Datasets:

sadat2307

/

MSciNLI

like 1

train_0

In Bradley & Murray (2011) an iterative stochastic procedure is proposed for finding such matrix.

we did not find it to work very well in an online optimization setting, and therefore stick to the original equilibration matrix D E . Although the original motivation for row equilibration is to prevent round-off errors, our interest is in how well it is able to reduce the condition number.

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NeurIPS

train_1

In the most general case, the encoder can update A t and B t in an unconstrained fashion at each timestep t. From the decoder side, we do not know C and therefore we cannot know F C, an estimate of which is needed by the user to update the encoder.

the decoder sets F and can predict updates to [CA CB] directly, instead of to [A B] as the actual encoder does (equation 15).

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train_2

Note that the proposed RankGAN has a Nash Equilibrium when the generator G θ simulates the humanwritten sentences distribution P h , and the ranker R φ cannot correctly estimate rank between the synthetic sentences and the human-written sentences.

as also discussed in the literature [8,9], it is still an open problem how a non-Bernoulli GAN converges to such an equilibrium.

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train_3

We give a problem for which the sample complexity of the trivial algorithm is logarithmic in n, whereas it is linear in n for the natural class of algorithms that predicts with the linear combination of instances.

why should we consider learning problems that pick columns out of a random matrix?

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train_4

The disadvantage of using the binary SVM is that, in general, the 0-1 loss is a poor approximation for the AP loss.

the quality of the approximation is not uniformly poor for all samples, but depends heavily on their separability.

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train_5

In most cases, especially with a very large vocabulary, V is significantly larger than D, and the additional computation cost is negligible.

as V decreases, the portion of the overhead increases.

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train_6

2 From the table it is seen that all the samplers perform equally well when the number of items is small (N = 8).

as the number of items increases SM significantly outperforms all other samplers.

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train_7

Given Φ, we can compute the joint distribution over the prediction range for each time series analytically, as this joint distribution is a multivariate Gaussian.

in practice it is often more convenient to represent the forecast distribution in terms of K Monte Carlo samples, In order to generate prediction samples from a state space model, one first computes the posterior of the latent state p(l T |z 1:T ) for the last time step T in the training range, and then recursively applies the transition equation and the observation model to generate prediction samples.

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train_8

Besides the superiority of finding good clusters, AP exhibits the surprising ability of handling large-scale data.

aP is computationally expensive to acquire clusters when the number of clusters is set in advance.

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train_9

Two classes of computable Stein discrepancies-the graph Stein discrepancy [10,12] and the kernel Stein discrepancy (KSD) [7,11,19,21]-have since been developed to assess and tune Markov chain Monte Carlo samplers, test goodness-of-fit, train generative adversarial networks and variational autoencoders, and more [7, 10-12, 16-19, 27].

in practice, the cost of these Stein discrepancies grows quadratically in the size of the sample being evaluated, limiting scalability.

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NeurIPS

train_10

[10] study k-way clustering and show that the eigenvectors of the graph Laplacian are stable in 2-norm under small perturbations.

this justifies the use of k-means in the perturbed subspace since ideally without noise, the spectral embedding by the top k eigenvectors of the graph Laplacian reflects the true cluster memberships, closeness in 2-norm does not translate into a strong bound on the total number of errors made by spectral clustering.

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train_11

In fact, it can be shown that a more rudimentary form of reweighted 1 applied to this model in [19] amounts to performing exactly one such iteration.

repeated execution of ( 9) is cheap computationally since it scales as O nm x (k+1) 0 , where typically x (k+1) 0 ≤ n, and is substantially less intensive than the subsequent 1 step given by (3).

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train_12

Point processes are popular models of neural spiking behavior as they provide a statistical distribution over temporal sequences of spikes and help to reveal the complexities underlying a series of recorded action potentials.

the most common neural point process models, the Poisson process and the gamma renewal process, do not capture interactions and correlations that are critical to modeling populations of neurons.

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NeurIPS

train_13

Complementary techniques such as exact covariance thresholding [13,19], and the divide and conquer approach of [8], have also been proposed to speed up the solvers.

as noted in [8], the above methods do not scale to problems with more than 20, 000 variables, and typically require several hours even for smaller dimensional problems involving ten thousand variables.

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train_14

Batch normalization tackles the issue by normalizing the output of neurons to zero mean and unit variance and then performing dropout independently 1 .

our proposed evolutional dropout tackles this issue from another perspective by exploiting a distribution-dependent dropout, which adapts the sampling probabilities to the evolving distribution of a layer's outputs.

