Thus, to find temporally correlated pixels all we need to do is run a clustering algorithm on the rows of the design matrix A.

we used the k-means algorithm on the rows of the matrix A but any method of Factor Analysis can be used.


These two types of vectors govern conditional probabilities that relate each word to its surrounding context.

the conditional probability of a word combines its embedding and the context vectors of its surrounding words.


More broadly, relative to our rich and detailed understanding of regularization via penalization (e.g., see the books [17,33,32,37] and papers [2,20] for details), the theory for early stopping regularization is still not as well developed.

for penalized estimators, it is now wellunderstood that complexity measures such as the localized Gaussian width, or its Rademacher analogue, can be used to characterize their achievable rates [2,20,32,37].


Here we show that an efficient coding model that incorporates biologically realistic ingredients -input and output noise, nonlinear response functions, and a metabolic cost on the firing rate -predicts receptive fields and response nonlinearities similar to those observed in the retina.

we develop numerical methods for simultaneously learning the linear filters and response nonlinearities of a population of model neurons, so as to maximize information transmission subject to metabolic costs.


We build upon the approximate naturalgradient method proposed in [19] and modify it to estimate structured covariance-approximations.

we extend the Variational Online Gauss-Newton (VOGN) method [19].


For instance, we can construct the following problem for standard HMC: Recall that the joint likelihood is p(θ, p) ∝ exp(−H(θ, p)), H(•, •) being the energy from (1).

then, we use correct samplers that preserve the desired densities in the E step of a Monte Carlo EM (MCEM) framework, and use the obtained samples of p in the corresponding M step to perform suitable updates for the mass M . to wrap the standard HMC sampler in our framework, we perform the generalized leapfrog steps [1,15] to obtain proposal updates for θ, p followed by Metropolis-Hastings corrections in the E step, and use the obtained p values in the M step.


Existing deep attentive trackers [8,27] mainly use additional attention modules to generate feature weights.

attention schemes are implemented by performing feature selection.


It is interesting to mention that, for many variants of boosting, general convergence rates were known.

once it was revealed that boosting is trying to be not only correct but also have large margins [12], much work was invested into methods which explicitly maximized the margin [13], or penalized variants focused on the inseparable case [14,15].


Recent work [16,3,7] studied the problem of automatically synthesizing If-Then programs from their descriptions.

lSTM-based sequence-to-sequence approaches [7] and an approach of ensembling a neural network and logistic regression [3] were proposed to deal with this problem.


Otherwise, the increasing dimensionality of α brings consistency into question, and in fact certain negative results are known for a related model, known as the p 1 exponential random graph model [20].

[21] reports that the maximum likelihood estimator for the p 1 model exhibits bias with magnitude equal to its variance.


We evaluate our method via 10-fold cross-validation, where we have sub-sampled the training folds in order to (a) keep the computational burden as low as possible and (b) show that we can learn sensible parameterizations based upon relatively low requirements in terms of the preferences seen on previous users.

we have subsampled 50 training users and selected about 5 training pairwise preferences drawn from each of the N = 10 available items.


If the set X of signals is well-specified, one can usually compute a (nearly) minimax on X linear estimator in a closed form.

if X is a class of smooth signals, such as a Hölder or a Sobolev ball, then the corresponding estimator is given by the kernel estimator with the properly set bandwidth parameter [16] and is minimax among all possible estimators.


The appearance terms and IDPR terms depend on the image patches.

a local image patch I(l i ) not only gives evidence for the presence of a part i, but also about the relationship t ij between it and its neighbors j ∈ N(i), where j ∈ N(i) if, and only if, (i, j) ∈ E. This requires us to learn distribution for the state variables i, t ij conditioned on the image patches I(l i ).


Then for a generic choice of input vector v, Algorithm 2 will demonstrate order r−1 convergence.

if G is κ 3 , then we obtain quadratic convergence; and if G is κ 4 , we obtain cubic convergence.


In order to address such scalability issues, Monteiro [2003, 2005] restrict the search to the set of matrices of rank at most k by factorizing X as X = Y Y * , with Y ∈ K n×k .

it has been shown that if the search space C (1) is compact, then (SDP) admits a global optimum of rank at most r, where dim S r×r ≤ m [Barvinok, 1995, Pataki, 1998, with dim S r×r = r(r+1) for K = R and dim S r×r = r 2 for K = C. restricting C to the space of matrices with rank at most k with dim S k×k > m does not change the optimal value.


In the GMMK, the kernel between two objects x and y is defined via px and py , which are estimated probabilistic models of x and y, respectively.

a probabilistic model px is learned for each example and used as a surrogate to construct the kernel between those examples.


If there is no additional structure imposed, beyond what we have stated above, then up to scaling, in the noiseless case, Outlier Pursuit can recover from as many outliers (i.e., the same fraction) as any possible algorithm with arbitrary complexity.

it is easy to see that if the rank of the matrix L 0 is r, and the fraction of outliers satisfies γ ≥ 1/(r + 1), then the problem is not identifiable, i.e., no algorithm can separate authentic and corrupted points.


µ, we would prefer step sizes that are µ-independent, thus giving rates that adapt to the local curvature (see [9]).

it turns out that by choosing a step size that maximizes min K ρ(γ)/ρ * (K), we obtain a K-agnostic step size with rate off by at most 1/2: To gain more insights into the trade-offs for these fixed large universal step sizes, the following corollary details the range of rates obtained: we have for the choice γ = 1 5L that ρ = min{ 1 3 q n , 1 5 µ L } (roughly matching the rate given in [4] for q = 1).


Such specialized structures can potentially capture the constraints in certain applications better than simple low-rankness.

we discuss in detail, a non-trivial example of the spectral k-support norm introduced by McDonald et al.


is convex for every y ∈ Y, and if the evaluation metric is the AP, the ERR or the PD, then there are distributions and sequences of scoring functions for which (1) does not hold.

we show that learning-to-rank algorithms that define their objective through a convex surrogate loss cannot provably optimize any of these evaluation metrics.


We first consider the estimation problem of Fréchet means [10] on manifolds.

the manifold under consideration is the sphere in which we wish to estimate both the extrinsic and intrinsic mean [3].


The idea of replacing uniformly sampled mini-batches with nested mini-batches is quite general, and applicable to other mini-batch algorithms.

we believe that the sparse dictionary learning algorithm of Mairal et al.


We study the fundamental problems of variance and risk estimation in high dimensional statistical modeling.

we consider the problem of learning a coefficient vector θ 0 ∈ R p from noisy linear observations y = Xθ 0 + w ∈ R n (p > n) and the popular estimation procedure of solving the 1 -penalized least squares objective known as the LASSO or Basis Pursuit DeNoising (BPDN).


For bounded input functions F, G, let ρ(F, G)=max t |F(t) − G(t)|, and let U be the class of all unimodal distributions.

then the dip statistic of a distribution function F is given by: the dip statistic computes the minimum among the maximum deviations observed between the cdf F and the cdfs from the class of unimodal distributions.


Two classes of functions that are covered by continuous submodularity are the Lovasz extensions [28] and multilinear extensions [6] of submodular set functions.

multilinear extensions of submodular set functions are also continuous DR-submodular [3], but with the special property that they are coordinate-wise linear.


Some of these predictions could be generalized to the case of large neural populations.

these analytical results may provide a theoretical basis for the predominant segregation into ONand OFF-cells in early visual processing areas.


The EWMA parameter is tuned via 10-fold cross-validation.

moreover, we also listed the performance of EWMA with projection, which is the method that runs EWMA, but with each model f k replaced by model f k is the best linear scaling of f k to predict y.


A µ efficiently to P µ A µ , we would end up with the optimal (overall) rate O( 1/ ), even though we approximate the nonsmooth function f by the proximal average A µ .

approximation itself does not lead to an inferior rate.


It maps the continuous representation e x of an architecture x into its performance s x measured by dev set accuracy.

f first conducts mean pooling on e x = {h 1 , • • • , h T } to obtain e x = 1 T T t h t , and then maps e x to a scalar value using a feed-forward network as the predicted performance.


We now consider the case where the designer has direct access to the source tasks (i.e., it is possible to choose how many samples to draw from each source) and can define an arbitrary proportion λ.

we propose a method that adapts λ at each iteration so as to minimize the transfer error E λ .


