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The generic third order Newton’s Method—also known as Halley’s method—to compute roots f(x)=0𝑓𝑥0f(x)=0italic_f ( italic_x ) = 0 numerically
improves solutions xi→xi+1=xi+Δx→subscript𝑥𝑖subscript𝑥𝑖1subscript𝑥𝑖Δ𝑥x_{i}\rightarrow x_{i+1}=x_{i}+\Delta xitalic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → ita... | \prime\prime}(x)+\frac{(\Delta x)^{3}}{3!}f^{\prime\prime\prime}(x)\approx 0.italic_f ( italic_x + roman_Δ italic_x ) ≈ italic_f ( italic_x ) + roman_Δ italic_x italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) + divide start_ARG ( roman_Δ italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG s... | \frac{f^{\prime\prime}(x)}{f^{\prime}(x)}\right)roman_Δ italic_x = - divide start_ARG italic_f ( italic_x ) end_ARG start_ARG italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) end_ARG / ( 1 - divide start_ARG italic_f ( italic_x ) end_ARG start_ARG 2 italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ... | {\prime}(x)}\left(h_{0}(x)\frac{f(x)}{f^{\prime}(x)}+h_{1}(x)\right)\right].roman_Δ italic_x = - divide start_ARG italic_f ( italic_x ) end_ARG start_ARG italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) end_ARG / [ 1 + divide start_ARG 1 end_ARG start_ARG 2 italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCR... | ^{2}}{6}\frac{f^{\prime\prime\prime}(x)}{f^{\prime}(x)}\approx 0,1 + divide start_ARG roman_Δ italic_x end_ARG start_ARG 2 end_ARG divide start_ARG italic_f start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x ) end_ARG start_ARG italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) end_ARG + divide sta... | B |
There are several well-known generating sets for classical groups. For example, special linear groups are generated by the subset of all transvections [21, Theorem 4.3] or by two well chosen matrices, such as the Steinberg generators [19]. Another generating set which has become important in algorithms and application... | One important task in this context is writing elements of classical groups as words in standard generators using SLPs. This is done in Magma [14] using the results of Elliot Costi [6] and in GAP using the results of this paper see Section 6. Other rewriting algorithms also exist, for example Cohen et al. [26] present a... | Note that a small variation of these standard generators for SL(d,q)SL𝑑𝑞\textnormal{SL}(d,q)SL ( italic_d , italic_q ) are used in Magma [14] as well
as in algorithms to verify presentations of classical groups, see [12], where only the generator v𝑣vitalic_v is slightly different in the two scenarios when d𝑑ditali... |
There are several well-known generating sets for classical groups. For example, special linear groups are generated by the subset of all transvections [21, Theorem 4.3] or by two well chosen matrices, such as the Steinberg generators [19]. Another generating set which has become important in algorithms and application... | The LGO generating set offers a variety of advantages. In practice it is the generating set produced by the constructive recognition algorithms from [10, 11] as implemented in MAGMA. Consequently, algorithms in the composition tree data structure, both in MAGMA and in GAP, store elements in classical groups as words in... | D |
To show the existence and uniqueness of solutions for (21), we proceed by parts. The existence of solution for the first equation follows from Lemma LABEL:l:lrmsystem. Solving the second equation is equivalent to (22), and such system is well-posed due to the coercivity of (⋅,T⋅)∂𝒯H(\cdot,T\cdot)_{{\partial\mathcal{T}... |
It is essential for the performing method that the static condensation is done efficiently. The solutions of (22) decay exponentially fast if w𝑤witalic_w has local support, so instead of solving the problems in the whole domain it would be reasonable to solve it locally using patches of elements. We note that the ide... |
Above, and in what follows, c𝑐citalic_c denotes an arbitrary constant that does not depend on H𝐻Hitalic_H, ℋℋ{\mathscr{H}}script_H, hℎhitalic_h, 𝒜𝒜\mathcal{A}caligraphic_A, depending only on the shape regularity of the elements of 𝒯Hsubscript𝒯𝐻{\mathcal{T}_{H}}caligraphic_T start_POSTSUBSCRIPT italic_H end_POST... | Except for (ii), all steps above above can be performed efficiently as the matrices involved are sparse and either local or independent of hℎhitalic_h. Solving (25) on the other hand involves computing the hℎhitalic_h-dependent, global operator P𝑃Pitalic_P, leading to a dense matrix in (25). From now on, we concentrat... |
The key to approximate (25) is the exponential decay of Pw𝑃𝑤Pwitalic_P italic_w, as long as w∈H1(𝒯H)𝑤superscript𝐻1subscript𝒯𝐻w\in{H^{1}({\mathcal{T}_{H}})}italic_w ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( caligraphic_T start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ) has local support. That al... | C |
The difference is mainly due to the degenerate case (where a chord of P𝑃Pitalic_P is parallel to an edge of P𝑃Pitalic_P) and the float issue of both programs.
Our implementations of Alg-K and Alg-CM have logical difference in handling degenerate cases. |
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.) | The difference is mainly due to the degenerate case (where a chord of P𝑃Pitalic_P is parallel to an edge of P𝑃Pitalic_P) and the float issue of both programs.
Our implementations of Alg-K and Alg-CM have logical difference in handling degenerate cases. | Moreover, Alg-A is more stable than the alternatives.
During the iterations of Alg-CM, the coordinates of three corners and two midpoints of a P-stable triangle (see Figure 37) are maintained. These coordinates are computed somehow and their true values can differ from their values stored in the computer. Alg-CM uses 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. | C |
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... | 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.
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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... | 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... |
As observed in [19, 20], rumor features are very prone to change during an event’s development. In order to capture these temporal variabilities, we build upon the Dynamic Series-Time Structure (DSTS) model (time series for short) for feature vector representation proposed in [20]. We base our credibility feature on t... | A |
\prime}\left(u\right)=0roman_lim start_POSTSUBSCRIPT italic_u → ∞ end_POSTSUBSCRIPT roman_ℓ ( italic_u ) = roman_lim start_POSTSUBSCRIPT italic_u → ∞ end_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u ) = 0), a β𝛽\betaitalic_β-smooth function, i.e. its derivative is β𝛽\betaitalic_β-Lipsh... | 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... | Assumption 1 includes many common loss functions, including the logistic, exp-loss222The exp-loss does not have a global β𝛽\betaitalic_β smoothness parameter. However, if we initialize with η<1/ℒ(𝐰(0))𝜂1ℒ𝐰0\eta<1/\mathcal{L}(\mathbf{w}(0))italic_η < 1 / caligraphic_L ( bold_w ( 0 ) ) then it is straightforward to... | 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... | loss function (Assumption 1) with an exponential
tail (Assumption 3), any stepsize η<2β−1σmax−2(𝐗 )𝜂2superscript𝛽1superscriptsubscript𝜎2𝐗 \eta<2\beta^{-1}\sigma_{\max}^{-2}\left(\text{$\mathbf{X}$ }\right)italic_η < 2 italic_β start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT roman_max ... | B |
The time period of a rumor event is sometimes fuzzy and hard to define. One reason is a rumor may have been triggered for a long time and kept existing, but it did not attract public attention. However it can be triggered by other events after a uncertain time and suddenly spreads as a bursty event. E.g., a rumor999htt... |
Given a tweet, our task is to classify whether it is associated with either a news or rumor. Most of the previous work (castillo2011information, ; gupta2014tweetcred, ) on tweet level only aims to measure the trustfulness based on human judgment (note that even if a tweet is trusted, it could anyway relate to a rumor)... |
For this task, we developed two kinds of classification models: traditional classifier with handcrafted features and neural networks without tweet embeddings. For the former, we used 27 distinct surface-level features extracted from single tweets (analogously to the Twitter-based features presented in Section 3.2). Fo... | the idea on focusing on early rumor signals on text contents; which is the most reliable source before the rumors widely spread. Specifically, we learn a more complex representation of single tweets using Convolutional Neural Networks, that could capture more hidden meaningful signal than only enquiries to debunk rumor... | We consider two types of Ensemble Features: features accumulating crowd wisdom and averaging feature for the Tweet credit Scores.
The former are extracted from the surface level while the latter comes from the low dimensional level of tweet embeddings; that in a way augments the sparse crowd at early stage. | B |
Evaluating methodology.
For RQ1, given an event entity e, at time t, we need to classify them into either Breaking or Anticipated class. We select a studied time for each event period randomly in the range of 5 days before and after the event time. In total, our training dataset for AOL consists of 1,740 instances of b... |
Results. The baseline and the best results of our 1stsuperscript1𝑠𝑡1^{st}1 start_POSTSUPERSCRIPT italic_s italic_t end_POSTSUPERSCRIPT stage event-type classification is shown in Table 3-top. The accuracy for basic majority vote is high for imbalanced classes, yet it is lower at weighted F1. Our learned model achie... | RQ3. We demonstrate the results of single models and our ensemble model in Table 4. As also witnessed in RQ2, SVMall𝑆𝑉subscript𝑀𝑎𝑙𝑙SVM_{all}italic_S italic_V italic_M start_POSTSUBSCRIPT italic_a italic_l italic_l end_POSTSUBSCRIPT, will all features, gives a rather stable performance for both NDCG and Recall... |
RQ2. Figure 4 shows the performance of the aspect ranking models for our event entities at specific times and types. The most right three models in each metric are the models proposed in this work. The overall results show that, the performances of these models, even better than the baselines (for at least one of the ... | 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... | A |
RL [Sutton and Barto, 1998] has been successfully applied to a variety of domains,
from Monte Carlo tree search [Bai et al., 2013] and hyperparameter tuning for complex optimization in science, engineering and machine learning problems [Kandasamy et al., 2018; Urteaga et al., 2023], | SMC weights are updated based on the likelihood of the observed rewards:
wt,a(m)∝pa(yt|xt,θt,a(m))proportional-tosuperscriptsubscript𝑤𝑡𝑎𝑚subscript𝑝𝑎conditionalsubscript𝑦𝑡subscript𝑥𝑡superscriptsubscript𝜃𝑡𝑎𝑚w_{t,a}^{(m)}\propto p_{a}(y_{t}|x_{t},\theta_{t,a}^{(m)})italic_w start_POSTSUBSCRIPT italic_t , it... | the fundamental operation in the proposed SMC-based MAB Algorithm 1
is to sequentially update the random measure pM(θt,a|ℋ1:t)subscript𝑝𝑀conditionalsubscript𝜃𝑡𝑎subscriptℋ:1𝑡p_{M}(\theta_{t,a}|\mathcal{H}_{1:t})italic_p start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_θ start_POSTSUBSCRIPT italic_t , itali... | The techniques used in these success stories are grounded on statistical advances on sequential decision processes and multi-armed bandits.
The MAB crystallizes the fundamental trade-off between exploration and exploitation in sequential decision making. | we propagate forward the sequential random measure pM(θt,a|ℋ1:t)subscript𝑝𝑀conditionalsubscript𝜃𝑡𝑎subscriptℋ:1𝑡p_{M}(\theta_{t,a}|\mathcal{H}_{1:t})italic_p start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_θ start_POSTSUBSCRIPT italic_t , italic_a end_POSTSUBSCRIPT | caligraphic_H start_POSTSUBSCRIPT 1 : ... | C |
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... | Likewise, the daily number of measurements taken for carbohydrate intake, blood glucose level and insulin units vary across the patients.
The median number of carbohydrate log entries vary between 2 per day for patient 10 and 5 per day for patient 14. | 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... | 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... | Median number of blood glucose measurements per day varies between 2 and 7. Similarly, insulin is used on average between 3 and 6 times per day.
In terms of physical activity, we measure the 10 minute intervals with at least 10 steps tracked by the google fit app. | C |
Weight values from the ASPP module and decoder were initialized according to the Xavier method by Glorot and Bengio (2010). It specifies parameter values as samples drawn from a uniform distribution with zero mean and a variance depending on the total number of incoming and outgoing connections. Such initialization sc... | Various measures are used in the literature and by benchmarks to evaluate the performance of fixation models. In practice, results are typically reported for all of them to include different notions about saliency and allow a fair comparison of model predictions Kümmerer et al. (2018); Riche et al. (2013). A set of nin... | 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... | Table 2 demonstrates that we obtained state-of-the-art scores for the CAT2000 test dataset regarding the AUC-J, sAUC, and KLD evaluation metrics, and competitive results on the remaining measures. The cumulative rank (as computed above) suggests that our model outperformed all previous approaches, including the ones ba... | We normalized the model output such that all values are non-negative with unit sum. The estimation of saliency maps can hence be regarded as a probability distribution prediction task as formulated by Jetley et al. (2016). To determine the difference between an estimated and a target distribution, the Kullback-Leibler ... | D |
We observe that the reduction from MinCutwidth to MinLoc from Section 4.1 combined with the reduction from MinLoc to MinPathwidth from Section 5.2 gives a reduction from MinCutwidth to MinPathwidth. Moreover, this reduction is approximation preserving; thus, it carries over approximations for MinPathwidth (e. g., [21,... | In the following, we obtain an approximation algorithm for the locality number by reducing it to the problem of computing the pathwidth of a graph. To this end, we first describe another way of how a word can be represented by a graph. Recall that the reduction to cutwidth from Section 4 also transforms words into grap... |
We observe that the reduction from MinCutwidth to MinLoc from Section 4.1 combined with the reduction from MinLoc to MinPathwidth from Section 5.2 gives a reduction from MinCutwidth to MinPathwidth. Moreover, this reduction is approximation preserving; thus, it carries over approximations for MinPathwidth (e. g., [21,... | One of the main results of this section is a reduction from the problem of computing the locality number of a word α𝛼\alphaitalic_α to the probem of computing the pathwidth of a graph. This reduction, however, does not technically provide a reduction from the decision problem Loc to Pathwidth, since the constructed gr... | Pathwidth and cutwidth are classical graph parameters that play an important role for graph algorithms, independent from our application for computing the locality number. Therefore, it is the main purpose of this section to translate the reduction from MinCutwidth to MinPathwidth that takes MinLoc as an intermediate s... | D |
This model compared with vanilla conv-deconv and u-net performs better by an average of 5% in terms of Dice.
Patravali et al.[140] trained a model based on u-net using Dice combined with cross entropy as a metric for LV/RV and myocardium segmentation. | The model was designed to accept a stack of image slices as input channels and the output is predicted for the middle slice.
Based on experiments they conducted, it was concluded that three input slices were optimal as an input for the model, instead of one or five. | Autoencoders (AEs) are neural networks that are trained with the objective to copy the input x𝑥xitalic_x to the output in such a way that they encode useful properties of the data.
It usually consists of an encoding part that downsamples the input down to a linear feature and a decoding part that upsamples to the orig... | A common AE architecture is Stacked Denoised AE (SDAE) that has an objective to reconstruct the clean input from an artificially corrupted version of the input[20] which prevents the model from learning trivial solutions.
