As shown in (3), this allows for a modular representation of Q π in which the MDP's dynamics are decoupled from its rewards, which are captured by the weights w. One potential benefit of having such a decoupled representation is that only the relevant module must be relearned when either the dynamics or the reward changes, which may serve as an argument in favor of adopting SFs as a general approximation scheme for RL.

in this paper we focus on a scenario where the decoupled value-function approximation provided by SFs is exploited to its full extent, as we discuss next.


The granularity of partitions in [10] is controlled by a user-specified threshold on the minimum and maximum membership probabilities that are reachable within the boundaries of a node in the KD-tree.


Understanding why a model makes a certain prediction can be as crucial as the prediction's accuracy in many applications.

the highest accuracy for large modern datasets is often achieved by complex models that even experts struggle to interpret, such as ensemble or deep learning models, creating a tension between accuracy and interpretability.


Similarly, under the setting of domain adaptation, standard binary adversarial cost Lempitsky, 2015, Ganin et al., 2016] and central moment discrepancy (CMD) [Zellinger et al., 2017] have been utilized to learn features that are domain invariant.

all these invariance inducing criteria suffer from a similar drawback, which is they are defined to measure the divergence between a pair of distributions.


In order to utilize the auxiliary function of the estimating function for the gNG, we defined an auxiliary function on the criterion of the near optimality of the estimating function, by minimizing the distance between the immediate reward as the regressand and the subspace of the regressors of the gNG at the solution of the gNG.

it may be possible to use different criterion, such as the optimality on the Fisher information matrix metric instead of the Euclidean metric.


More importantly, [11] proposes a specific initialization using which the alternating minimization method is shown to converge linearly in noise-free PR and CPR.

the number of measurements required to establish this convergence is effectively quadratic in k. In [12] and [13] the 1 -regularized form of the trace minimization is proposed for the CPR problem.


The previous section shows how the performance of the joint regression model ( 1) can be improved by incorporating the centrality, when this topology information is available.

when this vector is unknown, it can be estimated from the data, using an independent algorithm, and then incorporated to the optimization in model ( 2).


Then it combines the solutions of these smaller problems and uses rank-one modification to solve the original problem.

this method can only be used for tridiagonal matrices and it is unclear how to extend it to general sparse matrices.


For instance, in their setting, there are statistical barriers so that no algorithm can achieve an optimization error below some fixed threshold, no matter how many samples are taken.

in the current Byzantine setting, the adversarial corruptions can only occur in a fraction of the machines (as opposed to each machine having some adversarial corruptions).


The hybrid sp-tree search algorithm is much more efficient than the traditional MT-DFS algorithm.

this speed-up becomes less pronounced when the dimension of a dataset becomes high (say, over 30).


These can be chosen systematically, using cross-class calibration, where one trains with a subset of the background classes, say half of them, then tests on the remaining unseen background classes.

this adds complexity and computational cost.


All numerical results are based on this estimator.

analyzing the sensitivity of the performance with respect to the estimation error in α t,u is quite challenging, and for a theoretical analysis, we assume we have access to an oracle that provides the exact value of α t,u , replacing Algorithm 3.


On the contrary, randomized coordinate descent can only converge to a sub-optimal solution in finite time when there are infinite number of dimensions.

in practice, only a very limited class of basis functions allow the greedy step in (24) to be performed exactly.


Scoring a detector: Template classifiers are often scored using missed detections versus falsepositives-per-window statistics.

recent analysis suggests such measurements can be misleading [9].


Prior to this work, Lipschitzian error bounds for composite function minimization are available for cases where the non-smooth part either has a polyhedral epigraph (such as the ℓ 1 -norm) [23] or is the (sparse) group LASSO regularization [22,25].

the question of whether a similar bound holds for trace norm regularization has remained open, despite its apparent similarity to ℓ 1 -norm regularization.


This can be intuitively understood as a safeguard against overfitting.

overfitting in BLBF is more nuanced than in conventional supervised learning.


it is expected that the extra effort on "hard cases" will cause the algorithm to converge faster to good density models.


Racing algorithms (e.g., [10,12]) and action-elimination algorithms [7] address this problem under a constraint on the accuracy in identifying the best arm and they minimize the budget needed to achieve that accuracy.

uCB-E and Successive Rejects are designed for a single bandit problem, and as we will discuss later, cannot be easily extended to the multi-bandit case studied in this paper.


The reason is because the objective function in ( 6) is a linear function in p. Hence, we can simply obtain s k,DUDE (c, •) = arg max s p(c) s , where p(c) s stands for the s-th coordinate of p(c).

as seen in the previous section, DUDE utilizes the estimated loss matrix L, which does not depend on the clean sequence x n . the main drawback of DUDE is that, as can be seen in ( 5), it treats each context c independently from others.


Traditionally, machine learning approaches relied on user-defined heuristics to extract features encoding structural information about data.

recently there has been a surge in approaches that learn how to encode the data automatically in a low dimensional space.


For example, the method in Fearnhead and Rigaill [12] only produces point estimates and is limited to fitting a piecewise constant function to univariate data.

bOCPD can be applied to multivariate data and a set of models M while quantifying uncertainty about these models, their parameters and potential CPS, but is not robust.


So we have: £ 1 a mx@t £ 1 x; 0A; £ 2 a mx@x t £ 1 w; 0A A v £ a min @mx@@x t £ 1A=; 0A ; wA : This expresses v £ as a function of the unknown optimal dual variable t £ .

for the simple case of 2-potentials, we can solve for t £ explicitly and get a closed form expression: 2 @e kl C 1 @xk xlA@e k e l AA if jxk xlj < in general to find t £ we note that v £ must satisfy v £ ¡ 1 a y.


Semi-supervised learning is a principled framework for combining these different label sources.

it scales polynomially with the number of images, making it impractical for use on gigantic collections with hundreds of millions of images and thousands of classes.


It has long been observed that such oscillations provide a substrate for representing analog information in the firing phases of neurons relative to the underlying population rhythm.

it has become increasingly clear that it is essential for neural populations to represent uncertainty about the information they capture, and the substantial recent work on neural codes for uncertainty has omitted any analysis of oscillatory systems.


Additional assumptions such as strong convexity or uniform convexity provide respectively linear [Nesterov, 2013b] and faster polynomial [Juditski and Nesterov, 2014] rates of convergence.

these assumptions are often too restrictive to be applied.


[24] uses the shortest path distance, and [29] uses the diffusion distance.

these methods can be applied only to non-attributed (unlabeled) graphs, which largely limits their applications in the real world.


We mostly use the general notation of feedforward networks with shared weights throughout the paper as this will be more general and simplifies the development and notation.

when focusing on RNNs, it is helpful to discuss them using a more familiar notation which we briefly introduce next.


This combines the rule enumeration and confidence assignment.

the parameterization in Equation 5is not sufficiently expressive, as it assumes that all rules are of the same length.


Ambiguous visual input can therefore be represented appropriately.

convolutional networks use one position for each feature as representation.


In recent work, Chan et al [CDSS14] presented an approach employing an intricate combination of dynamic programming and linear programming which yields a poly(k/") time algorithm for the above problem.

the running time of the [CDSS14] algorithm is ⌦(k 3 ) even for constant values of ", making it impractical for applications.


Moreover it has to be remarked that this tightening of the bound does not appear to deliver any reduction in the capabilities to select a good model (such a case would imply that we can predict more accurately a bigger error rate, but our bound is able to predict accurately the same error rate as the PAC-Bayes Bound).


Previously this game has been solved asymptotically as both T and K tend to ∞: asymptotically the upper bound on the performance of the state-of-theart Multiplicative Weights Algorithm (MWA) for the learner matches the optimal multiplicative constant of the asymptotic minimax optimal regret rate � (T /2) log K [3].

for finite K, this asymptotic quantity actually overestimates the finite-time value of the game.


The distinguishing feature of judgments relative to other kinds of decision problems is that they are asymmetric; if I apply for a loan and I get the loan, I am satisfied, and do not particularly care for an explanation; even the bank may not care as long as on aggregate the algorithm makes the bank money.

i very much care if the algorithm denies my mortgage application.


Similar to the blind source separation case, our metric clearly does not necessarily converge to the Hessian and so it is not necessarily asymptotically efficient (ie does not attain a second order convergence rate).

in general, the Hessian will not be positive definite and so the curvature it provides could be of little use until B is close to a local maxima.


[2017a], who propose to 'de-render' an image with an encoder-decoder framework that uses a neural network as the encoder and a graphics engine as the decoder.

their method cannot back-propagate the gradients from the graphics engine or generalize to a new environment, and the results were limited to simple game environments such as Minecraft.


A more uniform averaging effectively allows to aggregate information from many neighbors at once.

the more distinct neighbors obtained with a low temperature allow to first non-linearly process the information before eventually fusing it.


Since models with larger empirical reward are preferred, ideally one hopes to maximize empirical reward during training.

since empirical reward is not amenable to numerical optimization, one often considers optimizing alternative differentiable objectives.


We also considered a control model ("Time Decay"), described in Section 2, that produces temporal dynamics by learning time constants on the activations at each layer, rather than by learning connectivity between units.

this model did not perform any better than the basenet, implying that ConvRNN performance is not a trivial result of outputting a dynamic time course of responses.


Roughly speaking, the more the labeled data, the better the learning performance will be.

in many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming.


For every θ there exists (ǫ, γ), and a numerical constant K, such that This first result implies in particular that G can be reconstructed with polynomial complexity for any bounded ∆.

the degree of such polynomial is pretty high and non-uniform in ∆.


For instance, exemplars 0111 and 0001 were given lower ratings in the common-cause and common-effect conditions , respectively (39.4 and 26.3) than in the control condition (67.6 and 67.0) presumably because in these exemplars correlations are broken (effect features are present even though their causes are absent).

exemplar 1111 received a significantly higher rating in the common-cause and common-effect conditions than in the control condition (90.8 and 9l.0 vs. 75.6) , presumably because in both conditions all correlations are preserved.