Online tree search approximations underlie a varied and popular class of Bayesian RL techniques [9][10][11][12][13][14][15][16].

the Bayes-adaptive Monte-Carlo planning (BAMCP) algorithm [16] has been shown empirically to outperform the other techniques in this category.


Compared with the setting in [6], we add the constraint π Θ = 0.

the reason is due to Since Rπ 1 2 = 0, there exists a c×(c−1) orthogonal matrix ∆, the columns of which are the eigenvectors of R. ∆ satisfies ∆ ∆ = I c−1 and ∆ π ] is the c×c matrix of the orthonormal eigenvectors of R. Since for an arbitrary class scoring matrix Θ, its rank is c−1, we have Θ = ΘΥ where Υ is some (c−1)×(c−1) orthonormal matrix.


However, despite the amount of work dedicated to this statistical learning theory-many authors improved the initial results [8,21,25,30,35] and/or generalized them for various machine learning setups [4,12,15,20,28,31,32,33]-it is mostly used as a frequentist method.

under the assumptions that the learning samples are i.i.d.-generated by a data-distribution, this theory expresses probably approximately correct (PAC) bounds on the generalization risk.


The potential benefit of acceleration to mini-batching has been briefly noted in [4], but here we study this issue in much more depth.

we make the following contributions: • We develop novel convergence bounds for the standard gradient method, which refines the result of [10,4] by being dependent on L(w ), the expected loss of the best predictor in our class.


3 Learning the parameters Different loss functions that capture the mismatch between predictions and observations have been used in the inverse optimization literature.

the (squared) distance between the observed decision and the predicted decision enjoys a direct physical meaning, and thus is most widely used [14,15,16,7].


They may downweight walks of lengths 2 and more, so much so that the similarity score is ultimately completely dominated by walks of length 1.

they are almost identical to a simple comparison of edges and vertices, which ignores any topological information in the graph beyond single edges.


Thus, the main point arising from our results is that, in contrast to previous conception, there is no inherent generalization problem with training using large mini batches.

model training using large mini-batches can generalize as well as models trained using small mini-batches.


Using the hinge loss yields support vector machines [5].

structured support vector machines [27] employ a convex approximation of the multivariate loss over a training dataset D using the hinge loss function: linear parameters θ for feature functions Φ(•, •) are desired that make the example label y (i) have a potential value θ • Φ(x (i) , y (i) ) that is better than all alternative labels y by at least the multivariate loss between y and y (i) , denoted ∆(y , y (i) ).


Recall that g := The matrix C is referred to as the self-adjoint dilation of the matrix g [20].

given the SVD of g = U⌃V > with no repeated singular values, the eigen-decomposition of C is given as the top-k left and right singular vectors of C  As in Section 2, we take the convex hull of the constraint set to get a convex relaxation to Problem (11).


Now that we know how to achieve optimal reconstruction for a given dictionary, we examine how to learn a good dictionary, that is, a dictionary that balances between reconstruction error, reconstruction complexity, overall complexity relative to the given training set.

we seek a learning method that facilitates both induction of new dictionary words and the removal of dictionary words with low predictive power.


We approximate it by the outcome of the subgradient method run for M steps, with a step-size proportional to 2/(m + 2) as suggested in [23].

initialized with θ0 i = θ t i , it performs the iterations We thus replace the step (b ) by running M steps of the subgradient method to obtain θM i . Theorem 4.


Based on this building block, we design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in the problem's dimensionality) under a bounded variance condition of stochastic gradients than previous first-order stochastic algorithms.

we develop a general framework of first-order stochastic algorithms with a secondorder convergence guarantee based on our new technique and existing algorithms that may only converge to a first-order stationary point.


As a result, we cannot distinguish g * from g * + t without labels.

pairwise comparisons do not help in improving label complexity when we are learning a threshold function on R, where all instances are in the natural order.


While extending the SM kernel to the multi-output setting via the LMC framework (i.e., the SM-LMC kernel) provides a powerful modeling framework, the SM-LMC kernel does not intuitively represent the data.

the SM-LMC kernel encodes the cross-amplitude spectrum (square root of the cross power spectral density) between every pair of channels, but provides no crossphase information.


In this work, given an image and a question about the image, we first identify useful sub-graphs of a large knowledge graph such as DBpedia [3] and then use GCNs to produce representations encoding node and neighborhood features that can be used for answering the question.

we propose a model that retrieves the most relevant facts to a question-answer pair based on GloVe features.


The general result, Theorem 1, can be applied to the adaptive GAMP algorithm with arbitrary estimation and adaptation function.

the result can be used to rigorously justify the SE analysis of the EM-GAMP presented in [11,12].


to achieve the error of 0.1, MC needs 10 6 samples in 27.8 hours, but HYBRID only needs 14.4 minutes with 10 3 samples.


The other 97 features f 4 , f 5 , ..., f 100 are conditionally independent of y given the three features f 1 , f 2 and f 3 .

f 4 , ..., f 33 are all generated by the rule f j = 3 * f 1 + ξ; f 34 , ..., f 72 are all generated by the rule f j = sin(10 * f 2 ) + ξ; and the remaining features (f 73 , ..., f 100 ) simply follow a uniform distribution in [0, 1].


The goal of a sketch is to estimate such quantities without explicitly storing the elements or the counts.

in the classical count-min sketch, N hash functions h n : V → [J], where [J] := {1, . . . , J} and |V| J, are chosen uniformly at random from a pairwise independent hash family H. a random h ∈ H has the property that for all v 1 , v 2 ∈ V, such that v 1 = v 2 , the probability that v 1 and v 2 hash to values j 1 , j 2 ∈ [J] is The sketch data structure C = [c n,j ] n∈[N ],j∈ [J] is an array of counts of size N × J.


We also give the model information about the distribution of notes in real musical scores by providing it with an external corpus of symbolic music data.

the following information is available to the model during inference: 1) the total spectrogram for each song, X (r) , which is the input, 2) the event parameters, µ, which we estimate by collecting statistics on note occurrences in the external corpus of symbolic music, and 3) truncated normal priors on the envelopes and spectral profiles, α and σ, which we extract from the MIDI samples.


That is, The consequence of this property is that taking a sample from q(f L i ) is straightforward, and furthermore we can perform the sampling using only univariate unit Gaussians using the 're-parameterization trick' (Rezende et al., 2014;Kingma et al., 2015).

we first sample l i ∼ N (0, I D l ) and then recursively draw the sampled variables where the terms in ( 15) are D l -dimensional and the square root is element-wise.


In addition to being analytically tractable and popular, the usage of multivariate Gaussian distribution can also be motivated using Bayesian asymptotics for large datasets.

for parameters in R d and under regularity conditions, if the size of the subset D i is large, the Bernstein-von Mises Theorem shows that the local posterior distribution is well approximated by a multivariate Gaussian; hence the EP approximation by an exponential family density will be very good.


This paper's goal is to understand the upper and lower bounds on the player's regret when he faces these adversaries.

we focus on how the expected regret depends on the number of rounds, T , with either full-information or bandit feedback.


Hence, to find the closest object to q on every level, we build the same heap structure, the only difference is that instead of repeating this procedure n times in each level, since there is just one query point, we need to ask at most O(D 3 log 2 n + D log 2 n log log n D 3 ) questions totally.

the closest object at level L = log 2D (n) will be q's nearest neighbor w.h.p.


With these noun-phrases as the basis, it then proceeds to construct the caption through recursive composition until a complete caption is obtained.

at each step of the composition, a higher-level phrase is formed by joining two selected sub-phrases via a connecting phrase.


Although such networks are not used in practice, they capture important aspects of the non-convexity associated with training residual networks.

hardt and Ma [2016] showed that linear residual networks with the squared loss have no spurious local minima (namely, every local minimum is also a global one).


This is consistent with the backward branch circuit of the Laguerre lattice filter, Figure 2, being different then that of the forward branch (which results in different transfer function).

a combination of different glutamate receptors such as AMPA and NMDA, as well as GABA receptors are thought to be responsible for observed responses in lagged cells [32].


Specifically, consider making a prediction using the modified inequality test: Here, the vector of parameters θ = (θ 1 , θ 2 ) controls the relative "weighting" between the message subjects and bodies, and will be fit discriminatively.

we will model the class posteriors, which we denote by Pθ to make explicit the dependence on θ, as: 3 Pθ (y|x) = P (y) P (x 1 |y) We had previously motivated our model as assigning different weights to different parts of the document.


