the histograms of the participants’ responses in all questions/tasks. The completion times between the two groups were very sim... | On the other hand, t-viSNE obtained consistently higher scores for Tool Supportiveness, with a higher average in all the proposed tasks. The bulk of the distributions of the supportiveness scores from the two groups overlap little, mostly near outliers (the “N/A” option was chosen three times, all in the GEP group).
Wh... | Overall Accuracy
We start by executing a grid search and, after a few seconds, we are presented with 25 representative projections. As we notice that the projections lack high values in continuity, we choose to sort the projections based on this quality metric for further investigation. Next, as the projections are q... |
A quick visual inspection of the two tables already hints at t-viSNE having superior scores than GEP in all components, with all cells being green-colored (as opposed to GEP’s table, which contains many red-colored cells). Indeed, the smallest score for t-viSNE was 4.75, while GEP got many scores under 4 (or even unde... | B |
After reviewing the algorithms and both taxonomies, we have identified several key learned lessons which serve as recommendations to deal with in forthcoming years for that is working on nature- and bio-inspired optimization. The learned lessons gained from the taxonomies and research outlined in [1] form the foundati... |
The prior related work reviewed above indicates that the community widely acknowledges (with more emphasis in recent times) the need for properly organizing the plethora of bio- and nature-inspired algorithms in a coherent taxonomy. However, the majority of them are only focused on the natural inspiration of the algor... | In summary, although in the last years many nature-inspired algorithms have been proposed by the community and their number grows steadily every year, more than half of the proposals reviewed in our work are incremental, minor versions of only three very classical algorithms (PSO, DE, and GA). We, therefore, conclude t... | The behavior is more relevant than the natural inspiration: As was exposed in Section 5, the current literature is flooded with a huge number of nature- and bio-inspired algorithms. However, as has been spotted by our proposed taxonomies, several algorithms belonging to categories with different sources of inspiration ... | Considering the previous examples, it is clear that the real behavior of the algorithm is much more informative than its natural or biological inspiration. Even more, we have observed that in our first proposed taxonomy, built upon the review of 518 proposals, the huge diversity of inspirational sources does not corres... | C |
TABLE IV: k0subscript𝑘0k_{0}italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT: initial sparsity; γ𝛾\gammaitalic_γ: learning rate; tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT: number of iterations to update GAE; struct: neurons of each layer used in AdaGAE.
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Figure 1: Framework of AdaGAE. k0subscript𝑘0k_{0}italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the initial sparsity. First, we construct a sparse graph via the generative model defined in Eq. (7). The learned graph is employed to apply the GAE designed for the weighted graphs. After training the GAE, we update ... |
Figure 2: Visualization of the learning process of AdaGAE on USPS. Figure 2(b)-2(i) show the embedding learned by AdaGAE at the i𝑖iitalic_i-th epoch, while the raw features and the final results are shown in Figure 2(a) and 2(j), respectively. An epoch corresponds to an update of the graph. | In our experiments, the encoder consists of two GCN layers. If the input dimension is 1024, the first layer has 256 neurons and the second layer has 64 neurons. Otherwise, the two layers have 128 neurons and 64 neurons respectively. The activation function of the first layer is set as ReLU while the other one employs t... | To illustrate the process of AdaGAE, Figure 2 shows the learned embedding on USPS at the i𝑖iitalic_i-th epoch. An epoch means a complete training of GAE and an update of the graph. The maximum number of epochs, T𝑇Titalic_T, is set as 10. In other words, the graph is updated 10 times. Clearly, the embedding becomes mo... | D |
Although the agents provide the optimal setup for testing filtering, with control over the packets that can be crafted and sent from both sides, as we explain in Related Work Section 2, this approach is limited only to networks that deploy agents on their networks. In contrast, SMap provides better coverage since it i... | Since the Open Resolver and the Spoofer Projects are the only two infrastructures providing vantage points for measuring spoofing - their importance is immense as they facilitated many research works analysing the spoofability of networks based on the datasets collected by these infrastructures. Nevertheless, the studi... | Agents Active Measurements. Agents with active probes found 608 ASes that were found not to be enforcing ingress filtering using the agents approach of the Spoofer Project (these include duplicates with the traceroute loops measurements). Those contain some of the duplicates from traceroute measurements: together both ... |
These findings show that SMap offers benefits over the existing methods, providing better coverage of the ASes in the Internet and not requiring agents or conditions for obtaining traceroute loops, hence improving visibility of networks not enforcing ingress filtering. |
SMap (The Spoofing Mapper). In this work we present the first Internet-wide scanner for networks that filter spoofed inbound packets, we call the Spoofing Mapper (SMap). We apply SMap for scanning ingress-filtering in more than 90% of the Autonomous Systems (ASes) in the Internet. The measurements with SMap show that ... | C |
All neural networks in this section were trained using stochastic gradient descent with momentum [24] on the loss function ℒℒ\mathcal{L}caligraphic_L. The learning rate was set to 10−3superscript10310^{-3}10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT and the momentum factor to 0.90.90.90.9. Networks were trained fo... | First, the effect of sensor drift on classification accuracy is demonstrated using classifiers trained on a single batch. For each batch 1111 through 10101010, a feedforward model was trained on that batch. Training of a new model was repeated 30 times on each batch. The accuracy of all classifiers were evaluated on ev... |
In order to improve performance, Vergara et al. [7] employed an ensemble technique on the SVM classifiers (Fig. 2B). The same technique was reimplemented and tested on the modified dataset in this paper. The ensemble meant to generalize to batch T𝑇Titalic_T was constructed by training a collection of single-batch cla... |
The skill network approach incorporates all available data into a single training set, disregarding the sequential structure between batches of the dataset. For each batch T𝑇Titalic_T, a network was trained using batches 1111 through T−1𝑇1T-1italic_T - 1 as the training set and evaluated on batch T𝑇Titalic_T. | Figure 2: Neural network architectures. (A.) The batches used for training and testing illustrate the training procedure. The first T−1𝑇1T-1italic_T - 1 batches are used for training, while the next unseen batch T𝑇Titalic_T is used for evaluation. When training the context network, subsequences of the training data a... | C |
For the second change, we need to take another look at how we place the separators tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.
We previously placed these separators in every second nonempty drum σi:=[iδ,(i+1)δ]×Balld−1(δ/2)assignsubscript𝜎𝑖𝑖𝛿𝑖1𝛿superscriptBall𝑑1𝛿2\sigma_{i}:=... | It would be interesting to see whether a direct proof can be given for this fundamental result.
We note that the proof of Theorem 2.1 can easily be adapted to point sets of which the x𝑥xitalic_x-coordinates of the points need not be integer, as long as the difference between x𝑥xitalic_x-coordinates of any two consecu... | Finally, we will show that the requirements for Lemma 5.7 hold, where we take 𝒜𝒜\mathcal{A}caligraphic_A to be the algorithm described above.
The only nontrivial requirement is that T𝒜(Pλ)⩽T𝒜(P)subscript𝑇𝒜subscript𝑃𝜆subscript𝑇𝒜𝑃T_{\mathcal{A}}(P_{\lambda})\leqslant T_{\mathcal{A}}(P)italic_T start_POSTSUBS... | We generalize the case of integer x𝑥xitalic_x-coordinates to the case where the drum [x,x+1]×Balld−1(δ/2)𝑥𝑥1superscriptBall𝑑1𝛿2[x,x+1]\times\mathrm{Ball}^{d-1}(\delta/2)[ italic_x , italic_x + 1 ] × roman_Ball start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ( italic_δ / 2 ) contains O(1)𝑂1O(1)italic_O ( ... | However, in order for our algorithm to meet the requirements of Lemma 5.7, we would like to avoid having a point enter a drum after the x𝑥xitalic_x-coordinates are multiplied by some factor λ>1𝜆1\lambda>1italic_λ > 1.
Furthermore, since the proof of Lemma 4.3 requires every drum to be at least δ𝛿\deltaitalic_δ wide,... | D |
Then mapping all elements of Sisubscript𝑆𝑖S_{i}italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to e𝑒eitalic_e for i<n𝑖𝑛i<nitalic_i < italic_n gives the homomorphisms
ϕisubscriptitalic-ϕ𝑖\phi_{i}italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in the hypothesis of 12. | While our main result significantly relaxes the hypothesis for showing that the free product of self-similar semigroups (or automaton semigroups) is self-similar (an automaton semigroup), it does not settle the underlying question whether these semigroup classes are closed under free product. It is possible that there ... | from one to the other, then their free product S⋆T⋆𝑆𝑇S\star Titalic_S ⋆ italic_T is an automaton semigroup (8). This is again a strict generalization of [19, Theorem 3.0.1] (even if we only consider complete automata).
Third, we show this result in the more general setting of self-similar semigroups111Note that the c... |
There is a quite interesting evolution of constructions to present free groups in a self-similar way or even as automaton groups (see [15] for an overview). This culminated in constructions to present free groups of arbitrary rank as automaton groups where the number of states coincides with the rank [18, 17]. While t... | The construction used to prove Theorem 6 can also be used to obtain results which are not immediate corollaries of the theorem (or its corollary for automaton semigroups in 8). As an example, we prove in the following theorem that it is possible to adjoin a free generator to every self-similar semigroup without losing ... | D |
It is also interesting to note that the drop in training accuracy is lower with this regularization scheme as compared to the state-of-the-art methods. Of course, if any model was actually visually grounded, then we would expect it to improve performances on both train and test sets. We do not observe such behavior in ... | While our results indicate that current visual grounding based bias mitigation approaches do not suffice, we believe this is still a good research direction. However, future methods must seek to verify that performance gains are not stemming from spurious sources by using an experimental setup similar to that presented... |
The VQA-CP dataset Agrawal et al. (2018) showcases this phenomenon by incorporating different question type/answer distributions in the train and test sets. Since the linguistic priors in the train and test sets differ, models that exploit these priors fail on the test set. To tackle this issue, recent works have ende... | Some recent approaches employ a question-only branch as a control model to discover the questions most affected by linguistic correlations. The question-only model is either used to perform adversarial regularization Grand and Belinkov (2019); Ramakrishnan et al. (2018) or to re-scale the loss based on the difficulty o... |
As expected of any real world dataset, VQA datasets also contain dataset biases Goyal et al. (2017). The VQA-CP dataset Agrawal et al. (2018) was introduced to study the robustness of VQA methods against linguistic biases. Since it contains different answer distributions in the train and test sets, VQA-CP makes it nea... | A |
Readability. Readability of a text can be defined as the ease of understanding or comprehension due to the style of writing (Klare et al., 1963). Along with length, readability plays a role in internet users’ decisions to either read or ignore a privacy policy (Ermakova et al., 2015). While prior studies on readability... |
To satisfy the need for a much larger corpus of privacy policies, we introduce the PrivaSeer Corpus of 1,005,380 English language website privacy policies. The number of unique websites represented in PrivaSeer is about ten times larger than the next largest public collection of web privacy policies Amos et al. (2020)... |
Topic Modelling. Topic modelling is an unsupervised machine learning method that extracts the most probable distribution of words into topics through an iterative process (Wallach, 2006). We used topic modelling to explore the distribution of themes of text in our corpus. Topic modelling using a large corpus such as P... | Topic modelling showed the distribution of themes of privacy practices in policies, corresponding to the expectations of legal experts in some ways, but differing in others. The positive relationship between PageRank of a domain and the number of topics covered in its policy indicates that more popular domains have a s... |
It is likely that the divergence between OPP-115 categories and LDA topics comes from a difference in approaches: the OPP-115 categories represent themes that privacy experts expected to find in privacy policies, which diverge from the actual distribution of themes in this text genre. Figure 2 shows the percentage of ... | B |
T3: Manage the performance metrics for enhancing trust in the results. Many performance or validation metrics are used in the field of ML. For each data set, there might be a different set of metrics to measure the best-performing stacking. Controlling the process by alternating these metrics and observing their influe... |
G4: Facilitate human interaction and offer guidance. During development of any VA tool, it is key to decide on concrete visual representations and interaction technologies between multiple coordinated views. It is not uncommon to find gaps between visualization design guidelines and their applicability in implemented ... | and (v) we track the history of the previously stored stacking ensembles in StackGenVis: Alignment of Data, Algorithms, and Models for Stacking Ensemble Learning Using Performance Metrics(b) and compare their performances against the active stacking ensemble—the one not yet stored in the history—in StackGenVis: Alignme... |
T4: Compare the results of two stages and receive feedback to guide interaction. To assist the knowledge generation, a comparison between the currently active stack against previously stored versions is important. In general, this includes monitoring the historical process of the stacking ensemble, facilitating intera... |
Figure 1: Knowledge generation model for ensemble learning with VA derived from the model by Sacha et al. [44]. On the left, it illustrates how a VA system can enable the exploration of the data and the models with the use of visualization. On the right, a number of design goals assist the human in the exploration, ve... | C |
We thus have 3333 cases, depending on the value of the tuple
(p(v,[010]),p(v,[323]),p(v,[313]),p(v,[003]))𝑝𝑣delimited-[]010𝑝𝑣delimited-[]323𝑝𝑣delimited-[]313𝑝𝑣delimited-[]003(p(v,[010]),p(v,[323]),p(v,[313]),p(v,[003]))( italic_p ( italic_v , [ 010 ] ) , italic_p ( italic_v , [ 323 ] ) , italic_p ( italic_v... | {0¯,1¯,2¯,3¯,[013],[010],[323],[313],[112],[003],[113]}.¯0¯1¯2¯3delimited-[]013delimited-[]010delimited-[]323delimited-[]313delimited-[]112delimited-[]003delimited-[]113\{\overline{0},\overline{1},\overline{2},\overline{3},[013],[010],[323],[313],%
[112],[003],[113]\}.{ over¯ start_ARG 0 end_ARG , over¯ start_ARG 1 end... | p(v,[013])=p(v,[313])=p(v,[113])=1𝑝𝑣delimited-[]013𝑝𝑣delimited-[]313𝑝𝑣delimited-[]1131p(v,[013])=p(v,[313])=p(v,[113])=1italic_p ( italic_v , [ 013 ] ) = italic_p ( italic_v , [ 313 ] ) = italic_p ( italic_v , [ 113 ] ) = 1.