Another AE-like architecture is u-net[4], which is of special interest to the biomedical community... | Another three models were trained using the signals as 1D.
The first model was a FNN with dropout, the second a three layer 1D CNN and the third a 2D CNN same as the first but trained with a stacked version of the signal (also trained with data augmentation). | A |
The primary evaluation in our experiments studies the sample efficiency of SimPLe, in comparison with state-of-the-art model-free deep RL methods in the literature. To that end, we compare with Rainbow (Hessel et al., 2018; Castro et al., 2018), which represents the state-of-the-art Q-learning method for Atari games, ... |
While SimPLe is able to learn more quickly than model-free methods, it does have limitations. First, the final scores are on the whole lower than the best state-of-the-art model-free methods. This can be improved with better dynamics models and, while generally common with model-based RL algorithms, suggests an import... | The iterative process of training the model, training the policy, and collecting data is crucial for non-trivial tasks where random data collection is insufficient. In a game-by-game analysis, we quantified the number of games where the best results were obtained in later iterations of training. In some games, good pol... | Figure 1: Main loop of SimPLe. 1) the agent starts interacting with the real environment following the latest policy (initialized to random). 2) the collected observations will be used to train (update) the current world model. 3) the agent updates the policy by acting inside the world model. The new policy will be eva... | The results in these figures are generated by averaging 5555 runs for each game.
The model-based agent is better than a random policy for all the games except Bank Heist. Interestingly, we observed that the best of the 5555 runs was often significantly better. For 6666 of the games, it exceeds the average human score (... | D |
This is achieved with the use of multilayer networks, that consist of million parameters [1], trained with backpropagation [2] on large amount of data.
Although deep learning is mainly used in biomedical images there is also a wide range of physiological signals, such as Electroencephalography (EEG), that are used for ... | 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... | This is achieved with the use of multilayer networks, that consist of million parameters [1], trained with backpropagation [2] on large amount of data.
Although deep learning is mainly used in biomedical images there is also a wide range of physiological signals, such as Electroencephalography (EEG), that are used for ... | For the spectrogram module, which is used for visualizing the change of the frequency of a non-stationary signal over time [18], we used a Tukey window with a shape parameter of 0.250.250.250.25, a segment length of 8888 samples, an overlap between segments of 4444 samples and a fast Fourier transform of 64646464 sampl... | 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. | A |
The track tip positioning was the key parameter controlled during the creation of these climbing gaits. To assure seamless locomotion, trajectories for each joint of the robot were defined through a fifth-order polynomial along with their first and second derivatives. The trajectory design took into account six constra... | 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 ... |
Figure 10: The Cricket robot tackles a step of height h using rolling locomotion mode, negating the need for a transition to the walking mode. The total energy consumed throughout the entire step negotiation process in rolling locomotion stayed below the preset threshold value. This threshold value was established bas... |
The whole-body climbing gait involves utilizing the entire body movement of the robot, swaying forwards and backwards to enlarge the stability margins before initiating gradual leg movement to overcome a step. This technique optimizes stability during the climbing process. To complement this, the rear-body climbing ga... | The evaluation of energy consumption for the walking locomotion mode encompassed the entire step negotiation process, from the commencement of the negotiation until its completion. Fig. 8 reveals minimal discrepancies in energy consumption for the whole-body climbing gait, which can be attributed to the thoughtful desi... | C |
The algorithm classifies items according to their size. Tiny items have their size in the range (0,1/3]013(0,1/3]( 0 , 1 / 3 ], small items in (1/3,1/2]1312(1/3,1/2]( 1 / 3 , 1 / 2 ], critical items in (1/2,2/3]1223(1/2,2/3]( 1 / 2 , 2 / 3 ], and large items in (2/3,1]231(2/3,1]( 2 / 3 , 1 ]. In addition, the algorithm... |
Intuitively, Rrc works similarly to Reserved-Critical except that it might not open as many critical bins as suggested by the advice. The algorithm is more “conservative” in the sense that it does not keep two thirds of many (critical) bins open for critical items that might never arrive. The smaller the value of α𝛼\... | The algorithm classifies items according to their size. Tiny items have their size in the range (0,1/3]013(0,1/3]( 0 , 1 / 3 ], small items in (1/3,1/2]1312(1/3,1/2]( 1 / 3 , 1 / 2 ], critical items in (1/2,2/3]1223(1/2,2/3]( 1 / 2 , 2 / 3 ], and large items in (2/3,1]231(2/3,1]( 2 / 3 , 1 ]. In addition, the algorithm... | The worst case is reached when tiny items form a subsequence (1/6,ϵ,1/6,ϵ,…)16italic-ϵ16italic-ϵ…(1/6,\epsilon,1/6,\epsilon,\ldots)( 1 / 6 , italic_ϵ , 1 / 6 , italic_ϵ , … ), while there is no critical item. In this case, all critical bins are filled up to a level slightly more than 1/6161/61 / 6. Hence, untrusted adv... | First, if γ≤α𝛾𝛼\gamma\leq\alphaitalic_γ ≤ italic_α, by Lemma 10, the competitive ratio will be at most 1.5+152k/2+11.515superscript2𝑘211.5+\frac{15}{2^{k/2+1}}1.5 + divide start_ARG 15 end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_k / 2 + 1 end_POSTSUPERSCRIPT end_ARG. Next, assume α<γ𝛼𝛾\alpha<\gammaitalic_α < ... | C |
Since ⊕1subscriptdirect-sum1\oplus_{1}⊕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the addition, instead of processing the whole document again, we could update the already computed vector, (0.15,3.65,2.0,0.15)0.153.652.00.15(0.15,3.65,2.0,0.15)( 0.15 , 3.65 , 2.0 , 0.15 ), by adding it to the new sentence confidence v... | 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 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... | However, this is a vital aspect, especially when the task involves sensitive or risky decisions in which, usually, people are involved. In Figure 9 is shown an example of a piece of what could be a visual description of the classification process for the subject 9579292929Note that this is the same subject who was prev... | We could make use of this “dynamic information” to apply certain policies to decide when to classify subjects as depressed.
For example, one of such a policy would be “classify a subject as positive when the accumulated positive value becomes greater than the negative one” —in which case, note that our subject would be... | A |
Since 𝒞(𝐞t+12,k)𝒞subscript𝐞𝑡12𝑘\mathcal{C}({\bf e}_{t+\frac{1}{2},k})caligraphic_C ( bold_e start_POSTSUBSCRIPT italic_t + divide start_ARG 1 end_ARG start_ARG 2 end_ARG , italic_k end_POSTSUBSCRIPT ) is sparse, 𝐰t+1−𝐰tsubscript𝐰𝑡1subscript𝐰𝑡{\bf w}_{t+1}-{\bf w}_{t}bold_w start_POSTSUBSCRIPT italic_t + 1 ... | There are some other ways to combine momentum and error feedback. For example, we can put the momentum term on the server. However, these ways lead to worse performance than the way adopted in this paper. More discussions can be found in Appendix A.
| Recently, parameter server (Li et al., 2014) has been one of the most popular distributed frameworks in machine learning. GMC can also be implemented on the parameter server framework.
In this paper, we adopt the parameter server framework for illustration. The theories in this paper can also be adapted for the all-red... | The error feedback technique keeps the compressed error into the error residual on each worker and incorporates the error residual into the next update.
Error feedback based sparse communication methods have been widely adopted by recent communication compression methods and achieved better performance than quantizatio... |
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... | A |
These results suggest that reconstruction error by itself is not a sufficient metric for decomposing data in interpretable components.
Trying to solely achieve lower reconstruction error (such as the case for the Identity activation function) produces noisy learned kernels, while using the combined measure of reconstru... | Comparing the differences of φ¯¯𝜑\bar{\varphi}over¯ start_ARG italic_φ end_ARG between the Identity, the ReLU and the rest sparse activation functions in Fig. 4LABEL:sub@subfig:flithos_m we notice that the latter produce a minimum region in which we observe interpretable kernels.
| During validation we selected the models with the kernel size that achieved the best φ¯¯𝜑\bar{\varphi}over¯ start_ARG italic_φ end_ARG out of all epochs.
During testing we feed the test data into the selected model and calculate CR−1𝐶superscript𝑅1CR^{-1}italic_C italic_R start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIP... |
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. | These results suggest that reconstruction error by itself is not a sufficient metric for decomposing data in interpretable components.
Trying to solely achieve lower reconstruction error (such as the case for the Identity activation function) produces noisy learned kernels, while using the combined measure of reconstru... | A |
Typical wireless protocol 802.11b/g only provides limited channels for users, which is far more than enough for high-quality communication services [18]. To reduce the load in central system, making use of distributed available resources in networks turns out to be an ideal solution. Underlay Device-to-Device (D2D) co... |
Game theory provides an efficient tool for the cooperation through resource allocation and sharing [20][21]. A computation offloading game has been designed in order to balance the UAV’s tradeoff between execution time and energy consumption [25]. A sub-modular game is adopted in the scheduling of beaconing periods fo... | We propose a novel UAV ad-hoc network model with the aggregative game which is compatible with the large-scale highly dynamic environments, in which several influences are coupled together. In the aggregative game, the interference from other UAVs can be regarded as the integral influence, which makes the model more pr... | Since the UAV ad-hoc network game is a special type of potential game, we can apply the properties of the potential game in the later analysis. Some algorithms that have been applied in the potential game can also be employed in the UAV ad-hoc network game. In the next section, we investigate the existing algorithm wit... |
In post-disaster scenarios, a great many of UAVs are required to support users [4]. Therefore, we propose aggregative game theory into such scenarios and permit UAV to learn in the constrained strategy sets. Because the aggregative game can integrate the impact of all other UAVs on one UAV, it reduces the complexity o... | A |
Equation 5.16 can be solved for the constant fIsubscript𝑓𝐼f_{I}italic_f start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT if
fPisubscript𝑓subscript𝑃𝑖f_{P_{i}}italic_f start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT is temporarily set to zero at the fixed-point nodes along | where hIsubscriptℎ𝐼h_{I}italic_h start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT is the height of the rectangular cross-section of the
insulating wall, and routsubscript𝑟𝑜𝑢𝑡r_{out}italic_r start_POSTSUBSCRIPT italic_o italic_u italic_t end_POSTSUBSCRIPT and rinsubscript𝑟𝑖𝑛r_{in}italic_r start_POSTSUBSCRIPT it... | fI(t)=−1L~ins+L~intΔi[=1]NnΣ(fP0i(t)si3ri)f_{I}(t)=\>\frac{-1}{\tilde{L}_{ins}+\tilde{L}{}_{int\Delta}}\stackrel{{%
\scriptstyle[}}{{i}}=1]{N_{n}}{\Sigma}\left(\frac{f_{P0_{i}}(t)s_{i}}{3r_{i}}\right)italic_f start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG - 1 end_ARG start_ARG o... | the inner wall of the insulator (see figure 10),
so that fP→fP0→subscript𝑓𝑃subscript𝑓𝑃0f_{P}\rightarrow f_{P0}italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT → italic_f start_POSTSUBSCRIPT italic_P 0 end_POSTSUBSCRIPT. Equation 5.16 is modified | Equation 5.16 can be solved for the constant fIsubscript𝑓𝐼f_{I}italic_f start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT if
fPisubscript𝑓subscript𝑃𝑖f_{P_{i}}italic_f start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT is temporarily set to zero at the fixed-point nodes along | C |
When using the framework, one can further require reflexivity on the comparability functions, i.e. f(xA,xA)=1A𝑓subscript𝑥𝐴subscript𝑥𝐴subscript1𝐴f(x_{A},x_{A})=1_{A}italic_f ( italic_x start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) = 1 start_POSTSUBSCRIP... | Indeed, in practice the meaning of the null value in the data should be explained by domain experts, along with recommendations on how to deal with it.
Moreover, since the null value indicates a missing value, relaxing reflexivity of comparability functions on null allows to consider absent values as possibly | fA(u,v)=fB(u,v)={1if u=v≠nullaif u≠null,v≠null and u≠vbif u=v=null0otherwise.subscript𝑓𝐴𝑢𝑣subscript𝑓𝐵𝑢𝑣cases1if 𝑢𝑣null𝑎formulae-sequenceif 𝑢null𝑣null and 𝑢𝑣𝑏if 𝑢𝑣null0otherwise.f_{A}(u,v)=f_{B}(u,v)=\begin{cases}1&\text{if }u=v\neq\texttt{null}\\
a&\text{if }u\neq\texttt{null},v\neq\texttt{null}... | Intuitively, if an abstract value xAsubscript𝑥𝐴x_{A}italic_x start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT of ℒAsubscriptℒ𝐴\mathcal{L}_{A}caligraphic_L start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is interpreted as 1111 (i.e., equality)
by hAsubscriptℎ𝐴h_{A}italic_h start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT... | When using the framework, one can further require reflexivity on the comparability functions, i.e. f(xA,xA)=1A𝑓subscript𝑥𝐴subscript𝑥𝐴subscript1𝐴f(x_{A},x_{A})=1_{A}italic_f ( italic_x start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) = 1 start_POSTSUBSCRIP... | A |
The sources of DQN variance are Approximation Gradient Error(AGE)[23] and Target Approximation Error(TAE)[24]. In Approximation Gradient Error, the error in gradient direction estimation of the cost function leads to inaccurate and extremely different predictions on the learning trajectory through different episodes b... |
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... |
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... | To that end, we ran Dropout-DQN and DQN on one of the classic control environments to express the effect of Dropout on Variance and the learned policies quality. For the Overestimation phenomena, we ran Dropout-DQN and DQN on a Gridworld environment to express the effect of Dropout because in such environment the optim... |
The results in Figure 3 show that using DQN with different Dropout methods result in better-preforming policies and less variability as the reduced standard deviation between the variants indicate to. In table 1, Wilcoxon Sign-Ranked test was used to analyze the effect of Variance before applying Dropout (DQN) and aft... | C |
Figure 5: Top: An illustration of the SegNet architecture. There are no fully connected layers, and hence it is only convolutional. Bottom: An illustration of SegNet and FCN (Long et al., 2015) decoders. a,b,c,d𝑎𝑏𝑐𝑑a,b,c,ditalic_a , italic_b , italic_c , italic_d correspond to values in a feature map. SegNet uses ... |
Milletari et al. (2016) proposed a similar architecture (V-Net; Figure 7) which added residual connections and replaced 2D operations with their 3D counterparts in order to process volumetric images. Milletari et al. also proposed optimizing for a widely used segmentation metric, i.e., Dice, which will be discussed in... | V-Net (Milletari et al., 2016) and FCN (Long et al., 2015). Sinha and Dolz (2019) proposed a multi-level attention based architecture for abdominal organ segmentation from MRI images. Qin et al. (2018) proposed a dilated convolution base block to preserve more detailed attention in 3D medical image segmentation. Simil... | To perform image segmentation in real-time and be able to process larger images or (sub) volumes in case of processing volumetric and high-resolution 2D images such as CT, MRI, and histopathology images, several methods have attempted to compress deep models. Weng et al. (2019a) applied a neural architecture search met... |
The standard CE loss function and its weighted versions, as discussed in Section 4, have been applied to numerous medical image segmentation problems (Isensee et al., 2019; Li et al., 2019b; Lian et al., 2018; Ni et al., 2019; Nie et al., 2018; Oktay et al., 2018; Schlemper et al., 2019). However, Milletari et al. (20... | A |
Each fold is, in turn, selected as the test set, while the remaining 9 folds become the training set.