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train_15

One direct way to address this problem is to score workers using their past performance on similar problems.

this is not always practical, since historical records are hard to maintain for anonymous workers, and their past tasks may be very different from the current one.

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train_16

Ideally, we would perform maximum likelihood estimation on the parameters, p ✓ (x r , z r )dz r , and compute the posterior p ✓ (z|x).

under a fLDS neither the p ✓ (z|x) nor p ✓ (x) are computationally tractable (both due to the noise model P and the nonlinear observation model f (•)).

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train_17

The reconstruction error for each instance can be used as an anomaly score.

the reconstruction errors are not reliable because they are calculated from parameters that are estimated using data with anomalies by assuming that all of the instances are non-anomalous.

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NeurIPS

train_18

In particular, we expect that with random initialization, general stochastic gradient descent will need exponential time to escape saddle points in the worst case.

if we add perturbations per iteration or the inherent randomness is non-degenerate in every direction (so the covariance of noise is lower bounded), then polynomial time is known to suffice [Ge et al., 2015].

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train_19

In fact, the empirical error with a finite margin is shown to converge to zero if , is sufficiently large.

the existence of a weak learner with error 1/2 -, is not always useful in terms of generalization error, since it applies even to the extreme case where the binary labels are drawn independently at random with equal probability at each point, in which case we cannot expect any generalization.

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train_20

how do I move through the door without bumping into it [11,16,17,20,33,34].

our work focuses on the longer time-scale problem of path following e.g.

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train_21

Finding large independent sets is a fundamental problem in algorithm design and analysis and computing ALPHA(G) is a classic NP-hard problem which is even very hard even to approximate [11].

the Lovász function ϑ(G) gives a tractable upper-bound and since then Lovász ϑ function has been extensively used in solving a variety of algorithmic problems e.g.

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NeurIPS

train_22

In recent years, the language model Latent Dirichlet Allocation (LDA), which clusters co-occurring words into topics, has been widely applied in the computer vision field.

many of these applications have difficulty with modeling the spatial and temporal structure among visual words, since LDA assumes that a document is a "bag-of-words".

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train_23

We also observe this phenomenon on English→German translation.

we show that by using value network, such shortage can be largely avoided.

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train_24

In the more general setting, an instantiation x can change hyperparameters all through the network, leading to different weights.

we believe that a single data instance will not usually lead to a dramatic change in the distributions.

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train_25

In this case, it is not possible to correctly identify all the features with large probability.

we can show that FoBa can still select part of the features reliably, with good parameter estimation accuracy.

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train_26

Thus, the eigen-distortion test reveals generalization failures in the CNN and VGG16 architectures that are not exposed by traditional methods of cross-validation.

the models with architectures that mimic biology (On-Off, LGG, LG) are constrained in a way that enables better generalization.

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train_27

All of the above methods are designed for the batch setting, where all of the data is collected in advance and used at once.

if the training dataset is extremely large or the data are streaming and encountered in sequence, we may want to incrementally update the approximate posterior of the latent function f . Early work by Csató and Opper [6] proposed an online version of GPR, which greedily performs moment matching of the true posterior given one sample instead of the posterior of all samples.

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train_28

Assuming that EP already yields a good approximation, the computation of a small number of these terms maybe sufficient to obtain the most dominant corrections.

when the leading corrections come out large or do not sufficiently decrease with order, this may indicate that the EP approximation is inaccurate.

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train_29

All that is needed now is to show a separation of their "-optimal sets for 0 < " < 1 60 d 3 2 , and this is done by showing a separation of the more manageable sets S 1 and S 2 .

indeed, fix 0 < " < 1 60 d 3 2 and observe that for any w 2 S 1 we have and so, for d 4, for any w 2 S 2 we have We see that no w can exist in both S 1 and S 2 , so these sets are disjoint.

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train_30

The most efficient message update schedule for tree structured models is a two-pass procedure where messages are first sent from the leaves to the root node, and then propagated backwards from the root to the leaves.

as with other message-passing algorithms, for tree structured instances the algorithm will converge with either a sequential or a parallel update schedule, with any initial condition for the messages.

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NeurIPS

train_31

1, we provide NeuralFDR, a practical end-to-end algorithm to the multiple hypotheses testing problem where the hypothesis features can be continuous and multi-dimensional.

the currently widely-used algorithms either ignore the hypothesis features (BH [3], Storey's BH [21]) or are designed for simple discrete features (group BH [13], IHW [15]).