To overcome this limitations, Li et al., propose to learn the parametric distributions from retrieved similar HR images [12] via block matching, and obtain state-of-the-art SR performance for landmark images.

for general LR images, for which similar HR images cannot be found, the sparse prior (β i , θ i ) cannot be learned.


In order for the complement generator to work, according to condition (3) in Assumption 1, the discriminator needs to have strong true-fake belief on unlabeled data, i.e., max K k=1 w k f (x) > 0.

the objective function of the discriminator in [16] does not enforce a dominant class.


Unlike algorithms that find and track features in an image, the delay-and-correlate EMD does not measure true image velocity independent of the spatial structure of the image.

recent work has shown that for natural scenes, these Reichardt EMDs give reliable estimates of image velocity [15].


A reasonable concern is that the performance increase comes from the variance reduction due to increasing the batch size.

experiments done with a batch size of 2000 for vanilla networks actually deteriorates the test accuracy.


Recovering low-dimensional structure from a corrupted matrix is related to robust PCA [4,9,33].

robust PCA focuses on recovering low-rank part X from complete or incomplete matrix, and the target is different from attacks detection (which is our task).


Kuznetsov and Mohri [2014] also give generalization bounds in terms of discrepancy.

unlike the result of Kuznetsov and Mohri [2014], our analysis does not require any mixing assumptions which are hard to verify in practice.


It turns out that, within a class, the correlation coefficient between most pairs of dimensions is fairly low.

for related auditory features (similar time and frequency band) correlations are high and also for related visual features.


Tantalizingly, [12] derived the optimal paths for some toy models in the context of path sampling and showed that they vastly outperformed geometric averages.

as choosing an optimal path is generally intractable, geometric averages still predominate.


Rather, they only ensure that L(γ ) ≤ L(γ ) at all points.

the MacKay updates do not even guarantee cost function decrease.


In the case of GRAVITAR, even though both A3C and A2C, which are policy gradient methods, have a lower score than valuebased DQN, G2AC achieves a higher score with less frames of training.

for SEAQUEST, it seems that G2AC is not able to deviate from the local minima, scoring a similar rewards as A2C.


On the one hand, they need to train a lot of iterations since DSDH updates the hash code bit-by-bit, while HashNet requires to increase β iteratively.

dCC which is used to solve the discrete optimization by DSDH, can only be applied to the standard binary quadratic program (BQP) problem with limited applications, while HashNet is still distracted by quantization error as β cannot increase infinitely.


However, in certain subclasses of MRF, an optimal or close-to-optimal assignment can be found very efficiently using combinatorial optimization algorithms: certain MRFs with mutual exclusion constraints can be solved using bipartite matching, and MRFs with regular potentials can be solved using minimum cut methods.

these solutions do not apply to the many MRFs that contain such tractable components as sub-networks, but also other non-complying potentials.


In addition, as analyzed in Section 2, the approximation in [1,14] drops the rank constraint of the matrix y y, which does not lead to a tight approximation.

our prediction function f * implements the conjugate of conjugate of the prediction function f (x), which is the convex envelope of f (x) [7].


Thematically, this question is similar to the pursuit of compressed sensing.

as we explain in Section 2, standard compressed sensing does not apply under this setting.


Both of the reformulated problems do not require backpropagation algorithm at all and are easy to be parallelized.

neither of  Figure 1: Illustrations of the backward pass of the backpropagation algorithm (BP) [30], decoupled neural interface (DNI) [13] and decoupled parallel backpropagation (DDG) [11].


Then it may be able to compute g in , assuming we allow multiplicative units in the hidden layer in the hidden layer to use powers of their inputs in the product they compute (which we allow here for the proof to be more generic).

it will require at least as many of these units as the number of unique products that can be found in the expansion of g in .


This symmetry condition and the orthogonality condition, WTW = I, uniquely determine the coding optimal transform (COT) W. The COT reduces to the PCA transform when the data is Gaussian.


The feature conjunctions that are not collinear remain more unstructured and appear to react highly to blob-like stimuli.

both component types are quite different from ordinary linear detectors for edges, bars and blobs, since their conjunctive nature makes them much more selective.


Note that the cost function in ( 14) depends on U explicitly through p(y d |✓ d,L , U, ), which can be computed directly from its definition in Section 2.

the cost function in (14) depends on implicitly through ✓ d,L .


Evasion attacks happen at test time -a clean target instance is modified to avoid detection by a classifier, or spur misclassification.

these attacks do not map to certain realistic scenarios in which the attacker cannot control test time data.


Their method, as ours, is based on replacing the standard Dirichlet prior over word-topic probabilities.

unlike our approach that is entirely data-driven, it appears that their method relies on interactive feedback from the user or on the careful selection of words within an ontological concept.


Robust algorithms have been exhibited for mean and covariance estimation and clustering (Diakonikolas et al., 2016;Lai et al., 2016;Charikar et al., 2017), classification (Klivans et al., 2009;Awasthi et al., 2014), regression (Nasrabadi et al., 2011;Nguyen and Tran, 2013;Chen et al., 2013;Bhatia et al., 2015) and crowdsourced data aggregation (Steinhardt et al., 2016).

these bounds only hold for specific (sometimes quite sophisticated) algorithms and are focused on good asymptotic performance, rather than on giving good numerical error guarantees for concrete datasets/defenses.


4  provements are significant at about 98% level.

multiple ordinal classification did not show significant improvement over multiple classification, except for the artificial dataset.


Changing the number of states through adding and removing jumps or switching the states of segments is sufficient to guarantee that the sampler converges to the posterior density.

the sampler is very unlikely to reduce the number of states through these actions, if all states are used in multiple segments, so that convergence might take a very long time in this case.


The accuracies of these two methods depend on the given training data.

we argue that Boosting has the following practical advantages.


In a sense, the argument above demonstrates the fact that in most "interesting" IDS games (those in which each player is a true participant, and can have their behavior swayed by that of the overall population), there are two trivial pure NE (all invest and none invest).

we are also interested in finding NE in which some players are choosing to invest and others not to (even though no player has a dominant strategy).


As in Section 3 we define the stochastic policy π by Equation 3, The results of Section 3.1 apply to the policy optimization problem.

note that the data are now assumed to be drawn from the execution of a sequence of T policies π 1 , .


Such simple penalization has been shown to be capable of obtaining models with a high degree of disentanglement in image datasets.

it is not made explicit why penalizing KL(q(z|x)||p(z)) with a factorial prior can lead to learning latent variables that exhibit disentangled transformations for all data samples.


Speech conveys different yet mixed information ranging from linguistic to speaker-specific components, and each of them should be exclusively used in a specific task.

it is extremely difficult to extract a specific information component given the fact that nearly all existing acoustic representations carry all types of speech information.


When the time required to lock memory for writing is dwarfed by the gradient computation time, this method results in a linear speedup, as the errors induced by the lag in the gradients are not too severe.

we note that in many applications of interest in machine learning, gradient computation time is incredibly fast, and we now demonstrate that in a variety of applications, HOGWILD!


If the two domains are close to each other, η s and d H∆H (D s , D t ) are expected to be small.

if the domains are far apart, these terms will be big and the use of extra source samples may not help in learning a better target hypothesis.


However, obtaining a root-n consistent estimator and performing a Newton-Raphson step can be demanding computationally.

our initialization doesn't need to be root-n consistent, thus a small portion of data suffices to initialize.


We make three contributions towards this goal: (i) we propose a novel recurrent neural network architecture that can simultaneously predict the responses of thousands of cortical neurons while accounting for neural variability caused by eye movements or brain state changes related to measurable factors such as running state or pupil dilations; (ii) we demonstrate that training our model on natural movies allows some extent of domain transfer and recovers neurons' tuning properties, such as orientation tuning, direction tuning, and receptive field structure, under artificial stimuli; and (iii) we analyze the limits of domain transfer to show that models trained directly on the target domain always outperform those trained on other domains in terms of predictive performance.

we also demonstrate that the nonlinear feature representation learned on natural movies can transfer well to the noise domain by fine-tuning the last layer's weights, while the converse is not true (i. e. generalization from noise to movies).


Similar to svms, the solution of lmnn is also obtained through a convex optimization problem that maximizes a large margin between input vectors from different classes.

instead of positioning a separating hyperplane, lmnn learns a Mahalanobis metric.


This can be paired with a loss that maximises the entropy of the average distribution across each batch: L diversity = −H(q) = i qi log qi .

we found that using a ReLU nonlinearity [33] followed by divisive normalisation 3 , paired with an L 2 diversity loss, L diversity = (k • q − 1) 2 , tends to converge more robustly.


APPROXFACTORBIAS-OUT performs the best on real world data, possibly due to the direct use of samples from the sampling distribution rather than a sampling distribution estimate.

this method comes without performance guarantees and does not exploit the knowledge of the full sample space.


As the seemingly distinct problems of count and mixture modeling are united under the NB process framework, new opportunities emerge for better data fitting, more efficient inference and more flexible model constructions.

neither [5] nor [6] explore the properties of the NB distribution deep enough to achieve fully tractable closed-form inference.


[7] show that Laplace mechanism is asymptotically optimal for the mini-max error.

the communication cost per user in Laplace mechanism is Ω(log m) bits, and our experiments show it also leads to larger error compared to our 1-bit mechanism.


Consider a formula ¬R(x, y) ∨ S(y, z) where each variable has domain-size equal to d. The data-complexity of computing the satisfied groundings in this formula is clearly d 3 .

for any specific value of y, say y = A, the satisfied groundings in this formula can be computed in closed form as, n 1 d + n 2 d − n 1 n 2 , where n 1 is the number of false groundings of R(x, A) and n 2 is the number of true groundings in S(A, z).