Similarly, when f=[112]𝑓delimited-[]112f=[112]italic_f = [ 112 ], | By using the pairwise adjacency of (v,[112])𝑣delimited-[]112(v,[112])( italic_v , [ 112 ] ), (v,[003])𝑣delimited-[]003(v,[003])( italic_v , [ 003 ] ), and
(v,[113])𝑣delimited-[]113(v,[113])( italic_v , [ 113 ] ), we can confirm that in the 3333 cases, these | Then, by using the adjacency of (v,[013])𝑣delimited-[]013(v,[013])( italic_v , [ 013 ] ) with each of
(v,[010])𝑣delimited-[]010(v,[010])( italic_v , [ 010 ] ), (v,[323])𝑣delimited-[]323(v,[323])( italic_v , [ 323 ] ), and (v,[112])𝑣delimited-[]112(v,[112])( italic_v , [ 112 ] ), we can confirm that | C |
In text classification experiment, we use accuracy (Acc) to evaluate the classification performance.
In dialogue generation experiment, we evaluate the performance of MAML in terms of quality and personality. We use PPL and BLEU [Papineni et al., 2002] to measure the similarity between the reference and the generated r... | In this paper, we take an empirical approach to systematically investigating these impacting factors and finding when MAML works the best. We conduct extensive experiments over 4 datasets. We first study the effects of data quantity and distribution on the training strategy:
RQ1. Since the parameter initialization lear... | The finding suggests that parameter initialization at the late training stage has strong general language generation ability, but performs comparative poorly in task-specific adaptation.
Although in the early training stage, the performance improves benefiting from the pre-trained general language model, if the languag... | In text classification experiment, we use accuracy (Acc) to evaluate the classification performance.
In dialogue generation experiment, we evaluate the performance of MAML in terms of quality and personality. We use PPL and BLEU [Papineni et al., 2002] to measure the similarity between the reference and the generated r... |
To answer RQ1, we compare the changing trend of the general language model and the task-specific adaptation ability during the training of MAML to find whether there is a trade-off problem. (Figure 1) We select the trained parameter initialization at different MAML training epochs and evaluate them directly on the met... | D |
where SNRthsubscriptSNRth\text{SNR}_{\text{th}}SNR start_POSTSUBSCRIPT th end_POSTSUBSCRIPT is a certain SNR threshold. If an arbitrary link between the t-UAV and the r-UAV is in the outage, there is an outage in the network. Hence, the outage probability is determined by the minimum SNR among K𝐾Kitalic_K t-UAVs. Fig... | Given the received signal in (4), the signal-to-interference-plus-noise ratio (SINR) 111When the transmit signals of t-UAVs have similar AOAs at the r-UAV, the combiners may not separate them correctly and the inter-UAV interference exists in the UAV networks. Due to the constraint on the minimum distance between UAVs,... | and the CCA scheme achieves higher SE than the UPA scheme obviously with different t-UAV number K𝐾Kitalic_K. The main reason is that the UPA with DREs can only receive/transmit the signal within a limited angular range at a certain time slot while the CCA does not have such limitation. It is also shown that the gap be... | As shown in Fig. 11, the SE of the CCA codebook scheme and the traditional codebook scheme is compared. The proposed DRE-covered CCA codebook is used in the CCA codebook scheme. In the traditional codebook scheme, the codebook without subarray partition is used. The CCA on the r-UAV is equally partitioned into K𝐾Kital... |
In this paper, we mainly focus on the analog beam tracking without considering the inter-UAV interference. The sum SE calculated by (11) and (28) with different numbers of t-UAVs and the given transmit power is shown in Fig. 10, respectively, to verify the influence of the inter-UAV interference. It is shown that the ... | D |
There are other logics, incomparable
in expressiveness with 𝖥𝖮Pres2subscriptsuperscript𝖥𝖮2Pres\mathsf{FO}^{2}_{\textup{Pres}}sansserif_FO start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Pres end_POSTSUBSCRIPT, where periodicity of the spectrum has been proven [17]. The | The paper [4] shows decidability for a logic with incomparable expressiveness: the quantification allows a more powerful
quantitative comparison, but must be guarded – restricting the counts only of sets of elements that are adjacent to a given element. | There are other logics, incomparable
in expressiveness with 𝖥𝖮Pres2subscriptsuperscript𝖥𝖮2Pres\mathsf{FO}^{2}_{\textup{Pres}}sansserif_FO start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Pres end_POSTSUBSCRIPT, where periodicity of the spectrum has been proven [17]. The | In addition, to make the main line of argument clearer, we consider only the finite graph case in the body of the paper,
which already implies decidability of the finite satisfiability of 𝖥𝖮Pres2subscriptsuperscript𝖥𝖮2Pres\mathsf{FO}^{2}_{\textup{Pres}}sansserif_FO start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_... | Related one-variable fragments in which we have only a
unary relational vocabulary and the main quantification is ∃Sxϕ(x)superscript𝑆𝑥italic-ϕ𝑥\exists^{S}x~{}\phi(x)∃ start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT italic_x italic_ϕ ( italic_x ) are known to be decidable (see, e.g. [2]), and their decidability ... | A |
*}\cdot(1-\gamma)^{-1}\cdot\alpha^{-1},≤ 1 / 2 ⋅ ( 1 - italic_γ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⋅ over¯ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT + italic_C start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ⋅ ( 1 - italic_γ ) st... |
In this paper, we study temporal-difference (TD) (Sutton, 1988) and Q-learning (Watkins and Dayan, 1992), two of the most prominent algorithms in deep reinforcement learning, which are further connected to policy gradient (Williams, 1992) through its equivalence to soft Q-learning (O’Donoghue et al., 2016; Schulman et... |
We first introduce the assumptions for our analysis. In §4.1, we establish the global optimality and convergence of the PDE solution ρtsubscript𝜌𝑡\rho_{t}italic_ρ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT in (3.4). In §4.2, we further invoke Proposition 3.1 to establish the global optimality and convergence of ... |
Contribution. Going beyond the NTK regime, we prove that, when the value function approximator is an overparameterized two-layer neural network, TD and Q-learning globally minimize the mean-squared projected Bellman error (MSPBE) at a sublinear rate. Moreover, in contrast to the NTK regime, the induced feature represe... | In this section, we extend our analysis of TD to Q-learning and policy gradient. In §6.1, we introduce Q-learning and its mean-field limit. In §6.2, we establish the global optimality and convergence of Q-learning. In §6.3, we further extend our analysis to soft Q-learning, which is equivalent to policy gradient.
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We implemented our approach based on the Neutron implementation of the Transformer Xu and Liu (2019). To show the effects of depth-wise LSTMs on the 6-layer Transformer, we first conducted experiments on the WMT 14 English to German and English to French news translation tasks to compare with the Transformer baseline ... |
To test the effectiveness of depth-wise LSTMs in the multilingual setting, we conducted experiments on the challenging massively many-to-many translation task on the OPUS-100 corpus Tiedemann (2012); Aharoni et al. (2019); Zhang et al. (2020). We tested the performance of 6-layer models following the experiment settin... | For machine translation, the performance of the Transformer translation model Vaswani et al. (2017) benefits from including residual connections He et al. (2016) in stacked layers and sub-layers Bapna et al. (2018); Wu et al. (2019b); Wei et al. (2020); Zhang et al. (2019); Xu et al. (2020a); Li et al. (2020); Huang et... |
We examine whether depth-wise LSTM has the ability to ensure the convergence of deep Transformers and measure performance on the WMT 14 English to German task and the WMT 15 Czech to English task following Bapna et al. (2018); Xu et al. (2020a), and compare our approach with the pre-norm Transformer in which residual ... | We applied joint Byte-Pair Encoding Sennrich et al. (2016) with 32k32𝑘32k32 italic_k merging operations on all data sets to address the unknown word issue. We only kept sentences with a maximum of 256256256256 subword tokens for training. For fair comparison, we did not tune any hyperparameters but followed Vaswani e... | D |
ψ⊇Pn≜∃x0,…,xn−1.⋀i<j¬(xi=xj)∧⋀0≤i<n−1E(xi,xi+1)formulae-sequence≜subscript𝜓subscript𝑃𝑛absentsubscript𝑥0…subscript𝑥𝑛1subscript𝑖𝑗subscript𝑥𝑖subscript𝑥𝑗subscript0𝑖𝑛1𝐸subscript𝑥𝑖subscript𝑥𝑖1\psi_{\supseteq P_{n}}\triangleq\exists x_{0},\dots,x_{n-1}.\bigwedge_{i<j}% | X≜{x→∈∏i∈IXi∣∀i≤j∈I,xj=fi,j(xi)}≜𝑋conditional-set→𝑥subscriptproduct𝑖𝐼subscript𝑋𝑖formulae-sequencefor-all𝑖𝑗𝐼subscript𝑥𝑗subscript𝑓𝑖𝑗subscript𝑥𝑖X\triangleq\left\{\vec{x}\in\prod_{i\in I}X_{i}\mid\forall i\leq j\in I,x_{j}=%
f_{i,j}(x_{i})\right\}italic_X ≜ { over→ start_ARG italic_x end_ARG ∈ ∏ start_POST... | \neg(x_{i}=x_{j})\wedge\bigwedge_{0\leq i<n-1}E(x_{i},x_{i+1})\;italic_ψ start_POSTSUBSCRIPT ⊇ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≜ ∃ italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT . ⋀ start_POSTSUBSCRIPT italic_i... | _{i}^{1})_{1\leq i\leq n};(y\models\psi_{i}^{2})_{1\leq i\leq n}\right)=1\;.∀ italic_x ∈ italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y ∈ italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_f ( italic_x , italic_y ) ⊧ italic_φ ⇔ italic_β ( ( italic_x ⊧ italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSU... | \,n-1})italic_ψ start_POSTSUBSCRIPT ⊇ italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≜ ∃ italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT . ⋀ start_POSTSUBSCRIPT italic_i < italic_j end_POSTSUBSCRIPT ¬ ( italic_x start_POSTSUBS... | B |
Qualitative Comparison: To qualitatively show the performance of different learning representations, we visualize the 3D distortion distribution maps (3D DDM) derived from the ground truth and these two schemes in Fig. 8, in which each pixel value of the distortion distribution map represents the distortion level. Sinc... | Figure 13: Qualitative evaluations of the rectified distorted images on real-world scenes. For each evaluation, we show the distorted image and corrected results of the compared methods: Alemán-Flores [23], Santana-Cedrés [24], Rong [8], Li [11], and Liao [12], and rectified results of our proposed approach, from left ... | Figure 12: Qualitative evaluations of the rectified distorted images on people (left) and challenging (right) scenes. For each evaluation, we show the distorted image, ground truth, and corrected results of the compared methods: Alemán-Flores [23], Santana-Cedrés [24], Rong [8], Li [11], and Liao [12], and rectified re... | We visually compare the corrected results from our approach with state-of-the-art methods using our synthetic test set and the real distorted images. To show the comprehensive rectification performance under different scenes, we split the test set into four types of scenes: indoor, outdoor, people, and challenging scen... |
Figure 11: Qualitative evaluations of the rectified distorted images on indoor (left) and outdoor (right) scenes. For each evaluation, we show the distorted image, ground truth, and corrected results of the compared methods: Alemán-Flores [23], Santana-Cedrés [24], Rong [8], Li [11], and Liao [12], and rectified resul... | D |
Furthermore, researchers in [19] argued that the extrapolation technique is suitable for large-batch training and proposed EXTRAP-SGD.