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train_32

The running time of the algorithm is O nM(r)d where r = min(r 1 , r 2 ).

at smaller scales, where M(r) is comparable with n, it is O n 2 d . since the variance of the estimate also tends to be smaller at smaller scales, the algorithm iterates less for the same accuracy.

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train_33

The most common way around this is to encourage sparsity during training by way of a penalty function on the expected conditional hidden unit activations given data [10].

this training-time procedure is a heuristic and does not guarantee sparsity at test time.

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train_34

The metric is often taken to be Euclidean, Manhattan or χ 2 distance.

it is well known that in many cases these choices are suboptimal in that they do not exploit statistical regularities that can be leveraged from labeled data.

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train_35

Alike AT, β of FT is set to 10 3 in ImageNet and PASCAL VOC 2007.

we set it to 5 × 10 2 in CIFAR-10 and CIFAR-100 because a large β hinders the convergence.

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NeurIPS

train_36

The technique for estimating bandable covariance matrices proposed in [6] is shown to achieve the optimal rate of convergence.

no such theoretical guarantees have been shown for the bandable precision estimator proposed in recent work for estimating sparse and smooth precision matrices that arise from cosmological data [15].

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NeurIPS

train_37

In the literatures of manifold learning, many methods have been proposed to construct these adjacency matrices locally, e.g., via heat kernel function [2].

in the context of manifold alignment, there might be partial alignment cases, in which some points on one manifold might not be corresponded to any points on the other manifold.

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train_38

With the increasing popularity of machine learning techniques, it has become common to see prediction algorithms operating within some larger process.

the criteria by which we train these algorithms often differ from the ultimate criteria on which we evaluate them.

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train_39

This establishes minimax optimality of the DS/LOO estimators for functionals of two distributions when s ≥ d/2.

when s < d/2 there is a gap between our upper and lower bounds.

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train_40

Corel hist 20,000 histograms (64-dimensional) of color thumbnail-sized images taken from the COREL STOCK PHOTO library.

of the 64 dimensions, only 44 of them contain non-zero entries.

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train_41

Interestingly, because log |Σ| ≤ tr(Σ)−M , the trace norm is the convex envelope for the log-determinant, and thus the two minimization problems are somehow doing similar things.

the framework introduced in this paper goes beyond the two methods by introducing an informative kernel Ω between tasks.

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NeurIPS

train_42

With the notation z i = L • q(i) − ψ i and z 0 = −ψ 0 we re-write the minimax quantity as Observe that without the constraint that p t (i) ≤ 1/L for i > 0 we would put all the probability mass on the maximum of the z i .

with the constraint the maximizer put as much probability mass as allowed on the maximum coordinate argmax i∈{0,...,K} z i and continues to the next highest quantity.

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train_43

Their approach augmented a traditional supervised learning algorithm with distribution information made available from the unlabeled data.

this paper considers a method for augmenting a traditional unsupervised learning problem with the addition of equivalence classes.

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train_44

The fixed order of the 441 features can be considered acceptable since any input-output mapping can in principle be learned, assuming we have sufficient training data (and an appropriate network architecture).

if the amount of training data is limited then a better-structured, more compact representation might be of great advantage as opposed to requiring to see most of the possible configurations of co-evolution.

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NeurIPS

train_45

The existence of shift parameters seem to require extra additions/subtractions (see ( 2) and ( 8)).

the binarization operation with a shift parameter can be implemented as a comparator where the shift parameter is the number for comparison, e.g., H v (R) = ⇢ 1, R 0.5 v; 1, R < 0.5 v. (0.5 v is a constant), so no extra additions/subtractions are involved.

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train_46

There also exists a few works that use GAN to impute the missing values [46].

what these works focused is non-sequential dataset and they have not adopted pertinent measures to process the temporal relations.

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train_47

As feedback SNR increases, we expect the BER to decrease.

as shown in Figure 4 (Left), the C-L scheme, which is designed for noisy feedback, and S-K scheme are very sensitive to noise in the feedback, and reliability is almost independent of feedback quality.

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train_48

If this can be achieved, the resulting GIS procedure will be unbiased via the arguments of the previous section.

the G -weights must not only satisfy the constraint (1), they must also be efficiently calculable from a given sample.

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train_49

For all i ∈ N , all θ i ∈ Θ i and all a i ∈ A i,θi , we have u i (σ Standard BN inference methods could be used to compute E[U ai |θ i ].

such standard algorithms do not take advantage of structure that is inherent in BAGGs.