For each different train/test split, we set aside 10% of the training data as validation set, which is used for early stopping, i.e., we interrupt the training procedure after the loss on the validation set does not de... | The LSTM baseline generally achieves a better accuracy than Dense, since it captures the sequential ordering of the words in the reviews, which also helps to prevent overfitting on training data.
Finally, the TCN baseline always outperforms LSTM, both in terms of accuracy and computational costs. | Additional baselines are the Weisfeiler-Lehman (WL) graph kernel [47], a GNN with only MP layers (Flat), and a network with only dense layers (Dense).
The comparison with Flat helps to understand whether pooling operations are useful for a given task. | Interestingly, the Dense architecture achieves the best performance on MUTAG, indicating that in this case, the connectivity of the graps does not carry useful information for the classification task.
The performance of the Flat baseline indicates that in Enzymes and COLLAB pooling operations are not necessary to impro... | Interestingly, the GNNs configured with GRACLUS and NDP always achieve better results than the Dense network, even if the latter generates the word embeddings used to build the graph on which the GNN operates. This can be explained by the fact that the Dense network immediately overfits the dataset, whereas the graph s... | C |
NRFI with and without the original data is shown for different network architectures. The smallest architecture has 2222 neurons in both hidden layers and the largest 128128128128. For NRFI (gen-ori), we can see that a network with 16161616 neurons in both hidden layers (NN-16-16) is already sufficient to learn the dec... | Current state-of-the-art methods directly map random forests into neural networks. The number of parameters of the resulting network is evaluated on all datasets with different numbers of training examples. The overall performance is shown in the last column.
Due to the stochastic process when training the random fores... | Here, we additionally include decision trees, support vector machines, random forests, and neural networks in the comparison. The evaluation is performed on all nine datasets, and results for different numbers of training examples are shown (increasing from left to right). The overall performance of each method is summ... | NRFI introduces imitation instead of direct mapping. In the following, a network architecture with 32323232 neurons in both hidden layers is selected.
The previous analysis has shown that this architecture is capable of imitating the random forests (see Figure 4 for details) across all datasets and different numbers of... | First, we analyze the performance of state-of-the-art methods for mapping random forests into neural networks and neural random forest imitation. The results are shown in Figure 4 for different numbers of training examples per class.
For each method, the average number of parameters of the generated networks across all... | B |
In a more practical setting, the agent sequentially explores the state space, and meanwhile, exploits the information at hand by taking the actions that lead to higher expected total rewards. Such an exploration-exploitation tradeoff is better captured by the aforementioned statistical question regarding the regret or ... | The policy improvement step defined in (3.2) corresponds to one iteration of NPG (Kakade, 2002), TRPO (Schulman et al., 2015), and PPO (Schulman et al., 2017). In particular, PPO solves the same KL-regularized policy optimization subproblem as in (3.2) at each iteration, while TRPO solves an equivalent KL-constrained s... | To answer this question, we propose the first policy optimization algorithm that incorporates exploration in a principled manner. In detail, we develop an Optimistic variant of the PPO algorithm, namely OPPO. Our algorithm is also closely related to NPG and TRPO. At each update, OPPO solves a Kullback-Leibler (KL)-regu... |
We study the sample efficiency of policy-based reinforcement learning in the episodic setting of linear MDPs with full-information feedback. We proposed an optimistic variant of the proximal policy optimization algorithm, dubbed as OPPO, which incorporates the principle of “optimism in the face of uncertainty” into po... | step with α→∞→𝛼\alpha\rightarrow\inftyitalic_α → ∞ corresponds to one step of policy iteration (Sutton and Barto, 2018), which converges to the globally optimal policy π∗superscript𝜋\pi^{*}italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT within K=H𝐾𝐻K=Hitalic_K = italic_H episodes and hence equivalently induces... | B |
The challenge is to reduce the number of bits as much as possible while at the same time keeping the prediction accuracy close to that of a well-tuned full-precision DNN.
Subsequently, we provide a literature overview of approaches that train reduced-precision DNNs, and, in a broader view, we also consider methods that... | Knowledge distillation is an approach where a small student DNN is trained to mimic the behavior of a larger teacher DNN, which has been shown to yield improved results compared to training the small DNN directly.
The idea of weight sharing is to use a small set of weights that is shared among several connections of a ... | In recent years, the STE (Bengio et al., 2013) (see Section 2.6) became the method of choice to compute an approximate gradient for training DNNs with weights that are represented using a very small number of bits.
Such methods typically maintain a set of full-precision weights that are quantized during forward propaga... | By injecting additive noise to the deterministic weights before rounding, one can compute probabilities of the weights being rounded to specific values in a predefined discrete set.
Subsequently, these probabilities are used to differentiably round the weights using the Gumbel-softmax approximation (Jang et al., 2017). | The two works of Höhfeld and Fahlman (Höhfeld and Fahlman, 1992a, b) rounded the weights during training to fixed-point format with different numbers of bits.
They observed that training eventually stalls as small gradient updates are always rounded to zero. | D |
by inequality (6) and Remark 9.2. Again, as in the first item, J=(0,l23]𝐽0superscript𝑙23J=\left(0,\frac{l^{2}}{3}\right]italic_J = ( 0 , divide start_ARG italic_l start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 3 end_ARG ]. Note that the existence of the interval (0,l23]0superscript𝑙23\left(0,\frac{l^... |
In this section, we will see one such example which arises from the interplay between the hyperbolicity of the geodesic metric space X𝑋Xitalic_X and its tight span E(X)𝐸𝑋E(X)italic_E ( italic_X ) (see Example 3.1 to recall the definition of tight span). |
Let X𝑋Xitalic_X be the metric gluing of a loop of length l2subscript𝑙2l_{2}italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and an interval of length l1subscript𝑙1l_{1}italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (glued to the circle at one of its endpoints). Then, by Proposition 9.1, I≤spread(X)𝐼spread𝑋I\le... |
Motivated by Example 9.2 above, in the proposition below we will clarify the relationship between the persistence barcode and the multiset consisting of all I(ω,s)subscript𝐼𝜔𝑠I_{(\omega,s)}italic_I start_POSTSUBSCRIPT ( italic_ω , italic_s ) end_POSTSUBSCRIPT. |
An example similar to the one described in the previous item arises from Figure 3. Consider the tube connecting the two blobs to be large: in that case the standard spread of the space will be large yet the lifetime of the individual H2subscriptH2\mathrm{H}_{2}roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT classes wi... | D |
In their tool, Coimbra et al. [42] support interactive exploration of 3-D projections using adapted biplots and different widgets for viewpoint selection. Our tool is similar to theirs from the perspective of providing a collection of interconnected views for projection exploration, but they focus on projection-agnosti... |
Most similarly to one of our proposed interactions (the Dimension Correlation, Subsection 4.4), in AxiSketcher [47] (and its prior version InterAxis [48]) the user can draw a polyline in the scatterplot to identify a shape, which results in new non-linear high-dimensional axes to match the user’s intentions. Since the... |
Adaptive Parallel Coordinates Plot Our first proposal to support the task of interpreting patterns in a t-SNE projection is an Adaptive PCP [59], as shown in Figure 1(k). It highlights the dimensions of the points selected with the lasso tool, using a maximum of 8 axes at any time, to avoid clutter. The shown axes (... | Fujiwara et al. [44] proposed the contrasting clusters in PCA (ccPCA) method to find which dimensions contributed more to the formation of a selected cluster and why it differs from the rest of the dataset, based on information on separation and internal vs. external variability. We have similar goals, but approach the... | Adaptive PCP vs. PCP
Although it is not uncommon to find tools that use PCP views together with DR-based scatterplots (e.g., iPCA [69]) with various schemes for re-ordering and prioritizing the axes (e.g., [70, 71]), the arrangement and presentation of these PCP’s are usually static in order to reflect attributes of ... | C |
The complete list of reviewed algorithms in this category is provided in Tables 9 and 10 (physics-based algorithms) and Table 11 (chemistry-based methods). In this category we can find some well-known algorithms reported in the last century such as Simulated Annealing [79], or one of the most important algorithms in ph... |
Algorithms falling in this category are inspired by human social concepts, such as decision-making and ideas related to the expansion/competition of ideologies inside the society as ideology (Ideology Algorithm, IA, [466]), or political concepts such as the Imperialist Colony Algorithm (ICA, [467]). This category also... | Tables 18, 19, 20, 21, 22, 23 and 24 show the different algorithms in this subcategory. An exemplary algorithm of this category that has been a major meta-heuristic solver in the history of the field is PSO [80]. In this solver, each solution or particle is guided by the global current best solution and the best soluti... |
In this same line of reasoning, the largest subcategory of the second taxonomy (Differential Vector Movements guided by representative solutions) not only contains more than half of the reviewed algorithms (almost 60%), but it also comprises algorithms from all the different categories in the first taxonomy: Social Hu... | The complete list of reviewed algorithms in this category is provided in Tables 9 and 10 (physics-based algorithms) and Table 11 (chemistry-based methods). In this category we can find some well-known algorithms reported in the last century such as Simulated Annealing [79], or one of the most important algorithms in ph... | A |
(1) Via extending the generative graph models into general type data, GAE is naturally employed as the basic representation learning model and weighted graphs can be further applied to GAE as well. The connectivity distributions given by the generative perspective also inspires us to devise a novel architecture for dec... | To study the impact of different parts of the loss in Eq. (12), the performance with different λ𝜆\lambdaitalic_λ is reported in Figure 4.
From it, we find that the second term (corresponding to problem (7)) plays an important role especially on UMIST. If λ𝜆\lambdaitalic_λ is set as a large value, we may get the trivi... | (3) AdaGAE is a scalable clustering model that works stably on different scale and type datasets, while the other deep clustering models usually fail when the training set is not large enough. Besides, it is insensitive to different initialization of parameters and needs no pretraining.
| As well as the well-known k𝑘kitalic_k-means [1, 2, 3], graph-based clustering [4, 5, 6] is also a representative kind of clustering method.
Graph-based clustering methods can capture manifold information so that they are available for the non-Euclidean type data, which is not provided by k𝑘kitalic_k-means. Therefore,... | Classical clustering models work poorly on large scale datasets. Instead, DEC and SpectralNet work better on the large scale datasets. Although GAE-based models (GAE, MGAE, and GALA) achieve impressive results on graph type datasets, they fail on the general datasets, which is probably caused by the fact that the graph... | B |
Recent work showed that even TCP traffic gets fragmented under certain conditions (Dai et al., 2021b). Fragmentation has long history of attacks which affect both the UDP and TCP traffic (Kent and Mogul, 1987; Herzberg and Shulman, 2013; Shulman and Waidner, 2014).
| Identifying servers with global IPID counters. We send packets from two hosts (with different IP addresses) to a server on a tested network. We implemented probing over TCP SYN, ping and using requests/responses to Name servers and we apply the suitable test depending on the server that we identify on the tested networ... |
The challenge here is to accurately probe the increments rate of the IPID value (caused by the packets from other sources not controlled by us), in order to be able to extrapolate the value that will have been assigned to our second probe from a real source IP. This allows us to infer if the spoofed packets incremente... | Methodology. The core idea of the Path MTU Discovery (PMTUD) based tool is to send the ICMP Packet too Big (PTB) message from a spoofed source IP address, belonging to the tested network, and in the 8 bytes payload of the ICMP to insert the real IP address belonging to the prober. If the network does not enforce ingres... |
Methodology. We use services that assign globally incremental IPID values. The idea is that globally incremental IPID [RFC6864] (Touch, 2013) values leak traffic volume arriving at the service and can be measured by any Internet host. Given a server with a globally incremental IPID on the tested network, we sample the... | D |
The purpose of this study was to demonstrate that explicit representation of context can allow a classification system to adapt to sensor drift. Several gas classifier models were placed in a setting with progressive sensor drift and were evaluated on samples from future contexts. This task reflects the practical goal... |
Second, skill NN and context+skill NN models were compared. The context-based network extracts features from preceding batches in sequence in order to model how the sensors drift over time. When added to the feedforward NN representation, such contextual information resulted in improved ability to compensate for senso... |
For each batch T𝑇Titalic_T from 3 through 10, the batches 1,2,…,T−112…𝑇11,2,\ldots,T-11 , 2 , … , italic_T - 1 were used to train skill NN and context+skill NN models for 30 random initializations of the starting weights. The accuracy was measured classifying examples from batch T𝑇Titalic_T (Fig. 3A, Table 1, Skill... | The context+skill NN model builds on the skill NN model by adding a recurrent processing pathway (Fig. 2D). Before classifying an unlabeled sample, the recurrent pathway processes a sequence of labeled samples from the preceding batches to generate a context representation, which is fed into the skill processing layer.... |
While context did introduce more parameters to the model (7,57575757{,}5757 , 575 parameters without context versus 14,3151431514{,}31514 , 315 including context), the model is still very small compared to most neural network models, and is trainable in a few hours on a CPU. When units were added to the “skill” layer ... | A |
Now we can define the tables A(1)superscript𝐴1A^{(1)}italic_A start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT, A(2)superscript𝐴2A^{(2)}italic_A start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT and A(3)superscript𝐴3A^{(3)}italic_A start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT that our algorithm uses.