Then the margin condition (7) is satisfied if and only if there exists a constant C > 0 such that, for all ✏ > 0, (`, F ✏ , P ⇤ ) is ⌘-stochastically mixable for ⌘ = C✏ ( 1)/ .

if the margin condition is satisfied with constants  and c 0 , we can take C = min V ( 1)/ .


Finally, we show that Neural LP can perform well in answering partially structured queries, where the query is posed partially in natural language.

neural LP also obtains state-of-the-art results on the KB version of the WIKIMOVIES dataset [16] for question-answering against a knowledge base.


Unfortunately, this maximization problem can be degenerate.

note that if ✓ t = 0 for ↵ t < 1, then the corresponding term in P (✓) will take the value 1, overwhelming the likelihood term.


We then focus on applying MGOMP to high-dimensional multivariate time series analysis problems.

we propose a novel application of multivariate regression methods with variable selection, namely that of inferring key influencers in online social communities, a problem of increasing importance with the rise of planetary scale web 2.0 platforms such as Facebook, Twitter, and innumerable discussion forums and blog sites.


From the point of view of online learning, we will see that the optimization problem of the saddle point type can be solved by playing two online convex optimization algorithms against each other (henceforth called Players I and II).

assume that Player I produces a sequence f 1 , .


The drift term in SAMD η,a,s is identical to the deterministic case, and the volatility term −η(t)σ(X(t), t)dB(t) represents the noise in the gradient.

we note that the learning rate η(t) multiplies σ(X(t), t)dB(t), to capture the fact that the gradient noise is scaled by the learning rate η.


We generate 100 problem instances from y = Ax 0 + e, where the solutions vectors have been sampled from the proposed prior using the kernel Σ i,j = 50 exp(−||i − j|| 2 2 /(2 • 10 2 )), but constrained to have a fixed sparsity level of the K/D = 0.25.

each solution x 0 has the same number of non-zero entries, but different sparsity patterns.


h1 guarantees the uniqueness of the decomposition in the LORIS model (3).


We then have ⌦ rL(M ⇤ ), where the last inequality follows from Lemma 4.3.

we see that both convex relaxation and non-convex optimization give the same rate.


3) shrinker η * for the singular values of Y given by We also discover that for each contamination model, there is a critical signal-to-noise cutoff, below which X cannot be reconstructed from the singular values and vectors of Y .

let η 0 be the zero singular value shrinker, η 0 (y) ≡ 0, so that Xη0 (Y ) ≡ 0.


Finally in Figure 2(c) we demonstrate the advantage of AttentiveChrome over LSTM-α model by printing out the β j weights for genes with differential expressions across the three cell types.

we select genes with varying ON(+1)/OFF(−1) states across the three chosen cell types using a heatmap.


Because the two sounds originate from the same signal, the binaural stimulus has a specific structure, which should result in synchrony patterns in the encoding neurons.

we modelled the response of monaural neurons by a linear filtering of the sound followed by a spiking nonlinearity.


Bernoulli variables with where ρ ∈ [0, 1] is the average fraction of stimuli in the receptive field.

we then assume that, given the vector ξ of receptive field indicators, the filter weight coefficients are independent with distribution the linear weight coefficients are zero outside the receptive field and Gaussian within the receptive field.


HMMs due to a per-iteration complexity linear in T . Unfortunately, despite the simple chain-based dependence structure, applying a minibatch-based method is not obvious.

there are two potential issues immediately arising in sampling subchains as minibatches: (1) the subsequences are not mutually independent, and (2) updating the latent variables in the subchain ignores the data outside of the subchain introducing error.


Therefore, we propose to use the Metropolis-Hastings algorithm, a popular Markov chain Monte Carlo (MCMC) method, to sample from the posterior.


We found that for different models different modular upper/lower bounds gave the best results.

for FLID we used the upper bound given by Algorithm 1 to bound each summand F l (A) in the facility location term separately.


We also discuss comparisons to [9] in Appendix C. We can observe that all methods that use matrix factorization clearly outperform their counterparts that do not, even though the noise mechanism does not obey common noise assumptions in matrix factorization literature.

the Mahalanobis matching (Match) benefits the most from matrix factorization that simultaneously alleviates the measurement noise and reduces the dimension.


The statement of the theorem is written in a conditional form, because nonvanishing errors e(x) prevent us from making a stronger statement.

once the iterates enter a region where the residual norm falls below the error threshold, the behavior of x k may be arbitrary.


SCG again outperforms gPb in all the cases.

we are much better for depth-only contours, for which gPb is not designed.


Next, we consider the high dimensional case, and show that if the loss function is a convex generalized linear function, then an -LDP algorithm is achievable with its risk bound depending only on n and the Gaussian Width of C, which is much smaller than the one in [19].

if C is an 1 norm ball or a distribution simplex, the risk bound depends only on n and log p, instead of p. techniques to other problems.


We develop a new online algorithm, the regularized dual averaging (RDA) method, that can explicitly exploit the regularization structure in an online setting.

at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the running average of all past subgradients of the loss functions and the whole regularization term, not just its subgradient.


As a result the utility of a mixture of strategies is just u(mix While the techniques in this paper apply to general extensive games, our empirical results will focus on the domain of poker.

we look at heads-up limit Texas Hold'em, the game used in the AAAI Computer Poker Competition [ZL06].


To prove this theorem, we present a more careful analysis on the problem structure.

let T = supp ∇F (x t−1 , r) , J t = S t−1 ∪T , and consider the elements of ∇F (x t−1 ).


geometry and generate new data points from the manifold.

we focus on the case where the points in X are unevenly sampled from M, and aim to generate a set of M new data points Y = {y 1 , ..., y M } ⊆ R D such that 1. the new points Y approximately lie on the manifold M, and 2. the distribution of points in the combined dataset Z X ∪ Y is uniform.


If we encode x as a binary indicator vector z(x) over the bins, z(x) z(y) = 1 if x and y fall in the same bin and zero otherwise, so Now consider shift-invariant kernels that can be written as convex combinations of hat kernels on a compact subset of R × R: k(x, y) = ∞ 0 k hat (x, y; δ)p(δ) dδ.

if the pitch δ of the grid is sampled from p, z again gives a random map for if the pitch δ of the grid is sampled from p, and the shift u is drawn uniformly from [0, δ] the probability that x and y are binned together is k(x, y).


In this paper, our goal is to provide a quantitative description of the content of a subject's visual memory for the display after the delay interval.

we want to characterize a subject's belief state about s. We suggest that a subject's belief state can be expressed as a random variable ŝ = [ŝ 1 , ŝ2 , .


However, using the conditional autoregression (CAR) representation of Gaussian Markov random fields [16], we can write where −i ≡ {1 . . . p}\i and C = [c i j ] has the same non-zero pattern as the adjacency matrix of the graph A, and amounts to a proper probability distribution for X.

by Brook's Lemma [16]  Taking limit as ζ → 0, it follows that L and L + correspond to Ω and Σ, respectively.


The concatenation of the modules' representations also enables the interpretability of the reasoning behavior of our model.

by observing which elements of the vector g are most active we can infer which modules ReasonNet used in the reasoning process and thus explain the reasons for its behavior.


Such a measurement arises when the tracker receives evidence that a specific track y j corresponds with high likelihood to a specific object x i .

the measurement likelihood of this correspondence is α, and the error probability is 1 − α.


We point out in addition that when w, w t ∈ B and ▽φ i x ⊤ i • is constant on B the effects of sampling φ i cancels out and v t i = ▽F ( w).

we can set p t i = 0 with no loss of information.


Our goal is to find a strategy that guarantees low regret for all data sequences.

this paper is concerned with the minimax strategy s * , which is the strategy that minimizes the worst case regret over all possible covariate and outcome sequences in some constraint set, i.e.


~S k Y mf on pn pn q is a Nash equilibrium f if each individual policy is a best response to the others.

for all ¢ q h , ' q ¡ and any individual policy , where S d k is the joint policy of all agents except agent ¢ .


frequentist coverage) of adaptive credible sets in nonparametric regression may extend to our setting [16].

[ Theorem 1,16] shows that -under some extrapolation conditions-adapting a GP prior via risk-based empirical Bayes guarantees honest credible sets: investigating the validity of these results in our setting is an interesting topic for future research.


However, a disadvantage of the method is that the randomness introduces variance -this is caused by the fact that g t (w (t−1) , ξ t ) equals the gradient ∇P (w (t−1) ) in expectation, but each g t (w (t−1) , ξ t ) is different.

if g t (w (t−1) , ξ t ) is large, then we have a relatively large variance which slows down the convergence.