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train_50

As noted earlier, the running time of the batch algorithm goes up as t increases (as it has to optimize over the entire past).

the running time of the online algorithm is independent of the past and only depends on the number of documents introduced in each timestep (which in this case is always 1000).

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train_51

Both of these algorithms were also compared with the SVM equipped with a RBF kernel of variance σ 2 and a soft margin parameter C. Each SCM algorithm used the L 2 metric since this is the metric present in the argument of the RBF kernel.

in contrast with , each SCM was constrained to use only balls having centers of the same class (negative for conjunctions and positive for disjunctions).

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train_52

While discussing learning the kernel, we showed that L 1 and L 2 cannot be updated simultaneously in a CCCP-style iteration since g is not convex over (S 1 , S 2 ).

it can be shown that g is geodesically convex over the Riemannian manifold of positive definite matrices, which suggests that deriving an iteration which would take advantage of the intrinsic geometry of the problem may be a viable line of future work.

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train_53

However, a limitation of SDPs is their computational complexity [1], which has restricted their application to small scale problems [6].

an important special case of SDPs are quadratically constrained quadratic programs (QCQP) which are computationally more efficient.

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train_54

[45] used a deep network to project dense trajectory features from different views into a canonical view.

most of the previous methods require access to 3D human pose information (e.g.

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train_55

Deep learning has become a ubiquitous technology to improve machine intelligence.

most of the existing deep models are structurally very complex, making them difficult to be deployed on the mobile platforms with limited computational power.

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train_56

Over the years, several methods have been proposed to solve coupled tensor completion (Acar et al., 2014;Ermis et al., 2015).

many of these methods are non-convex models leading to local optimal solutions.

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train_57

Also purely feed-forward network architectures were proposed [3].

such networks become unfeasibly large for practical applications.

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train_58

TiMT, however, follows by a slim margin.

tiWnet explains the similarities exclusively by adding more (unnecessary) edges, which is reflected in its increased, but strongly consistent false positive rate.

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train_59

At first sight, the difference between move-making algorithms and the LP relaxation appears to be the standard accuracy vs. speed trade-off.

for some special cases of distance functions, it has been shown that appropriately designed move-making algorithms can match the theoretical guarantees of the LP relaxation [14,15,20].

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train_60

The problem of estimating the sparse precision matrix in Gaussian graphical models has been studied by a large body of literature [23,29,12,28,6,34,37,38,33].

the real world data may not follow a sparse GGM, especially when some of the variables are unobservable.

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train_61

The introduction of local max-pooling layers in CNNs has helped to satisfy this property by allowing a network to be somewhat spatially invariant to the position of features.

due to the typically small spatial support for max-pooling (e.g.

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train_62

Although dimensionality reduction can be done posthoc using PCA, [10] shows that this doesn't lead to performance improvement.

we show in §4 that selecting k v can improve the performance of SRM beyond that attained by HA.

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train_63

The difference of the learner's total reward and the total reward of the optimal strategy is called the (Audibert et al., 2009) of the algorithm and it can be formally written as As compared to the regret, the pseudo-regret has the same expected value, but lower variance because the additive noise ⌘ t is removed.

the omitted quantity is uncontrollable, hence we have no interest in including it in our results (the omitted quantity would also cancel, if ⌘ t was a sequence which is independently selected of X 1:t .) In what follows, for simplicity we use the word regret instead of the more precise pseudo-regret in connection to R n . The goal of the algorithm is to keep the regret R n as low as possible.

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train_64

In the MAB literature, several recent works consider multi-player MAB scenarios in which players actually compete with each other, either on arm-pulls resources [15] or on the rewards received [19].

we study a collaborative multi-player problem and investigate how sharing observations helps players achieve their common goal.

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train_65

On the one hand, it is usually acceptable to output a policy that is only locally optimal with respect to the optimization objective.

in many application scenarios where constraints encode safety requirements or the amount of available resources, violating the constraint even by a small amount may have significant consequences.

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train_66

Column normalization can be viewed as a principled first step towards solving challenging sparse estimation problems.

when non-convex sparse regularizers are used for the image penalty, e.g., p norms with p < 1, then local minima can be a significant problem.

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train_67

In simple cases, topic models can be used to cluster local textural elements, coarsely representing categories via a bag of visual features [1,2].

spatial structure plays a crucial role in general scene interpretation [3], particularly when few labeled training examples are available.