Prominent work on these issues has been done in the domain of flat representations [4,3], where each class is modeled as an immediate aggregate of local features.

there is an increasing literature consensus, that hierarchies provide a more suitable form of multi-class representation [7,8,9,10,11,12].


Choosing a MAP state as the starting point has two advantages: 1) we are guaranteed that x M AP is an interior point and 2) it is the point with the highest probability density and therefore highest probability mass in a small local neighborhood.

starting from a MAP state elevates the importance of the second question, since the MAP state often lies exactly on the boundary of the polytope and therefore we are likely to start the sampling algorithm from a vertex of the polytope.


The two key findings in this work are that 1) the SGLD algorithm converges weakly to the target distribution π, 2) the optimal rate of convergence to equilibrium scales as n −1/3 where n is the number of iterations, see [31,Section 5].

in most of the applications, constant rather than decreasing step sizes are used, see [1,8,18,22,30,32].


In the classic domain adaptation setup we usually have large amounts of data for solving a source task characterized by the density function p s (x, y).

only a partial or reduced sample is available for solving a target task with density p t (x, y).


Let p be the solution of (1), then fj = p(X j ) for j = 1, . . . , m. The kernel gram matrix K is always positive semi-definite.

if K is not full rank (singular), then the correct interpretation of , where I m×m is the m × m identity matrix.


Thus, we will need to sample some variables from this set.

exact inference over {V 7 , ..., V 10 } is easy.


Our analysis is based on an extension of the sparse regression result by [10] that requires exact projections.

a critical challenge in the case of overlapping groups is the projection onto the set of group-sparse vectors is NP-hard in general.


This might explain the relatively poor practical performance of variational methods in deep learning.

simple incremental encoding methods yield excellent compression values on deep networks, vindicating Solomonoff's approach.


In particular, we have DM = L 2 max ξ(κ(F ) √ rRQ max ), ξ(z) = z 2 + z, where L max denotes the maximal value of all elements in L, κ(F ) is the condition number of the matrix F and r in an upper bound on the rank of F. The constants R and Q max come from the kernel ASGD setup and, importantly, depend only on the data distribution, but not on the loss L or score matrix F . Note that for a given subspace F, the choice of matrix F is arbitrary and it can always be chosen as an orthonormal basis of F giving a κ(F ) of one.

such F can lead to inefficient prediction (1), which makes the whole framework less appealing.


In such cases, we can still apply the Gaussian copula (and other copulas derived from elliptically symmetric distributions), since it is parametrized by the same covariance matrix as a n-dimensional Gaussian.

the Gaussian copula becomes prohibitively expensive to fit in high dimensions, since evaluating the likelihood requires an exponential number of evaluations of the multivariate Gaussian cdf, which itself must be computed numerically.


Compounded by a data independent latent prior, the resulting tracker gets easily confused: it jumps across loops and/or remains put, trapped in local optima.

lELVM is stable and follows tightly a 1D manifold (see videos).


for x ∈ R and z ∈ R, the variational distribution q θ (z) = N (z|θ, 1) for θ ∈ R, and its reparameterization f θ ( ) = + θ and q( ) = N ( |0, 1) for ∈ R. For an observed value x 0 = 0, the joint density 1), where c 1 = N (0|5, 1) and c 2 = N (0|−2, 1).

notice that r is non-differentiable only at z = 0 and {0} is a null set in R. For any θ, ∇ θ ELBO θ is computed as follows: Since log(r(z)/q θ (z RepGrad θ is computed as follows: After reparameterizing z into , we have and we obtain RepGrad θ = −θ.


Note that e A T e B is not the desired rank-r approximation of A T B.

[1] is a recent sampling method with very good performance guarantees.


Because of this, it is sufficient for P 2 to use a Markov policy, and this can be calculated in polynomial time in the number of states and actions.

there is a polynomial reduction from stochastic games to MVDPs (Lemma 1), and since Letchford et al.


This is the approach chosen in [5] (with SVMs).

our goal is to learn jointly D and the model parameters θ.


Note that in [18], the QP-RL relaxation was described using the variable y = 1+x 2 .

the above formulation can easily be shown to be equivalent to the one presented in [18].


the compute cost would void all computational benefits of the iterative (stochastic) methods over full gradient methods.

it is often possible to efficiently compute approximations of p k or p k instead.


In terms of RMSE on the training set (which is proportional to the log likelihood), both models fit the training data very well, with the non-invariant model even fitting marginally better.

the invariant model generalises much better, as points on one half of the input inform the function on the other half.


The main culprit here is that interchanging differentiation and integration is sometimes invalid: for The RepGrad θ term in Theorem 1 can be easily estimated by the standard Monte Carlo: We write RepGrad θ for this estimate.

estimating the other BouContr θ term is not that easy, because of the difficulties in estimating surface integrals in the term.


In this case, the prediction should correspond to the most likely outcome that can be learned by training on similar video.

if multiple outcomes are present in the training set then minimizing the L 2 distance to these multiple outcomes induces the network to predict the average outcome.


Moreover, it can be shown that this inner product always is a strong kernel.

the concept of strong kernels is more general: there are strong kernels k such that the associated kernel metric d k (x, y) = φ(x) − φ(y) is not an ultrametric.


The classical approaches are based on backpropagation through time [22,6].

apart from convergence issues, backpropagation through time is not a feasible method for biological systems.


Moreover, these analyses do not guarantee a consistent estimate of the optimal regressor when the measurements have zero-mean random noise.

we provide results under a more general RSC/RSS condition, which is weaker [20], and provide crisp rates for the error bounds when the noise in measurements is random.


A good data fit can be achieved by generating arbitrarily many evidences for all classes as long as the ground-truth class is assigned the majority of them.

in order to perform proper uncertainty modeling, the model also needs to learn variances that reflect the nature of the observations.


One possible approach is to learn a dynamics model in the feature representation of a higher layer of a deep neural network trained to perform image classification (say on ImageNet) (Vondrick et al., 2016).

this is not a general way of learning task relevant features and it is unclear whether features adept at object recognition are also optimal for object manipulation.


1, we can clearly see that for all the experiments, large batches have a noticeably larger Hessian spectrum both in the dominant eigenvalue as well as the rest of the 19 eigenvalues.

note that curvature is a very local measure.


Although several works have focused on integrating these weakly labeled information that are complementary to the labeled or unlabeled training data into existing algorithms, these approaches are usually computational expensive.

in our framework we have the opposite behavior: the more prior knowledge we exploit to construct the candidate set, the better the performance and the faster the algorithm will be.


Typically d is much larger than n which suggests that statistical clustering methods will perform poorly.

it may be the case that there are only a small number of relevant genes in which case we might expect better behavior by focusing on this small set of relevant genes.


• Add a value: Choose a rule r m ∈ R [t] , a condition c k ∈ r m and then a candidate value • Remove a condition: Choose a rule r m ∈ R [t] and a condition c k ∈ r m , then where we use ν(•) to access the feature in a condition.

if the misclassified example is negative, it means R [t] covers more than it should and therefore needs to reduce the coverage by randomly choosing one of the following actions.


Soft correspondence techniques that achieve this using a single nonlinear transformation [2,3] perform well on some challenging problems.

the smoothness constraints used to control the nonlinearity of the transformation will prevent these techniques from selecting the discontinuous transformations associated with part-based movements.


Table 1 shows that dynamic modeling again has the best performance.

the gains over the holistic attribute classifier are smaller than in Weizmann.


One may improve the sampling procedure to get better run time.

the time spent on selecting the most related tasks is small compared to the other operations when dealing with many tasks.


Note that if the state transition distribution p(x ′ |x, u) is known, we can easily derive parameter θ analytically or numerically.

p(x ′ |x, u) is generally unknown, and it is a non-trivial problem to identify this distribution in large-scale applications.


For additional comparisons, we provide the results presented in the recent work on joint object detection and segmentation of [23], from which we also provide evaluation results of the Implicit Shape Model (ISM) [16].

please note that the results of [23] are based on a different baseline implementation.


Fluorescent calcium imaging provides a potentially powerful tool for inferring connectivity in neural circuits with up to thousands of neurons.

a key challenge in using calcium imaging for connectivity detection is that current systems often have a temporal response and frame rate that can be orders of magnitude slower than the underlying neural spiking process.


Several recent works have also trained deep networks for the task of frame prediction [12][14] [13].

unlike other works that focus on prediction as a final objective, in this work prediction is regarded as a proxy for learning representations.


Given the prominence of Nash equilibrium as a solution concept in game theory (compared to coarser notions such as correlated equilibria or the Hannan set), this problem lies at the heart of multi-agent learning [4].

convergence to a Nash equilibrium is, in the words of [4], "considerably more difficult" than attaining a no-regret state for all players (which leads to weaker notion of coarse correlated equilibrium in finite games).


Many approaches have been proposed for classifier adaptation; e.g., feature space transformations [12], model adaptation approaches [13,14] and joint feature and model adaptation [15,16].

even the joint learning models are not able to modify the feature extraction process and so are limited to shallow adaptation techniques.


At these higher values, nearly all discriminative information in the question encoding is lost -with the VQA model sacrificing its own performance to lower that of the question-only model.

we observe that for reasonable values of λ H , the strength of the question-only adversary can be varied over a much wider range with less dramatic losses (blue curve in Figure 2).


As in [LS05], noise tolerance is facilitated by the fact that the path through the network is not affected by altering the label of an example.

balancing the distribution before passing it to the weak learner, which was needed to use a standard weak learner, may disturb the independence between the event that an example is noisy, and the random draw of x.


Our technique is analogous to the 'Fastfood' approach to accelerating random Fourier features using fast Hadamard transforms [LSS13].

our runtime scales with nnz(A), which can be significantly smaller than the Õ(nd) runtime given by Fastfood when A is sparse.