However, experimental implementations of these methods still require additional training tricks, such as warm-up, which may make the results inconsistent with the theory. | In this paper, we first review the convergence property of MSGD, one of the most widely used variants of SGD, and analyze the failure of MSGD in large-batch training from an optimization perspective. Then, we propose a novel method, called
stochastic normalized gradient descent with momentum (SNGM), for large-batch tra... | We compare SNGM with four baselines: MSGD, ADAM [14], LARS [34] and LAMB [34]. LAMB is a layer-wise adaptive large-batch optimization method based on ADAM, while LARS is based on MSGD.
The experiments are implemented based on the DeepCTR 888https://github.com/shenweichen/DeepCTR-Torch framework. | If we avoid these tricks, these methods may suffer from severe performance degradation.
For LARS and its variants, the proposal of the layer-wise update strategy is primarily based on empirical observations. Its reasonability and necessity remain doubtful from an optimization perspective. | SGD and its variants are iterative methods. In the t𝑡titalic_t-th iteration, these methods randomly
choose a subset (also called a mini-batch) ℐt⊂{1,2,…,n}subscriptℐ𝑡12…𝑛{\mathcal{I}}_{t}\subset\{1,2,\ldots,n\}caligraphic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ⊂ { 1 , 2 , … , italic_n } and compute the | C |
5555-approximation for homogeneous 2S-MuSup-Poly, with |𝒮|≤2m𝒮superscript2𝑚|\mathcal{S}|\leq 2^{m}| caligraphic_S | ≤ 2 start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT and runtime poly(n,m,Λ)poly𝑛𝑚Λ\operatorname*{poly}(n,m,\Lambda)roman_poly ( italic_n , italic_m , roman_Λ ).
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We follow up with 3333-approximations for the homogeneous robust outlier MatSup and MuSup problems, which are slight variations on algorithms of [6] (specifically, our approach in Section 4.1 is a variation on their solve-or-cut methods). In Section 5, we describe a 9-approximation algorithm for an inhomogeneous MatSu... | If we have a ρ𝜌\rhoitalic_ρ-approximation algorithm for AlgRW for given 𝒞,ℱ,ℳ,R𝒞ℱℳ𝑅\mathcal{C},\mathcal{F},\mathcal{M},Rcaligraphic_C , caligraphic_F , caligraphic_M , italic_R, then we can get an efficiently-generalizable (ρ+2)𝜌2(\rho+2)( italic_ρ + 2 )-approximation algorithm for the corresponding problem 𝒫𝒫\m... | The other three results are based on a reduction to a single-stage, deterministic robust outliers problem described in Section 4; namely, convert any ρ𝜌\rhoitalic_ρ-approximation algorithm for the robust outlier problem into a (ρ+2)𝜌2(\rho+2)( italic_ρ + 2 )-approximation algorithm for the corresponding two-stage sto... | We now describe a generic method of transforming a given 𝒫𝒫\mathcal{P}caligraphic_P-Poly problem into a single-stage deterministic robust outlier problem. This will give us a 5-approximation algorithm for homogeneous 2S-MuSup and 2S-MatSup instances nearly for free; in the next section, we also use it obtain our 11-a... | C |
That is, the mean square error at the next time can be controlled by that at the
previous time and the consensus error. However, this can not be obtained for the case with the linearly growing subgradients. Also, different from [15], the subgradients are not required to be bounded and the inequality (28) in [15] does n... | I. The local cost functions in this paper are not required to be differentiable and the subgradients only satisfy the linear growth condition.
The inner product of the subgradients and the error between local optimizers’ states and the global optimal solution inevitably exists in the recursive inequality of the conditi... | (Lemma 3.1).
To this end, we estimate the upper bound of the mean square increasing rate of the local optimizers’ states at first (Lemma 3.2). Then we substitute this upper bound into the Lyapunov function difference inequality of the consensus error, and obtain the estimated convergence rate of mean square consensus (... | As a result, the existing methods are no longer applicable. In fact, the inner product of the subgradients and the error between local optimizers’ states and the global optimal solution inevitably exists in the recursive inequality of the conditional mean square error, which leads the nonegative supermartingale converg... |
III. The co-existence of random graphs, subgradient measurement noises, additive and multiplicative communication noises are considered. Compared with the case with only a single random factor, the coupling terms of different random factors inevitably affect the mean square difference between optimizers’ states and an... | C |
\lceil 1/\delta\rceil$}\end{cases}.{ start_ROW start_CELL ∑ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT > italic_k ⋅ italic_m italic_a italic_x ( italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSC... | For instance, since the random output tables in Figure 3 comply with 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG-probability, for any QI value whose corresponding column has at least one probability greater than 0, there are at least 2 records can carry the QI value.
| Specifically, there are three main steps in the proposed approach. First, MuCo partitions the tuples into groups and assigns similar records into the same group as far as possible. Second, the random output tables, which control the distribution of random output values within each group, are calculated to make similar ... |
Additionally, differing from traditional principles that directly confine the values in microdata, we propose a δ𝛿\deltaitalic_δ-probability principle to control random output tables so as to limit the probability of any QI value being used to re-identify a target person. For instance, the random output tables in Fig... | This experiment measures the effectiveness of privacy preservation for MuCo. We assume that an adversary has all the QI values of the target person, and each QI value is combined to match the target person with the probability of Pmatchsubscript𝑃𝑚𝑎𝑡𝑐ℎP_{match}italic_P start_POSTSUBSCRIPT italic_m italic_a ital... | A |
We implement PointRend using MMDetection Chen et al. (2019b) and adopt the modifications and tricks mentioned in Section 3.3. Both X101-64x4d and Res2Net101 Gao et al. (2019) are used as our backbones, pretrained on ImageNet only. SGD with momentum 0.9 and weight decay 1e-4 is adopted. The initial learning rate is set... | As shown in Table 3, all PointRend models achieve promising performance. Even without ensemble, our PointRend baseline, which yields 77.38 mAP, has already achieved 1st place on the test leaderboard. Note that several attempts, like BFP Pang et al. (2019) and EnrichFeat, give no improvements against PointRend baseline,... | Bells and Whistles. MaskRCNN-ResNet50 is used as baseline and it achieves 53.2 mAP. For PointRend, we follow the same setting as Kirillov et al. (2020) except for extracting both coarse and fine-grained features from the P2-P5 levels of FPN, rather than only P2 described in the paper. Surprisingly, PointRend yields 62.... | Deep learning has achieved great success in recent years Fan et al. (2019); Zhu et al. (2019); Luo et al. (2021, 2023); Chen et al. (2021). Recently, many modern instance segmentation approaches demonstrate outstanding performance on COCO and LVIS, such as HTC Chen et al. (2019a), SOLOv2 Wang et al. (2020), and PointRe... | Table 3: PointRend’s performance on testing set (trackB). “EnrichFeat” means enhance the feature representation of coarse mask head and point head by increasing the number of fully-connected layers or its hidden sizes. “BFP” means Balanced Feature Pyramid. Note that BFP and EnrichFeat gain little improvements, we guess... | A |
(0log0:=0assign0000\log 0:=00 roman_log 0 := 0). The base of the log\logroman_log does not really matter here. For concreteness we take the log\logroman_log to base 2222. Note that if f𝑓fitalic_f has L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT norm 1111 then the sequence {|f^(A)|2}A⊆[n]subsc... |
In version 1 of this note, which can still be found on the ArXiv, we showed that the analogous version of the conjecture for complex functions on {−1,1}nsuperscript11𝑛\{-1,1\}^{n}{ - 1 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT which have modulus 1111 fails. This solves a question raised by Gady Kozma s... |
where for A⊆[n]𝐴delimited-[]𝑛A\subseteq[n]italic_A ⊆ [ italic_n ], |A|𝐴|A|| italic_A | denotes the cardinality of A𝐴Aitalic_A. This object, especially for boolean functions, is a deeply studied one and quite influential (but this is not the reason for the name…) in several directions. We refer to [O] for some info... | For the significance of this conjecture we refer to the original paper [FK], and to Kalai’s blog [K] (embedded in Tao’s blog) which reports on all significant results concerning the conjecture. [KKLMS] establishes a weaker version of the conjecture. Its introduction is also a good source of information on the problem.