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train_68

For example, Node A concerns washbasins for infants, and has two polarized children nodes: reviewers take a positive perspective when their children enjoy the product (Node B: "loves", "splash", "play") but have negative reactions when it leaks (Node C: "leak(s/ed/ing)").

the lowest topics in the hierarchy are often polarized; one child topic of "router" focuses on upgradable firmware such as "tomato" and "ddwrt" (Node E, positive) while another focuses on poor "tech support" and "customer service" (Node F, negative).

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train_69

The teaching set of a concept c ∈ C is a set of indices (or examples) X ⊆ [n] that uniquely identifies c from C. Formally, given a concept class C ⊆ {0, 1} n (a set of binary strings of length n), X ⊆ [n] is a teaching set for a concept c ∈ C (a binary string in C) if X satisfies c| X = c | X , for all other concepts c ∈ C. The teaching dimension of a concept class C is the smallest number t such that every c ∈ C has a teaching set of size no more than t [GK95, SM90].

teaching dimension does not always capture the cooperation in teaching and learning (as we will see in Example 2), and a more optimistic and realistic notion of recursive teaching dimension has been introduced and studied extensively in the literature [Kuh99, DSZ10, ZLHZ11, WY12, DFSZ14, SSYZ14, MSWY15].

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train_70

(2013) and references therein).

recent work (Dauphin et al., 2014;Choromanska et al., 2014) has brought theoretical and empirical evidence suggesting that local minima are with high probability not the main obstacle to optimizing large and deep neural networks, contrary to what was previously believed: instead, saddle points are the most prevalent critical points on the optimization path (except when we approach the value of the global minimum).

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train_71

Recent progress in deep generative models has led to tremendous breakthroughs in image generation.

while existing models can synthesize photorealistic images, they lack an understanding of our underlying 3D world.

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train_72

As expected, Hotelling, Edist and MMD perform best for differences in the Gaussian distribution (column 4).

in all other settings Hotelling's test has poor power, and our approach with minP as the univariate test has more power than Edist and MMD in columns 5-7.

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train_73

By its definition, the sampler q * (u j ) is computed based on the hidden feature h (l) (u j ) that is aggregated by its neighborhoods in previous layers.

under our top-down sampling framework, the neural units of lower layers are unknown unless the network is completely constructed by the sampling.

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train_74

In general, the exact form of this pooling function is determined by the complex interaction between the MSR and agent utility, and a closed form of p i from (4) might not be attainable in many cases.

given a paticular MSR, we can venture to identify agent utility functions which give rise to well-known opinion pools.

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train_75

In particular, it should be possible to increase the number of limited capacity units in a population to form a more precise representation of the sensory signal.

to the best of our knowledge, such a code has not been characterized analytically, even in the simplest case.

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train_76

[19] use a simple linear algorithm for this step, which takes O(|P|) time.

we propose a more efficient algorithm to maximize δ j (•), which exploits the special structure of this discrete function.

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train_77

On the one hand, we want to be expressive, and learn all the transitions possible from every o within a horizon h. When o is a high dimensional image observation, this typically requires mapping the image to an extensive feature space [30,12].

however, we want to plan efficiently, which generally requires either low dimensional state spaces or well-structured representations.

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train_78

by D-softmax is restricted from the start, and may therefore be lacking in terms of expressiveness.

our algorithm first trains words with a full-length vector and dynamically limits the dimension during evaluation.

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train_79

When constrained to only two labels, their results provide a bipartite SBM.

in the bipartite SBM case, [17] has two drawbacks compared to the results presented here: (1) The data-generating process in [17] rules out certain nested structures of the sets V i .

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train_80

Discriminator D. In traditional GANs, the discriminator distinguishes between real groundtruth images and fake generated images (which is generated from random noise).

in our conditional network, G2 takes the condition image I A instead of a random noise as input.

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train_81

For the second category of global landscape analysis, the typical result is that every local minimum is a global minimum.

even for single-layer networks, strong assumptions such as over-parameterization, very special neuron activation functions, fixed second layer parameters and/or Gaussian data distribution are often needed in the existing works.

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train_82

All prior methods select the local neighborhood based on proximity, and they typically fix its size.

our idea is to predict the set of training instances that will produce an effective discriminative model for a given test instance.

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train_83

Recent work has shown that randomized value functions can implement something similar to Thompson sampling without the need for an intractable exact posterior update.

this work is restricted to linearly-parameterized value functions [16].