In order to have a fair comparison on generating the trimap in terms of interaction time, we asked 10 users to generate a trimap for each image from scratch.

users were first asked to draw scribbles to indicate foreground/background and a bounding box of the foreground object.


This viewpoint supports several contributions: • Revisiting the two scenarios proposed in [2], we articulate a natural causal criterion that formally distinguishes them.

we show that observational criteria are unable to determine if a protected attribute has direct causal influence on the predictor that is not mitigated by resolving variables.


For convenience, we assume this analog quantity to be the membrane potential, but, via a non-linear transformation [7], it could equally well be an analog firing rate.

we assume that network computations require the evolution of the membrane potential of a neuron to be a function of the membrane potentials of its presynaptic partners.


It does not provide for a 1/T 2 accelerated optimization rate, but even worse, compared to non-accelerated SGD it suffers a slowdown quadratic in the delay, compared to the linear slowdown we would expect.

the guarantee ( 14) only allows maximum delay of ⌧ = O(T 1/4 ) in order to attain the optimal statistical rate ⇥(L/ p T ), whereas the lower bound allows a delay up to ⌧ = O(T 3/4 ).


( 1) with a similarly constructed vector that uses a fixed-window moving average of the previous sufficient statistics.

we replace the sufficient statistics with an appropriate scaled sum, L−1 j=0 Ŝi−j .


V and W, which essentially involves solving for their maximum-a-posteriori (MAP) estimates, which are available in closed form.

as shown in [20], estimating w k requires solving a linear system which, in our case, is of the form where S k = XΩ k X + Γ −1 , d k = Xκ k , Ω k and κ k are defined as in Section 3.1, except that the Pólya-Gamma random variables are replaced by their expectations given by Eq.


[14,9] for an overview) or functional analysis in a Reproducing Kernel Hilbert Space.


Suppose we run Algorithm 1 in the non-realizable case with inputs example oracle U , labelling oracle O, hypothesis class H, confidence-rated predictor P , target excess error � and target confidence δ.

then with probability 1 − δ, for all k = 1, 2, . . . , k 0 + 1, and for all h ∈ V k , err D (h) ≤ err D (h * (D)) + � k . the ĥ returned at the end of the algorithm satisfies err D ( ĥ) ≤ err D (h * (D)) + �.


• Conductance Testing in Graphs: MSTP-estimation is an essential primitive for conductance testing in large Markov chains [1].

in regular undirected graphs, Kale et al [6] develop a sublinear bidirectional estimator based on counting collisions between walks in order to identify 'weak' nodes -those which belong to sets with small conductance.


Each sample k in the training set is a triplet containing (1) the RGB input patch x k , (2) the binary mask corresponding to the input patch m k (with m ij k ∈ {±1}, where (i, j) corresponds to a pixel location on the input patch) and (3) a label y k ∈ {±1} which specifies whether the patch contains an object.

a patch x k is given label y k = 1 if it satisfies the following constraints: (i) the patch contains an object roughly centered in the input patch (ii) the object is fully contained in the patch and in a given scale range Otherwise, y k = −1, even if an object is partially present.


Using the definitions of A, b and ρ from the previous section, we observe that in our case λ * = 1.

there exists a solution such that Aρ = b.


We construct an arm selection policy whose regret improves upon previous result for a large class of problems.

our results imply that if X is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally Hölder with a known exponent, then the expected regret is bounded up to a logarithmic factor by √ n, i.e., the rate of the growth of the regret is independent of the dimension of the space.


At the outset we make no special assumptions on the order or form of the functions f i .

an adversary may choose to order or generate them in response to our previous choices of x.


We instead approximate the global MLE by a twostage procedure that calculates the local MLEs separately for each sub-sample, then sends the local MLEs to the fusion center and combines them.

the k-th sub-sample's local MLE is θk = arg max and we want to construct a combination function f ( θ1 , .


We also study relationships between the properties, independent of any particular algorithm.

we strengthen Kleinberg's impossibility result [8] using a relaxation of one of the properties that he proposed.


For the ease of notation, assume we are given a single example Y belonging to a certain category C, and our goal is to learn to sample additional examples X belonging to the same category.

we would like to learn to sample from the class posterior: P(X|C, Y ).


Our network is also able to preform this task, without even requiring supervision.

we extract the motion vector, z, from two reference frames using our motion encoder (Figure 3(a)).


Our analysis can be trivially reduced to semi-bandits by conditioning on the event of observing all items.

let  and analyze our problem under the assumption that all items in A t are observed.


[14]) it is easily seen that Ψ * (S) is also a spectral function.


The Gauss map takes a point θ on the curve and maps it to the unit velocity vector v(θ) = v(θ)/ v(θ) • v(θ).

the natural metric on S N −1 induces a Gauss metric on the curve, given by g G (θ) = (∂ θ v(θ)) • (∂ θ v(θ)), which measures how quickly the unit tangent vector v(θ) changes as θ changes.


We formulate the problem of learning with prior knowledge of the ground metric, and propose the Wasserstein loss as an alternative to traditional information divergence-based loss functions.

we focus on empirical risk minimization (ERM) with the Wasserstein loss, and describe an efficient learning algorithm based on entropic regularization of the optimal transport problem.


These differences between the cases dim(H) = ∞ and dim(H) < ∞ are sharp and manifest themselves in the concrete formulas for the Log-Hilbert-Schmidt metric which we obtain in Sections 4.2 and 5.

the formulas for the case dim(H) = ∞ are not obtainable from their corresponding finite-dimensional versions when dim(H) → ∞.


We show that we can use algorithm A to violate property 5 above.

letting p = µ/ µ 1 for µ a random measure taken from either D or D , we note that with high probability p 2 = Θ(n −1/2 ) and p 3 = Θ(n −2/3 ).


When the sampling rate δ satisfies δ ≥ log 4 m m , the authors show that with probability 1 − exp(− log 4 m), This performance guarantee is derived from Lemma 1.1 and Theorem 1.4 in [AM07].

to compare (2) and (3), note that our assumptions on the bounded entries of M ensures that: we see that the worst case bound for (3) is √ nm which is always lower than the worst case bound for (2): , our bound is only larger by a logarithmic term in m compared to [AM07].


We evaluate both recognition and verification accuracy.

we consider verification setting where there are different number of negative examples from different classes, and determine as success only when the positive example is closer to the query example than any other negative example.


Most existing MF approaches [2,14,16,23,26] adapt graph-based approaches to handle temporal dependencies.

the dependencies are described by a weighted similarity graph and incorporated through a Laplacian regularizer [18].


. . , a K }; then given h and current values of the c i,t 's, it samples every ancestor a k . To improve the mixing rate, we sample the inheritance variables one block at a time.

every time we sample δ consecutive states c t+1 , .


In this paper we strengthen these theoretical results and, motivated by these, make additional algorithmic and empirical contributions.

we focus on latent Gaussian models (LGM) whose latent variables are normally distributed.


Policies with high expected reward are often desired for uncertain decision processes with which little experience exists.

we consider the setting in which only a limited number of trajectories from a sub-optimal control policy through a decision process are available.


. . , d n can be chosen as −e l (l ∈ S) and e l (l ∈ V \ S).

now we focus on the case where |S| = k. Define a neighbor S (i,j) of S as the neighbor S (i,j) is given by replacing one of the elements of S with that of V \ S. Note that I S (i,j) − I S = e j − e i for any neighbor S (i,j) of S. Let S (i * ,j * ) be a neighbor that maximizes f (S (i,j) ) among all neighbors of S. Since a subset S of size ) and S * ← S (i * ,j * ) .


There exists a PTAS for the offline K-MAX problem.

for any constant > 0, there is a polynomial-time (1 − )-approximation algorithm for the offline K-MAX problem.


We propose a method that attempts to overcome these deficiencies.

we propose an energy-based model that scores segmentation quality using a deep neural network that flexibly integrates shape and image information: Combinatorial Energy Learning for Image Segmentation (CELIS).


For example, a large margin in the input space implies immunity to input perturbations.

if a classifier attains margin of γ, i.e.


However, as the demand for scalability keeps increasing, the scalability of existing LP solvers has become unsatisfactory.

most algorithms in machine learning targeting large-scale data have a complexity linear to the data size [8,9,10], while the complexity of state-of-the-art LP solvers (i.e.