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Here we give an embarrassingly simple presentation of an example of such a function (although it can be shown to be a version of the example in the previous version of this note). As was written in the previous version, an anonymous referee of version 1 wrote that the theorem was known to experts but not published. Ma... | C |
From Figure 1, we see LSVI-UCB-Restart with the knowledge of global variation drastically outperforms all other methods designed for stationary environments , in both abruptly-changing and gradually-changing environments, since it restarts the estimation of the Q𝑄Qitalic_Q function with knowledge of the total variatio... |
Figure 2 shows that the running times of LSVI-UCB-Restart and Ada-LSVI-UCB-Restart are roughly the same. They are much less compared with MASTER, OPT-WLSVI, LSVI-UCB, Epsilon-Greedy. This is because LSVI-UCB-Restart and Ada-LSVI-UCB-Restart can automatically restart according to the variation of the environment and th... | The last relevant line of work is on dynamic regret analysis of nonstationary MDPs mostly without function approximation (Auer et al., 2010; Ortner et al., 2020; Cheung et al., 2019; Fei et al., 2020; Cheung et al., 2020). The work of Auer et al. (2010) considers the setting in which the MDP is piecewise-stationary and... |
Compared to OPT-WLSVI and MASTER, our proposed algorithms achieve comparable empirical performance. More specifically, MASTER outperforms our proposed algorithm which agrees with its dynamic regret upper bound. However, the variance of MASTER is larger due to the random scheduling of multiple base algorithms. Our algo... | Using the master algorithm to select the window size is reminiscent of model selection approaches to online RL (Agarwal et al., 2017; Pacchiano et al., 2020; Lee et al., 2021; Abbasi-Yadkori et al., 2020). Typically model selection approaches achieve worse rates compared to the best base algorithm. However, in our sett... | C |
A series of 1-5 Likert scale questions (1: strongly disagree, 5: strongly agree) were presented to the respondents (in SeenFake-57) to further gain insights into their views on fake news. Respondents feel that the issue of fake news will remain for a long time (M=4.33,SD=0.831formulae-sequence𝑀4.33𝑆𝐷0.831M=4.33,SD=... | While fake news is not a new phenomenon, the 2016 US presidential election brought the issue to immediate global attention with the discovery that fake news campaigns on social media had been made to influence the election (Allcott and Gentzkow, 2017). The creation and dissemination of fake news is motivated by politic... | Singapore is a city-state with an open economy and diverse population that shapes it to be an attractive and vulnerable target for fake news campaigns (Lim, 2019). As a measure against fake news, the Protection from Online Falsehoods and Manipulation Act (POFMA) was passed on May 8, 2019, to empower the Singapore Gover... | Many studies worldwide have observed the proliferation of fake news on social media and instant messaging apps, with social media being the more commonly studied medium. In Singapore, however, mitigation efforts on fake news in instant messaging apps may be more important. Most respondents encountered fake news on inst... |
In general, respondents possess a competent level of digital literacy skills with a majority exercising good news sharing practices. They actively verify news before sharing by checking with multiple sources found through the search engine and with authoritative information found in government communication platforms,... | C |
In Table 11, we present the entity alignment results concerning various embedding-sizes. Our findings indicate that decentRL with an embedding size of 256 is adequate to outperform AliNet with an embedding size of 512. Furthermore, even a modestly sized decentRL (with an embedding size of 64) can surpass vanilla GAT wi... |
The performance of decentRL at the input layer notably lags behind that of other layers and AliNet. As discussed in previous sections, decentRL does not use the embedding of the central entity as input when generating its output embedding. However, this input embedding can still accumulate knowledge by participating i... | Table 4 presents the results of conventional entity alignment. decentRL achieves state-of-the-art performance, surpassing all others in Hits@1 and MRR. AliNet [39], a hybrid method combining GCN and GAT, performs better than the methods solely based on GAT or GCN on many metrics. Nonetheless, across most metrics and da... | Figure 4 shows the experimental results. decentRL outperforms both GAT and AliNet across all metrics. While its performance slightly decreases compared to conventional datasets, the other methods experience even greater performance drops in this context. AliNet also outperforms GAT, as it combines GCN and GAT to aggreg... |
We conduct an analysis of the training time for decentRL and AliNet with varying hidden-sizes on a V100 GPU, as detailed in Table 12. We employ a two-layer AliNet (each layer comprising one GCN and one GAT) and a four-layer decentRL. The two methods exhibit comparable running times per epoch. AliNet runs marginally fa... | D |
9: Taking stochastic gradient ascent tvdmsubscript𝑡vdmt_{\rm vdm}italic_t start_POSTSUBSCRIPT roman_vdm end_POSTSUBSCRIPT times to maximize LVDMsubscript𝐿VDML_{\rm VDM}italic_L start_POSTSUBSCRIPT roman_VDM end_POSTSUBSCRIPT and update parameters (φ,ψ,θ)𝜑𝜓𝜃(\varphi,\psi,\theta)( italic_φ , italic_ψ , italic_θ... |
In this section, we conduct experiments to compare the proposed VDM with several state-of-the-art model-based self-supervised exploration approaches. We first describe the experimental setup and implementation detail. Then, we compare the proposed method with baselines in several challenging image-based RL tasks. The ... |
In this section, we introduce VDM for exploration. In section III-A, we introduce the theory of VDM based on conditional variational inference. In section III-B, we present the detail of the optimizing process. In section III-C, we analyze the result of VDM used in ‘Noisy-Mnist’ that models the multimodality and stoch... |
To validate the effectiveness of our method, we compare the proposed method with the following self-supervised exploration baselines. Specifically, we conduct experiments to compare the following methods: (i) VDM. The proposed self-supervised exploration method. (ii) ICM [10]. ICM first builds an inverse dynamics mode... | Upon fitting VDM, we propose an intrinsic reward by an upper bound of the negative log-likelihood, and conduct self-supervised exploration based on the proposed intrinsic reward. We evaluate the proposed method on several challenging image-based tasks, including 1) Atari games, 2) Atari games with sticky actions, which... | A |
In summary: We answer Questions 1–2 by establishing an efficient m𝑚mitalic_mD interpolation scheme that can approximate a generic class of functions and,
at least empirically, reaches the proposed exponential approximation rate for strongly varying Trefethen functions, such as the Runge function f(x)=1/(1+10‖x‖2)𝑓�... | As far as we recognize, tensorial Chebyshev interpolation [32, 83, 84] best answers these questions among all state-of-the-art approaches.
However, Chebyshev interpolation is done on a (full) tensorial grid, which suffers from the curse of dimensionality by requiring |PAm,n|∈𝒪(nm)subscript𝑃subscript𝐴𝑚𝑛𝒪superscri... | Furthermore, so far none of these approaches is known to reach the optimal Trefethen approximation rates when requiring the number of nodes of the underlying tensorial grids to
scale sub-exponential with space dimension. As the numerical experiments in Section 8 suggest, we believe that only non-tensorial grids are abl... | If one only requires the nodes PAsubscript𝑃𝐴P_{A}italic_P start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT to be unisolvent, then they do not have to be given by a (sub) grid at all. The nodes used for the present m𝑚mitalic_mD Newton interpolation are given by a sub-grid,
but that sub-grid is neither symmetric nor ten... | convergence rates for the Runge function, as a prominet example of a Trefethen function. We show that the number of nodes required scales sub-exponentially with space dimension. We therefore believe that the present generalization of unisolvent nodes to non-tensorial grids is key to lifting the curse of dimensionality.... | B |
A two-sample test is performed to decide whether to accept the null hypothesis H0:μ=ν:subscript𝐻0𝜇𝜈H_{0}:~{}\mu=\nuitalic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : italic_μ = italic_ν or the general alternative hypothesis H1:μ≠ν:subscript𝐻1𝜇𝜈H_{1}:~{}\mu\neq\nuitalic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : ... | The test power for different methods is presented in Fig. 1, in which the results are averaged for 100100100100 independent trials.
The first two plots in Fig. 1 show receiver operating characteristic (ROC) curves for mean-shifted Gaussian distributions, where Fig. 1a) examines the test power for different choices of s... | For instance, in anomaly detection [1, 2, 3], the abnormal observations follow a different distribution from the typical distribution.
Similarly, in change-point detection [4, 5, 6], the post-change observations follow a different distribution from the pre-change one. | 3(a)) Illustration of the projection mapping trained on two collections of samples generated from two different target distributions with m=n=100𝑚𝑛100m=n=100italic_m = italic_n = 100.
Here the red and blue points are generated from Gaussian distributions with two different covariance matrix. | When under H0subscript𝐻0H_{0}italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, we set target distributions μ𝜇\muitalic_μ and ν𝜈\nuitalic_ν to be the uniform distribution on [−1,1]dsuperscript11𝑑[-1,1]^{d}[ - 1 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT.
When under H1subscript𝐻1H_{1}italic_H start_POS... | B |
Learning disentangled factors h∼qϕ(H|x)similar-toℎsubscript𝑞italic-ϕconditional𝐻𝑥h\sim q_{\phi}(H|x)italic_h ∼ italic_q start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_H | italic_x ) that are semantically meaningful representations of the observation x𝑥xitalic_x is highly desirable because such interpreta... | Prior work in unsupervised DR learning suggests the objective of learning statistically independent factors of the latent space as means to obtain DR. The underlying assumption is that the latent variables H𝐻Hitalic_H can be partitioned into independent components C𝐶Citalic_C (i.e. the disentangled factors) and corre... | Specifically, we apply a DGM to learn the nuisance variables Z𝑍Zitalic_Z, conditioned on the output image of the first part, and use Z𝑍Zitalic_Z in the normalization layers of the decoder network to shift and scale the features extracted from the input image. This process adds the details information captured in Z𝑍Z... | While the aforementioned models made significant progress on the problem, they suffer from an inherent trade-off between learning DR and reconstruction quality. If the latent space is heavily regularized, not allowing enough capacity for the nuisance variables, reconstruction quality is diminished. On the other hand, i... |
The model has two parts. First, we apply a DGM to learn only the disentangled part, C𝐶Citalic_C, of the latent space. We do that by applying any of the above mentioned VAEs111In this exposition we use unspervised trained VAEs as our base models but the framework also works with GAN-based or FLOW-based DGMs, supervise... | A |
We will look at the inputs through 18 test cases to see if the circuit is acceptable. Next, it verifies with DFS that the output is possible for the actual pin connection state. As mentioned above, the search is carried out and the results are expressed by the unique number of each vertex. The result is as shown in Tab... | We will look at the inputs through 18 test cases to see if the circuit is acceptable. Next, it verifies with DFS that the output is possible for the actual pin connection state. As mentioned above, the search is carried out and the results are expressed by the unique number of each vertex. The result is as shown in Tab... |
DFS (Depth First Search) verifies that the output is possible for the actual Pin connection state. As described above, the output is determined by the 3-pin input, so we will enter 1 with the A2 and A1 connections, the B2 and B1 connections (the reverse is treated as 0), and the corresponding output will be recognized... | The structure-based computer mentioned in this paper are based on Boolean Algebra, a system commonly applied to digital computers. Boolean algebra is a concept created by George Boole (1815-1854) of the United Kingdom that expresses the True and False of logic 1 and 0, and mathematically describes digital electrical si... |
Exploration based on previous experiments and graph theory found errors in structural computers with electricity as a medium. The cause of these errors is the basic nature of electric charges: ‘flowing from high potential to low’. In short, the direction of current, which is the flow of electricity, is determined only... | D |
where x∈𝔽n𝑥superscript𝔽𝑛x\in\mathbb{F}^{n}italic_x ∈ blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is the state and A∈𝔽n×n𝐴superscript𝔽𝑛𝑛A\in\mathbb{F}^{n\times n}italic_A ∈ blackboard_F start_POSTSUPERSCRIPT italic_n × italic_n end_POSTSUPERSCRIPT is the state transition map represented as ... | Irrespective of whether the dynamics (2) being linear or not, the Koopman operator 𝐊𝐊\mathbf{K}bold_K is a linear operator over the function space ℱ(𝔽n)ℱsuperscript𝔽𝑛\mathcal{F}(\mathbb{F}^{n})caligraphic_F ( blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ). This linearity of the Koopman operator... | When the dynamics is non-linear, the computation of the cycle set is a computationally hard problem. Apart from brute force computations, the work [26] gives an algorithmic procedure to estimate the cycle set of a non-linear dynamical system over finite fields by using the Koopman operator and constructing a reduced Ko... | The first statement of Theorem 3 does not imply an equivalence between the cycle structure of the permutation polynomial and the cycle set of the linear dynamics (19), and the former is a subset of the latter. This is because the linear dynamics evolve over a larger set 𝔽Nsuperscript𝔽𝑁\mathbb{F}^{N}blackboard_F star... | Initially, the Koopman operator framework was used extensively for dynamics over reals (or complex) state space, and the function space is infinite-dimensional, which leads to resorting to finite-dimensional numerical approximations of the Koopman operator [28, 29] for practical computations. In our setting of dynamica... | B |
We use the same software as described in Section 4.2. All cross-validation loops used for parameter tuning are nested within the outer loop used for evaluating classification performance. We again use the recommendations of Hofner \BOthers. (\APACyear2015) for choosing the parameters, by specifying q𝑞qitalic_q and a ... |