The CSP spatial filters extracted as above are optimized for the calibration measurement.

in online operation of the BCI system different non task-related modulations of brain signals may occur which are not suppressed by the CSP filters.


In the transduction setting, this is not an issue, it is even an advantage as one can use the unlabeled data in the computation of the bound.

in the induction setting, this is a drawback.


Generalization has been one of the major challenges for learning dynamics models in model-based reinforcement learning.

previous work on actionconditioned dynamics prediction focuses on learning the pixel-level motion and thus does not generalize well to novel environments with different object layouts.


Mutual information estimators have been recently proposed in [2,3,16] based on local likelihood maximization.

they involve heuristic choices of hyper-parameters or solving elaborate optimization and numerical integrations, which are far from being easy to implement.


This matches the lower bound for degree d bounded graphs in [16].

note that Theorem 4 is stronger in the sense that the bound holds for a smaller class of graphs i.e.


Testing the accuracy of approximate inference is very difficult, because, for large graphical models, the true posterior distribution is intractable.

we can probe the reliability of the model on a synthetic data set.


At each time step, new cars enter the grid with probability p arrive from each of the four directions.

the total number of cars at any given time is limited to N max = 10.


The first convolution further improves the results, in line with previous findings about convolutions as feature extractors before recurrent layers [36].

applying a second convolution significantly degrades accuracy.


For both video recognition tasks (e.g., action classification) and video generation tasks (e.g., future prediction), a model of how scenes transform is needed.

creating a model of dynamics is challenging because there is a vast number of ways that objects and scenes can change.


Note that the nonlinearity of the transformation makes the SMS approach capable of recovering a fairly wide set of changes, including non-monotonic ones.

because of the smoothness constraint imposed on the location-scale transformation, it might not recover some extreme cases where the scale or location change is non-smooth/discontinuous.


It is worth pointing out that the proof will hold for a larger class of priors on K, as long as the prior shrinks at least exponentially fast (meaning that it is bounded from above by ae −bK for constants a, b > 0).

a prior at this exponential limit will require tuning, because the optimal a and b will depend on K(f 0 , Ω EB ).


When the amount of training data available is very small, adding positivity constraints for enforcing metric properties is useful for reducing overfitting and improving generalization.

when sufficient data is available, as in many modern applications, adding positive semi-definitiveness constraints is very costly, and its benefit in terms of generalization may be limited.


Importantly, we also concerned about multiple output systems.

[9] proposes an efficient inference procedure for Bayesian Online Change Point Detection (BOCPD) in which the underlying predictive model (UPM) is a GP.


This is also verified by the bad performance of TopEig in separating out the first two (shown in Section 5).

in Figure 2, we can see that the Citeseer networks are different from HEP-Th in the sense that they are not as strongly connected inside as HEP-Th.


A straightforward idea is using the optimized decisions under certain penalty to supervise the decision network.

for a backbone CNN with m layers, the time complexity of collecting the supervised signal is O( m i=1 n m ), which is NP-hard and unacceptable for prevalent very deep architecture such as VGG [37] and ResNet [3].


As a consequence, the BCM approximation is calculated when the inputs to the test data are known.

inductive methods (RRA, OGP, SRM) build a model solely on basis of information from the training data.


Furthermore, this test is almost-surely consistent under very general alternatives.

it is not clear what sort of assumptions would allow finite sample analysis of the power of their test.


With reference to the constraint in ( 5), if we only retrain the -th layer, the output of the retrained layer is in the -neighborhood of that before retraining.

when all the layers are retrained through (5), an immediate question would be whether the retrained network produces an output which is controllably close to the initially trained model.


Similarly, in (3), p(y | x) does not appear (this type of behavior is particularly useful if we only have access to a simulator of a system, but not access to the actual likelihood function).

(4) has a direct path from ✓ to f , which contributes a term to the gradient estimator.


As Table 1 shows, the grounded language has the communication success rate similar to natural language.


Each cluster can still be represented by choosing one of the given HMMs, e.g., the HMM which the spectral clustering procedure maps the closest to each spectral clustering center.

this may be suboptimal for various applications of HMM clustering, e.g., in hierarchical estimation of HMM mixtures.


In these and many other applications, it has been widely noted that by incorporating temporal information and allowing for the possibility of time-varying models, significant improvements are possible in practice.

despite the reported superior empirical performance of these dynamic models over their static counterparts, there is limited theoretical justification for introducing these more complex models.


This method is comparable to CD-MCBoost in the sense that it does not require multiclass weak learners.

there are no guarantees that the weak learners in common use are able to discriminate the complex classes of the augmented binary problem.


RR falls in the family of "shrinkage" estimators as it shrinks the weight parameter towards zero.

pCA is a "keep-or-kill" estimator as it kills components with smaller eigenvalues.


We suspect that this difference in the bias is caused by the rather extreme parameter values in the original alarm network model, which leads to a relatively large signal-to-noise ratio even in small data sets.

gene-expression data is known to be extremely noisy.


This behavior is in agreement with our earlier discussion, that sharpness is sensitive to scaling of the parameters and is not a capacity control measure as it can be artificially changed by scaling the network.

combined with the norm, sharpness does seem to provide a capacity measure.


For one specific dataset D, a natural way to quantify support would be the posterior probability of a model given the data, Pr(θ|D).

randomly generating a large number of datasets so as to approximate the expected value of Pr(θ|D) over all datasets, in the spirit of previous work on model identifiability [23], becomes intractable for complex models such as ours.


Alternating optimization algorithms have been used in both of the early methods [2,14] to solve Problem (1) which alternate between solving two subproblems: solve for β t with fixed c; solve for c with fixed β t . The convergence property of such an alternating algorithm has been analyzed in [2] that it converges to a local minimizer.

both subproblems in the existing methods can only be solved using iterative algorithms such as gradient descent, linear or quadratic program solvers.


The aleatoric uncertainty models add negligible compute.

epistemic models require expensive Monte Carlo dropout sampling.


During dropout, the error of any classifier h is characterized by two measures.

in the end, we are interested in the usual generalization error (expected risk) of h where x is drawn from the underlying data-generating measure: Err (h) since dropout training works on the corrupted data x (see (3)), in the limit of infinite data, the dropout estimator will converge to the minimizer of the generalization error with respect to the dropout measure over x: Err (h) The main difficulty in analyzing the generalization of dropout is that classical theory tells us that the generalization error with respect to the dropout measure will decrease as n !


We know how many times the users have listened to each of the songs in the dataset, but they have not explicitly rated them.

we can assume that users will probably listen to songs more often if they enjoy them.


Intuitively, this instability is quite undesirable.

the previous large margin learning bounds seemed to have dismissed this concern.


Recent years have seen significant progress in using machine learning to reason with relational data [16], especially within the context of knowledge graph embeddings [6,23,18,28,29,45], probabilistic soft logic [3], and differentiable tensor-based logic [11,33].

existing work in this area primarily focuses on using logical reasoning to improve edge prediction in knowledge graphs [14,13,27], for example, by using logical rules as regularization [15,20,35,37].


The memory layers are stacked in the following way: • The input to layers above the first is the sum of the output o k and the input u k from layer k (different ways to combine o k and u k are proposed later): • Each layer has its own embedding matrices A k , C k , used to embed the inputs {x i }.

as discussed below, they are constrained to ease training and reduce the number of parameters.


Still, the parametric approach does not directly generalize to more than one nonlinearity, e.g., g(U ) = i g i (U ∩ W i ) for sets W i ⊆ U.

algorithm 1 (with the summarization) can handle all of these cases.


This is somewhat surprising as this implies that maintaining price coherence does not offer strong informational advantages (at least when traders are individually coherent, as assumed here).

while there is little difference between the two costs in terms of accuracy, there is a difference in terms of the worst-case loss.


Some other works on conditional graphical model estimation [20,4] are based on similar structural sparsity assumptions.

our model does not assume any specific structure on the weight vectors, and by explicitly modeling the covariance structure of the weight vectors, learns the appropriate underlying structure from the data.


Since CFR only needs to store values at each information set, its space requirement is O(|I|).

as previously mentioned vanilla CFR requires a complete traversal of the game tree on each iteration, which prohibits its use in many large games.


While NMT with beam search has been proved to be successful, it has several obvious issues, including exposure bias [9], loss-evaluation mismatch [9] and label bias [16], which have been studied.

we observe that there is still an important issue associated with beam search of NMT, the myopic bias, which unfortunately is largely ignored, to the best of our knowledge.


The DDM and DEPC variants, further, replicate the finding that the buildup of neural firing depends approximately linear on coherence (normalised mean square error of a corresponding linear model was 0.04 and 0.03, respectively).

the CONST variant exhibited an inverse relation between coherence and buildup of predicted neural response, i.e., buildup was larger for small coherences.


They are related to Hopfield networks, which have been used in the context of models of hallucinations before [11].

whereas the latter model some abstract memory system, DBMs (and the related Deep Belief Nets) learn hierarchical representations of data [7], thus capturing aspects of the visual cortex [12], the locus where visual hallucinations are ultimately realized [1,13].


While impressive results have been shown, each different restrictions of the feasible set may require a tailored algorithm.

we propose to include probabilistic equalities among the model beliefs, which permits derivation of an algorithm that is generally applicable.


Until recently, neural-symbolic systems were not able to fully represent, reason and learn expressive languages other than propositional and fragments of first-order logic (Cloete & Zurada, 2000).

in (d'Avila Garcez et al., 2002b;d'Avila Garcez et al., 2002c;d'Avila Garcez et al., 2003), a new approach to knowledge representation and reasoning in neural-symbolic systems based on neural networks ensembles has been introduced.