We compare to a variety of classical and learned alternate approaches, and outperform them.

our experimental results show the value of end-to-end learning the entire path-following process.


Random CSPs exhibit sharp phase transitions between satisfiable and unsatisfiable instances and are an important model to analyze the average hardness of CSPs, both in theory and practice [25,26].

we consider two such CSPs: k-SAT where constraints are restricted to disjunctions involving exactly k (possibly negated) variables [3] and XORSAT which substitutes disjunctions in k-SAT for XOR constraints of fixed length.


Lemma 3.1 Given the log-transformed observation x t = log |z t | generated by Equations ( 6), if the assumptions A 1 ∼ A 2 hold, by solving simple linear equations involving the autocovariances of x t , the covariance Σ η and AB τ can be uniquely determined, where A = (I − A 0 ) −1 ; furthermore, A and B τ can be identified up to some rotation transformations.

suppose that two models with parameters (A, Non-Gaussianity of the innovations ϵ t makes the model fully identifiable, as seen in the following theorem.


Secondly, we find that Vapnik's key inequalities become equalities once "the misclassification error" is replaced by "the margin error" (defined as the rate of data instances incurring positive costs).

we find that for margin errors the following three expectations: (i) of the empirical risk, (ii) of the the true risk and (iii) of the leave-one-out risk estimator are equal to each other.


Task: We considered the task of learning complex non-linear functions from a teacher.

we chose as family F of tasks a class of continuous functions of two real-valued variables (x 1 , x 2 ).


Although any shift invariant kernel can be represented using random Fourier features, currently these are limited to Gaussian kernels or to kernels with analytical inverse Fourier transforms.

ω needs to be sampled from the inverse Fourier transform of the corresponding shift invariant kernel.


The main insight is that similar results to the setting with bounded feature functions can be obtained, however, under additional assumptions on the distribution of the observations.

the proof of Theorem 2 has shown that the sequence of observations needs to satisfy conditions under which Hoeffding-type concentration inequalities can be obtained.


Often, sparsity based methods are the key towards finding interpretable models in real-world problems.

regularization based on 1 type penalties is a powerful approach for dealing with the problem of variable selection, since it provides sparse solutions by minimizing a convex functional.


In the adversarial formulation, it is usually assumed that the losses are chosen by an all-powerful adversary that has full knowledge of our algorithm.

the loss sequences need not comply with any distributional assumptions.


The main issue is to find that appropriate direction f ∈ T θ1 (D) such that a geodesic from θ 1 in that direction passes through θ 2 at time t = 1.

the problem is to solve for an f ∈ T θ1 (D) such that Ψ(θ 1 , 0, f ) = θ 1 and Ψ(θ 1 , 1, f ) = θ 2 .


Convolutional neural networks (CNNs) [15] have proven extremely successful for a wide range of computer vision problems and other applications.


From the table, we can clearly observe the dynamic loss functions outputted via our teacher model can guide the student model to have superior performance compared with other specially designed loss functions.

with a shallow student model LSTM-1, we improve the BLEU score by more than 2.0 points compared with predefined crossentropy loss.


This is a cardinality potential that depends only on the number of variables y i assigned to a particular label.

ignoring constant terms, the lower bound can be written as a summation of cardinality factors (one for each previous solution y ∈ S): d lb (y|S) = With this approximation, the greedy step means performing MAP inference in a factor graph augmented with cardinality potentials: argmax y r(y) + λd lb (y|S).


LevinTS has to expand a large number of nodes for a small number of levels likely due to the training procedure used to derive the policy.

the policy is learned only from the 65% easiest levels solved after sampling single trajectories-harder levels are never solved during training.


This section proposes a scalable and thorough approach to solving the CONCORD objective function using recent advances in convex optimization and derives rates of convergence for such algorithms.

we use proximal gradient-based methods to achieve this goal and demonstrate the efficacy of such methods for the non-Gaussian graphical modeling problem.


Roughly speaking, such algorithms blend choices of exploration, aimed at acquiring knowledge, and exploitation that capitalizes on gained knowledge to accumulate rewards.

some experts algorithms can be interpreted as blending the testing of all experts and following those experts that observed to be more rewarding.


It is similarly clear what the best 3-clustering is, but this cannot be nested within the best 2-clustering.

the imposition of a hierarchical structure forces certain tradeoffs between the intermediate clusterings.


What we emphasize here is that by focusing only on the two highest components of ν(t), the decision maker loses information that might help him make a better selection of worker w(t).

chernoff's algorithm bases its decision largely on the workers' skills and thus does not behave as well as it should when these are not informative enough.


Performance Evaluation It can be observed from Tables 2 and 3 that in terms of both macro and micro F1 scores, the kernels integrated by either star or clique expansions achieve the highest performance on almost all of the data sets.

the integrated kernels outperform the best individual kernel significantly on all data sets.


when t xt ) t xt equals to the number of leaves of the decision tree ) S ud ~b y , the regularized risks are exactly equal to the "cost complexity" employed to prune decision trees by CART algorithm [4].

the basis of the pruning algorithm in CART is the minimisation of the regularised risk in the class of subtrees of the maximal tree, with the regularisation constant G selected by a heuristic applied to a validation set.


In their framework, a centralized network integrates information from multiple resources.

in their model, the improved decoding accuracy after combining input cues (i.e., the decreased uncertainty given by Eq.3) depends on the linear response nature of neurons, a feature in accordance with the statistics of Poisson spiking train.


Letting As shown in the graphical model, the local pseudo-data separate test and training latent functions.

the marginal posterior distribution of the local pseudo-data is then sufficient to obtain the approximate predictive distribution: once inference has been performed, prediction is local and therefore fast.


We followed [21] to develop our online data collection policy in our MOOC platform so as to prepare our offline evaluation data set.

any related questions with respect to a video will have an equal probability to be selected and displayed at all positions in this video.


One key observation of this work is that multiple restricted samplers can be combined to form an ergodic chain.

we consider a sampler that is restricted to only sample labels belonging to non-empty clusters.


. . , v Next, we conducted a series of experiments to confirm that MiME can indeed capture the relationship between Dx codes and treatment codes, thus producing robust performance in small datasets.

we created three small datasets D 1 , D 2 , D 3 from the original data such that each dataset consisted of patients with varying degree of Dx-treatment interactions (i.e.


Our known-L max algorithm is FTRL with regularizers ψ t (w) = ψ(w)/η t , where ψ(w) = ( w + 1) log( w + 1) − w and η t is a scale-factor that we adapt over time.


we assume that the neuronal responses obey multivariate Gaussian statistics governed by a stimulus-dependent correlation matrix.


As shown in [20], it is possible to re-write this objective of optimizing an infinite horizon discounted reward MDP (where the reward happens at each step) as one of optimizing an infinite mixture of finite horizon MDPs (where the reward only happens at the last time step).

we note that by introducing the trans-dimensional probability distribution on {k} × X k × A k × R + given by (2) we can easily rewrite V π µ (θ) as an infinite mixture model of finite horizon MDPs, with the reward happening at the last horizon step; namely at k. Specifically we have: for a randomized policy.


Our actual estimators, even in the isotropic case, do not directly correspond to the intuitive argument sketched in this section.

there is no partitioning of the data according to the sign of the label, and the unbiased estimator that we construct does not rely on any Gaussianity assumption.


An important characteristic of SRMs is that it allows the inclusion of certain prior knowledge of entities into the model.

the prior information is encoded as the prior covariance parameters, i.e.


Based on the Weisfeiler-Lehman algorithm, we define the following kernel function.

the Weisfeiler-Lehman kernel on two graphs G and G is defined as: where f is injective and the sets {f the Weisfeiler-Lehman kernel counts common multiset strings in two graphs.


For (predictive) log-likelihood, a positive value means that normal model fits better than P-L, whereas for AIC and BIC, a negative number means that normal model fits better than P-L. Predictive likelihood is different from likelihood in the sense that we compute the likelihood of the estimated parameters for a part of the data that is not used for parameter estimation.

3 we compute predictive likelihood for a randomly chosen subset of 100 votes.


There have been several recent work that address online algorithms for regularized learning problems, especially with ℓ -regularization; see, e.g., [14,15,16,5,17].

a forwardbackward splitting method (FOBOS) is studied in [17] for solving the same problems we consider.