In this article we investigate how the choice of meta-learner affects the view selection and classification performance of MVS. We compare the following meta-learners: (1) the interpolating predictor of Breiman (\APACyear1996), (2) nonnegative ridge regression (Hoerl \BBA Kennard, \APACyear1970; Le Cessie \BBA Van Hou... | The results of applying MVS with the seven different meta-learners to the colitis data can be observed in Table 2. In terms of raw test accuracy the nonnegative lasso is the best performing meta-learner, followed by the nonnegative elastic net and the nonnegative adaptive lasso. In terms of AUC and H, the best performi... | The results for the breast cancer data can be observed in Table 3. The interpolating predictor and the lasso are the best performing meta-learners in terms of all three classification measures, with the interpolating predictor having higher test accuracy and H, and the lasso having higher AUC. However, the interpolatin... |
The nonnegative lasso, utilizing only an L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT penalty, produced even sparser models than the elastic net. Interestingly, in our simulations this increased sparsity did not appear to have a substantial negative effect on accuracy, although some minor reduct... | B |
Computational costs of DepAD algorithms mainly come from the relevant variable selection and prediction model training phases. Different techniques have different time complexities. We summarize the average running times (in seconds) of the five relevant variable selection techniques and the five prediction models. It ... |
As shown in Figure 3(a), the two causal feature selection techniques, HITON-PC and FBED, show better performance than the other three techniques. HITON-PC has the best average results, followed by FBED, IEPC, MI and DC. From the p𝑝pitalic_p-values shown in the figure, HITON-PC is significatly better than MI and DC, a... | In Algorithm 1, the relevant variable selection is performed for each variable at Line 2, followed by the prediction model training at Line 3. The anomaly score generation phase occurs between Lines 5 and 14, where expected value estimation is carried out at Line 7, and the dependency deviations are computed at Line 8.... | The running times on the 32 datasets and their average values are shown in Table 10. Comparing the five methods, FBED is the most efficient, with an average running time of 2.7 seconds, followed by MI at 23 seconds, HITON-PC at 26 seconds, DC at 133 seconds, and IEPC being the most time-consuming at 1538 seconds. Notab... | Computational costs of DepAD algorithms mainly come from the relevant variable selection and prediction model training phases. Different techniques have different time complexities. We summarize the average running times (in seconds) of the five relevant variable selection techniques and the five prediction models. It ... | C |
Comparison with Oh & Iyengar [2019] The Thompson Sampling based approach is inherently different from our Optimism in the face of uncertainty (OFU) style Algorithm CB-MNL. However, the main result in Oh & Iyengar [2019] also relies on a confidence set based analysis along the lines of Filippi et al. [2010] but has a m... | In this paper, we build on recent developments for generalized linear bandits (Faury et al. [2020]) to propose a new optimistic algorithm, CB-MNL for the problem of contextual multinomial logit bandits. CB-MNL follows the standard template of optimistic parameter search strategies (also known as optimism in the face of... | Comparison with Faury et al. [2020] Faury et al. [2020] use a bonus term for optimization in each round, and their algorithm performs non-trivial projections on the admissible log-odds. While we do reuse the Bernstein-style concentration inequality as proposed by them, their results do not seem to extend directly to th... |
CB-MNL enforces optimism via an optimistic parameter search (e.g. in Abbasi-Yadkori et al. [2011]), which is in contrast to the use of an exploration bonus as seen in Faury et al. [2020], Filippi et al. [2010]. Optimistic parameter search provides a cleaner description of the learning strategy. In non-linear reward mo... | In this work, we proposed an optimistic algorithm for learning under the MNL contextual bandit framework. Using techniques from Faury et al. [2020], we developed an improved technical analysis to deal with the non-linear nature of the MNL reward function. As a result, the leading term in our regret bound does not suffe... | B |
Video self-stitching (VSS). For both datasets, VSS shows its effectiveness in improving short actions whether used with or without xGPN. For THUMOS, because most actions are short, the overall performance also has a boost with VSS. For ActivityNet, VSS sacrifices long actions since it reduces the bias towards long act... |
In this paper, to tackle the challenging problem of large action scale variation in the temporal action localization (TAL) problem, we target short actions and propose a multi-level cross-scale solution called video self-stitching graph network (VSGN). It contains a video self-stitching (VSS) component that generates ... |
Video self-stitching (VSS). For both datasets, VSS shows its effectiveness in improving short actions whether used with or without xGPN. For THUMOS, because most actions are short, the overall performance also has a boost with VSS. For ActivityNet, VSS sacrifices long actions since it reduces the bias towards long act... | Specifically, we propose a Video self-Stitching Graph Network (VSGN) for improving performance of short actions in the TAL problem. Our VSGN is a multi-level cross-scale framework that contains two major components: video self-stitching (VSS); cross-scale graph pyramid network (xGPN). In VSS, we focus on a short period... | Cross-scale graph pyramid network (xGPN). From Table 3 and 4, we can see that xGPN obviously improves the performance of short actions as well as the overall performance. On the one hand, xGPN utilizes long-range correlations in multi-level features and benefits actions of various lengths. On the other hand, xGPN enabl... | D |
This action is noticeable due to specific play/stop glyphs, as illustrated in VisEvol: Visual Analytics to Support Hyperparameter Search through Evolutionary Optimization(g), top-left corner.
There is one grid per different ML algorithm, plus one grid for the overall values (leftmost grid in Figure 2(d.1)), and all the... |
At this point, the importance of \raisebox{0.15pt}{\resizebox{!}{0.8ex}{\textbf{\textsf{C1}}}}⃝ and \raisebox{0.15pt}{\resizebox{!}{0.8ex}{\textbf{\textsf{C3}}}}⃝ is clear, so we decide to gradually scan for the in-depth connections of the models belonging to the remaining clusters and the data instances (Step 3 in Fi... | (2) project the models into a hyperparameter embedding according to the previous overall performance using DR methods; (3) compare the mean performance of all algorithms and models vs. a selection of models for every metric; and (4) analyze the predictive results for each instance and for all models against a selection... | If a data set contains fewer than 169 instances per class, then we display all of them in the grid.
Otherwise, in order to scale the visualization to larger data sets (e.g., our next use case), we first build a grid with a fixed number of cells (100). Then, we use K-means clustering to place the data set’s instances wi... | Each cell of the grid (as shown in Figure 2(d.2–d.4)) then presents the computed difference in predictive power for all its instances (from −--100% to +++100%) for the selected against all models. The color-encoding diverges from purple to green for negative to positive difference.
In case the K-means clustering functi... | D |
The current literature covers a broad spectrum of methodologies for Markov chain synthesis, incorporating both heuristic approaches and optimization-based techniques [4, 5, 6]. Each method provides specialized algorithms tailored to the synthesis of Markov chains in alignment with specific objectives or constraints.
Ma... | Consensus protocols, in contrast to Markov chains, operate without the limitations of non-negative nodes and edges or the requirement for the sum of nodes to equal one [18]. This broader scope enables consensus protocols to address a significantly wider range of problem spaces.
Therefore, there is a significant interes... | There are comprehensive survey papers that review the research on consensus protocols [19, 20, 21, 22]. In many scenarios, the network topology of the consensus protocol is a switching topology due to failures, formation reconfiguration, or state-dependence. There is a large number of papers that propose consensus prot... | Consensus protocols form an important field of research that has a strong connection with Markov chains [18].
Consensus protocols are a set of rules used in distributed systems to achieve agreement among a group of agents on the value of a variable [19, 20, 21, 22]. |
Markov chains and consensus protocols share a close relationship. The rich theory of Markov chains has proven to be valuable in analyzing specific consensus protocols. Notable works such as [23, 24, 25, 26] have leveraged Markov chain theory to provide insights and analysis for consensus protocols. | C |
In principle, any pairwise shape matching method can be used for matching a shape collection. To do so, one can select one of the shapes as reference, and then solve a sequence of pairwise shape matching problems between each of the remaining shapes and the reference. However, a major disadvantage is that such an appr... | Despite the exponential size of the search space, there exist efficient polynomial-time algorithms to solve the LAP [11]. A downside of the LAP is that the geometric relation between points is not explicitly taken into account, so that the found matchings lack spatial smoothness. To compensate for this, a correspondenc... |
We presented a novel formulation for the isometric multi-shape matching problem. Our main idea is to simultaneously solve for shape-to-universe matchings and shape-to-universe functional maps. By doing so, we generalise the popular functional map framework to multi-matching, while guaranteeing cycle consistency, both ... | Alternatively, one could solve pairwise shape matching problems between all pairs of shapes in the shape collection. Although this way there is no bias, in general the resulting correspondences are not cycle-consistent. As such, matching shape A via shape B to shape C, may lead to a different correspondence than matchi... |
In principle, any pairwise shape matching method can be used for matching a shape collection. To do so, one can select one of the shapes as reference, and then solve a sequence of pairwise shape matching problems between each of the remaining shapes and the reference. However, a major disadvantage is that such an appr... | C |
On the side of path graphs, we believe that, compared to algorithms in [3, 22], our algorithm is simpler for several reasons: the overall treatment is shorter, the algorithm does not require complex data structures, its correctness is a consequence of the characterization in [1], and there are a few implementation deta... | Directed path graphs are characterized by Gavril [9], in the same article he also gives the first recognition algorithms that has O(n4)𝑂superscript𝑛4O(n^{4})italic_O ( italic_n start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) time complexity. In the above cited article, Monma and Wei [18] give the second characterizati... |
On the side of directed path graphs, prior to this paper, it was necessary to implement two algorithms to recognize them: a recognition algorithm for path graphs as in [3, 22], and the algorithm in [4] that in linear time is able to determining whether a path graph is also a directed path graph. Our algorithm directly... | On the side of directed path graphs, at the state of art, our algorithm is the only one that does not use the results in [4], in which it is given a linear time algorithm able to establish whether a path graph is a directed path graph too (see Theorem 5 for further details). Thus, prior to this paper, it was necessary ... | We presented the first recognition algorithm for both path graphs and directed path graphs. Both graph classes are characterized very similarly in [18], and we extended the simpler characterization of path graphs in [1] to include directed path graphs as well; this result can be of interest itself. Thus, now these two ... | B |
UKfaculty: this network reflects the friendship among academic staff of a given Faculty in a UK university consisting of three separate schools UKfaculty . The original network contains 81 nodes, and the smallest group only has 2 nodes. The smallest group is removed for community detection in this paper.
| Before comparing these methods, we take some preprocessing to remove nodes that may have mixed memberships for community detection. For the Polbooks data, nodes labeled as “neutral” are removed. The smallest group with only 2 nodes in UKfaculty data is removed. Table 1 presents some basic information about the four dat... | published around the 2004 presidential election and sold by the online bookseller Amazon.com. In Polbooks, nodes represent books, edges represent frequent co-purchasing of books by the same buyers. Full information about edges and labels can be downloaded from http://www-personal.umich.edu/~mejn/netdata/. The original ... | UKfaculty: this network reflects the friendship among academic staff of a given Faculty in a UK university consisting of three separate schools UKfaculty . The original network contains 81 nodes, and the smallest group only has 2 nodes. The smallest group is removed for community detection in this paper.
| In this section, four real-world network datasets with known label information are analyzed to test the performances of our Mixed-SLIM methods for community detection. The four datasets can be downloaded from
http://www-personal.umich.edu/~mejn/netdata/. For the four datasets, the true labels are suggested by the origi... | A |
Here p(x|z)𝑝conditional𝑥𝑧p(x{\,|\,}z)italic_p ( italic_x | italic_z ) and p0subscript𝑝0p_{0}italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT are given in closed form.
In many statistical models, the computation of the integral in (3.8) is intractable. | Meanwhile, the right-hand side of (3.9) is known as the evidence lower bound, which avoids the integral in (3.8). The common practice of variational inference is to parameterize p𝑝pitalic_p by a finite-dimensional parameter and optimize such a parameter in (3.9), which, however, potentially leads to a bias in approxim... | See, e.g., Cheng et al. (2017); Cheng and Bartlett (2018); Xu et al. (2018); Durmus et al. (2019) and the references therein for the analysis of the Langevin MCMC algorithm.
Besides, it is shown that (discrete-time) Langevin MCMC can be viewed as (a discretization of) the Wasserstein gradient flow of KL[p(z),p(z|x))... | variational inference (Gershman and Blei, 2012; Kingma and Welling, 2019), policy optimization (Sutton et al., 2000; Schulman et al., 2015; Haarnoja et al., 2018), and GAN (Goodfellow et al., 2014; Arjovsky et al., 2017), and has achieved tremendous empirical successes.