Recall that for... |
A[i,B]:={A representative set containing pairs (M,x), where M is a perfect matching on B and x is a real number equal to the minimum total length of a path cover of P0∪⋯∪Pi−1∪B realizing the matching M.assign𝐴𝑖𝐵casesA representative set containing pairs (M,x), where M is a perfect matching on B and x is a real num... | A(1)[i,B]:={A representative set containing pairs (M,x), where M is a perfect matching on B∈ℬi(1) and x is a real number equal to the minimum total length of a path cover of P0∪⋯∪Pi−1∪B realizing the matching Massignsuperscript𝐴1𝑖𝐵casesA representative set containing pairs (M,x), where M is a perfect matching on B∈... | A(2)[i,B]:={A representative set containing pairs (M,x), where M is a perfect matching on B∈ℬi(2) and x is a real number equal to the minimum total length of a path cover of P0∪⋯∪Pi−1∪B realizing the matching M.assignsuperscript𝐴2𝑖𝐵casesA representative set containing pairs (M,x), where M is a perfect matching on B... |
A[i,B]:={A representative set containing pairs (M,x), where M is a perfect matching on B∈ℬi and x is a real number equal to the minimum total length of a path cover of P0∪⋯∪Pi−1∪B realizing the matching M.assign𝐴𝑖𝐵casesA representative set containing pairs (M,x), where M is a perfect matching on B∈ℬi and x is a re... | D |
We conclude this section by presenting a pair S,T𝑆𝑇S,Titalic_S , italic_T of semigroups without a homomorphism S→T→𝑆𝑇S\to Titalic_S → italic_T or T→S→𝑇𝑆T\to Sitalic_T → italic_S where S𝑆Sitalic_S and T𝑇Titalic_T possess typical properties of automaton semigroups, which makes them good candidates for also belong... |
The word problem of a semigroup finitely generated by some set Q𝑄Qitalic_Q is the decision problem whether two input words over Q𝑄Qitalic_Q represent the same semigroup element. The word problem of any automaton semigroup can be solved in polynomial space and, under common complexity theoretic assumptions, this cann... | A semigroup S𝑆Sitalic_S is generated by a set Q𝑄Qitalic_Q if every element s∈S𝑠𝑆s\in Sitalic_s ∈ italic_S can be written as a product q1…qnsubscript𝑞1…subscript𝑞𝑛q_{1}\dots q_{n}italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT of factors from Q𝑄Qitalic... | A semigroup arising in this way is called self-similar. Furthermore, if the generating automaton is finite, it is an automaton semigroup.
If the generating automaton is additionally complete, we speak of a completely self-similar semigroup or of a complete automaton semigroup. | The problem of presenting (finitely generated) free groups and semigroups in a self-similar way has a long history [15]. A self-similar presentation in this context is typically a faithful action on an infinite regular tree (with finite degree) such that, for any element and any node in the tree, the action of the elem... | A |
We compare the training accuracies to analyze the regularization effects. As shown in Table 1, the baseline method has the highest training results, while the other methods cause 6.0−14.0%6.0percent14.06.0-14.0\%6.0 - 14.0 % and 3.3−10.5%3.3percent10.53.3-10.5\%3.3 - 10.5 % drops in the training accuracy on VQA-CPv2 an... | We compare the baseline UpDn model with HINT and SCR-variants trained on VQAv2 or VQA-CPv2 to study the causes behind the improvements. We report mean accuracies across 5555 runs, where a pre-trained UpDn model is fine-tuned on subsets with human attention maps and textual explanations for HINT and SCR respectively. Fu... | As observed by Selvaraju et al. (2019) and as shown in Fig. 2, we observe small improvements on VQAv2 when the models are fine-tuned on the entire train set. However, if we were to compare against the improvements in VQA-CPv2 in a fair manner, i.e., only use the instances with visual cues while fine-tuning, then, the p... | We compare the training accuracies to analyze the regularization effects. As shown in Table 1, the baseline method has the highest training results, while the other methods cause 6.0−14.0%6.0percent14.06.0-14.0\%6.0 - 14.0 % and 3.3−10.5%3.3percent10.53.3-10.5\%3.3 - 10.5 % drops in the training accuracy on VQA-CPv2 an... |
We compare four different variants of HINT and SCR to study the causes behind the improvements including the models that are fine-tuned on: 1) relevant regions (state-of-the-art methods) 2) irrelevant regions 3) fixed random regions and 4) variable random regions. For all variants, we fine-tune a pre-trained UpDn, whi... | B |
Prior work in privacy and human-computer interaction establishes the motivation for studying these documents. Although most internet users are concerned about privacy (Madden, 2017), Rudolph et al. (2018) reports that a significant number do not make the effort to read privacy notices because they perceive them to be ... |
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)... | To build the PrivaSeer corpus, we create a pipeline concentrating on focused crawling Chakrabarti et al. (1999); Diligenti et al. (2000) of privacy policy documents. We used Common Crawl,222https://commoncrawl.org/ described below, to gather seed URLs to privacy policies on the web. We filtered the Common Crawl URLs to... |
URL Cross Verification. Legal jurisdictions around the world require organisations to make their privacy policies readily available to their users. As a result, most organisations include a link to their privacy policy in the footer of their website landing page. In order to focus PrivaSeer Corpus on privacy policies ... | We selected those URLs which had the word “privacy” or the words “data” and “protection” from the Common Crawl URL archive. We were able to extract 3.9 million URLs that fit this selection criterion. Informal experiments suggested that this selection of keywords was optimal for retrieving the most privacy policies with... | B |
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... | Visualization systems have been developed for the exploration of diverse aspects of bagging, boosting, and further strategies such as “bucket of models”.
Stacking, however, has so far not received comparable attention by the InfoVis/VA communities: actually, we have not found any literature describing the construction ... | The rest of this paper is organized as follows. In the next section, we discuss the literature related to visualization of ensemble learning.
Afterwards, we describe the knowledge generation model for ensemble learning with VA, design goals, and analytical tasks for attaching VA to ensemble learning. | In a bucket of models, the best model for a specific problem is automatically chosen from a set of available options. This strategy is conceptually different to the ideas of bagging, boosting, and stacking, but still related to ensemble learning.
Chen et al. [6] utilize a bucket of latent Dirichlet allocation (LDA) mod... |
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... | A |
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... | 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 | 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 | 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 ], | {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... | B |
where ℒDitrain(θ)subscriptℒsuperscriptsubscript𝐷𝑖𝑡𝑟𝑎𝑖𝑛𝜃\mathcal{L}_{D_{i}^{train}}(\theta)caligraphic_L start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t italic_r italic_a italic_i italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_θ ) and ℒDiv... | Task similarity. In Persona and Weibo, each task is a set of dialogues for one user, so tasks are different from each other. We shuffle the samples and randomly divide tasks to construct the setting that tasks are similar to each other. For a fair comparison, each task on this setting also has 120 and 1200 utterances o... | Model-Agnostic Meta-Learning (MAML) [Finn et al., 2017] is one of the most popular meta-learning methods. It is trained on plenty of tasks (i.e. small data sets) to get a parameter initialization which is easy to adapt to target tasks with a few samples. As a model-agnostic framework, MAML is successfully employed in d... |
In Experiment II: Dialogue Generation, we use Persona [Zhang et al., 2018] and Weibo, regarding building a dialogue model for a user as a task. Persona is a personalized dialogue dataset with 1137/99/100 users for meta-training/meta-validation/meta-testing. Each user has 121 utterances on average. Weibo is a personali... |
In Experiment I: Text Classification, we use FewRel [Han et al., 2018] and Amazon [He and McAuley, 2016]. They are datasets for 5-way 5-shot classification, which means 5 classes are randomly sampled from the full dataset for each task, and each class has 5 samples. FewRel is a relation classification dataset with 65/... | D |
The rest of this paper is as follows. In Section II, the system model is introduced. In Section III, the CCA codebook design and the codebook-based joint subarray partition and AWV selection algorithms are proposed. Next, the TE-aware codebook-based beam tracking with 3D beamwidth control is further proposed in Sectio... |
A CCA-enabled UAV mmWave network is considered in this paper. Here, we first establish the DRE-covered CCA model in Section II-A. Then the system setup of the considered UAV mmWave network is described in Section II-B. Finally, the beam tracking problem for the CA-enabled UAV mmWave network is modeled in Section II-C. | In addition, the AOAs and AODs should be tracked in the highly dynamic UAV mmWave network.
To this end, in Section IV we will further propose a novel predictive AOA/AOD tracking scheme in conjunction with tracking error treatment to address the high mobility challenge, then we integrate these operations into the codebo... | Note that directly solving the above beam tracking problem is very challenging, especially in the considered highly dynamic UAV mmWave network. Therefore, developing new and efficient beam tracking solution for the CA-enabled UAV mmWave network is the major focus of our work. Recall that several efficient codebook-base... |
The rest of this paper is as follows. In Section II, the system model is introduced. In Section III, the CCA codebook design and the codebook-based joint subarray partition and AWV selection algorithms are proposed. Next, the TE-aware codebook-based beam tracking with 3D beamwidth control is further proposed in Sectio... | A |
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 ... | 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. | D |
Deep reinforcement learning achieves phenomenal empirical successes, especially in challenging applications where an agent acts upon rich observations, e.g., images and texts. Examples include video gaming (Mnih et al., 2015), visuomotor manipulation (Levine et al., 2016), and language generation (He et al., 2015). Suc... | 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.
| Szepesvári, 2018; Dalal et al., 2018; Srikant and Ying, 2019) settings. See Dann et al. (2014) for a detailed survey. Also, when the value function approximator is linear, Melo et al. (2008); Zou et al. (2019); Chen et al. (2019b) study the convergence of Q-learning. When the value function approximator is nonlinear, T... |
Moreover, soft Q-learning is equivalent to a variant of policy gradient (O’Donoghue et al., 2016; Schulman et al., 2017; Nachum et al., 2017; Haarnoja et al., 2017). Hence, Proposition 6.4 also characterizes the global optimality and convergence of such a variant of policy gradient. |
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... | D |
As for the costs, the decoder depth has a strong impact on inference speed, as the decoder has to be computed once for each decoding step during auto-regressive decoding Kasai et al. (2021); Xu et al. (2021c), and the use of only deep encoders Bapna et al. (2018); Wang et al. (2019); Li et al. (2022a); Chai et al. (20... | 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... | For the convergence of deep Transformers, Bapna et al. (2018) propose the Transparent Attention mechanism which allows each decoder layer to attend weighted combinations of all encoder layer outputs. Wang et al. (2019) present the Dynamic Linear Combination of Layers approach that additionally aggregates shallow layers... | Multilingual translation uses a single model to translate between multiple language pairs Firat et al. (2016); Johnson et al. (2017); Aharoni et al. (2019). Model capacity has been found crucial for massively multilingual NMT to support language pairs with varying typological characteristics Zhang et al. (2020); Xu et ... |
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... | C |
introduce here the notation 𝒦∘(X)≜{U∈τ∣U is compact}≜superscript𝒦𝑋conditional-set𝑈τ𝑈 is compact\mathcal{K}^{\circ}\!\left(X\right)\triangleq\{U\in\uptau\mid U\text{ is %
compact}\}caligraphic_K start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ( italic_X ) ≜ { italic_U ∈ roman_τ ∣ italic_U is compact }. When the topol... | ⟨τ⊆i∩⟦𝖥𝖮[σ]⟧Struct(σ)⟩\langle\tau_{\subseteq_{i}}\cap\llbracket\mathsf{FO}[\upsigma]\rrbracket_{%
\operatorname{Struct}(\upsigma)}\rangle⟨ italic_τ start_POSTSUBSCRIPT ⊆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ ⟦ sansserif_FO [ roman_σ ] ⟧ start_POSTSUBSCRIPT roman_Struct ( roman_σ ) end_PO... | \rrbracket_{\operatorname{Struct}(\upsigma)}\right\rangle=\left\langle%
\llbracket\mathsf{EFO}[\upsigma]\rrbracket_{\operatorname{Struct}(\upsigma)}\right\rangle⟨ roman_τ start_POSTSUBSCRIPT ⊆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ ⟦ sansserif_FO [ roman_σ ] ⟧ start_POSTSUBSCRIPT roman_Struc... | topology ⟨τ⊆i∩⟦𝖥𝖮[σ]⟧Struct(σ)⟩\langle\uptau_{\subseteq_{i}}\cap\llbracket\mathsf{FO}[\upsigma]\rrbracket_{%
\operatorname{Struct}(\upsigma)}\rangle⟨ roman_τ start_POSTSUBSCRIPT ⊆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ ⟦ sansserif_FO [ roman_σ ] ⟧ start_POSTSUBSCRIPT roman_Struct ( roman_... | τ⊆i∩⟦𝖥𝖮[σ]⟧Struct(σ)\uptau_{\subseteq_{i}}\cap\llbracket\mathsf{FO}[\upsigma]\rrbracket_{%
\operatorname{Struct}(\upsigma)}roman_τ start_POSTSUBSCRIPT ⊆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ ⟦ sansserif_FO [ roman_σ ] ⟧ start_POSTSUBSCRIPT roman_Struct ( roman_σ ) end_POSTSUBSCRIPT | C |
Overall, the completed framework achieves the lowest error of distortion estimation as shown in Fig. 9, verifying the effectiveness of our proposed approach. For the optimization strategy, the BS-2 used ℒsmsubscriptℒ𝑠𝑚\mathcal{L}_{sm}caligraphic_L start_POSTSUBSCRIPT italic_s italic_m end_POSTSUBSCRIPT performs muc... | In this section, we first state the details of the synthetic distorted image dataset and the training process of our learning model. Subsequently, we analyze the learning representation for distortion estimation. To demonstrate the effectiveness of each module in our framework, we conduct an ablation study to show the ... |
Figure 1: Method Comparisons. (a) Previous learning methods, (b) Our proposed approach. We aim to transfer the traditional calibration objective into a learning-friendly representation. Previous methods roughly feed the whole distorted image into their learning models and directly estimate the implicit and heterogeneo... |
As listed in Table II, our approach significantly outperforms the compared approaches in all metrics, including the highest metrics on PSNR and SSIM, as well as the lowest metric on MDLD. Specifically, compared with the traditional methods [23, 24] based on the hand-crafted features, our approach overcomes the scene l... |
In this part, we compare our approach with the state-of-the-art methods in both quantitative and qualitative evaluations, in which the compared methods can be classified into traditional methods [23][24] and learning methods [8][11][12]. Note that our approach only requires a patch of the input distorted image to esti... | D |
We further conduct CTR prediction experiments to evaluate SNGM. We train DeepFM [8] on a CTR prediction dataset containing ten million samples that are sampled from the Criteo dataset 777https://ailab.criteo.com/download-criteo-1tb-click-logs-dataset/.
We set aside 20% of the samples as the test set and divide the rema... | 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. | We compare SNGM with four baselines: MSGD, LARS [34], EXTRAP-SGD [19] and CLARS [12]. For LARS, EXTRAP-SGD and CLARS, we adopt the open
source code 222https://github.com/NUS-HPC-AI-Lab/LARS-ImageNet-PyTorch 333http://proceedings.mlr.press/v119/lin20b.html 444https://github.com/slowbull/largebatch | We use a pre-trained ViT 555https://huggingface.co/google/vit-base-patch16-224-in21k [4] model and fine-tune it on the CIFAR-10/CIFAR-100 datasets.