Recent work on scaling kernel methods using random basis functions has shown that their performance on challenging tasks such as speech recognition can match closely those by deep neural networks [22,6,35].

research also highlighted two disadvantages of random basis functions.


θ = 0 would correspond to the best feed-back policy if the state was perfectly known.

moving in an direction orthogonal to the goal helps improving localization.


So long as we use a gift value g σ (I 1 , a ) ≤ CBV σ2 (I 1 ) − CBV σ2 (I 1 , a ), the resulting strategy will be safe.

using a lower bound further guarantees a reduction to exploitability when a P 1 best response reaches with positive probability an infoset I 1 ∈ S top that has positive margin, as proven in Theorem 1.


The empirical distribution provides a maximum likelihood estimate of the context distribution and the EALP and UCB-EALP algorithms achieve similar performance as ALP and UCB-ALP, respectively, as observed in numerical simulations.

a rigorous analysis for EALP and UCB-EALP is much more challenging due to the dependency introduced by the empirical distribution.


Note that the second condition depends on the solution trajectory X(t), and may be hard to check a priori.

we give one special case in which the condition trivially holds.


Overall, this analysis yields a dynamical equivalence between mAMP and bAMP as long as at each iteration time t, the optimal loss and regularizer for mAMP are chosen through the smoothing operation in ( 18), but using time-dependent smoothing parameters λ t η and λ t h whose evolution is governed by (7).

4 Determining optimal smoothing parameters via state evolution of AMP In the previous section, we have shown that mAMP and bAMP have the same dynamics, as long as, at each iteration t of mAMP, we choose a time dependent optimal loss L opt t and regularizer σ opt t through (18), where the time dependence is inherited from the time dependent smoothing parameters λ t η and λ t h . mAMP was motivated as an algorithmic solution to the M-estimation problem in (5) for a fixed loss and regularizer, while bAMP was motivated as a method of performing the Bayesian integral in (2).


This is since in online learning we compete against the best static action in hindsight, which does not move at all and hence has zero movement cost.

in a MTS, where one compete against the best dynamic sequence of actions, one needs both an upper a lower bound on the metric.


Exact probabilistic inference with the corrections described here again leads to intractable computations.

we can derive exact perturbation expansions involving a series of corrections with increasing computational complexity.


Since our current architectures solve complicated temporal tasks they must have some capacity for temporal relational reasoning.

it is unclear whether their inductive biases are limiting, and whether these limitations can be exposed with tasks demanding particular types of temporal relational reasoning.


Previous works [6,13,11] generally first develop a kind of approximation (or relaxation) for QP matching problem by ignoring the doubly stochastic constraint, and then aim to find the optimum of the relaxation problem by developing an algorithm.

our work focus on the general and challengeable QP problem with doubly stochastic constraint (Eq.


Naturally, this has inspired a lot of research and has given rise to new and currently very popular optimization methods such as Adam [9], AdaGrad [5], or RMSProp [22], which serve as competitive alternatives to classical stochastic gradient descent (SGD).

the current situation still causes huge overhead in implementations.


As can be seen in Figure 5, PeleeNet runs 1.8 times faster in FP16 mode than in FP32 mode.

the network that is built with depthwise separable convolution is hard to benefit from the TX2 half-precision (FP16) inference engine.


For the previous two examples, mixture importance sampling (IS) is nearly as accurate as the best multiscale methods (Gaussian seems ineffective).

in cases where all of the input densities have little overlap with the product density, mixture IS performs very poorly (see Figure 4(c)).


The visual question answering (VQA) problem is an excellent way to test such reasoning capabilities of an AI model and its multimodal representation learning.

the current VQA models are oversimplified deep neural networks, comprised of a long short-term memory (LSTM) unit for question comprehension and a convolutional neural network (CNN) for learning single image representation.


Although the empirical results are encouraging, the proposed method is based on heuristic measures and no theoretical analysis of its performance is provided.

in supervised learning a number of theoretical works investigated the effectiveness of transfer in reducing the sample complexity of the learning process.


number of transitions in a Markov chain, number of importance samples in SIR).

how well the sampling distribution approximates the posterior distribution for finite computation is typically difficult to analyze theoretically or estimate empirically with confidence.


This operation implicitly treats information from different agents equally.

information from various agents has different value for an agent to make decisions.


Therefore the distance should not be measured by the Frobenious norm as in (5).

our loss CoDA-PCA (X; A, V) is based on the KL divergence which locally agrees with the FIM [6].


They developed a linear programming approach for finding such compromise solutions; however, this is computationally impractical for most real-world problems.

we develop a more scalable game-theoretic approach for finding fairness solutions by exploiting structure in multi-agent factored MDPs and value function approximation.


We introduced this assumption to simplify the analysis.

most of the discussion can be rephrased in terms of the projected Bellman error, and we expect this approach to carry through and provide more general results (e.g.


Then, the following holds: The standard method for deriving generalization bounds from algorithmic stability results could be applied here to obtain a generalization bound for our Hankel matrix completion algorithm.

our goal is to give a generalization bound for the full HMC+SM algorithm.


A standard Algorithm 2: ADA-BARRONS 2 Run BARRONS with parameter β and η, where after each round t, if the following holds: with α t defined in Theorem 1 and then set β ← β 2 , and rerun BARRONS from Line 2 with time index reset to 1. doubling trick would suggest halving β whenever it is larger than α t (u * t ) and then restart the algorithm.

since α t (u * t ) is not monotone in t, standard analysis of doubling trick does not work.


The convergence graphs in Figure 1 plot NMI as a function of data passes, and show that our method converges to the (almost) correct solution in 1-2 data passes.

the batch algorithm takes 10 or more data passes to converge.


9,10 The tilt illusion has been explained by lateral interactions 11,12 in populations of orientationtuned units; and by calibration.

13 most models of this form cannot explain a number of crucial aspects of the data.


We show that the proposed measures can be estimated from data without any assumption on the underlying probability structure.

a distribution-free (non-parametric) approach often suffers from having free parameters, e.g.


We model each unseen class with a single signature (e.g., whales are big and gray).

it is straightforward to handle the case where a class has a multi-modal definition (e.g., whales are big and gray OR whales are big and black), by learning a zero-shot model per "mode".


Our hope is that the MAP estimate of basis coefficients ŵ produced for each input vector x will be useful for predicting a response y associated with x.

the sparse coding objective function only cares about reconstructing the input well, and does not attempt to make ŵ useful as input for any particular task.


Specifically, the standard asymptotic theory requires a large number N of words per document, compared to the number M of documents and the vocabulary size L. This may be reasonable in some collaborative filtering data such as the Last.FM data used in our experiments in Section 4.

l and/or M would be comparable to or larger than N in standard text analysis.


Our work can be viewed as an instance of meta learning [Andrychowicz et al., 2016, Santoro et al., 2016, Nichol and Schulman, 2018 in the sense that the intrinsic reward function module acts as a meta-learner that learns to improve the agent's objective (i.e., mixture of extrinsic and intrinsic reward) by taking into account how each gradient step of the agent affects the true objective (i.e., extrinsic reward) through the meta-gradient.

a key distinction from the prior work on meta learning for RL [Finn et al., 2017, Wang et al., 2016, Duan et al., 2016b is that our method aims to meta-learn intrinsic rewards within a single task, whereas much of the prior work is designed to quickly adapt to new tasks in a few-shot learning scenario.


containing interesting structure from unlabeled data.

of these only L 1 leads to an efficient, convex procedure for inference, and even this prior does not produce differentiable MAP estimates.


The above results suggest that the CAE is capturing important natural image priors.

it remains unclear whether these priors are sufficient to decode specific classes of natural images as accurately as decoding models that are tuned to incorporate class-specific priors.


In such a situation, we are free to choose any one; we broke ties arbitrarily in our implementation, and even as is, the algorithm performed as well or better than top-k PCA.

we expect that breaking the ties so as to optimize the ultimate objective would yield considerable additional benefit; this would also be non-target-agnostic.


In covariance estimation, the sample covariance in low-dimensional settings is close to the population covariance in spectral norm under mild conditions [Bunea and Xiao, 2015].

in most such applications it is not sufficient to settle for a spectral norm approximation.


It is argued that a higher mutual information can lead to better disentanglement [6], and some have even proposed to completely drop the penalty on this term during optimization [17,18].

recent investigations into generative modeling also claim that a penalized mutual information through the information bottleneck encourages compact and disentangled representations [3,19].


This weakens the theoretical guarantees of the algorithm, but in general did better than splitting gifts in a theoretically correct manner.


Until recently, neural data analysis techniques focused primarily upon the analysis of single neurons and small populations.

new experimental techniques enable the simultaneous recording of ever-larger neural populations (at present, hundreds to tens of thousands of neurons).


As a result, the combination of these methods is likely an effective model in a variety of problem settings because it should inherit both of these properties.

rather than focus on the statistical properties and empirical accuracy of this combination, we focus instead on how the resulting algorithm can be applied to problems where interpretability is a concern.


Alternatively, with a small step size, the low gradients will cause GD to miss the middle valley which has constant size of 1/2.

this exact function is quasi-convex and locally-Lipschitz, and hence the NGD algorithm provably converges to the optimum quickly.


This intuitively captures what we mean by a nonapproximable distribution.

one can easily construct a distribution with large weak dependencies such that representing this distribution exactly requires a network with large treewidth; however, if we are willing to sacrifice just a bit of accuracy, we get a very simple model.


The price we pay is in the increased dimensionality: the LP has an infinite dimensional decision variable.

combined with the geometric understanding of the the manifold of rank-1 tensors, we can identify the exact optimal solution.


First, assuming that X and Y have a joint Gaussian distribution.

we cannot use this method because one of the variables -a set of class labels -is discrete.