However, | To circumvent such intractability, variational inference turns to minimize the KL divergence between a variational posterior p𝑝pitalic_p and the true posterior p(z|x)𝑝conditional𝑧𝑥p(z{\,|\,}x)italic_p ( italic_z | italic_x ) in
(3.8) (Wainwright and Jordan, 2008; Blei et al., 2017), yielding the following distribu... | D |
Except MaxPressure analysed above, GeneraLight achieves the best in Hangzhou with the mixedl configuration, while performs poorly in other scenarios. The reason is that GeneraLight trains several models on diverse generated traffic flows, and select the model in testing by matching the flow. Hence, it limits the genera... |
3) MetaVIM outperforms Individual RL, MetaLight and PrssLight with 827, 423 and 411, respectively. The main reason is that they learn the traffic signal’s policy only using its own observation and ignore the influence of the neighbors, while MetaVIM considers the neighbors as the unobserved part of the current signal ... | We can obtain the following findings: 1) Among these 5 models, the performance of Baseline is the worst. The reason is that it is hard to learn the effective decentralized policy independently in the multi-agent traffic signal control task, where one agent’s reward and transition are affected by its neighbors. 2) Compa... | The most straightforward RL baseline considers each intersection independently and models the task as a single agent RL problem [12]. However, the observation, received reward and dynamics of each traffic signal are closely related to its neighbors, and the coordination between signals should be modeled. Hence, optimiz... |
To learn effective decentralized policies, there are two main challenges. Firstly, it is impractical to learn an individual policy for each intersection in a city or a district containing thousands of intersections. Parameter sharing may help. However, each intersection has a different traffic pattern, and a simple sh... | A |
𝔽nsuperscript𝔽𝑛\mathbbm{F}^{n}blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT with m>n𝑚𝑛m\,>\,nitalic_m > italic_n where 𝔽=ℂ𝔽ℂ\mathbbm{F}\,=\,\mathbbm{C}blackboard_F = blackboard_C or ℝℝ\mathbbm{R}blackboard_R.
A solution of an underdetermined system is always semiregular if the Jacobian | is identical to the nullity of Jacobian of 𝐟𝐟\mathbf{f}bold_f at (𝐮∗,𝐯∗)subscript𝐮subscript𝐯(\mathbf{u}_{*},\mathbf{v}_{*})( bold_u start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , bold_v start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ),
implying (𝐮∗,𝐯∗)subscript𝐮subscript𝐯(\mathbf{u}_{*},\mathbf{v}_{*})( bold_u start_POST... | Let 𝐱↦𝐟(𝐱)maps-to𝐱𝐟𝐱\mathbf{x}\,\mapsto\,\mathbf{f}(\mathbf{x})bold_x ↦ bold_f ( bold_x ) be a smooth mapping with a semiregular
zero 𝐱∗subscript𝐱\mathbf{x}_{*}bold_x start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT and r=𝓇𝒶𝓃𝓀(𝐟𝐱(𝐱∗))𝑟𝓇𝒶𝓃𝓀subscript𝐟𝐱subscript𝐱r\,=\,\mathpzc{rank}\left(\,\mathbf{f}_{... | is surjective or, equivalently, of full row rank.
For instance, let (𝐮∗,𝐯∗)subscript𝐮subscript𝐯(\mathbf{u}_{*},\mathbf{v}_{*})( bold_u start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , bold_v start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) be a zero of a smooth mapping | ϕ(𝐮)=(𝐮,𝐠(𝐮))italic-ϕ𝐮𝐮𝐠𝐮\phi(\mathbf{u})\,=\,(\mathbf{u},\mathbf{g}(\mathbf{u}))italic_ϕ ( bold_u ) = ( bold_u , bold_g ( bold_u ) ) for 𝐮∈Λ𝐮Λ\mathbf{u}\in\Lambdabold_u ∈ roman_Λ.
Furthermore, the Jacobian ϕ𝐮(𝐮∗)subscriptitalic-ϕ𝐮subscript𝐮\phi_{\mathbf{u}}(\mathbf{u}_{*})italic_ϕ start_POSTSUBSCRIPT ... | C |
Following the influential work (?), we refer to the competitive ratio of an algorithm with an error-free prediction as the consistency of the algorithm, and to the competitive ratio with an adversarial prediction as its robustness. Several online optimization problems have been studied in this learning-augmented settin... |
In this work, we focus on the online variant of bin packing, in which the set of items is not known in advance but is rather revealed in the form of a sequence. Upon the arrival of a new item, the online algorithm must either place it into one of the currently open bins, as long as this action does not violate the bin... |
In this section, we present an experimental evaluation of the performance of our algorithms111The code on which the experiments are based is available at https://github.com/shahink84/BinPackingPredictions.. Specifically, in Section 6.1 we describe the benchmarks and the input generation model; in Section 6.2, we expan... | We give the first theoretical and experimental study of online bin packing with machine-learned predictions. Previous work on this problem has assumed ideal and error-free predictions that must be provided by a very powerful oracle, without any learnability considerations, as we discuss in more detail in Section 1.2. I... | In terms of analysis techniques, we note that the theoretical analysis of the algorithms we present is specific to the setting at hand and treats items “collectively”. In contrast, almost all known online bin packing algorithms are analyzed using a weighting technique (?), which treats each bin “individually” and indep... | C |
Each function ϕisubscriptitalic-ϕ𝑖\phi_{i}italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT transforms an open set U∈ℝ2𝑈superscriptℝ2U\in\mathbb{R}^{2}italic_U ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT into a neighborhood Visubscript𝑉𝑖V_{i}italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRI... |
The proposed framework overcomes the limitations of previous methods. First, we theoretically solve the problem of stitching partial meshes since every chart is informed about its local neighborhood. Second, our method can easily fill the missing spaces in the final mesh by adding a new mapping for the region of inter... | Efficient 3D object representations are fundamental building blocks of many computer vision and machine learning applications, ranging from robotic manipulation (Kehoe et al., 2015) to autonomous driving (Yang et al., 2018a). Contemporary 3D registration devices, such as LIDARs and depth cameras, generate these represe... | Practically speaking, our approach transforms the embedding of point cloud obtained from the base model to parametrize the bijective function represented by the MLP network. This function aims to find a mapping between a canonical 2D patch to the 3D patch on the surface of the target mesh. We condition the positioning ... |
Theoretically, such an approach should reconstruct a single smooth mesh. However, the model operates on an arbitrary given number of k𝑘kitalic_k discrete functions, where a single function is responsible for generating a single patch. Consequently, it produces the discrete number of k𝑘kitalic_k patches that are disj... | D |
Finally, we show how the proposed method can be applied to prominent problem of computing Wasserstein barycenters to tackle the problem of instability of regularization-based approaches under a small value of regularizing parameter. The idea is based on the saddle point reformulation of the Wasserstein barycenter probl... |
We proposed a decentralized method for saddle point problems based on non-Euclidean Mirror-Prox algorithm. Our reformulation is built upon moving the consensus constraints into the problem by adding Lagrangian multipliers. As a result, we get a common saddle point problem that includes both primal and dual variables. ... | Our paper technique can be generalized to non-smooth problems by using another variant of sliding procedure [34, 15, 23]. By using batching technique, the results can be generalized to stochastic saddle-point problems [15, 23]. Instead of the smooth convex-concave saddle-point problem we can consider general sum-type s... | Paper organization. This paper is organized as follows. Section 2 presents a saddle point problem of interest along with its decentralized reformulation. In Section 3, we provide the main algorithm of the paper to solve such kind of problems. In Section 4, we present the lower complexity bounds for saddle point problem... | Now we show the benefits of representing some convex problems as convex-concave problems on the example of the Wasserstein barycenter (WB) problem and solve it by the DMP algorithm. Similarly to Section (3), we consider a SPP in proximal setup and introduce Lagrangian multipliers for the common variables. However, in t... | C |
And from the bijection we can deduce that
∩(Tw)<∩(Gw∧Ts)subscript𝑇𝑤subscript𝐺𝑤subscript𝑇𝑠\cap(T_{w})<\cap(G_{w}\wedge T_{s})∩ ( italic_T start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) < ∩ ( italic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ∧ italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) for so... |
The remainder of this section is dedicated to express the problem in the context of the theory of cycle bases, where it has a natural formulation, and to describe an application. Section 2 sets some notation and convenient definitions. In Section 3 the complete graph case is analyzed. Section 4 presents a variety of i... | In this section we present some experimental results to reinforce
Conjecture 14. We proceed by trying to find a counterexample based on our previous observations. In the first part, we focus on the complete analysis of small graphs, that is: graphs of at most 9 nodes. In the second part, we analyze larger families of g... |
The study of cycles of graphs has attracted attention for many years. To mention just three well known results consider Veblen’s theorem [2] that characterizes graphs whose edges can be written as a disjoint union of cycles, Maclane’s planarity criterion [3] which states that planar graphs are the only to admit a 2-ba... | necessarily complete) G=(V,E)𝐺𝑉𝐸G=(V,E)italic_G = ( italic_V , italic_E ) that admits a star spanning tree
Tssubscript𝑇𝑠T_{s}italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT. In the first part we present a formula to calculate ∩(Ts)subscript𝑇𝑠\cap(T_{s})∩ ( italic_T start_POSTSUBSCRIPT italic_s end_POSTSU... | B |
Fix a simplicial complex K𝐾Kitalic_K, a value δ∈(0,1]𝛿01\delta\in(0,1]italic_δ ∈ ( 0 , 1 ], and integers b≥1𝑏1b\geq 1italic_b ≥ 1 and m>μ(K)𝑚𝜇𝐾m>\mu(K)italic_m > italic_μ ( italic_K ). If ℱℱ\mathcal{F}caligraphic_F is a sufficiently large (K,b)𝐾𝑏(K,b)( italic_K , italic_b )-free cover such that πm(ℱ)≥δ(|ℱ|m)... | One immediate application of Theorem 1.2 is the reduction of fractional Helly numbers. For instance, it easily improves a theorem444[35, Theorem 2.3] was not phrased in terms of (K,b)𝐾𝑏(K,b)( italic_K , italic_b )-free covers but readily generalizes to that setting, see Section 1.4.1. of Patáková [35, Theorem 2.3] in... |
It is known that the Helly number of a (K,b)𝐾𝑏(K,b)( italic_K , italic_b )-free cover is bounded from above in terms of K𝐾Kitalic_K and b𝑏bitalic_b [18] 222The bound on Helly number of (K,b)-free cover directly follows from a combination of Proposition 30 and Lemma 26 in [18]., as is the Radon number [35, Proposit... |
Through a series of papers [18, 35, 22], the Helly numbers, Radon numbers, and fractional Helly numbers for (⌈d/2⌉,b)𝑑2𝑏(\lceil d/2\rceil,b)( ⌈ italic_d / 2 ⌉ , italic_b )-covers in ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT were bounded in terms of d𝑑ditalic_d and... |
Note that the constant number of points given by the (p,q)𝑝𝑞(p,q)( italic_p , italic_q )-theorem in this case depends not only on p𝑝pitalic_p, q𝑞qitalic_q, and d𝑑ditalic_d, but also on b𝑏bitalic_b. For the setting of (1,b)1𝑏(1,b)( 1 , italic_b )-covers in surfaces555By a surface we mean a compact 2-dimensional ... | A |
The ultimate goal is to exclude non-contributing features prior to extensive analyses within FeatureEnVi.
Finally, E3 discussed a recent example application in design materials [102] that he worked with, which could benefit from our tool. In particular, he said: “automatic ML can combine materials (i.e., features) with... | Visualization and interaction.
E1 and E2 were surprised by the promising results we managed to achieve with the assistance of our VA system in the red wine quality use case of Section 4. Initially, E1 was slightly overwhelmed by the number of statistical measures mapped in the system’s glyphs. However, after the interv... | Workflow.
All experts commented that the workflow of FeatureEnVi is straightforward, because it is mainly linear despite involving optional iterative steps. E2 stated that feature engineering is usually very time consuming, especially without the support of a system like ours. E3 also agreed with us that the features h... |
In FeatureEnVi, data instances are sorted according to the predicted probability of belonging to the ground truth class, as shown in Fig. 1(a). The initial step before the exploration of features is to pre-train the XGBoost [29] on the original pool of features, and then divide the data space into four groups automati... |
Figure 1: Selecting important features, transforming them, and generating new features with FeatureEnVi: (a) the horizontal beeswarm plot for manually slicing the data space (which is sorted by predicted probabilities) and continuously checking the migration of data instances throughout the process; (b) the table heat... | A |
The goal is to tune the parameters of the MPC-based planning unit without introducing any modification in the structure of the underlying control system.
We leverage the repeatability of the system, which is higher than the integrated encoder error of 3μm3𝜇𝑚3\mu m3 italic_μ italic_m, | MPC accounts for the real behavior of the machine and the axis drive dynamics can be excited to compensate for the contour error to a big extent, even without including friction effects in the model [4, 5]. High-precision trajectories or set points can be generated prior to the actual machining process following variou... | The physical system is a 2-axis gantry stage for (x,y)𝑥𝑦(x,y)( italic_x , italic_y ) positioning with industrial grade actuators and sensors [14].