The experiments are implemented based on the Transformers 666https://github.com/huggingface/transformers framework. We fine-tune the model with 20 epochs. | 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. | D |
{\mathcal{F}}roman_support ( caligraphic_D ) ⊆ 2 start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT × blackboard_R start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT and, in the black-box setting, |𝒟|𝒟|\mathcal{D}|| caligraphic_D | may be uncountably infinite.
| 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... | Stochastic optimization, first introduced in the work of Beale [4] and Dantzig [8], provides a way to model uncertainty in the realization of the input data. In this paper, we give approximation algorithms for a family of problems in stochastic optimization, and more precisely in the 2222-stage recourse model [27].
Our... | Clustering is a fundamental task in unsupervised and self-supervised learning. The stochastic setting models situations in which decisions must be made in the presence of uncertainty and are of particular interest in learning and data science. The black-box model is motivated by data-driven applications where specific ... | The most general way to represent the scenario distribution 𝒟𝒟\mathcal{D}caligraphic_D is the black-box model [24, 12, 22, 19, 25], where we have access to an oracle to sample scenarios A𝐴Aitalic_A according to 𝒟𝒟\mathcal{D}caligraphic_D. We also consider the polynomial-scenarios model [23, 15, 21, 10], where the ... | C |
Besides, the network graphs may change randomly with spatial and temporal dependency (i.e. Both the weights of different edges in the network graphs at the same time instant and the network graphs at different time instants may be mutually dependent.) rather than i.i.d. graph sequences as in [12]-[15],
and additive and... |
Motivated by distributed statistical learning over uncertain communication networks, we study the distributed stochastic convex optimization by networked local optimizers to cooperatively minimize a sum of local convex cost functions. The network is modeled by a sequence of time-varying random digraphs which may be sp... |
II. The structure of the networks among optimizers is modeled by a more general sequence of random digraphs. The sequence of random digraphs is conditionally balanced, and the weighted adjacency matrices are not required to have special statistical properties such as independency with identical distribution, Markovian... | 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... | We have studied the distributed stochastic subgradient algorithm for the stochastic optimization by networked nodes to cooperatively minimize a sum of convex cost functions.
We have proved that if the local subgradient functions grow linearly and the sequence of digraphs is conditionally balanced and uniformly conditio... | A |
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.
|
In this work, we propose a novel technique called the Mutual Cover (MuCo) to impede adversary from matching the combination of QI values while overcoming the above issues. The key idea of MuCo is to make similar tuples to cover for each other by randomizing their QI values according to random output tables. | The advantages of MuCo are summarized as follows. First, MuCo can maintain the distributions of original QI values as much as possible. For instance, the sum of each column in Figure 3 is shown by the blue polyline in Figure 2, and the blue polyline almost coincides with the red polyline representing the distribution i... |
This section presents the algorithm to implement the Mutual Cover (MuCo) framework111The code is available at https://github.com/liboyuty/Mutual-Cover.. We aim to achieve two goals. First, MuCo satisfies δ𝛿\deltaitalic_δ-probability to hinder the adversary from matching the combination of QI values. Second, the recor... |
For instance, suppose that we add another QI attribute of gender as shown in Figure 4, the mutual cover strategy first divides the records into groups in which the records in the same group cover for each other by perturbing their QI values. Then, the mutual cover strategy calculates a random output table on each QI a... | C |
HTC is known as a competitive method for COCO and OpenImage. By enlarging the RoI size of both box and mask branches to 12 and 32 respectively for all three stages, we gain roughly 4 mAP improvement against the default settings in original paper. Mask scoring head Huang et al. (2019) adopted on the third stage gains an... | 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... |
Due to limited mask representation of HTC, we move on to SOLOv2, which utilizes much larger mask to segment objects. It builds an efficient yet simple instance segmentation framework, outperforming other segmentation methods like TensorMask Chen et al. (2019c), CondInst Tian et al. (2020) and BlendMask Chen et al. (20... | 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.... | HTC is known as a competitive method for COCO and OpenImage. By enlarging the RoI size of both box and mask branches to 12 and 32 respectively for all three stages, we gain roughly 4 mAP improvement against the default settings in original paper. Mask scoring head Huang et al. (2019) adopted on the third stage gains an... | B |
I(f)<1,andH(|f^|2)>nn+1logn.formulae-sequence𝐼𝑓1and𝐻superscript^𝑓2𝑛𝑛1𝑛I(f)<1,\ \ {\mbox{and}}\ \ H(|\hat{f}|^{2})>\frac{n}{n+1}\log n.italic_I ( italic_f ) < 1 , and italic_H ( | over^ start_ARG italic_f end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) > divide start_ARG italic_n end_ARG start_ARG ita... |
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... |
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... | (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... | 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.
| B |
We propose a parameter-free algorithm called Ada-LSVI-UCB-Restart, an adaptive version of LSVI-UCB-Restart, and prove that it can achieve O~(B1/4d5/4H5/4T3/4)~𝑂superscript𝐵14superscript𝑑54superscript𝐻54superscript𝑇34\tilde{O}(B^{1/4}d^{5/4}H^{5/4}T^{3/4})over~ start_ARG italic_O end_ARG ( italic_B start_POSTSU... | We consider the setting of episodic RL with nonstationary reward and transition functions. To measure the performance of an algorithm, we use the notion of dynamic regret, the performance difference between an algorithm and the set of policies optimal for individual episodes in hindsight. For nonstationary RL, dynamic ... |
Bandit problems can be viewed as a special case of MDP problems with unit planning horizon. It is the simplest model that captures the exploration-exploitation tradeoff, a unique feature of sequential decision-making problems. There are several ways to define nonstationarity in the bandit literature. The first one is ... |
However, all of the aforementioned empirical and theoretical works on RL with function approximation assume the environment is stationary, which is insufficient to model problems with time-varying dynamics. For example, consider online advertising. The instantaneous reward is the payoff when viewers are redirected to ... | 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... | B |
There is a very strong, negative correlation between the media sources of fake news and the level of trust in them (ref. Figures 1 and 2) which is statistically significant (r(9)=−0.81𝑟90.81r(9)=-0.81italic_r ( 9 ) = - 0.81, p<.005𝑝.005p<.005italic_p < .005). Trust is built on transparency and truthfulness, and t... |
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,... |
There is a very strong, negative correlation between the media sources of fake news and the level of trust in them (ref. Figures 1 and 2) which is statistically significant (r(9)=−0.81𝑟90.81r(9)=-0.81italic_r ( 9 ) = - 0.81, p<.005𝑝.005p<.005italic_p < .005). Trust is built on transparency and truthfulness, and t... | 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... | A |
6: 𝐠i,𝐞i←ℳ(ei,Ni)←subscript𝐠𝑖subscript𝐞𝑖ℳsubscript𝑒𝑖subscript𝑁𝑖{\mathbf{g}}_{i},{\mathbf{e}}_{i}\leftarrow{\mathcal{M}}(e_{i},N_{i})bold_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ← caligraphic_M ( italic_e start_POSTSUBSCRIPT italic_i end... | The existing methods for KG embedding and word embedding exhibit even more similarities. As shown in Figure 1, the KG comprises three triplets conveying similar information to the example sentence. Triplet-based KG embedding models like TransE [11] transform the embedding of each subject entity and its relation into a ... |
We present the training procedure of decentRL for entity alignment in Algorithm 1. It is worth noting that decentRL does not rely on additional data such as pretrained KG embeddings or word embeddings. The algorithm first randomly initializes the DAN model, entity embeddings, and relation embeddings. The training proc... |
The results in Table 10 demonstrate that all variants of decentRL achieves state-of-the-art performance on Hits@1, empirically proving the superiority of using neighbor context as the query vector for aggregating neighbor embeddings. The proposed decentRL outperforms both decentRL w/ infoNCE and decentRL w/ L2, provid... |
Table 6 and Table 7 present the results for conventional entity prediction. decentRL demonstrates competitive or even superior performance when compared to state-of-the-art methods on the FB15K and WN18 benchmarks, showcasing its efficacy in entity prediction. While on the FB15K-237 and WN18RR datasets, the performanc... | B |
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 ... | We observe that our method performs the best in most of the games, in both the sample efficiency and the performance of the best policy. The reason our method outperforms other baselines is the multimodality in dynamics that the Atari games usually have. Such multimodality is typically caused by other objects that are ... |
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... | Conducting exploration without the extrinsic rewards is called the self-supervised exploration. From the perspective of human cognition, the learning style of children can inspire us to solve such problems. The children often employ goal-less exploration to learn skills that will be useful in the future. Developmental ... | We compare the model complexity of all the methods in Table I. VDM, RFM, and Disagreement use a fixed CNN for feature extraction. Thus, the trainable parameters of feature extractor are 0. ICM estimates the inverse dynamics for feature extraction with 2.21M parameters. ICM and RFM use the same architecture for dynamics... | B |
Finally, we observe that Floater-Hormann interpolation performs better than multivariate cubic splines. It is comparable to 5thsuperscript5𝑡ℎ5^{th}5 start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT-order splines,
but reaches an accuracy of 10−7superscript10710^{-7}10 start_POSTSUPERSCRIPT - 7 end_POSTSUPER... | Several improvements have been presented, including Floatman–Hormann interpolation [16, 38], that reach better approximation quality than splines.
However, all of them share the above weaknesses (A,B,C), as we demonstrate in the numerical experiments of Section 8. | Finally, we observe that Floater-Hormann interpolation performs better than multivariate cubic splines. It is comparable to 5thsuperscript5𝑡ℎ5^{th}5 start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT-order splines,
but reaches an accuracy of 10−7superscript10710^{-7}10 start_POSTSUPERSCRIPT - 7 end_POSTSUPER... | The observations made in 2D remain valid. However, Floater-Hormann becomes indistinguishable from 5thsuperscript5𝑡ℎ5^{th}5 start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT-order splines.
Further, when considering the amount of coefficients/nodes required to determine the interpolant, plotted in the right p... | In contrast to previous approaches, such as Chebfun [32], multivariate splines [26], and Floater-Hormann interpolation [38], the present
MIP algorithm achieves exponential approximation rates for the Runge function using only sub-exponentially many interpolation nodes. | C |
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. | However, the two-sample tests based on concentration inequalities in Section III give conservative results in practice. We examine the two-sample tests using the projected Wasserstein distance via the permutation approach.
Specifically, we permute the collected data points for Np=100subscript𝑁𝑝100N_{p}=100italic_N st... | In this paper, we consider non-parametric two-sample testing, in which no prior information about the unknown distribution is available.
Two-sample tests for non-parametric settings are usually constructed based on some metrics quantifying the distance between two distributions. | Several data-efficient two-sample tests [20, 21, 22] are constructed based on Maximum Mean Discrepancy (MMD), which quantifies the distance between two distributions by introducing test functions in a Reproducing Kernel Hilbert Space (RKHS).
However, it is pointed out in [23] that when the bandwidth is chosen based on ... | Classical tests (see, e.g., [12]) mainly follow the parametric approaches, which are designed based on prior information about the distributions under each class.
Examples in classical tests include the Hotelling’s two-sample test [13] and the Student’s t-test [14]. | D |
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... | 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... | 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... |
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... | 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... | D |
Graph described in Fig. 4 is an implementation of an XOR gate combining NAND and OR, expressed in 33 vertices and 46 mains. Graphs are expressed in red and blue numbers in cases where there is no direction of the main line (the main line that can be passed in both directions) and the direction of the main line (the ma... |
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... |
Fig. 3 is AND and/or gate consisting of 3-pin based logics, Fig. 3 also shows the connection status of the output pin when A=0, B=1 is entered in the AND gate. when A=0, B=1, or A is connected, and B is connected, output C is connected only to the following two pins, and this is the correct result for AND operation. |
The structural computer used an inverted signal pair to implement the reversal of a signal (NOT operation) as a structural transformation, i.e. a twist, and four pins were used for AND and OR operations as a series and parallel connection were required. However, one can think about whether the four pin designs are the... | 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... | A |
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... |
A finite field, by definition, is a finite set, and the set of all permutation polynomials over the finite field forms a group under composition. Given a finite subset of such permutations, we can compute a group generated by this set. In this paper, we propose a representation of such a group using the concept of lin... | Given a group G𝐺Gitalic_G of permutations over a finite set, the (group) representation represents the group action in terms of invertible matrices over a finite-dimensional vector space, and the group operation is replaced by matrix multiplication. Such representations are imperative in studying abstract groups as it... | This paper defines a linear representation of polynomial maps F𝐹Fitalic_F over finite fields 𝔽𝔽\mathbb{F}blackboard_F as matrices M𝑀Mitalic_M over 𝔽𝔽\mathbb{F}blackboard_F of smallest size N𝑁Nitalic_N. The number N𝑁Nitalic_N is defined as the Linear Complexity of F𝐹Fitalic_F over 𝔽𝔽\mathbb{F}blackboard_F. Th... |
A finite group, GFsubscript𝐺𝐹G_{F}italic_G start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT, can be generated from Fisubscript𝐹𝑖F_{i}italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT using composition as the group operation. In this section, we devise a procedure to compute the linear representation of the gro... | B |
Stacked penalized logistic regression (StaPLR) (Van Loon \BOthers., \APACyear2020) is a method specifically developed to tackle the joint classification and view selection problem. Compared with a variant of the lasso for selecting groups of features (the so-called group lasso (M. Yuan \BBA Lin, \APACyear2007)), StaPLR... | For this purpose, one would ideally like to use an algorithm that provides sparsity, but also algorithmic stability in the sense that given two very similar data sets, the set of selected views should vary little. However, sparse algorithms are generally not stable, and vice versa (Xu \BOthers., \APACyear2012).