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train_84

Note that the costs may be either positive or negative.

only their relative values are important.

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train_85

For GMRFs with n nodes indexing d-dimensional random subvectors, I(x R ; x A ) can be computed exactly in O((nd) 3 ) via Schur complements/inversions on the precision matrix J.

certain graph structures permit the computation via belief propagation of all local pairwise MI terms I(x i ; x j ), for adjacent nodes i, j ∈ V in O(n • d 3 ) -a substantial savings for large networks.

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train_86

This approach is appropriate when objects can be partitioned into relatively homogeneous subsets.

the properties of many objects are better captured by representing each object using multiple latent features.

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train_87

The partial derivatives take a standard log-sum-exp form, requiring expectations E p(y A naive computation of this expectation would require summing over D k= d y d configurations.

there are more efficient alternatives: the dynamic programming algorithms developed in the context of Poisson-Binomial distributions are applicable, e.g., the algorithm from [3] runs in O(Dk) time.

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train_88

In the fully supervised setting, one can sidestep the task of estimating conditional probabilities by directly learning a classifier in a discriminative fashion.

in unsupervised or semi-supervised settings, a reliable estimate of the conditional distributions becomes important.

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train_89

One possibility is to start at the position of the deleted point, θ i−1 , on the contour constraint, which is independent of the other points and not far from the bulk of the required uniform distribution.

if the Markov chain mixes slowly amongst modes, the new point starting at θ i−1 may be trapped in an insignificant mode.

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train_90

REINFORCE-OB contributes highly to reducing the variance especially when T is large, which also well agrees with our theory.

pGPE-OB still provides much smaller variance than REINFORCE-OB.

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train_91

Compared with [12] on subset evaluations, our method significantly improve over [12] on horse-riding and running detection.

[12] provides better detection results than ours on diving detection.

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train_92

Their runtime is independent of, or even decrease with, the number of training samples [5,6].

because of their simplicity, these methods have a slow convergence rate, and thus may require a large number of iterations.

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train_93

We find that a CNN augmented with an RN achieves an accuracy above 94% for both relational and non-relational questions.

a augmented with an MLP only reached this performance on the non-relational questions, plateauing at 63% on the relational questions.

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train_94

To see this, note that the sequence of partitions P 0,bad n becomes arbitrarily ill-balanced, which from (10) implies lim n→∞ Pcut G 0 n (P 0,bad n ) = 1.

the unperturbed graph G n grows in a self-similar fashion as n → ∞ and so the Product Cut of P n remains approximately a constant, say γ, for all n. Thus Pcut Gn (P n ) ≈ γ < 1 for n large enough, and Pcut G 0 n (P 0,good Comparing this upper-bound with the fact lim n→∞ Pcut G 0 n (P 0,bad n ) = 1, we see that the Product Cut of P 0,bad n becomes eventually larger than the Product Cut of P 0,good n . While we execute this program in full only for the example above, this line of argument is fairly general and similar stability estimates are possible for more general families of graphs.

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train_95

At the beginning of learning when Ja is of small magnitude, the diffusion term, ⟨F 2 a ⟩, has a large impact so that it greatly impedes learning for a large η case.

as the magnitude of the differences JA − JB increases, this effect weakens and the dependence of p A on η becomes quite small.

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train_96

3.1 Algorithm for Learning the Transitive Reduction of CBNs [13] show that by using O (log n) multiple-vertex interventions, one can recover the transitive reduction of a DAG.

in this case, each set of intervened variables has a size of O( n /2), which means that the method of [13] has to perform a total of O(2 n /2 log n) experiments, one for each possible setting of the O( n /2) intervened variables (see an example of this in Appendix D).

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train_97

Specifically, we set Q(x) to be sigmoidal with parameter λ (see Figure 2a): As λ → ∞, Q(x; λ) → 1{x > 0}, so R labelwise (w : D) approaches the objective function defined in (7).

r labelwise (w : D) is smooth for any finite λ > 0.

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NeurIPS

train_98

Any nonlinear Wiener functional, for instance, creates infinitely many correlations or cumulants of higher order, and often also of lower order.

a Wiener functional of order n produces only harmonic phase interactions up to order n + 1, but sometimes also of lower orders.

contrasting

NeurIPS

train_99

In small or medium networks, we can rely on well-known numerical methods to compute matrix exponentials [24].

in large networks, the explicit computation of Ψ(t) becomes intractable.

contrasting

NeurIPS