For simplicity, we assume that the deformations are restricted to an arbitrary rotation of the image, where the angle of rotation is not known beforehand.

this angle (which takes a value from a finite discrete set) is modeled as the hidden variable h. We specify the joint feature vector as Φ(x, y, h) = (0 y(m+1) ; θ h (x) 1; 0 (9−y)(m+1) ), where θ h (x) is the vector representation  of the image x rotated by the angle corresponding to h. the joint feature vector is the rotated image of the digit which is padded in the front and back with the appropriate number of zeroes.


Note that this procedure is inefficient in that it samples each particle φ k merely based on the first observation with label k. Therefore, we use this procedure for bootstrapping, and then run a Gibbs sampling scheme that iterates between parameter update and label update.

(Parameter update): We resample each particle ψ k from its source distribution conditioned on all samples with label k. for k ∈ [1, m] with r k > 0, we draw ψ k ∼ T (φ k , •)|{x i : l i = k}, and for k ∈ [m + 1, m], we draw ψ k ∼ p|{x i : l i = k}, where p = p qµ or p ν , depending which source ψ k was initially sampled from.


Thus, to maximize L with respect to parameters Θ of the encoding distribution p (r|x, Θ), we repeat the following steps until convergence: • For fixed Θ, find {q new (y|r) , q new (r)} = arg max {q(y|r),q(r)}∈{Q y|r ,Qr} L • For fixed q (y|r) and q (r), find Θ = arg max Θ L. We note that using a simple approximation for the decoding distribution, q(y|r), can carry additional benefits, besides rendering the IB algorithm tractable.

while an advantage of mutual information is its generality, in certain cases this can also be a drawback.


We use kernel descriptors (KDES) to capture low-level image statistics: color, shape and texture [3].

we use four types of kernel descriptors: color-based, normalized color-based, gradientbased, and local-binary-pattern-based descriptors 1 .


An instance of partial monitoring is defined by the loss matrix L∈R N×M and the feedback table H ∈Σ N×M , where N and M are the cardinality of the action set and the outcome set, respectively, while Σ is some alphabet of symbols.

if learner chooses action i while the outcome is j, the loss suffered by the learner is L[i,j], and the feedback received is For an action 1 ≤ i ≤ N, let i denote the column vector given by the i th row of L. Let ∆ M denote the M-dimensional probability simplex.


This paper proposes a new algorithm for the following task: given a sparse undirected unweighted graph, partition the nodes into disjoint clusters so that the density of edges within clusters is higher than the edges across clusters.

we are interested in settings where even within clusters the edge density is low, and the density across clusters is an additive (or small multiplicative) constant lower.


Another class of active learning approaches for classifiers is more selective than traditional queryby-committee.

they either generate or are given an unlabeled dataset and then use a heuristic to select the most promising example to query from the entire set.


. . , q n } in an arbitrary underlying metric space.

a is an m × n matrix for which A i,j = d(p i , q j ).


The eigenvalues of subgraphs with boundary conditions are studied in [13].

let S be any (connected) subnetwork of G , and denote by δ S the boundary of S in G . The Neumann boundary condition states that for every x ∈ δ S, ∑ y:{x,y}∈δ S ( f (x) − f (y)) = 0.


Let Σ d be the diagonal matrix formed from the top d singular values, and let U d and V d be the matrices produced by selecting the corresponding columns from U and V . The matrix is the matrix of rank d that best approximates the original matrix M , in the sense that it minimizes the approximation errors.

when using SVD, the dot-products between the rows of W = U d • Σ d are equal to the dot-products between rows of M d . In the context of word-context matrices, the dense, d dimensional rows of W are perfect substitutes for the very high-dimensional rows of M d . Indeed another common approach in the NLP literature is factorizing the PPMI matrix M PPMI with SVD, and then taking the rows of W SVD = U d • Σ d and C SVD = V d as word and context representations, respectively.


• We show for the first time how variational inference can be brought to bear upon the problem of semi-supervised classification.

we develop a stochastic variational inference algorithm that allows for joint optimisation of both model and variational parameters, and that is scalable to large datasets.


To train a network to predict the tags, we enforce a loss that encourages similar tags for detections from the same group and different tags for detections across different groups.

this tagging loss is enforced on candidate detections that coincide with the ground truth.


However, in cases where the aim is to optimize over one particular metric, one can do better.

this common tool suffers from several downsides.


We initialized the network to a random but self-predicting Here for computational efficiency we used a simplified network dynamics where the compartmental potentials are updated only in two steps before applying synaptic changes.

for each presented MNIST image, both pyramidal and interneurons are first initialized to their bottomup prediction state (3), u k = v B,k , starting from layer 1 up to the top layer N . Output layer neurons are then nudged towards their desired target u trgt N , yielding updated somatic potentials To obtain the remaining final compartmental potentials, the network is visited in reverse order, proceeding from layer k = N 1 down to k = 1.


5, one could attempt to estimate the distribution parameters Γ from a set of sampled particles, then apply the LR gradient using the estimated distribution q(x; Γ).

we will approximate the density as a Gaussian by estimating the mean μ = P i x i /P and variance Σ = P i (x i − μ) 2 /(P − 1).


To tackle such an issue as well as promote the quality of object attention, we introduce a simple yet effective Self-Erasing Network (SeeNet) to prohibit attentions from spreading to unexpected background regions.

seeNet leverages two self-erasing strategies to encourage networks to use reliable object and background cues for learning to attention.


With this understanding, we are able to characterize previously studied weak-learning conditions.

the condition implicitly used by AdaBoost.MH [19], which is based on a one-against-all reduction to binary, turns out to be strictly stronger than necessary for boostability.


Generality of Approximation: Our framework allows approximation of powerful classifiers such as RBF-SVM and Random Forests as shown in Figure 5 as red and black curves, respectively.

aDAPT-GBRT can well maintain high accuracy while reducing cost.


However, to maintain a correct stochastic optimization algorithm of the variational objective, the subsampling method must be valid.

the natural gradients estimated from the subsample must be unbiased estimates of the true gradients.


We can use this probability to estimate the number of highly confident connections.

we were able to find 50% more connections in the perturbed dataset than the spontaneous dataset, with a greater than 0.95 probability (1940 vs 1204); twice times as many highly confident connections with probability 0.99 or higher (1107 vs 535); and five times as many with the probability 0.999 or higher (527 vs 101).


The mean aggregator is nearly equivalent to the convolutional propagation rule used in the transductive GCN framework [17].

we can derive an inductive variant of the GCN approach by replacing lines 4 and 5 in Algorithm 1 with the following: 4 We call this modified mean-based aggregator convolutional since it is a rough, linear approximation of a localized spectral convolution [17].


We note a recent independent work of Bailey and Piliouras [2], who show the same result in a more general setting: first, they generalize to discrete Follow-The-Regularized-Leader (FTRL) dynamics, and second, they allow the step-sizes to be mildly decreasing.

they show that if a FTRL dynamic guarantees every update must stay in the interior of the state space (MW is an example of such FTRL dynamic), then the dynamic converges toward the boundary; otherwise, they show that the FTRL dynamic gets arbitrarily close to the boundary infinitely often (this is weaker than "convergence toward the boundary").


In addition, we provide a baseline method based on nearest neighbor (NN) search.

for each of the test image, we extract VGG feature from fc6 layer (4096-dim vector) [17] and retrieve the nearest training example using Euclidean distance in the feature space.


( 3) is attained by decomposing the log-likelihood into two terms, log p(y, , which can be easily verified by substituting the product, p(y, v, h v |h id ) = p(y, v |h id )p(h v |y, v), into the right hand side of the decomposition.

the first term is the KL-divergence [10] between the true posterior and the distribution q(h v ).


For now, we note that each update consists of multiplication by a factor.

it is straightforward to see that this multiplicative factor is unity when V = W H, so that perfect reconstruction is necessarily a fixed point of the update rules.


For continuous variables Z, under certain mild conditions we can reparameterize the variational distribution q φ (z) using some simple variables .

we can draw samples from some simple distribution p( ) and do the transformation z = g φ ( , x, y) to get the sample of the distribution q(z|x, y).


To this end, we introduce a new layer that encodes the notion of self-expressiveness.

let Θ denote the auto-encoder parameters, which can be decomposed into encoder parameters Θ e and decoder parameters Θ d . Furthermore, let Z Θe denote the output of the encoder, i.e., the latent representation of the data matrix X.