An exam... | In high-dimensional biomedical studies, a common goal is to create an accurate classification model using only a subset of the features (Y. Li \BOthers., \APACyear2018). A popular approach to this type of joint classification and feature selection problem is to apply penalized methods such as the lasso (Tibshirani, \AP... | In this article we investigated how different view-selecting meta-learners affect the performance of multi-view stacking. In our simulations, the interpolating predictor often performed worse than the other meta-learners on at least one outcome measure. For example, when the sample size was larger than the number of vi... | A particular challenge of the aforementioned joint classification and view selection problem is its inherent trade-off between accuracy and sparsity. For example, the most accurate model may not perform the best in terms of view selection. In fact, the prediction-optimal amount of regularization causes the lasso to sel... | D |
The proximity-based approach is mainstream in anomaly detection [8, 9, 10, 11], and operates on the assumption that anomalies are objects that exhibit significant distance or sparsity in their neighborhood compared to other objects. The anomalousness of an object is determined by its proximity to neighboring objects. P... | This example highlights the fundamental difference between proximity-based and dependency-based methods. Dependency-based methods focus on identifying anomalies based on underlying relationships between variables, whereas proximity-based methods rely on object similarity in terms of proximity. In cases like this, where... | Various anomaly detection methods have been developed to leverage the distinctive characteristics of anomalies that deviate from the norm in some manner. The typical process of anomaly detection involves assuming a specific aspect in which anomalies are considered abnormal and then assessing the anomalousness of object... |
Dependency-based approach is fundamentally different from proximity-based approach because it considers the relationship among variables, while proximity-based approach examines the relationship among objects. We use an example to explain the difference between the two approaches. |
The dependency-based approach works under the assumption that anomalies deviate from the normal dependency among variables, and the extend of anomalousness is evaluated based on this deviation. While the proximity-based approach that focuses on relationships among objects, the dependency-based approach emphasizes on t... | D |
At the start of the interaction, when no contexts have been observed, θ^tsubscript^𝜃𝑡\hat{\theta}_{t}over^ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is well-defined by Eq (5) when λt>0subscript𝜆𝑡0\lambda_{t}>0italic_λ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT > 0. Therefore, th... | Algorithm 1 follows the template of in the face of uncertainty (OFU) strategies [Auer et al., 2002, Filippi et al., 2010, Faury et al., 2020]. Technical analysis of OFU algorithms relies on two key factors: the design of the confidence set and the ease of choosing an action using the confidence set.
| 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 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... | where pessimism is the additive inverse of the optimism (difference between the payoffs under true parameters and those estimated by CB-MNL). Due to optimistic decision-making and the fact that θ∗∈Ct(δ)subscript𝜃subscript𝐶𝑡𝛿\theta_{*}\in C_{t}(\delta)italic_θ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ∈ italic_C star... | A |
Though many methods (e.g., [1, 3, 9, 20, 21, 24, 42, 43, 44, 46]) in recent years have been continuously breaking the record of TAL performance, a major challenge hinders its substantial improvement – large variation in action duration. An action can last from a fraction of a second to minutes in the real-world scenari... |
Recent temporal action localization methods can be generally classified into two categories based on the way they deal with the input sequence. In the first category, the works such as BSN [21], BMN [20], G-TAD [44], BC-GNN [3] re-scale each video to a fixed temporal length (usually a small length such as 100 snippets... | 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... |
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 ... |
Why are short actions hard to localize? Short actions have small temporal scales with fewer frames, and therefore, their information is prone to loss or distortion throughout a deep neural network. Most methods in the literature process videos regardless of action duration, which as a consequence sacrifices the perfor... | D |
To provide a holistic view on the performance of the models for the selected validation metrics, we use a UMAP [MHM18] projection, as seen in Figure 2(a), that consists of the 500 randomly-sampled models (MDS [Kru64] and t-SNE [vdMH08] are also available).
Each model uses a set of particular hyperparameters, and it is ... | Thus, groups of points represent clusters of models that perform similarly according to all the metrics.
The plot uses the Viridis colormap [LH18] to show the average performance of each model according to all selected metrics. This view provides the user with an overview of the hyperparameter space and ability to look... |
Figure 5: The exploration of clusters of interest that contain performant ML models. View (a) presents the user’s selection that drive the analyses performed in the remaining subfigures. (b.1) provides an overview of the performance, showing that \raisebox{0.15pt}{\resizebox{!}{0.8ex}{\textbf{\textsf{C3}}}}⃝ has under... | (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... |
At this phase, we want to confirm precisely the cluster affiliation and the relationship with the overall performance (here, the average of 4 validation metrics) for all the models. To achieve that, the beeswarm plots in Figure 2(b.1 and b.2) arrange the models according to the distinct algorithms in the x-axis and so... | A |
In the context of addressing the guidance problem for a large number of agents, considering the spatial distribution of swarm agents and directing it towards a desired steady-state distribution offers a computationally efficient approach. In this regard, both probabilistic and deterministic swarm guidance algorithms ar... | This algorithm treats the spatial distribution of swarm agents, called the density distribution, as a probability distribution and employs the Metropolis-Hastings (M-H) algorithm to synthesize a Markov chain that guides the density distribution toward a desired state.
The probabilistic guidance algorithm led to the dev... | Unlike the homogeneous Markov chain synthesis algorithms in [4, 7, 5, 6, 8, 9], the Markov matrix, synthesized by our algorithm, approaches the identity matrix as the probability distribution converges to the desired steady-state distribution. Hence the proposed algorithm attempts to minimize the number of state transi... | 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... | Building on this new consensus protocol, the paper introduces a decentralized state-dependent Markov chain (DSMC) synthesis algorithm. It is demonstrated that the synthesized Markov chain, formulated using the proposed consensus algorithm, satisfies the aforementioned mild conditions. This, in turn, ensures the exponen... | A |
We have proven that the IsoMuSh algorithm is convergent in the objective f(⋅,⋅)𝑓⋅⋅f(\cdot,\cdot)italic_f ( ⋅ , ⋅ ). However, we did not establish convergence of the variables U𝑈Uitalic_U and Q𝑄Qitalic_Q. In this context, we note that there are equivalence classes of U𝑈Uitalic_U and Q𝑄Qitalic_Q that lead to the sa... | Similar to the previous section, we want to impose cycle consistency on the pairwise functional maps 𝒞ijsubscript𝒞𝑖𝑗\mathcal{C}_{ij}caligraphic_C start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT.
We do so by defining a shape-to-universe functional map 𝒞isubscript𝒞𝑖\mathcal{C}_{i}caligraphic_C start_POSTS... |
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 ... | In contrast, HiPPI and our method require shape-to-universe representations. To obtain these, we use synchronisation to extract the shape-to-universe representation from the pairwise transformations. By doing so, we obtain the initial U𝑈Uitalic_U and Q𝑄Qitalic_Q. We refer to this method of synchronising the ZoomOut r... | A shortcoming when applying the mentioned multi-shape matching approaches to isometric settings is that they do not exploit structural properties of isometric shapes. Hence, they lead to suboptimal multi-matchings, which we experimentally confirm in Sec. 5. One exception is the recent work on spectral map synchronisati... | B |
The first three steps of algorithm RecognizePG are implied by the first part of Theorem 6. By following Theorem 6, we have to check that there are no full antipodal triangle in UpperCsubscriptUpper𝐶\text{Upper}_{C}Upper start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT (this is made in Step 4), and we have to find f:ΓC→[... | 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... |
The recognition algorithm RecognizePG for path graph is mainly built on path graphs’ characterization in [1]. This characterization decomposes the input graph G𝐺Gitalic_G by clique separators as in [18], then at the recursive step one has to find a proper vertex coloring of an antipodality graph satisfying some parti... | The paper is organized as follows. In Section 2 we present the characterization of path graphs and directed path graphs given by Monma and Wei [18], while in Section 3 we explain the characterization of path graphs by Apollonio and Balzotti [1]. In Section 4 we present our recognition algorithm for path graphs, we prov... | In this section we analyze all steps of algorithm RecognizePG. We want to explain them in details and compute the computational complexity of the algorithm. Some of these steps are already discussed in [22], anyway, we describe them in order to have a complete treatment.
| D |
Given (n,P,Θ,Π)𝑛𝑃ΘΠ(n,P,\Theta,\Pi)( italic_n , italic_P , roman_Θ , roman_Π ), we can generate a random adjacency matrix A𝐴Aitalic_A under DCMM. For convenience, we denote the DCMM model as DCMM(n,P,Θ,Π)𝐷𝐶𝑀𝑀𝑛𝑃ΘΠDCMM(n,P,\Theta,\Pi)italic_D italic_C italic_M italic_M ( italic_n , italic_P , roman_Θ , roman... | In this section, first, we investigate the performances of Mixed-SLIM methods for the problem of mixed membership community detection via synthetic data. Then we apply some real-world networks with true label information to test Mixed-SLIM methods’ performances for community detection, and we apply the SNAP ego-network... | In this section, we first introduce the main algorithm mixed-SLIM which can be taken as a natural extension of the SLIM (SLIM, ) to the mixed membership community detection problem. Then we discuss the choice of some tuning parameters in the proposed algorithm.
| In this paper, we extend the symmetric Laplacian inverse matrix (SLIM) method (SLIM, ) to mixed membership networks and call this proposed method as mixed-SLIM. As mentioned in SLIM , the idea of using the symmetric Laplacian inverse matrix to measure the closeness of nodes comes from the first hitting time in a random... |
This paper makes one major contribution: modified SLIM methods to mixed membership community detection under the DCMM model. When dealing with large networks in practice, we apply Mixed-SLIMapprosubscriptSLIM𝑎𝑝𝑝𝑟𝑜\mathrm{SLIM}_{appro}roman_SLIM start_POSTSUBSCRIPT italic_a italic_p italic_p italic_r italic_o ... | B |
For any functional F:ℳ→ℝ:𝐹→ℳℝF\colon\mathcal{M}\rightarrow\mathbb{R}italic_F : caligraphic_M → blackboard_R, we let gradFgrad𝐹\operatorname{{\mathrm{grad}}}Froman_grad italic_F denote the functional gradient of F𝐹Fitalic_F with respect to the Riemannian metric g𝑔gitalic_g.
| Here the statistical error is incurred in estimating the Wasserstein gradient by solving the dual maximization problem using functions in a reproducing kernel Hilbert space (RKHS) with finite data, which converges sublinearly to zero as the number of particles goes to infinity.
Therefore, in this scenario, variational ... | we prove that variational transport constructs a sequence of probability distributions that converges linearly to the global minimizer of the objective functional up to a statistical error due to estimating the Wasserstein gradient with finite particles. Moreover, such a statistical error converges to zero as the numbe... | To study optimization problems on the space of probability measures, we first introduce the background knowledge of the Riemannian manifold and the Wasserstein space. In addition, to analyze the statistical estimation problem that arises in estimating the Wasserstein gradient, we introduce the reproducing kernel Hilber... | Second, when the Wasserstein gradient is approximated using RKHS functions and the objective functional satisfies the PL condition, we prove that the sequence of probability distributions constructed by variational transport converges linearly to the global minimum of the objective functional, up to certain statistical... | C |
The evaluation scenarios come from four real road network maps of different scales, including Hangzhou (China), Jinan (China), New York (USA) and Shenzhen (China), illustrated in Fig. 6. The road networks and data of Hangzhou, Jinan and New York are from the public datasets222https://traffic-signal-control.github.io/.... |
Mixedh. The mixedh is a mixed high traffic flow with a total flow of 4770 in one hour, in order to simulate a heavy peak. The difference from the mixedl setting is that the arrival rate of vehicles during 1200-1800s increased from 0.33 vehicles/s to 4.0 vehicles/s. The data statistics are listed in Tab. II. | We run the experiments under three traffic flow configurations: real traffic flow, mixed low traffic flow and mixed high traffic flow. The real traffic flow is real-world hourly statistical data with slight variance in vehicle arrival rates, as shown in Tab. I. Since the real-world strategies tend to break down during ... |
Real. The traffic flows of Hangzhou (China), Jinan (China) and New York (USA) are from the public datasets444https://traffic-signal-control.github.io/, which are processed from multiple sources. The traffic flow of Shenzhen (China) is made by ourselves generated based on the traffic trajectories collected from 80 red-... |
Mixedl. The mixedl is a mixed low traffic flow with a total flow of 2550 in one hour, to simulate a light peak. The arrival rate changes every 10 minutes, which is used to simulate the uneven traffic flow distribution in the real world, the details of the vehicle arrival rate and cumulative traffic flow are shown in F... | B |
(\mathbf{z}_{\mathbf{b}})-\mathbf{b})\,=\,\mathbf{0}italic_ϕ start_POSTSUBSCRIPT bold_z end_POSTSUBSCRIPT ( bold_z start_POSTSUBSCRIPT bold_b end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT sansserif_H end_POSTSUPERSCRIPT ( italic_ϕ ( bold_z start_POSTSUBSCRIPT bold_b end_POSTSUBSCRIPT ) - bold_b ) = bold_0 and
thus (2.11) | 𝐯∈ℛ𝒶𝓃ℊℯ(ϕ𝐳(𝐳∗))⟂𝐯ℛ𝒶𝓃ℊℯsuperscriptsubscriptitalic-ϕ𝐳subscript𝐳perpendicular-to\mathbf{v}\,\in\,\mathpzc{Range}(\phi_{\mathbf{z}}(\mathbf{z}_{*}))^{\perp}bold_v ∈ italic_script_R italic_script_a italic_script_n italic_script_g italic_script_e ( italic_ϕ start_POSTSUBSCRIPT bold_z end_POSTSUBSCRIPT ( bold_... | Then, for every 𝐛∈Δ𝐛Δ\mathbf{b}\,\in\,\Deltabold_b ∈ roman_Δ, there exists a 𝐳𝐛∈Σ¯0subscript𝐳𝐛subscript¯Σ0\mathbf{z}_{\mathbf{b}}\,\in\,\overline{\Sigma}_{0}bold_z start_POSTSUBSCRIPT bold_b end_POSTSUBSCRIPT ∈ over¯ start_ARG roman_Σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT such that
‖ϕ(𝐳𝐛)−𝐛‖2=min𝐳∈... | from ℛ𝒶𝓃ℊℯ(ϕ𝐳(𝐳𝐛)𝖧)=ℛ𝒶𝓃ℊℯ(ϕ𝐳(𝐳𝐛)†)ℛ𝒶𝓃ℊℯsubscriptitalic-ϕ𝐳superscriptsubscript𝐳𝐛𝖧ℛ𝒶𝓃ℊℯsubscriptitalic-ϕ𝐳superscriptsubscript𝐳𝐛†\mathpzc{Range}(\phi_{\mathbf{z}}(\mathbf{z}_{\mathbf{b}})^{{\mbox{\tiny$%
\mathsf{H}$}}})\,=\,\mathpzc{Range}(\phi_{\mathbf{z}}(\mathbf{z}_{\mathbf{b}})% | 𝓇𝒶𝓃𝓀(ϕ𝐳(𝐳))≡𝓀𝓇𝒶𝓃𝓀subscriptitalic-ϕ𝐳𝐳𝓀\mathpzc{rank}\left(\,\phi_{\mathbf{z}}(\mathbf{z})\,\right)\,\equiv\,kitalic_script_r italic_script_a italic_script_n italic_script_k ( italic_ϕ start_POSTSUBSCRIPT bold_z end_POSTSUBSCRIPT ( bold_z ) ) ≡ italic_script_k for all 𝐳∈Λ1𝐳subscriptΛ1\mathbf{z}\in\La... | C |
=(1+ϵ)((1+(2+5ϵ)ηk+ϵ)λ+cA(1−λ))|Opt(σ)|,absent1italic-ϵ125italic-ϵ𝜂𝑘italic-ϵ𝜆subscript𝑐𝐴1𝜆Opt𝜎\displaystyle=(1+\epsilon)((1+(2+5\epsilon)\eta k+\epsilon)\lambda+c_{A}(1-%
\lambda))|\textsc{Opt}(\sigma)|,= ( 1 + italic_ϵ ) ( ( 1 + ( 2 + 5 italic_ϵ ) italic_η italic_k + italic_ϵ ) italic_λ + italic_c start... | To obtain the best theoretical performance, we can choose A𝐴Aitalic_A as the algorithm of the best known competitive ratio, that is Advanced Harmonic algorithm (?). However, as discussed in Section 2, such algorithms belong to a class that is tailored to worst-case competitive analysis, and do not tend to perform well... | In order to analyze the performance of an online algorithm, we will rely on the well-established framework of competitive analysis, which provides strict, theoretical performance guarantees against worst-case scenarios. In fact, as stated in (?), bin packing has served as “an early proving ground for this type of analy... | These algorithms are variants of the classic Harmonic algorithm (?), which places items of approximately equal sizes, according to a harmonic sequence, in the same bin.