The plant can be modeled as a mass-spring-damper system with two masses linked with a damper and a spring for capturing imperfection and friction in the transmitting movem... | To bring the model close to the real system, we unify the terms required for the contour control formulation with the velocity and acceleration for each axis from the identified, discretized state-space model from (4).
Also, we include the path progress sksubscript𝑠𝑘s_{k}italic_s start_POSTSUBSCRIPT italic_k end_POST... | which is an MPC-based contouring approach to generate optimized tracking references. We account for model mismatch by automated tuning of both the MPC-related parameters and the low level cascade controller gains, to achieve precise contour tracking with micrometer tracking accuracy. The MPC-planner is based on a combi... | B |
It is unknown how well the methods scale up to multiple sources of biases and large number of groups, even when they are explicitly annotated. To study this, we train the explicit methods with multiple explicit variables for Biased MNISTv1 and individual variables that lead to hundreds and thousands of groups for GQA ... | To test scalability on a natural dataset, we conduct four experiments per explicit method on GQA-OOD with the explicit bias variables: a) head/tail (2 groups), b) answer class (1833 groups), c) global group (115 groups), and d) local group (133328 groups). Unlike Biased MNISTv1, we do not test with combinations of thes... | We use the GQA visual question answering dataset [33] to highlight the challenges of using bias mitigation methods on real-world tasks. It has multiple sources of biases including imbalances in answer distribution, visual concept co-occurrences, question word correlations, and question type/answer distribution. It is u... | Results for GQA-OOD are similar, with explicit methods failing to scale up to a large number of groups, while implicit methods showing some improvements over StdM. As shown in Table 2, when the number of groups is small, i.e., when using a head/tail binary indicator as the explicit bias, explicit methods remain compara... |
where, |ai|subscript𝑎𝑖|a_{i}|| italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | is the number of instances for answer aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in the given group, μ(a)𝜇𝑎\mu(a)italic_μ ( italic_a ) is the mean number of answers in the group and β𝛽\betait... | A |
Different types of input have been explored to extract features. Kellnhofer et al. directly extract features from facial images [43]. Zhou et al. combine the feature extracted from facial and eye images [84]. Palmero et al. use facial images, binocular images and facial landmarks to generate the feature vectors [79].
D... | Conventional approaches typically estimate gaze using eye images. The Mnist [17] and GazeNet [49] methods employ eye images and head pose vector as input for gaze estimation.
Recent methods, i.e., the third-row methods, focus on estimating gaze from facial images. | These works usually use a three-stream network to extract features from face images, left and right eye images, respectively as shown in Fig. 4 (c) [42, 53, 73, 74, 75].
Besides, Deng et al. [76] decompose gaze directions into the head rotation and eyeball rotation. |
Decomposition of Gaze Direction. Human gaze can be decomposed into the head pose and the eyeball pose. Deng et al. use two CNNs to estimate head pose from facial images and eyeball pose from eye images. They integrate these two results into final gaze directions using geometric transformation [76]. | Some works seek to decompose the gaze into multiple related features and construct multi-task CNNs to estimate these feature. Yu et al. introduce a constrained landmark-gaze model for modeling the joint variation of eye landmark locations and gaze directions [119]. As shown in Fig. 9, they build a multi-task CNN to est... | C |
To evaluate the proposed method, we carried out experiments on very challenging masked face datasets. In the following, we present the datasets’ content and variations, the experimental results using the quantization of deep features obtained from three pre-trained models, and a comparative study with other state-of-t... | This deep quantization technique presents many advantages. It ensures a lightweight representation that makes the real-world masked face recognition process a feasible task. Moreover, the masked regions vary from one face to another, which leads to informative images of different sizes. The proposed deep quantization a... |
Real-World-Masked-Face-Dataset wang2020masked is a masked face dataset devoted mainly to improve the recognition performance of the existing face recognition technology on the masked faces during the COVID-19 pandemic. It contains three types of images namely, Masked Face Detection Dataset (MFDD), Real-world Masked F... | Experimental results are carried out on Real-world Masked Face Recognition Dataset (RMFRD) and Simulated Masked Face Recognition Dataset (SMFRD) presented in wang2020masked . We start by localizing the mask region. To do so, we apply a cropping filter in order to obtain only the informative regions of the masked face (... | The COVID-19 can be spread through contact and contaminated surfaces, therefore, the classical biometric systems based on passwords or fingerprints are not anymore safe. Face recognition is safer without any need to touch any device. Recent studies on coronavirus have proven that wearing a face mask by a healthy and in... | B |
⊢iy←oddsix::(y:streamA[i])\displaystyle\vdash^{i}y\leftarrow\operatorname{odds}\,i~{}x::(y:\operatorname%
{stream}_{A}[i])⊢ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_y ← roman_odds italic_i italic_x : : ( italic_y : roman_stream start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT [ italic_i ] ) |
That is, we assume that we have definitions that (1) append two lists together and (2) partitions one by a pivot. Then, at a high level, quicksort is a size-preserving definition with the input list length as its termination measure. For brevity, we nest patterns (boxed and highlighted yellow), which can be expanded i... |
As we mentioned in the introduction, we can make the SAX∞superscriptSAX\text{SAX}^{\infty}SAX start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT judgment arbitrarily rich to support more complex patterns of recursion. As long as derivations in that system can be translated to SAXωsuperscriptSAX𝜔\text{SAX}^{\omega}SAX start_... |
Our system is closely related to the sequential functional language of Lepigre and Raffalli [LR19], which utilizes circular typing derivations for a sized type system with mixed inductive-coinductive types, also avoiding continuity checking. In particular, their well-foundedness criterion on circular proofs seems to c... |
First, we define head and tail observations on streams of arbitrary depth. Since they are not recursive, we do not bother tracking the size superscript of the typing judgment, since they can be inlined. Moreover, we take the liberty to nest values (boxed and highlighted yellow), which can be expanded into SAX [PP20]. | D |
A watermarking technique being able to safeguard the user’s rights while maintaining the owner’s copyright is called Asymmetric Fingerprinting (AFP) [9, 10, 11, 12, 13, 14]. AFP mainly relies on cryptographic tools including public key cryptosystem and homomorphic encryption, in which the embedding operation is perform... |
As discussed above, AFP seems to solve Problems 2 and 3 perfectly. However, this is no longer the case when media contents are remotely hosted by the cloud since existing AFP schemes were designed without taking the cloud’s involvement into consideration. Thus it remains to be further explored how to develop a novel A... |
By delegating the management of the media content to the cloud, FairCMS-I and FairCMS-II can also be seen as an instantiation of privacy-preserving outsourcing of AFP, thereby solving the problem caused by insufficient local resources of the owner in media sharing. |
In this paper, facing these problems and challenges, we set out to solve them. First, to achieve data protection and access control, we adopt the lifted-ElGamal based PRE scheme, as discussed in [16, 17, 18, 19, 20], whose most prominent characteristic is that it satisfies the property of additive homomorphism. Then t... | In the user-side embedding AFP, since the encrypted media content shared with different users is the same, the encryption of the media content is only executed once. In contrast, due to the personalization of D-LUTs, once a new user initiates a request, the owner must interact with this user to securely distribute the ... | A |
Though based on graph spectral theory Bruna et al. (2013), the learning process of graph convolutional networks (GCN) Kipf and Welling (2017) also can be considered as a mean-pooling neighborhood aggregation.
GraphSAGE Hamilton et al. (2017) concatenates the node features and introduces three | At their core, GNNs learn node embeddings by iteratively aggregating features from the neighboring nodes, layer by layer. This allows them to explicitly encode high-order relationships between nodes in the embeddings. GNNs have shown great potential for modeling high-order feature interactions for click-through rate pr... |
To capture the diversified polysemy of feature interactions in different semantic subspaces Li et al. (2020) and also stabilize the learning process Vaswani et al. (2017); Veličković et al. (2018), we extend our mechanism to employ multi-head attention. | Due to the strength in modeling relations on graph-structured data, GNN has been widely applied to various applications like neural machine translation Beck et al. (2018), semantic segmentation Qi et al. (2017), image classification Marino et al. (2017), situation recognition Li et al. (2017), recommendation Wu et al. ... |
Currently, Graph Neural Networks (GNN) Kipf and Welling (2017); Hamilton et al. (2017); Veličković et al. (2018) have recently emerged as an effective class of models for capturing high-order relationships between nodes in a graph and have achieved state-of-the-art results on a variety of tasks such as computer vision... | C |
where Q𝑄Qitalic_Q is a symmetric positive definite matrix with log-normally distributed eigenvalues and φℝ+(⋅)subscript𝜑subscriptℝ⋅\varphi_{\mathbb{R}_{+}}(\cdot)italic_φ start_POSTSUBSCRIPT blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ⋅ ) |
Furthermore, with this simple step size we can also prove a convergence rate for the Frank-Wolfe gap, as shown in Theorem 2.6. More specifically, the minimum of the Frank-Wolfe gap over the run of the algorithm converges at a rate of 𝒪(1/t)𝒪1𝑡\mathcal{O}(1/t)caligraphic_O ( 1 / italic_t ). The idea of the proof is... | In practice, a halving strategy for the step size is preferred for the
implementation of the Monotonic Frank-Wolfe algorithm, as opposed to the step size implementation shown in Algorithm 1. This halving strategy, which is shown in Algorithm 2, helps | The stateless step-size does not suffer from this problem, however, because the halvings have to be performed at multiple iterations when using the stateless step-size strategy,
the per iteration cost of the stateless step-size is about three times that of the simple step-size. | The results are shown in Figure 7. On both of these instances, the simple step progress is slowed down or even seems stalled in comparison to the stateless
version because a lot of halving steps were done in the early iterations for the simple step size, which penalizes progress over the whole run. | D |
One that has attracted a lot of attention, especially in the past decade, is the graph stream model, which was introduced by Feigenbaum et al. [FKM+04, FKM+05, Mut05] in 2005.
In this model, the edges of the graph are not stored in the memory but appear in an arbitrary (that is, adversarially determined) sequential ord... |
In a new pass, for each edge e={u,v}𝑒𝑢𝑣e=\{u,v\}italic_e = { italic_u , italic_v } in the stream, the algorithm checks whether the structure containing u𝑢uitalic_u and the structure containing v𝑣vitalic_v, if such structures exist, can augment over e𝑒eitalic_e. If it is possible, via Augment-and-Clean the algori... |
It is known that finding an exact matching requires linear space in the size of the graph and hence it is not possible to find an exact maximum matching in the semi-streaming model [FKM+04], at least for sufficiently dense graphs. Nevertheless, this result does not apply to computing a good approximation to the maximu... | One that has attracted a lot of attention, especially in the past decade, is the graph stream model, which was introduced by Feigenbaum et al. [FKM+04, FKM+05, Mut05] in 2005.
In this model, the edges of the graph are not stored in the memory but appear in an arbitrary (that is, adversarially determined) sequential ord... | In particular, it is desirable that the number of passes is independent of the input graph size.
We call an algorithm a k𝑘kitalic_k-pass algorithm if the algorithm makes k𝑘kitalic_k passes over the edge stream, possibly each time in a different order [MP80, FKM+05]. | D |
We propose CPP – a novel decentralized optimization method with communication compression. The method works under a general class of compression operators and is shown to achieve linear convergence for strongly convex and smooth objective functions over general directed graphs. To the best of our knowledge, CPP is the... | In this paper, we proposed two communication-efficient algorithms for decentralized optimization over a multi-agent network with general directed topology. First, we consider a novel communication-efficient gradient tracking based method, termed CPP, that combines the Push-Pull method with communication compression. CP... | In this section, we compare the numerical performance of CPP and B-CPP with the Push-Pull/𝒜ℬ𝒜ℬ\mathcal{A}\mathcal{B}caligraphic_A caligraphic_B method [24, 25].