The allowable drift is restricted by ensuring that consecutive probability distributions are close in total variation distance.

this assumption allows for certain malicious choices of distribution sequences, which shift the probability mass into smaller and smaller regions where the algorithm is uncertain of the target's behavior, so that the number of mistakes grows linearly in the number of samples in the worst case.


Minimization of the conditional entropy of f i (x) can be used to push the predictor boundaries away from the high density regions [19,27,35].

this alone may result in overfitting to the unlabeled examples if the classifier has high capacity.


Previously, we assumed opp i = 0 and therefore pur i = 1.

if we imagine that there is an η fraction of examples on which the two views disagree, and that positive and negative regions expand uniformly at the same rate, then even if initially opp 0 = 0, it is natural to assume the following form of increase in cov and opp: (2) We show here how Definition 2 implies Definition 1.


For the ease of presentation we restrict ourselves to these balancing functions.

we can also handle the corresponding weighted cases e.g., Ŝ(C) = vol(C) = i∈C d i , where d i = n j=1 w ij , leading to the normalized cut [4].


The sublinear convergence rate obtained with this approach is up to constants identical to that in Theorem 2 for our MP variants, see [21].

as the correct scaling is unknown, one has to either take the risk of choosing τ too small and hence failing to recover an optimal solution of ( 4), or to rely on too large τ which can result in slow convergence.


Our Propagated Semantic Transfer, using the raw image descriptors to build a neighborhood structure, achieves 81.2% AUC and 40.5% Acc.

when propagating on the 85-dimensional attribute space, we improve over [11] and [9] to 83.7% AUC and 42.7% Acc.


But in dimension d > 1, following the Brownian sensing approach while sampling over the whole space would require generating M Brownian sheets (extension of Brownian motions to d > 1 dimensions) over X , and then building Assuming that the covariance matrix V X is invertible, this Brownian sensing matrix is also Gaussian and enjoys the same recovery properties as in the one-dimensional case.

in this case, estimating the stochastic integrals b m = � X f dB m using sensing points along a (d-dimensional) grid would provide an estimation error ε = b− � b that scales poorly with d since we integrate over a d dimensional space.


In the context of large margin classifiers, many techniques have been proposed for incorporating unlabeled data in an SVM, most of which are intuitively based on ensuring that large margins are also preserved on the unlabeled training data [8,2], just as in our case.

none of these previous proposals have formulated a convex optimization procedure that was guaranteed to directly maximize the margin, as we propose in Section 3.


[15], we formulate the Spatial-Model as an MRF-like model over the distribution of spatial locations for each body part.

the biggest drawback of their model is that the body part priors and the graph structure are explicitly hand crafted.


The Gaussian MRF negative log-likelihood Comparing Lemmas 1, 2 and 3, we see that the separability of the discriminative model loss depends weakly on the feature functions.

the separability for the generative model loss depends critically on the underlying sufficient statistics.


We know that 0 ≤ H(Π) ≤ log(|Π|) for any set of nodes Π, hence the result of Corollary 1 could be further manipulated to achieve a bound for the difference between BIC and BIC * of at most N log(min{|X|, |Π 1 |, |Π 2 |}).

corollary 1 is stronger and can still be computed efficiently as follows.


In essence, this process maximizes the likelihood of observing the image I given the shape z.

directly minimizing this loss often leads to noisy, unstable results (shown in Figure 2 in the supplementary document).


In detail, [14,8,9,10] view alternating minimization as a perturbed version of the power method.

their point of view relies on the closed form solution of each iteration of alternating minimization, which makes it hard to generalize to other algorithms, e.g., gradient-type methods.


That is, we maximize the data likelihood of q(x) (Eq.3) w.r.t φ, yielding the gradient: The second term involves estimating the expectation w.r.t an energy-based distribution E q(x) [•], which is in general very challenging.

we can exploit the special structure of q ∝ p θ exp{αf φ } for efficient approximation.


Note that a linear convergence rate for the training cost does not translate into a similar rate for the testing cost, and an appealing propertly of SG methods is that they achieve the optimal O(1/k) rate for the testing cost as long as every datapoint is seen only once.

as is common in machine learning, we assume that we are only given a finite training data set and thus that datapoints are revisited multiple times.


In experiments with nonlinear reward functions, GPIRL consistently outperformed prior methods, especially when generalizing the learned reward to new state spaces.

like many GP models, the GPIRL log likelihood is multimodal.


On a related note, a very recent paper [41] considered pseudo-monotone games (which strictly contain monotone games but also belong to concave games) and devised two distributed algorithms for computing Nash equilibria.

it is far from clear that the algorithms devised are no-regret when used in an online fashion.


Such hierarchical dense representation can be interpreted as a multi-layered abstraction of the original input and is believed to be critical for the success of deep models.

when F i is non-differentiable or even non-parametric, back-prop is no longer applicable since calculating the derivative of loss function with respect to its parameters is impossible.


That work showed that the likelihood of finding a causal relationship rose with the base rate peE) in non-contingent cases, suggesting that structural decisions are a relevant part of the present data.

the rating scale of the current task seems to encourage strength judgments rather, than purely structural decisions, because it is anchored at the endpoints by two qualitatively different causal strengths (strong generative, strong preventive).


The goal of any feature extractor is to (ideally) be invariant to within-class transformation while maximizing between-class discrimination.

most feature extractors are not trained to explicitly factor out specific transformations.


Because unrolling the structure necessary to represent distributions over all hypotheses has exponential blow-up, solutions are often derived from MCMC.

because of limitations in the design and parameterization of the jump function, these samplingbased methods suffer from local minima-the system must transition through lower-scoring configurations before arriving at a better MAP solution.


Unfortunately, it is somewhat difficult to relate this intuition to the above (quite cryptic) definition of submodularity, and therefore, a more friendly equivalent definition of submodularity is often used.

to present this equivalent definition in a simple form, we need some notation.


These linear type algorithms can also easily provide probabilistic (soft-decision) estimates for each symbol.

there is still a significant gap between the detection performance of the MMSE-SIC algorithm and the performance of the optimal ML detector.


Our framework is also similar to the Anytime Edge Evaluation (AEE*) framework [25] which deals with edge evaluation on a GBG.

our framework terminates once a single feasible path is found while AEE* continues to evaluation in order to minimize expected cumulative sub-optimality bound.


First, we will require that the objective is strongly concave in the discriminator parameter space at equilibrium (note that it is concave by default).

on the other hand, we cannot require the objective to be strongly convex in the generator parameter space as we saw that the objective is not convex-concave even in the nicest scenario, even arbitrarily close to equilbrium.


The Hamming based methods give poor results in terms of k τ but become the best choice when measuring the mean Hamming distance between predictions and ground truth (see Table 3 in the Supplementary).

the fact that the Lehmer embedding performs well for the optimization of the Kendall's τ distance highlights its practical relevance for label ranking.


This property makes them useful, for instance, in actually distorting the optimal policy in Markov decision problems to ensure that exploration is plmmed and executed in favor of exploitation.

they can be dangerous for exactly the same reason -and there are reports of them leading to incorrect behavior, making agents search too much.


This is identical to the mathematical definition of randomness given by Kolmogorov complexity.

the key point of this statistical approach is that we are not restricted to using R(X): we have a measure of the randomness of X for any choice of P (X|regular).


The number of possible scene configurations is exponential in the number of object hypotheses 1 .

not all cuboid and room subsets are compatible with each other.


Velocity can be estimated using a population of neurons tuned to different spatiotemporal frequencies [10] [11].

the output of a single motion energy neuron is an ambiguous indicator of velocity, since its output depends upon other stimulus dimensions in addition to motion, (e.g.


Let x t , 1 ≤ t ≤ N 0 be generated using a mixture of Gaussians (1) At a high level our analysis shows that at each step of the streaming updates, the error decreases on average.

due to the non-convexity of the objective function we can show such a decrease only if the current estimates of our centers lie in a small ball around the true centers of the gaussians.


It is also notable that the worst-case time complexity of S-AdaNCG is worse than that of a recent algorithm called Natasha2 proposed in [2], which has a state-of-the-art time complexity of O d/ 3.5 1 for finding an ( 1 , √ 1 ) second-order stationary point.

s-AdaNCG is much simpler than Natasha2, which involves many parameters and switches between several procedures.


This result guarantees that the sequence p k produced by ADMM globally converges under a mild assumption.

to establish the linear convergence rate, the key lies in estimating the other side inequality, Then one can combine these two results together to prove that sequence p k converges globally and linearly with , which further can be used to show the R−linear convergence of iterates x k , y k and z k . To estimate the constant γ, we first describe the geometry formed by the optimal primal solutions x * , y * and dual solutions z * of the LP (3).


Let H be the RKHS embedded with the dot product ⟨•, •⟩ H (we abbreviate ⟨•, •⟩ H as ⟨•, •⟩ for simplicity) and a positive definite kernel k. Additionally, let ϕ : M → H. Then, we define the Koopman operator on the feature map ϕ by ) Thus, the Koopman operator K H is a linear operator in H. Note that almost of the theoretical claims in this and the next sections do not necessarily require ϕ to be in RKHS (it is sufficient that ϕ stays in a Hilbert space).

this assumption should perform the calculation in practice (as described in the last parts of this and the next sections).


One proposed solution is to filter observations during policy execution [4].

without also predicting system trajectories w.r.t.


Previous MAPR approaches required a library of team activity sequences (team plans) be given as input.

collecting a library of team plans to ensure adequate coverage is often difficult and costly.


Deep neural networks continue to advance the state-of-the-art of image recognition tasks with various methods.

applications of these methods to multimodality remain limited.


with virtual support vectors (Decoste and Scholkopf, 2002).

as far as we know the NLMP performance is the best on the original USPS dataset among algorithms that do not use prior knowledge about images.


[15], that the bound would be 3 O(( If an upper bound U on u is known, we can use it to tune η to obtain an upper bound of the order of O(U √ T ).

we obtain for DFEG a bound of O( u log( u T + 1) T ), that is optimal bound, up to logarithmic terms, without knowing U .