The currently best algorithm is the Advanced Harmonic (AH) algorithm, which has a competitive ratio of 1.57829 (?), whereas the best-known lower bound ... | In this setting, the objective is to minimize the expected loss, defined as the difference between the number of bins opened by the algorithm, and the total size of all items normalized by the bin capacity.
Ideally, one aims for a loss that is as small as o(n)𝑜𝑛o(n)italic_o ( italic_n ), where n𝑛nitalic_n is the nu... | A |
Finally, we empirically show the proposed framework produces high-fidelity and watertight meshes. It means that it solves the initial problem of disjoint patches occurring in the original AtlasNet (Groueix et al., 2018). To evaluate the continuity of output surfaces, we propose to use the following metric. |
The above formulation alone causes that many of the produced patches have unnecessarily long edges, and the network folds them, so the patch fits the surface of an object. To mitigate the issue, we add an edge length regularization motivated by (Wang et al., 2018). If we assume that the reconstructed mesh has the form... | Watertigthness Typically, a mesh is referred to as being either watertight or not watertight. Since it is a true or false statement, there is no well-established measure to define the degree of discontinuities in the object’s surface. To fill this gap, we propose a metric based on a simple, approximate check of whether... |
In this experiment, we set N=105𝑁superscript105N=10^{5}italic_N = 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT. Using more rays had a negligible effect on the output value of WT𝑊𝑇WTitalic_W italic_T but significantly slowed the computation. We compared AtlasNet with LoCondA applied to HyperCloud (HC) and HyperFl... |
To leverage that knowledge, we express watertigthness as a ratio of rays that passed the parity test to the total number of all casted rays. Firstly, we sample N𝑁Nitalic_N points p∈S^𝑝^𝑆p\in\hat{S}italic_p ∈ over^ start_ARG italic_S end_ARG from all triangles of the reconstructed object S^^𝑆\hat{S}over^ start_ARG ... | B |
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... | 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... |
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. ... | 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 |
The set of cycles of a graph has a vector space structure over ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, in the case of undirected graphs, and over ℚℚ\mathbb{Q}blackboard_Q, in the case of directed graphs [5]. A basis of such a vector space is denoted cycle basis and its dimensio... |
Different classes of cycle bases can be considered. In [6] the authors characterize them in terms of their corresponding cycle matrices and present a Venn diagram that shows their inclusion relations. Among these classes we can find the strictly fundamental class. |
In the introduction of this article we mentioned that the MSTCI problem is a particular case of finding a cycle basis with sparsest cycle intersection matrix. Another possible analysis would be to consider this in the context of the cycle basis classes described in [6]. |
The length of a cycle is its number of edges. The minimum cycle basis (MCB) problem is the problem of finding a cycle basis such that the sum of the lengths (or edge weights) of its cycles is minimum. This problem was formulated by Stepanec [7] and Zykov [8] for general graphs and by Hubicka and Syslo [9] in the stric... | In the case that we can find some non-star spanning tree T𝑇Titalic_T of
G𝐺Gitalic_G such that ∩(T)<∩(Ts)𝑇subscript𝑇𝑠\cap(T)<\cap(T_{s})∩ ( italic_T ) < ∩ ( italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) then, we can “simplify” the instance by removing the interbranch cycle-edges with respect to T𝑇Tital... | A |
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)... |
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 ... |
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... | 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... | D |
Automatic feature transformation has been examined within the ML community with positive results in reinforcement learning. In the work by Khurana et al. [1], the authors conduct a performance-driven exploration of a transformation graph which systematically enumerates the space of given options. A single “best” measu... | Feature transformation usually denotes less sophisticated modifications over the features [14]. Some of the standard transformations also supported by our approach are: (1) rounding, (2) binning, (3) scaling, (4) logarithmic transformations, (5) exponential transformations, and (6) power functions. In this scenario, ML... | A VA system for regression analysis has been proposed by Mühlbacher and Piringer [15]. The system is more similar to our work as it also incorporates feature transformation in its design (specifically logarithmic, exponential, and power functions). The main difference between this work and ours is our focus on classifi... |
Automatic feature transformation has been examined within the ML community with positive results in reinforcement learning. In the work by Khurana et al. [1], the authors conduct a performance-driven exploration of a transformation graph which systematically enumerates the space of given options. A single “best” measu... | Several VA systems have been developed to explore and select subsets of features with the help of visualization. Finding which features to transform, and how, together with generating new features from different combinations, are some of the core phases that lack attention from the InfoVis/VA communities.
This section ... | B |
‖e^c‖∞subscriptnormsubscript^𝑒𝑐\|\hat{e}_{c}\|_{\infty}∥ over^ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT
‖e^c‖2subscriptnormsubscript^𝑒𝑐2\|\hat{e}_{c}\|_{2}∥ over^ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∥ st... |
Figure 5: Position, velocity, acceleration, and maximal contour error resulting from optimization of the MPC parameters, comparing unconstrained BO optimization (solid lines) to BO optimization with additional constraint on the maximal tracking error, for infinity (left) and octagon(center) geometries. The right panel... | 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... | To reduce the number of times this experimental “oracle” is invoked, we employ Bayesian optimization (BO) [16, 17], which is an effective method for controller tuning [13, 18, 19] and optimization of industrial processes [20]. The constrained Bayesian optimization samples and learns both the objective function and the ... | For the initialization phase needed to train the GPs in the Bayesian optimization, we select 20 samples over the whole range of MPC parameters, using Latin hypercube design of experiments. The BO progress is shown in Figure 5, right pannel, for the optimization with constraints on the jerk and on the tracking error. Af... | A |
Results. We find that implicit methods either improve or are comparable with StdM, but most explicit methods fail when asked to generalize to multiple bias variables and a large number of groups, even when the bias variables are explicitly provided. As shown in Fig. 4, all explicit methods are below StdM on Biased MNI... | Results.
In Fig. 3(a), we present the MMD boxplots for all bias variables, comparing cases when the label of the variable is either explicitly specified (explicit bias), or kept hidden (implicit bias) from the methods. Barring digit position, we observe that the MMD values are higher when the variables are not explicit... | 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... |
Results. We find that implicit methods either improve or are comparable with StdM, but most explicit methods fail when asked to generalize to multiple bias variables and a large number of groups, even when the bias variables are explicitly provided. As shown in Fig. 4, all explicit methods are below StdM on Biased MNI... | B |
ℒEuclidean=‖𝒑−𝒑^‖2,subscriptℒEuclideansubscriptnorm𝒑bold-^𝒑2\mathcal{L}_{\mathrm{Euclidean}}=||\boldsymbol{p}-\boldsymbol{\hat{p}}||_{2},caligraphic_L start_POSTSUBSCRIPT roman_Euclidean end_POSTSUBSCRIPT = | | bold_italic_p - overbold_^ start_ARG bold_italic_p end_ARG | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , |
The calibration problem can be considered as domain adaption problems, where the training set is the source domain and the test set is the target domain. The test set usually contains unseen subjects or unseen environment. Researchers aim to improve the performance in the target domain using calibration samples. | We also convert the two definitions with post-processing methods following Sec. 4.2.2.
We respectively conduct benchmarks for 2D PoG and 3D gaze estimation. The 3D gaze estimation also are divided into within-dataset and cross-dataset evaluation. We mark the top three performance in all benchmarks with underlines. | Two kinds of evaluation protocols are commonly used for deep-learning based gaze estimation methods, including within-dataset and cross-dataset evaluation.
The within-dataset evaluation assesses the model performance on the unseen subjects from the same dataset. The dataset is divided into training and test set accordi... | It is the most popular dataset for appearance-based gaze estimation methods. It contains a total of 213,659 images collected from 15 subjects. The images are collected in daily life over several months and there is no constraint for the head pose. MPIIGaze dataset provides both 2D and 3D gaze annotation. It also provid... | C |
The images of the used dataset are already cropped around the face, so we don’t need a face detection stage to localize the face from each image. However, we need to correct the rotation of the face so that we can remove the masked region efficiently. To do so, we detect 68 facial landmarks using Dlib-ml open-source l... | he2016deep has been successfully used in various pattern recognition tasks such as face and pedestrian detection mliki2020improved . It containing 50 layers trained on the ImageNet dataset. This network is a combination of Residual network integrations and Deep architecture parsing. Training with ResNet-50 is faster d... |
The images of the used dataset are already cropped around the face, so we don’t need a face detection stage to localize the face from each image. However, we need to correct the rotation of the face so that we can remove the masked region efficiently. To do so, we detect 68 facial landmarks using Dlib-ml open-source l... | The next step is to apply a cropping filter in order to extract only the non-masked region. To do so, we firstly normalize all face images into 240 ×\times× 240 pixels. Next, we partition a face into blocks. The principle of this technique is to divide the image into 100 fixed-size square blocks (24 ×\times× 24 pixels ... | 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 (... | C |
Note: this is an extended version of an eponymous paper that appeared in FSCD 2022 that includes further examples (Examples 1, 1, and 1), a more straightforward presentation of the metatheory (Section 4) based on Kripke logical relations [Plo73], and a representative set of the corresponding proofs (Sections 3 and 4).
| Adding (co)inductive types and terminating recursion (including productive corecursive definitions) to any programming language is a non-trivial task, since only certain recursive programs constitute valid applications of (co)induction principles. Briefly, inductive calls must occur on data smaller than the input and, ... | Sized types are a type-oriented formulation of size-change termination [LJBA01] for rewrite systems [TG03, BR09]. Sized (co)inductive types [BFG+04, Bla04, Abe08, AP16] gave way to sized mixed inductive-coinductive types [Abe12, AP16]. In parallel, linear size arithmetic for sized inductive types [CK01, Xi01, BR06] was... |
One solution that avoids syntactic checks is to track the flow of (co)data size at the type level with sized types, as pioneered by Hughes et al. [HPS96] and further developed by others [BFG+04, Bla04, Abe08, AP16]. Inductive and coinductive types are indexed by the height and observable depth of their data and codata... |
Moreover, some prior work, which is based on sequential functional languages, encodes recursion via various fixed point combinators that make both mixed inductive-coinductive programming [Bas18] and substructural typing difficult, the latter requiring the use of the ! modality [Wad12]. Thus, like Fωcopsuperscriptsubsc... | A |
Afterwards, Bianchi et al. [10] proposed a LUT-based AFP scheme without involving a Trusted Third Party (TTP) based on homomorphic encryption, which also implements AFP within the user-side framework.
Despite the fact that Problems 2 and 3 are solved in these works, Problem 1 is not mentioned. | Thirdly, there are also studies that deal with both privacy-protected access control and traitor tracing. Xia et al. [26] introduced the watermarking technique to privacy-protected content-based ciphertext image retrieval in the cloud, which can prevent the user from illegally distributing the retrieved images. However... |
This paper solves the three problems faced by cloud media sharing and proposes two schemes FairCMS-I and FairCMS-II. FairCMS-I gives a method to transfer the management of LUTs to the cloud, enabling the calculation of each user’s D-LUT in the ciphertext domain and its subsequent distribution. However, utilizing the s... | Moreover, FairCMS-I does not perform any processing on the encrypted media content stored in the cloud, but only performs homomorphic operations and re-encryption operations on the encrypted LUT and fingerprint that are much smaller in size, which results in outstanding cloud-side efficiency. In contrast, the two schem... | The owner-side efficiency and scalability performance of FairCMS-II are directly inherited from FairCMS-I, and the achievement of the three security goals of FairCMS-II is also shown in Section VI. Comparing to FairCMS-I, it is easy to see that in FairCMS-II the cloud’s overhead is increased considerably due to the ado... | A |
To capture the impact of these feature interactions, a model might consider a 3-order cross feature such as (Genre = fiction, Director = Christopher Nolan, Starring = Leonardo DiCaprio) or (Language = Chinese, Genre = action, Starring = Bruce Lee) as potentially indicating higher user preferences.
| In this work, we disclose the relationship between FM and GNN, and seamlessly combine them to propose a novel model GraphFM for feature interaction learning.
The proposed model leverages the strength of FM and GNN and also solve their respective drawbacks. |
Factorization machine (FM) Rendle (2010, 2012) are a popular and effective method for modeling feature interactions, which involve learning a latent vector for each one-hot encoded feature and modeling the pairwise (second-order) interactions between them through the inner product of their respective vectors. FM has b... | Modeling feature interactions is a crucial aspect of predictive analytics and has been widely studied in the literature. FM Rendle (2010) is a popular method that learns pairwise feature interactions through vector inner products. Since its introduction, several variants of FM have been proposed, including Field-aware ... | (2) By treating features as nodes and their pairwise feature interactions as edges, we bridge the gap between GNN and FM, and make it feasible to leverage the strength of GNN to solve the problem of FM.
(3) Extensive experiments are conducted on CTR benchmark and recommender system datasets to evaluate the effectivenes... | B |
For clarity we want to stress that any linear rate over polytopes has to depend also on the ambient dimension of the polytope; this applies to our linear rates and those in Table 1 established elsewhere (see Diakonikolas et al. [2020]).