This is a forced interpretation, but it is useful in understanding the implicit prior over counts that is imposed when using LR.

if p(k) is defined as be a particular function of θ (known as a Poisson-Binomial distribution [2]): , where Z(θ) = k Z k (θ), then the joint probability p(y, k; θ) becomes equivalent to a LR model in the following sense.


We would like to have similar labels at a pair of sites for which the observed data supports such a hypothesis.

we are interested in learning a pairwise discriminative model as the interaction potential.


In this section, we present our DCRF that incorporates plane and edge predictions for depth and surface normal regularization.

the field of variables we optimize are depths, D = {d i } K i=1 , where K is number of the pixels, and normals, N = {n i } K i=1 , where n i = [n ix , n iy , n iz ] T indicates the 3D normal direction.


To achieve this goal, one part of the model needs to discriminate the generated samples from the training data, and another part should mislead the former part, analogous to the generator and discriminator in GANs.

we select the approximate inference model (or encoder) of VAEs as the discriminator of GANs and the generator model of VAEs as the generator of GANs.


We prove that there is a tight connection between a stable basis and a TLP limit solution.

given a stable basis, one can find a TLP limit solution in polynomial time (all proofs are presented in Appendix A): Theorem 5.


Passing messages through this function requires summing over all edge configurations that correspond to each possible degree, d i , and summing over d i .

the message, µ Ii→e ik (e ik ), sent from function I(d i , j e ij ) to edge variable e ik is given by where µ eij →Ii (e ij ) is the message sent from e ij to function I(d i , j e ij ).


These patterns are specified by groups defined over both input and output variables.

mGOMP can handle cases where the set of relevant features may differ from one response (group) to another, and is thus more general than simultaneous variable selection procedures (e.g.


This problem recently has been explored by a number of studies in the literature.

meyers and Leskovec inferred the diffusion network by learning the infection probability between two nodes using a convex programming, called CONNIE [14].


It is defined, for every matrix A, as where ψ is a gauge function, namely a function which is symmetric and invariant under permutations.

if ψ is the ℓ 1 norm then problem (7) corresponds to tensor trace norm regularization, whereas if ψ = ω * * α it implements the proposed regularizer.


Further, we identify a few simple statistics about generated samples to act as proxies for sample quality and diversity.

we report the number of unique n-grams as a proxy for the latter and the fraction of "valid" n-grams as a proxy for former (see Supplementary   Table 8: Samples from our GAN trained to conditionally transform a given hypothesis and a label as either (E)ntailment, (C)ontradiction or (N)eutral into a premise that satisfies the specified relationship Let P be the data distribution and Q be our model's distribution.


MAPLE provides an elegant solution to this problem by using the local training distributions.

we can determine if we can apply a particular exemplar explanation to a proposed test point by evaluating how likely the proposed point is under the exemplar explanation's local training distribution.


In practice, neural data is binned such that a spike train is represented as a sequence of spike counts per time bin.

generalized Linear Models (GLMs) are built on this representation.


Our first experiment compares the performance of the subgame-solving techniques when applied to information abstraction (which is card abstraction in the case of poker).

we solve NLFH with no information abstraction on the preflop.


An easy application of L'Hôpital's rule shows that as (N log N )/T , we see that the right-hand side of (5) tends to 2 p T N log N . However the choice ↵ !

1 is clearly not the optimal choice, as we show in the following statement, which directly follows from the theorem once we see that the choice of ↵ = 1 2 gives a regret of no more than 4 p NT .


One problem, however, is that not every spike might be of equal importance.

if a spike train differs only in one spike, it might nevertheless represent a completely different stimulus.


Our work falls in between: we leverage properties of a specific kind of parallelisation-data parallelisation-for a fairly broad class of models.

we describe a parallel inference procedure that allows nonparametric Bayesian models based on the Indian Buffet Process to be applied to large datasets.


between intervals k m−1 + 1 and k m (both inclusive).

we model the spiketrains by an inhomogeneous Bernoulli process with piecewise constant probabilities.


The first component of our system is an approximate inverse graphics module for physical object and scene understanding, as shown in Figure 2-I.

the system sequentially computes object proposals, recognizes objects and estimates their physical state, and recovers the scene layout.


We have presented some applications of this framework in image understanding.

these ideas lead to a novel statistical theory of shapes of planar objects with powerful tools for shape analysis.


The "full-exponential-family-ness" is formally captured by the statistical curvature originally defined by Efron (1975), and is a measure of the minimum loss of Fisher information when summarizing the data using first order efficient estimators (e.g., Fisher, 1925, Rao, 1963.

we show that arbitrary combinations of the local MLEs on the local datasets can approximate the global MLE on the whole dataset at most up to an asymptotic error rate proportional to the square of the statistical curvature.


It can be seen that the ambiguity region mainly lies on the boundary between the two classes.

they cover those points which appear to be closer to the class other than the one they really belong to.


If we can write this covariance matrix as A T A for some (n+1) × (n+1) matrix A, then each entry in the covariance matrix is the dot product of two (not necessarily distinct) columns of A.

the mapping from the random variables Z, X 1 , .


We need such an orthogonal transformation, because cos ⇥ E r , U s 2 F can be expressed as Since the sequence {z sh , U s } 1 s=1 forms a discrete Markov process, we can apply diffusion approximations to establish global convergence of SGD.

by a continuous time interpolation, we construct continuous time processes U ⌘ (t) and X ⌘ (t) such that U ⌘ (t) = U bt/⌘c+1 and X ⌘ (t) = X bt/⌘c+1 .


First, the proposed algorithm does not require any training images of the target object before the tracking task starts.

our tracker learns a low dimensional eigenspace representation on-line and incrementally updates it as time progresses (We assume, like most tracking algorithms, that the target region has been initialized in the first frame).


The proof of Theorem 2 is given in the arXiv version [34].

when the incoherence is Õ(1), the constant L ∞ does not grow with dimension: This includes the separable case when the X i vectors are orthogonal and have unit magnitude.


Exact maximum likelihood learning in this model is intractable, but efficient approximate learning can be carried out by using mean-field inference to estimate data-dependent expectations, and an MCMC based stochastic approximation procedure to approximate the model's expected sufficient statistics [1].

during the inference step, we approximate the true posterior P (h|v; θ), where v = {v m , v t }, with a fully factorized approximating distribution over the five sets of hidden units {h m , h t , h (3) }: tl |v) where µ = {µ m , µ t , µ t , µ (3) } are the mean-field parameters with q(h Learning proceeds by finding the value of µ that maximizes the variational lower bound for the current value of model parameters θ, which results in a set of the mean-field fixed-point equations.


The crux of our proof argument lies in showing that t − t+1 ≥ J t+1 (w t+1 ) − J t (w t ) (see Theorem 4) is sufficiently bounded away from 0.

since t bounds the distance from the optimality, at every step Algorithm 1 makes sufficient progress towards the optimum.


Our heuristic deal with this is to compute a quantity ↵ 0 i,j,p which is analogous to the ↵ i,j defined above, but is defined for every pixel p. Then we compute ↵ i,j for a given patch by averaging ↵ 0 i,j,p over all pixels in the patch.

we compute ↵ i,j for patch i as the mean over all pixels p in that patch of the following quantity: Where Ov(p) denotes the set of features for positive patches that contain the pixel p. It is admittedly difficult to analyze how well these heuristics approximate the adaptive bandwidth approach of [4], and even there the setting of the bandwidth for each datapoint has heuristic aspects.


where {ψ i , ψ α } are (given) non-negative functions, the so-called factors; F is a collection of subsets (each α j is a subset of {1, 2, . . . , n} with |α j | ≥ 2); z α is the projection of z onto dimensions included in α.


Since we assume access to a database of only clean, noiseless images, we implicitly specify the desired image processing task by integrating a noise process into the training procedure.

we assume a noise process n(x) that operates on an image x i drawn from a distribution of natural images X.


P Ψ ) log-likelihood ratio Q of the observed labels under P Φ and P Ψ .

we show that, roughly, log(n/E Φ [ε π (n)]) must be smaller than E Ψ [Q] for n large enough.


As a consequence, consists of conjunctions and disjunctions of linear (in)equalities; standard tools exist for checking whether such formulas are satisfiable [12].

the controller is correct if and only if ¬ is unsatisfiable, since a satisfying assignment to ¬ is a counterexample showing that does not always hold.