In the machine learning setting, [26] and the later work of [27] have provided additional insights into these types of methods.

these papers study the case of smooth objective functions of a sum structure, and so do not directly apply to general case we consider here.


[2015] combined Q-learning with a deep neural network in an algorithm called DQN, which impressively learned to play many games using a single set of hyper-parameters.

as discussed above, to handle the different reward magnitudes with a single system all rewards were clipped to the interval [ 1,1].


(28) Since the kernel ω is strictly non-negative, the only steady-state solution is Z ≡ 0, which means that the output signals of original nonlocal blocks tend to be damped out (diffused) along iterations.

the original nonlocal operation with positive eigenvalues is unstable in finite time.


The user only needs to know when to stop.

both LMNN and PSDBoost have parameters to cross validate.


This is similar to the trend observed for FedAvg algorithm while reducing the Client fraction (C) which makes the (stochastic) agent interaction matrix sparser.

from the plot of the variance of accuracy values over agents (a smooth version using moving average filter), it can be observed that the level of consensus is more stable for denser networks compared to that for sparser networks.


Efficient inference algorithms with provable guarantees have been proposed by applying recent advances in tensor decomposition methods [14,15], directly clustering the users [16,17], or using sampling methods [18].

this mixture PL is still restrictive, and both clustering and tensor based approaches rely heavily on the fact that the distribution is a "mixture" and require additional incoherence assumptions on Θ * .


One way to merge topics is to perform bipartite matching across the two topic sets, using the Hungarian algorithm.

performing this topic matching step imposes a computational penalty as the number of topics increases.


The latter is ensured by requiring the additional counting numbers and potentials to equal zero.

we note that the program will change since the constraint set is augmented.


Since C is complete, the intrinsic mean as defined above always exists.

there may be collections of shapes for which µ is not unique.


First, OTMANN shares similarities with Wasserstein (earth mover's) distances which also have an OT formulation.

it is not a Wasserstein distance itself-in particular, the supports of the masses and the cost matrices change depending on the two networks being compared.


When initialized from one, the Prior sampler creates no additional features.

starting from one, our SM+DD moves rapidly identify a diverse set of 33 behaviors.


The proposed probabilistic population coding (PPC) framework provides a statistically efficient neural representation of probability distributions that is both broadly consistent with physiological measurements and capable of implementing some of the basic operations of probabilistic inference in a biologically plausible way.

these experiments and the corresponding neural models have largely focused on simple (tractable) probabilistic computations such as cue combination, coordinate transformations, and decision making.


In fact, as shown by Corollary 4, converging to a reasonably good arm suffices.

standard MAB algorithms do identify better arms very well.


In the case with a binary dependency, structure learning recovers the necessary dependency with few samples, and has similar performance to Coral.

in the second and third settings with high-order dependencies, structure learning struggles to recover the relevant dependency, and performs worse than Coral even as more training data is provided.


Triple GAN can be implemented by specifying both p x (x|y) and p y (y|x) to be distributions with explicit density form, e.g., Gaussian distributions.

the performance can be bad since it fails to capture the multi-modality of p x (x|y) and p y (y|x).


When the output space can be assumed to be a vector space, many ideas from regression can be extended, think for example to multivariate [20] or functional regression [32].

a lack of a natural vector structure is a feature of many practically interesting problems, such as ranking [18], quantile estimation [26] or graph prediction [38].


These advantages are paid for by the rate (3), which can be arbitrarily high.


In Growth, RMH is slightly more accurate.

it tends to select more variables than the other methods.


The inversion can therefore also be expanded according to the Neumann series; i.e., θ = ∞ i=0 [γλ] i r. Similar to other kernel-based policy evaluation methods [23,24], NPDP has a computational complexity of O(n 3 ) when performed naively.

by taking advantage of sparse matrix computations, this complexity can be reduced to O(nz), where z is the number of non-zero elements in (I − γλ).


This is much more practical, because in real-world applications communication is restricted by bandwidth and/or range and incurs additional cost, and thus it may not be possible or cost too much to maintain full connectivity among all the agents.

dynamic communication can keep the agent from receiving useless information compared to full connectivity.


Using steerable pyramid coefficients allowed the algorithm to classify edges at multiple orientations and scales.

the steerable pyramid decomposition has a low-frequency residual component that cannot be classified.


However, for certain language pairs, there can be an important decrease in performance.

when comparing the performance of BAE-tr with the use of 5-sentences merges, no substantial impact is observed.


For example, gradient computation for L 2 smoothing requires sorting factors rather than just maximizing over their values, incurring worst-case cost of O(W r log W r ) per factor instead of just O(W r ) for soft-max gradient.

one can use partitioning around a pivot value instead of sorting, yielding O(W r ) cost in expectation [3], and caching the pivot can also speed-up the runtime considerably.


A naive simple Monte Carlo estimator of the KL divergence between the output distributions of two inference algorithms requires evaluating output probabilities for both algorithms.

it is typically intractable to compute output probabilities for sampling-based inference algorithms like MCMC and SMC, because that would require marginalizing over all possible values that the random variables drawn during the algorithm could possibly take.


The rationale is that more rapid sampling can reduce the variance of a Monte Carlo estimate and hence outweigh the bias introduced.

the added flexibility introduces new challenges for sampler and parameter selection, since standard sample quality measures, like effective sample size, asymptotic variance, trace and mean plots, and pooled and within-chain variance diagnostics, presume eventual convergence to the target [1] and hence do not account for asymptotic bias.


This possibly indicates that soft attention models are approximating latent alignment models.

training with latent alignments and testing with soft attention performed badly.


Previous work has also considered declarative specifications of agreement constraints among classifiers (e.g., [19]) or strategies for a classification learning [9] and recommendation (e.g., [15].

these prior systems have not considered declarative specification of SSL constraints.


A GMM is learned in the low-dimensional output space for each class, and now, instead of comparing samples against each other, comparing samples against the components of the GMMs suffices.

as the parameters of the transform are being learned iteratively, the ¦ C © will change at each iteration, and the GMMs need to be estimated again.


Some common approaches for obtaining an approximate solution include the well-known EM [8] and CCCP algorithms [9,23,24].

these approaches are prone to getting stuck in a bad local minimum with high training and generalization error.


This characteristic severely limits the ability of the IPM or Simplex technique to solve these problems.

first-order methods have received extensive attention recently due to their ability to deal with large data sets.


These mechanisms achieve IC via comparing the reports from the to-be-scored worker, against those from a randomly selected reference worker, to bypass the challenge of no ground-truth verification.

we note several undesirable properties of these methods.


The expert's help consists in simply specifying a reward function.

in many practical problems, even specifying a reward function is not easy.


For setting (i), where the data lies in R D , the estimation rate has a dependence on the dimension D, which may be unattractive in high-dimensional situations: this is known as the curse of dimensionality, suffered by density-based procedures in general.

when the data has low intrinsic dimension in (ii), it turns out that, remarkably, without any changes to the procedure, the estimation rate depends on the lower dimension d and is independent of the ambient dimension D. in realistic situations, the data may not lie exactly on a lower-dimensional manifold, but near one.


In the context of sparse tensor graphical model with a general K, [9] shows the existence of a local optimum with desired rates, but does not prove whether there exists an efficient algorithm that provably attains such a local optimum.

we prove that our alternating minimization algorithm achieves an estimator with desired statistical rates.


Here we have only studied variants of CAL (and its noise-robust generalization).

one could also apply this style of analysis to other methods, to investigate analogous questions of how the label complexities of such methods degrade as the batch sizes increase, or how such methods might be modified to account for a sublinear cost function, and what results one might obtain on the total cost of learning with these modified methods.


There has been a surge of interest on adversarial attacks against supervised learning [12,15].

little is known on adversarial attacks against stochastic multi-armed bandits (MABs), a form of online learning with limited feedback.


This decoder also outperforms the linear decoder that acts on the firing rate of the neuron.

the main utility of this decoder is that it comes with verifiable bounds on the error of decoding as we record more spikes.


For any positive ε > 0, there exists a positive numerical constant C such that if entries are revealed with probability , where σ max max σ and σ min min σ , then the following holds with probability at least 1 − n −5 log 2 (4 √ r T F /ε): • the problem (3) has a unique optimal solution; and .

for simplicity of notations and to emphasize key points of our proof, we only focus on 3-mode tensors in Section 2.3.


On the other hand, off-policy reinforcement learning (RL) concerns the problem of learning a good (or possibly better) policy with data collected from a behavior policy [Sutton and Barto, 1998].

to the best of our knowledge, previous methods do not have satisfiable performance or are not directly applicable in a complex environment as ours with continuous state and hybrid action space.


Let δ t be the step size at iteration t. The algorithm performs the following updates: The above updates require computation of K 1/2 which is expensive for large high-rank matrices.

using elementary linear algebra we can show that K and the learned kernel function can be computed efficiently without computing K 1/2 by maintaining S = K −1/2 KK −1/2 from step to step.


without the need for additional diagonal dominance conditions.

the averaging has a desirable built-in balancing effect.


For illustration we assume the queries are noiseless: Generally speaking, it is impossible to reconstruct the coefficients from these queries.

since the coefficients are sparse, then the observations are reduced to The observations are captured by a bipartite graph, which consists of s = 4 left nodes and 8 right nodes (see Fig.


Our problem then becomes: Since the log likelihood scales linearly with the size of the data, and the singular values scale with the square root of the size, we also add a factor of √ nT in front of the nuclear norm term.

in the examples in this paper, we assume spikes are drawn from a Poisson distribution: this method can be used with any point process with a log-concave likelihood.


This structure is very similar to multi-task learning [4,3], in which different supervised tasks share the same input, as well as some ILR.

collaborative learning has the same supervised tasks for all heads.