In contrast, the 𝒪(1/ε)𝒪1𝜀\mathcal{O}(1/\varepsilon)caligraphic_O ( 1 / italic_... | the second-order step size and the LLOO algorithm from Dvurechensky et al. [2022] (denoted by GSC-FW and LLOO in the figures) and the Frank-Wolfe and the Away-step Frank-Wolfe algorithm with the backtracking stepsize of Pedregosa et al. [2020],
denoted by B-FW and B-AFW respectively. | We show that a small variation of the original Frank-Wolfe algorithm [Frank & Wolfe, 1956] with an open-loop step size of the form γt=2/(t+2)subscript𝛾𝑡2𝑡2\gamma_{t}=2/(t+2)italic_γ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 2 / ( italic_t + 2 ), where t𝑡titalic_t is the iteration count is all that is needed ... | We note that the LBTFW-GSC algorithm from Dvurechensky et al. [2022] is in essence the Frank-Wolfe algorithm with a modified version of the backtracking line search of Pedregosa et al. [2020]. In the next section, we provide improved convergence guarantees for various cases of interest for this algorithm, which we refe... |
After publication of our initial draft, in a revision of their original work, Dvurechensky et al. [2022] added an analysis of the Away-step Frank-Wolfe algorithm which is complementary to ours (considering a slightly different setup and regimes) and was conducted independently; we have updated the tables to include th... | D |
For the rest of the graph, [EKMS12] show that it is enough to store the length of the shortest alternating path that has reached each matched edge. This length is called label.
In the first challenge, we considered the possibility that a vertex γ𝛾\gammaitalic_γ “blocks” the DFS exploration of α𝛼\alphaitalic_α and dis... | If the alternating path Pγsubscript𝑃𝛾P_{\gamma}italic_P start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT starting from γ𝛾\gammaitalic_γ was of length i′>isuperscript𝑖′𝑖i^{\prime}>iitalic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_i, then it could be that γ𝛾\gammaitalic_γ did not find β𝛽\betaitalic_β si... | See the red path in Figure 3 for an illustration.
This, in turn, brings us to trouble since we cannot use the observation from the first challenge (in which α𝛼\alphaitalic_α and γ𝛾\gammaitalic_γ could augment), as there might not be any other free vertex to find an augmentation to. | Nodes α𝛼\alphaitalic_α, β𝛽\betaitalic_β, and γ𝛾\gammaitalic_γ are free. The black single-segments are unmatched and black (full) double-segments are matched edges. The path P′superscript𝑃′P^{\prime}italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT corresponding to a DFS branch of γ𝛾\gammaitalic_γ is shown by th... | For the rest of the graph, [EKMS12] show that it is enough to store the length of the shortest alternating path that has reached each matched edge. This length is called label.
In the first challenge, we considered the possibility that a vertex γ𝛾\gammaitalic_γ “blocks” the DFS exploration of α𝛼\alphaitalic_α and dis... | B |
The n𝑛nitalic_n agents are connected through a general directed network and only communicate directly with their immediate neighbors.
The problem (1) has received much attention in recent years due to its wide applications in distributed machine learning [1, 2, 3], multi-agent target seeking [4, 5], and wireless netwo... | Recently, several compression methods have been proposed for distributed and federated learning, including [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].
Recent works have tried to combine the communication compression methods with decentralized optimization. | For example, the rapid development of distributed machine learning involves data whose size is getting increasingly large, and they are usually stored across multiple computing agents that are spatially distributed. Centering large amounts of data is often undesirable due to limited communication resources and/or priva... | The n𝑛nitalic_n agents are connected through a general directed network and only communicate directly with their immediate neighbors.
The problem (1) has received much attention in recent years due to its wide applications in distributed machine learning [1, 2, 3], multi-agent target seeking [4, 5], and wireless netwo... |
In decentralized optimization, efficient communication is critical for enhancing algorithm performance and system scalability. One major approach to reduce communication costs is considering communication compression, which is essential especially under limited communication bandwidth. | B |
20: uxmk+1=δkuxmk+(1−δk)xmk+1subscriptsuperscript𝑢𝑘1subscript𝑥𝑚superscript𝛿𝑘subscriptsuperscript𝑢𝑘subscript𝑥𝑚1superscript𝛿𝑘superscriptsubscript𝑥𝑚𝑘1u^{k+1}_{x_{m}}=\delta^{k}u^{k}_{x_{m}}+(1-\delta^{k})x_{m}^{k+1}italic_u start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ita... | Our first two methods make several iterations between communications when λ𝜆\lambdaitalic_λ is small (or vice versa, for big λ𝜆\lambdaitalic_λ make some communications between one local iteration). The following method (Algorithm 3) is also sharpened on the alternation of local iterations and communications, but it m... |
We adapt the proposed algorithm for training neural networks. We compare our algorithms: type of sliding (Algorithm 1) and type of local method (Algorithm 3). To the best of our knowledge, this is the first work that compares these approaches in the scope of neural networks, as previous studies were limited to simpler... | To the best of our knowledge, this paper is the first to consider decentralized personalized federated saddle point problems, propose optimal algorithms and derives the computational and communication lower bounds for this setting. In the literature, there are works on general (non-personalized) SPPs. We make a detaile... | Unlike (2), the formulation (1) penalizes not the difference with the global average, but the sameness with other connected local nodes. Thereby the decentralized case can be artificially created in centralized architecture, e.g., if we want to create the network and W𝑊Witalic_W matrix to connect only some clients bas... | A |
σ∗=b−CATα∗+Cβ∗.superscript𝜎𝑏𝐶superscript𝐴𝑇superscript𝛼𝐶superscript𝛽\sigma^{*}=b-CA^{T}\alpha^{*}+C\beta^{*}.italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_b - italic_C italic_A start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_α start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT + italic_C it... | MG(C)CE can provide solutions in general-support and, similar to MECE, MG(C)CE permits a scalable representation when the solution is full-support. Under this scenario, the distribution inequality constraint variables, β𝛽\betaitalic_β, are inactive, are equal to zero, can be dropped, and the α𝛼\alphaitalic_α variable... |
The full-support assumption states that all joint probabilities have some positive mass, σ>0𝜎0\sigma>0italic_σ > 0. In this scenario, the dual variable vector corresponding to the non-negative probability constraint is zero, β=0𝛽0\beta=0italic_β = 0. Therefore we can define simplified primal and dual objectives. | The primal objective that we wish to optimize is minσmaxα,β,λL(σ,α,β,λ)=Lσα,β,λsubscript𝜎subscript𝛼𝛽𝜆𝐿𝜎𝛼𝛽𝜆superscriptsubscript𝐿𝜎𝛼𝛽𝜆\min_{\sigma}\max_{\alpha,\beta,\lambda}L(\sigma,\alpha,\beta,\lambda)=L_{%
\sigma}^{\alpha,\beta,\lambda}roman_min start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT roman_max... | There are two important solution concepts in the space of CEs. The first is Maximum Welfare Correlated Equilibrium (MWCE) which is defined as the CE that maximises the sum of all player’s payoffs. An MWCE can be obtained by solving a linear program, however the MWCE may not be unique and therefore does not fully solve ... | B |
Given η>0𝜂0\eta>0italic_η > 0 and a query q𝑞qitalic_q, the Gaussian mechanism with noise parameter η𝜂\etaitalic_η returns its empirical mean q(s)𝑞𝑠{q}\left(s\right)italic_q ( italic_s ) after adding a random value, sampled from an unbiased Gaussian distribution with variance η2superscript𝜂2\eta^{2}italic_η start... |
In order to leverage Lemma 3.5, we need a stability notion that implies Bayes stability of query responses in a manner that depends on the actual datasets and the actual queries (not just the worst case). In this section we propose such a notion and prove several key properties of it. Missing proofs from this section ... | 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... | In this section, we give a clean, new characterization of the harms of adaptivity. Our goal is to bound the distribution error of a mechanism that responds to queries generated by an adaptive analyst.
This bound will be achieved via a triangle inequality, by bounding both the posterior accuracy and the Bayes stability ... | 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... | C |
Our algorithmic results are based on a combination of graph reduction and color coding [6] (more precisely, its derandomization via the notion of universal sets). We use reduction steps inspired by the kernelization algorithms [12, 46] for Feedback Vertex Set to bound the size of 𝖺𝗇𝗍𝗅𝖾𝗋𝖺𝗇𝗍𝗅𝖾𝗋\mathsf{antler... |
Our algorithmic results are based on a combination of graph reduction and color coding [6] (more precisely, its derandomization via the notion of universal sets). We use reduction steps inspired by the kernelization algorithms [12, 46] for Feedback Vertex Set to bound the size of 𝖺𝗇𝗍𝗅𝖾𝗋𝖺𝗇𝗍𝗅𝖾𝗋\mathsf{antler... | To motivate the use of an FPT algorithm to find antlers, we start by presenting the hardness results mentioned in the introduction. As these results apply to the simplest types of antlers this also forms an introduction to their properties. The hardness results presented in this section apply to the type of antlers as ... | As described in Section 1, our algorithm aims to identify vertices in antlers using color coding. To allow a relatively small family of colorings to identify an entire antler structure (C,F)𝐶𝐹(C,F)( italic_C , italic_F ) with |C|≤k𝐶𝑘|C|\leq k| italic_C | ≤ italic_k, we need to bound |F|𝐹|F|| italic_F | in terms of... | The remainder of the paper is organized as follows. After presenting preliminaries on graphs and sets in Section 2, we prove the mentioned hardness results in Section 3. We present structural properties of antlers and how they combine in Section 4. In Section 5 we show how color coding can be used to find a large feedb... | D |
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... | Some examples in DESOBA dataset are exhibited in the second row in Fig. 14, in which we show the composite image without foreground shadow, foreground object mask, and ground-truth image with foreground shadow. As mentioned in [52], manual shadow removal is extremely expensive.
| ARShadowGAN [92] released a rendered dataset named Shadow-AR by inserting a foreground object into real background image and generating its corresponding shadow with rendering technique. Shadow-AR dataset contains 3,00030003,0003 , 000 quintuples, in which each quintuple consists of a composite image without foreground... | Figure 14: In the first row, we show two examples from Shadow-AR dataset [92], which is constructed based on rendered images. In the second row, we show two examples from DESOBA dataset [52], which is constructed based on real images. From left to right in each example, we show the composite image without foreground sh... |
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... | B |
Our data collection covers a total of 7 cities, namely Beijing, Shanghai, Shenzhen, Chongqing, Xi’an111The original data were obtained from the HKUST-DiDi Joint Research Laboratory. Some of the data can be made available upon request after undergoing a process of desensitization., Chengdu††footnotemark: and Hong Kong... | Comprehensiveness: Fig. 1(a), illustrates that CityNet comprises three types of raw data (mobility data, geographical data, and meteorological data) collected from seven different cities. Furthermore, we have processed the raw data into several sub-datasets (as shown in Fig. 1(b)) to to capture a wider range of urban p... | In order to facilitate a clear understanding of the data used in this study, we have classified all taxi-related mobility data (including flow, pickup, and idle driving and traffic speed data) as service data, as they pertain to the operational states of transport service providers. Accordingly, all other data have bee... | In addition to the collection and processing of data, it is essential to identify and quantify the correlations between sub-datasets in CityNet to gain insights into the effective utilization of the multi-modal data. In this section, we leverage data mining tools to explore and visualize the relationships between servi... | Table I provides details on the properties of the collected data, including data range, size, and availability. It is important to note that due to limitations in data availability, not all types of data are accessible for each city. For ease of reference, we have compiled a list of notations used in this paper in Tabl... | D |
In this and the following section some of the models introduced above are experimentally investigated. They are evaluated and compared based on some general performance measures. Moreover, some general conclusions that can be used in future applications or research are derived.
| To see the influence of the training-calibration split on the resulting prediction intervals, two smaller experiments were performed where the training-calibration ratio was modified. In the first experiment the split ratio was changed from 50/50 to 75/25, i.e. more data was reserved for the training step. The average ... | 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... | An optimal interval estimator should satisfy some conditions. To assess the quality of the models, the HQ principle from Section 3.3 is adopted. First of all a model ought to be valid (or calibrated) in the sense of Eq. (2). The more a model deviates from being well calibrated, the less reliable it becomes since the re... | 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... | C |
These constitute the main ideas of the CP representation \parencitehsiao21aaai,
which has at least the following two advantages over its REMI counterpart: 1) the number of time steps needed to represent a MIDI piece is much reduced, since the tokens are merged into a “super token” (a.k.a. a “compound word” \parencitehs... |
To train Transformers, it is required that all input sequences have the same length. For both REMI and CP, we divide the token sequence for each entire piece into a number of shorter sequences with equal sequence length 512, zero-padding those at the end of a piece to 512 with an appropriate number of Pad tokens. | In addition to REMI, we experiment with the “token grouping” idea of the compound word (CP) representation \parencitehsiao21aaai, to reduce the length of the token sequences.
We depict the two adopted token representations in Fig. 1 and provide some details below. | To study whether the accuracy gain comes simply from a longer musical context enjoyed by CP, we also train “our model (performance)+++CP” with a sequence of length 128, obtaining 95.43, 80.32 and 64.04 accuracies for three-class melody classification, style classification and emotion classification, respectively.
We no... | For fine-tuning, we create training, validation and test splits for each of the three datasets of the downstream tasks with the 8:1:1 ratio at the piece level (i.e., all the 512-token sequences from the same piece are in the same split).
With the same batch size of 12, we fine-tune the pre-trained our model for each ta... | A |
Now, observe that if the block to the left is also of type A, then a respective block from Z(S)𝑍𝑆Z(S)italic_Z ( italic_S ) is (0,1,0)010(0,1,0)( 0 , 1 , 0 ) – and when we add the backward carry (0,0,1)001(0,0,1)( 0 , 0 , 1 ) to it, we obtain the forward carry to the rightmost block. And regardless of the value of t... | Finally, note that the aforementioned forward carry resulting from backward carry appears in the block which has to be equal to (0,0,1)001(0,0,1)( 0 , 0 , 1 ) (as it has to be the second case above), so it turns it into (1,0,1)101(1,0,1)( 1 , 0 , 1 ) and it does not generate any future carries.
| In any way, the forward carry to the (i+1)𝑖1(i+1)( italic_i + 1 )-th block cannot exceed (1,1,0)110(1,1,0)( 1 , 1 , 0 ). However, since the (i+1)𝑖1(i+1)( italic_i + 1 )-th blocks of Z(S)𝑍𝑆Z(S)italic_Z ( italic_S ) and Z(S2)𝑍subscript𝑆2Z(S_{2})italic_Z ( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) are (0,... |
Therefore, the only possible backward carry from the block of type A to the block of type B has to be in the form (0,0,1)001(0,0,1)( 0 , 0 , 1 ). However, this will be combined with a block (0,1,0)010(0,1,0)( 0 , 1 , 0 ) from Z(S)𝑍𝑆Z(S)italic_Z ( italic_S ) – thus, the sum of the blocks from Z(S)𝑍𝑆Z(S)italic_Z (... |
Now, observe that if the block to the left is also of type A, then a respective block from Z(S)𝑍𝑆Z(S)italic_Z ( italic_S ) is (0,1,0)010(0,1,0)( 0 , 1 , 0 ) – and when we add the backward carry (0,0,1)001(0,0,1)( 0 , 0 , 1 ) to it, we obtain the forward carry to the rightmost block. And regardless of the value of t... | A |
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