In the experiments, we equip CPP and B-CPP with different compression operators and consider different graph topologies. |
We consider an asynchronous broadcast version of CPP (B-CPP). B-CPP further reduces the communicated data per iteration and is also provably linearly convergent over directed graphs for minimizing strongly convex and smooth objective functions. Numerical experiments demonstrate the advantages of B-CPP in saving commun... | In the second part of this paper, we propose a broadcast-like CPP algorithm (B-CPP) that allows for asynchronous updates of the agents: at every iteration of the algorithm, only a subset of the agents wake up to perform prescribed updates. Thus, B-CPP is more flexible, and due to its broadcast nature, it can further sa... | C |
Possible interesting areas for further research are related to the practical features that arise in the federated learning setup, such as asynchronous transmissions and information compression to minimize communication costs, among other issues. It is worth considering the use of the variance reduction technique in acc... | The inclusion of noise ymsubscript𝑦𝑚y_{m}italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT in the optimization process adds an interesting dimension to the standard loss function of a machine learning model. While the primary objective is still the minimization of the loss, the maximization of the noise term pl... | Discussions. We compare algorithms based on the balance of the local and global models, i.e. if the algorithm is able to train well both local and global models, then we find the FL balance by this algorithm. The results show that the Local SGD technique (Algorithm 3) outperformed the Algorithm 1 only with a fairly fre... |
This work was supported by a grant for research centers in the field of artificial intelligence, provided by the Analytical Center for the Government of the Russian Federation in accordance with the subsidy agreement (agreement identifier 000000D730324P540002) and the agreement with the Moscow Institute of Physics and... | For this case we present Algorithm 2. This algorithm is the Tseng method [44] with a resolvent/proximal operator calculation (4). Here, as in Algorithm 1, the proximal operator is computed inexactly. Note that we need to communicate with other devices only when we solve the problem (4) and need to multiply by the matri... | C |
There is a rich polytope of possible equilibria to choose from, however, an MS must pick one at each time step. There are three competing properties which are important in this regard, exploitation, robustness, and exploration. For exploitation, maximum welfare equilibria appear to be useful. However, to prevent JPSRO... | We have shown that JPSRO converges to an NF(C)CE over joint policies in extensive form and stochastic games. Furthermore, there is empirical evidence that some MSs also result in high value equilibria over a variety of games. We argue that (C)CEs are an important concept in evaluating policies in n-player, general-sum ... | PSRO has proved to be a formidable learning algorithm in two-player, constant-sum games, and JPSRO, with (C)CE MSs, is showing promising results on n-player, general-sum games. The secret to the success of these methods seems to lie in (C)CEs ability to compress the search space of opponent policies to an expressive an... | In this work we propose using correlated equilibrium (CE) (Aumann, 1974) and coarse correlated equilibrium (CCE) as a suitable target equilibrium space for n-player, general-sum games333We mean games (also called environments) in a very general sense: extensive form games, multi-agent MDPs and POMDPs (stochastic games)... |
In Section 2 we provide background on a) correlated equilibrium (CE), an important generalization of NE, b) coarse correlated equilibrium (CCE) (Moulin & Vial, 1978), a similar solution concept, and c) PSRO, a powerful multi-agent training algorithm. In Section 3 we propose novel solution concepts called Maximum Gini ... | A |
The second part is a direct result of the known variational representation of total variation distance and χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divergence, which are both f𝑓fitalic_f-divergences (see Equations 7.88 and 7.91 in Polyanskiy and Wu (2022) for more details). | Our Covariance Lemma (3.5) shows that there are two possible ways to avoid adaptivity-driven overfitting—by bounding the Bayes factor term, which induces a bound on |q(Dv)−q(D)|𝑞superscript𝐷𝑣𝑞𝐷\left|{q}\left(D^{v}\right)-{q}\left(D\right)\right|| italic_q ( italic_D start_POSTSUPERSCRIPT italic_v end_POSTSUPERSC... |
These results extend to the case where the variance (or variance proxy) of each query qisubscript𝑞𝑖q_{i}italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is bounded by a unique value σi2superscriptsubscript𝜎𝑖2\sigma_{i}^{2}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_PO... | Since achieving posterior accuracy is relatively straightforward, guaranteeing Bayes stability is the main challenge in leveraging this theorem to achieve distribution accuracy with respect to adaptively chosen queries. The following lemma gives a useful and intuitive characterization of the quantity that the Bayes sta... | Using the first part of the lemma, we guarantee Bayes stability by bounding the correlation between specific q𝑞qitalic_q and K(⋅,v)𝐾⋅𝑣{K}\left(\cdot,v\right)italic_K ( ⋅ , italic_v ) as discussed in Section 6. The second part of this Lemma implies that bounding the appropriate divergence is necessary and sufficient... | D |
We have taken the first steps into a new direction for preprocessing which aims to investigate how and when a preprocessing phase can guarantee to identify parts of an optimal solution to an 𝖭𝖯𝖭𝖯\mathsf{NP}sansserif_NP-hard problem, thereby reducing the running time of the follow-up algorithm. Aside from the techni... |
This line of investigation opens up a host of opportunities for future research. For combinatorial problems such as Vertex Cover, Odd Cycle Transversal, and Directed Feedback Vertex Set, which kinds of substructures in inputs allow parts of an optimal solution to be identified by an efficient preprocessing phase? Is i... | The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly [26, 31]. In a nutshell, this new direction can be summarized as: how can an algorithm identify part of an optimal solution in an efficient prepro... |
As the first step of our proposed research program into parameter reduction (and thereby, search space reduction) by a preprocessing phase, we present a graph decomposition for Feedback Vertex Set which can identify vertices S𝑆Sitalic_S that belong to an optimal solution; and which therefore facilitate a reduction fr... | We have taken the first steps into a new direction for preprocessing which aims to investigate how and when a preprocessing phase can guarantee to identify parts of an optimal solution to an 𝖭𝖯𝖭𝖯\mathsf{NP}sansserif_NP-hard problem, thereby reducing the running time of the follow-up algorithm. Aside from the techni... | A |
To alleviate the burden of manually annotating masks and removing shadows, [96] design an automatic pipeline to construct shadow generation dataset and contributed a larger-scale dataset DESOBAv2. Specifically, [96] employ the pretrained object-shadow detection model [163] to predict object-shadow masks and employ the... | To evaluate the quality of generated composite images with foreground shadows, existing shadow generation works [189] without paired data adopt Frechet Inception Distance (FID) [50] and Manipulation Score (MS) [12] to measure the realism of generated shadow images. For the works [92, 52] with paired data, they adopt St... | In real-world application scenarios, there exist no ground-truth images for a pair of foreground and background, so we cannot calculate the distance between generated image and ground-truth image. Therefore, previous works [183, 141, 16, 205] used FID [50] to measure the discrepancy between generated images and real im... | To evaluate the quality of generated composite images, previous object placement works usually adopt the following three schemes: 1) Some works measure the similarity between real images and composite images. For example, Tan et al. [145] score the correlation between the distributions of predicted boxes and ground-tru... |
Similar to image harmonization in Section IV, composite images without foreground shadows can be easily obtained. Nonetheless, it is very difficult to obtain paired data, i.e., a composite image without foreground shadow and a ground-truth image with foreground shadow, which are required by supervised deep learning me... | A |
To evaluate the performance of these methods, we implement a simulator based on historical request data using the approach presented in [6, 22]. This simulator allows us to simulate the operation of the taxi system and assess the effectiveness of each method. | Problem Statement. To address the taxi dispatching task, we learn a real-time dispatching policy based on historical passenger requests. At every timestamp τ𝜏\tauitalic_τ, we use this policy to dispatch available taxis to current passengers, with the aim of maximizing the total revenue of all taxis in the long run. To... |
Efficient taxi allocation is crucial for the passenger transportation services in smart cities. To address this challenge, we leverage the data available in CityNet and present benchmarks for the taxi dispatching task. In this task, operators are responsible for dispatching available taxis to waiting passengers in rea... | Table VII presents the results of our inter-city transfer learning experiments. Specifically, we report the results obtained by training our models using both full and 3-day target data, which correspond to the lower and upper bounds of errors, respectively. Furthermore, we also include the results of fine-tuning and R... | Table VIII presents the taxi dispatching results for Chengdu, where the completion rate denotes the ratio of completed requests within all requests, and accumulated revenue represents the total revenue earned by all taxis throughout the day. Based on the experimental results, we draw the following conclusions:
| D |
Although a variety of methods was considered, it is not feasible to include all of them. The most important omission is a more detailed overview of Bayesian neural networks (although one can argue, as was done in the section on dropout networks, that some common neural networks are, at least partially, Bayesian by nat... | In Fig. 1, both the coverage degree, average width and R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-coefficient are shown. For each model, the data sets are sorted according to increasing R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-coefficient (averaged over th... | The choice of data sets in this comparative study was very broad and no specific properties were taken into account a priori. After comparing the results of the different models, it did become apparent that certain assumptions or properties can have a major influence on the performance of the models. The main examples ... | Most of the data sets were obtained from the UCI repository Dua2019 . Specific references are given in Table 2. This table also shows the number of data points and (used) features and the skewness and (Pearson) kurtosis of the response variable. All data sets were standardized (both features and target variables) befor... | For each of the selected models, Fig. 4 shows the best five models in terms of average width, excluding those that do not (approximately) satisfy the coverage constraint (2). This figure shows that there is quite some variation in the models. There is not a clear best choice. Because on most data sets the models produc... | B |
While genre classification categorises music based on shared musical attributes and conventions, style classification seeks to capture the nuanced stylistic variations within either a specific genre, composer or performer, accounting for the diverse artistic choices and performance practices that shape musical expressi... | In what follows, we use ‘our model (score)’ to indicate the result when MIDI scores are considered and similarly ‘our model (performance)’ for MIDI performances.
Since MIDI performance contains velocity information, we do not evaluate on the velocity prediction task for fairness. We note that, while ‘our model (score)’... |
As an additional baseline for style and emotion classification, we implement the ResNet50-based CNN model from \textcitelee20ismirLBD, which represents the state-of-the-art for composer classification, based on the authors’ code.151515https://github.com/KimSSung/Deep-Composer-Classification | Deep learning-based composer classification in MIDI has been attempted by \textcitelee20ismirLBD and \textcitekong2020largescale, both treating MIDI pieces as 2D-representation matrices (via the piano-roll representation) and using CNN classifiers. Our work differs from theirs in that: 1) we encode MIDI pieces as token... | Second, we aim at establishing a benchmark for symbolic music classification and include not only sequence-level but also note-level tasks. Furthermore, the labelled data we employ for our downstream tasks is comparatively modest, with each dataset containing fewer than 1,000 annotated pieces. This differs from MusicBE... | C |
In this paper, we turn our attention to the special case when the graph is complete (denoted Knsubscript𝐾𝑛K_{n}italic_K start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT) and its backbone is a (nonempty) tree or a forest (which we will denote by T𝑇Titalic_T and F𝐹Fitalic_F, respectively).
Note that it has a natural in... | This description draws a comparison e.g. to L(k,1)𝐿𝑘1L(k,1)italic_L ( italic_k , 1 )-labeling problem (see e.g. [10] for a survey), where the colors of any two adjacent vertices have to differ by at least k𝑘kitalic_k and the colors of any two vertices within distance 2222 have to be distinct.
| First, we note that Z(S2)𝑍subscript𝑆2Z(S_{2})italic_Z ( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) by the property (A)𝐴(A)( italic_A ) of the Zeckendorf representation does not have two consecutive ones. Thus, the only combinations available when we sum the rightmost blocks of type A (i.e. the ones which do... | We will color F𝐹Fitalic_F by assigning colors to Y1subscript𝑌1Y_{1}italic_Y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, B1subscript𝐵1B_{1}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and R1subscript𝑅1R_{1}italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT first, and then to Y2subscript𝑌2Y_{2}italic_Y start_POSTSUBS... |
Since all vertices in c𝑐citalic_c have different colors, it is true that |Y|≤l𝑌𝑙|Y|\leq l| italic_Y | ≤ italic_l. Moreover, the optimality of c𝑐citalic_c implies that both R𝑅Ritalic_R and B𝐵Bitalic_B are non-empty. From the fact that c𝑐citalic_c is a coloring of Knsubscript𝐾𝑛K_{n}italic_K start_POSTSUBSCRIPT ... | A |
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