If available budget allows for multiple observations, our the goal is to determine an optimal set of variables to observe.

performing such nonmyopic analyses is prohibitively expensive for many active learning heuristics [7].


One way to see that is to look at the problem of approximating the L 1 distance between D 1 and D 2 using samples S 1 and S 2 .

one possible way to do that is to define this estimate is almost surely going to be 1 if the underlying distributions are absolutely continuous.


Note that, since we are choosing permutations in the support with replacement, there could be repetitions.

for large N and K N !, this happens with a vanishing probability.


Moreover, local features across different scales can be fused to encode global context [19].

all previous methods still reside in a regular 2D feature map with the exception of [20].


In this study, the steganalysis networks were trained and used after the complete encoding system was created.

using pre-trained and/or simultaneously trained steganalysis networks in an adversarial learning framework, such as Generative-Adversarial-Networks (GAN) [30], during the training of steganography nets provides a method to incorporate an obfuscation based error metric.


Language comprehension in humans is significantly constrained by memory, yet rapid, highly incremental, and capable of utilizing a wide range of contextual information to resolve ambiguity and form expectations about future input.

most of the leading psycholinguistic models and fielded algorithms for natural language parsing are non-incremental, have run time superlinear in input length, and/or enforce structural locality constraints on probabilistic dependencies between events.


However, such analysis will rely on a bound on the variance of the local gradients (as apposed to the bound in equation ( 2)), which is a strong assumption that ignores the effect of averaging over the P nodes.

assumption 1 allows for a more elegant analysis that provides better convergence rates, which are due to the averaging of the local gradients at every step of the TopK algorithm.


That mutual information in arbitrary measure spaces can indeed be computed as a maximum over quantization is a classical result [18,36,38].

the choice of quantization is complicated and while some quantization schemes are known to be consistent when there is a joint density [11], the mixed case is complex.


The problem of image translation between domains differs since when mapping between domains, part of the content is replaced with new content that matches the target domain and not just the style.

the distinction is not sharp, and many of the cross-domain mapping examples in the literature can almost be viewed as style transfers.


This situation is somehow undesirable: when the dimensionality is high and the target error is low, the dependency on α −(p+1) could make the sample size quite large.

several results have already shown that for some specific loss functions, the exponential dependency on the dimensionality can be avoided.


Lemma 1 states that, in the population setting, one can identify the terminal vertex, and therefore the causal ordering, just by assuming causal minimality (Assumption 1).

to identify terminal vertices from a finite number of samples, one needs additional assumptions.


Our framework is similar to dependency tree kernels [4] since both methods are using the notion of smoothing for structured objects.

our method is interested in the problem of smoothing the count of structured objects.


One might argue this only holds for this contrived distribution.

in general this is a fairly common occurrence, in the sense that the ambiguity criterion works to nudge the decision surfaces so they better fit a relatively small set of labeled examples.


The probability distribution over the elements y nj is defined as previously in the limit construction of the IBP: ).

the lower binary variables z nk are also defined as Bernoulli distributed random quantities: here the probability that z nk = 1 is a function of the upper binary variables y n,: and the kth column of the weight matrix V , with probabilities V jk ∈ [0, 1] connecting y nj to z nk .


in presence of correlations between the output tuples, the problem becomes non-trivial.


The integrality of the bipartite LP relaxation is not guaranteed, however it can be enforced by adding the so-called blossom inequalities [22]: where L ⊂ 2 V is a collection of odd cycles in G, called blossoms, and δ(S) is a set of edges between S and V \ S. It is known that if L is the collection of all the odd cycles in G, then LP (2) always has an integral solution.

notice that the number of odd cycles is exponential in |V |, thus solving LP ( 2) is computationally intractable.


Even slight random shifts in the pixel-derived representation of less than 0.25 • of visual angle (on the order of micro-saccades) dramatically reduced the fits to 0.25 (G) and 0.21 (S).

the same random shifts did not change the average fit numbers for the combination of dorsal and ventral models (0.53 for G, 0.39 for S).


In order to select a feature h k , we compute the following score Notice that due to the block structure of the tableau, working with a reduced set of columns also eliminates a large number of inequalities (rows).

the large set of q • r inequalities for the parallel reductions is still prohibitive.


For a matrix M, we define a risk function R : When X has covariance matrix Σ, R(w; Σ) = Var(w T X) is the variance of the portfolio return, w T X, and is employed as the objected function in the GMV formulation.

estimating Σ is difficult due to the heavy tails of asset returns.


While the exact solution in the Gaussian case can be computed in polynomial time, estimating an Ising model via maximum likelihood in general is N P-hard .

as a consequence of applying the screening rule, the blockwise structure of an 1 -regularized Ising model can be determined as easily as the blockwise structure of a Gaussian graphical model, despite the fact that within each block, exact learning of a sparse Ising model could still be challenging.


Let O = S 1 ∩ S 2 be the indices for which both sensors are available, and By using the samples in S 1 and S 2 as two independent training sets, we may obtain two separate classifiers (denoted w 1 and w 2 ).

we can coordinate the information from both sensors by using an idea known as co-training [4]: on the O U samples, classifiers w 1 and w 2 should agree as much as possible.


T π is defined as Taking again a pessimistic approach, the one-step Bellman-error of F is defined as Typically by increasing F, E 1 (F, Π) can be made smaller (this is discussed at some length in [3]).

it also holds for both Π and F that making them bigger will increase their capacity (pseudo-dimensions) which leads to an increase of the estimation errors.


Wavelet denoising, under various choices of wavelet bases, also achieves the minimax rate.


(2014) provides an algorithm that converges to a local minimum of the Fréchet function by successive iterations of the Hungarian algorithm.

the Hungarian algorithm does not scale well with the size of diagrams, and non-convexity yields potentially convergence to bad local minima.


In our case, interconnected brain regions can be considered as nodes of a probabilistic graphical model and interactivity relationships between regions are modeled by probability values on the arcs (or the lack of) between these nodes.

the major challenge in such a machine learning approach is the choice of a particular structure that models connectivity and interactivity between brain regions in an accurate and efficient manner.


If the coin lands Heads, it is considered lucky and P 1 receives an EV of $0.50 for choosing Sell.

if the coin lands Tails, it is considered unlucky and P 1 receives an EV of −$0.50 for action Sell.


An alternative solution is to use the Shannon entropy of each column of the matrix factor as sparsity regularizer [7], since a vector with unit 1 norm and low entropy implies that only a few of its components are significant.

the entropic prior based regularizer does not afford a closed form solution, and an iterative fixed point algorithm is described based on the Lamber's W-function in [7].


The value of dR(I) can be clearly used to measure the degree of relevance of image I, assuming that small values of dR(I) are related to very relevant images.

the hypothesis that image I is relevant to the user's query can be supported by a high value of dNR(I).


This means an advantage of the Bayesian learning.

in the Bayesian learning, one computes the stochastic complexity or the predictive distribution by integrating over the posterior distribution, which typically cannot be performed analytically.


Similarly, the rate function η(t) is given by η(t)dt := dM(t).

therefore η(t) = E[λ(t)] which serves as a measure of activity in the network.


The most recent version of dPCA [8] is a general method with relatively weak modeling assumptions, arbitrary dimensionality, and a fast estimation algorithm based on low-rank regression [12].

dPCA requires that all observed neurons display firing rates for all possible combinations of task variables, a condition that may be too strict to be applicable for complex experiments.


Solving this optimization problem leads to the solution that put weights equally at the 4 corners of the convolution kernel while leaving everywhere else 0.

using this solution to do random weight initialization is too aggressive, and leaving a lot of weights to 0 makes learning slow.


Others [19] have previously tried to use the training data to "fix" the hierarchy, as a preprocessing step to classification.

it is unclear whether it is statistically permissible to reuse the training data twice: once to fix the hierarchy and then again in the actual learning procedure.


We show that considerable gains in accuracy can be obtained this way.

methods such as [12] can suffer from an exponential increase in running time when larger clusters are considered.


Both spectrum-k and mismatch(k,m) kernel directly extract string features based on the observed sequence, X.

the profile kernel, proposed by Kuang et al.


Second, since the data X(k) might not satisfy the linear ICA assumption, the optimal linear transform might not transform X(k) into independent coordinates.

it does transform X (k) into the least dependent coordinates, since Further more, if the linear transform A is constrained to be orthogonal, then finding the least dependent coordinates is equivalent to finding the marginally most non-Gaussian coordinates, since (notice that the negentropy is invariant under orthogonal transforms).


By construction, the distribution of these signals over the whole experiment, P(81, 82), is Gaussian.

each time we see a spike we get a sample from the distribution P(81' 82Ispike@to), leading to the picture in fig.


For general M σ , an analytical expression of λ min (τ ) in ( 6) is not available.

it is straightforward to compute λ min (τ ) numerically since all quantities in (7) are observable.


We believe investigating this question may yield insight into the design of predictive models for computer vision.

since annotating this knowledge is both expensive and ambiguous, we instead seek to learn it directly from large amounts of in-the-wild, unlabeled video.


This property is shared with the Sliced Wasserstein kernel [Carriere et al., 2017].

neither Persistence Scale Space kernel [Reininghaus et al., 2015] nor Persistence Weighted Gaussian kernel [Kusano et al., 2016] has this property.


The super-pixel approach has been applied with segmentation, stereo and object recognition [15].

if the grouping heuristic merges variables that should have a different label in the MAP solution, only an approximate labeling is computed.


In some cases, there is very close agreement between anatomical and functional classes, such as the (α,β) and (Y,X) cells in the cat.

the link between anatomy and function is not always so clear.


10 −20 ) prevents the whole denominator from vanishing.

if we drop the classes based on purely random Bernoulli trials, we may exclude the classes that are important for classification.