This paper applies graphical models to the problem of fusing low-res depth images with high-res camera images.

we propose a Markov Random Field (MRF) method for integrating both data sources.


The rows of the basis matrix M correspond to φ(s), and the approximation space is generated by the columns of the matrix.

the basis matrix M , and the value function v are represented as: and is transitive-feasible when v ≥ Lv.


It is the large scale data sets where our method exhibits superior performance.

for a huge relation extraction data set (TAC-KBP) that has millions redundant features and samples, we outperform all existing methods in accuracy and time, in addition to generating plausible features (in fact, many competing methods could not finish the execution).


Results in Table 1 show that on our tasks, the FuN modifications do not learn well, and other than Ant Gather are significantly out-performed by HIRO.

it is worth noting that the use of learned representations, rather than observation goals, leads to almost no improvement on the tasks.


In this work we analyze Stochastic Gradient Descent (SGD) with k-sparsification or compression (for instance top-k or random-k) and show that this scheme converges at the same rate as vanilla SGD when equipped with error compensation (keeping track of accumulated errors in memory).

communication can be reduced by a factor of the dimension of the problem (sometimes even more) whilst still converging at the same rate.


We show a general reduction from the problem of searching for a violated comparison S 2 C to the problem of agnostic learning over the corresponding family of boolean functions.

recall that for a collection of comparisons C ✓ 2 X ⇥X , we can also think of C as a family of boolean concepts, where for each S 2 C, there is an associated boolean function c We frame this search problem as an agnostic learning problem, where we design a set of "labels" for each pair (x, x 0 ) 2 X ⇥ X such that any function that is highly correlated with these labels encodes a way to update the parameters towards feasibility.


[12] performed local orthonormalization instead.

only the features (rows of W ) that share a location (e.g., only the red nodes in figure 2) were orthonormalized using symmetric orthogonalization.


It is useful to contrast these multiplicative updates with those arising from gradient descent [14].

a simple additive update for H that reduces the squared distance can be written as (6) If 'flatt are all set equal to some small positive number, this is equivalent to conventional gradient descent.


Third, we give a formal separation between the power of online prediction with selfish versus honest experts by proving lower bounds for both IC and non-IC algorithms.

with selfish experts and the absolute loss function, there is no (randomized) algorithm for online prediction-IC or otherwise-with asymptotically vanishing regret.


In this paper, we carefully study the receptive field of deep CNNs, focusing on problems in which there are many output unites.

we discover that not all pixels in a receptive field contribute equally to an output unit's response.


The idea for choosing these partial sums is pictured in Figure 1: For a function s(t) that returns an integer smaller than t, we take f t = f s(t)+1:t + f s(s(t))+1:s(t) + • • • + f 0:0 .

s(t) is determined by writing t in binary, then flipping the rightmost "one" bit to zero.


Although ARC-I enjoys the flexibility brought by the convolutional sentence model, it suffers from a drawback inherited from the Siamese architecture: it defers the interaction between two sentences (in the final MLP) to until their individual representation matures (in the convolution model), therefore runs at the risk of losing details (e.g., a city name) important for the matching task in representing the sentences.

in the forward phase (prediction), the representation of each sentence is formed without knowledge of each other.


Results: Our results, summarized in Table 1, show that our proposed methods substantially outperform all baseline methods.

we see that the error is reduced by up to 45% over baseline methods which have fixed context definitions.


Our results can be generalized to all functions that are "close" to a mixture of log-concave distributions.

more precisely, assume the function 𝑓 satisfies the following properties: where ∇𝑓 0 (0) = 0, and 𝑓 is within a 𝑒 Δ multiplicative factor of an (unknown) mixture of log-concave distributions.


With the above definitions, we show in the following theorem that when X and Y are both full row rank, (G 0 , E 0 ) is the unique solution to Eq.

( 2) with high probability as long as there are O(r log N ) observed components in F. with a sampling rate of O(r log N ), our method can fully recover both E 0 and G 0 with a high probability when X and Y are full row rank.


Below we describe an alternate mapping in place of pred * MAP which can be computed efficiently, and show that under certain conditions on the probability distribution, the surrogate ψ * MAP together with this mapping is still calibrated for � MAP .

define pred MAP : R r(r+1)/2 →S r as follows: Clearly, pred MAP (u) can be implemented efficiently by simply sorting the 'diagonal' elements u ii for i ∈ [r].


The variational distribution q φ (z t |z t−1 , d t:T , x t:T ) needs to approximate the dependence of the true posterior p θ (z t |z t−1 , d t:T , x t:T ) on d t:T and x t:T , and as alluded to in (7), this is done by running a RNN with inputs d t:T and x t:T backwards in time.

we initialize the hidden state of the backward-recursive RNN in Figure 2b as a T +1 = 0, and recursively compute a t = g φa (a t+1 , [ d t , x t ]).


(that is, we do alternating optimization of the bits over the blocks).

to optimize over the bits in block B, ignoring the constants, we can rewrite equation ( 6) in the standard form for the GraphCut algorithm as: where v nm = a nm , u nm = 2 m ∈B a nm z mi − µh i (x n ).


There are two shortcomings of SMC: (i) it does not handle inference over hyper-parameters; (ii) despite the fact that the sampler targets the joint smoothing distribution, it does in general not provide an accurate approximation of the full joint distribution due to path degeneracy.

the successive resampling steps cause the particle diversity to be very low for time points t far from the final time instant T . To address these issues, we propose to use a particle Markov chain Monte Carlo (PMCMC, [3,13]) sampler.


Note this boundedness implies that > 0 or that there exists a zero-cost state which is stochastically asymptotically stable [15].

we consider the case where the immediate cost is not known a priori but is sequentially observed.


Substituting yields Note that 1) ρ is the empirical distribution over the observations, and so converges almost surely (a.s.) to the true generating distribution, and 2) the convergence properties of P n are determined by the quantity between the square brackets in (2).

let L ∞ is commonly referred to as the likelihood ratio, familiar from classical binary hypothesis testing.


However, such tools are not well understood for edit distance [16,17,30,38,46], even though many methods have been proposed [15,27,39,48,54].

no published system has demonstrated the potential to handle billions of reads, and no efficient algorithms have experimental or theoretical results supporting that they would achieve high enough accuracy on DNA storage datasets.


A particularly remarkable feature is that the choice of H k is much easier for the problems that we consider than in the binary classification problem.

if (for some fixed k) the classifier that achieves the minimal (Bayes) error for the classification problem is not in H k , then obviously the error of an empirical risk minimizer will not tend to zero, no matter how much data we have.


A limitation of the horseshoe is that it does not consider dependencies in the feature selection process.

the fact that one feature is actually relevant for prediction has no impact at all in the prior relevancy or irrelevancy of other features.


By generalizing linear response methods from statistical physics [9][10][11][12] to exponential family variational posteriors, we develop a methodology that augments MFVB to deliver accurate uncertainty estimates for model variables-both for individual variables and coherently across variables.

as we elaborate in Section 2, when the approximating posterior in MFVB is in the exponential family, MFVB defines a fixed-point equation in the means of the approximating posterior, and our approach yields a covariance estimate by perturbing this fixed point.


( 2) to be robust against variations of the mean and covariance matrix within these sets: inf Pr{aTy::; b} 2 a V(y,~y) E y.

we would like to guarantee a worst-case misclassification probability for all distributions which have unknown-but-bounded mean and covariance matrix, but which are otherwise arbitrary.


. . , d K (i, j)] for all training samples 4: Construct Pareto fronts on set of all dyads until each dyad is in a front Testing phase: Calculate dissimilarities between test sample X and all training samples in criterion l 4: nb ← [nb, nb l ] {append neighbors to list} 6: Create s new dyads D new i between X and training samples in nb 7: . . , D new s }, corresponding to the connections between X and the union of the k i nearest neighbors in each criterion i.

we create a dyad between X and X j if X j is among the k i nearest neighbors 1 of X in any criterion i.


The Burer-Monteiro approach (BM) is implemented in complex form directly using Manopt, a toolbox for optimization on manifolds [Boumal et al., 2014].

a Riemannian Trust-Region method (RTR) is used [Absil et al., 2007].


Then, using the right singular vectors V n of H , the next step consists of computing a weighted automaton A Z = α, β, {A a } as follows: The fact that the spectral method is based on a singular value decomposition justifies in part the use of a Schatten p-norm as a regularizer in (HMC-H).

two very natural choices are p = 1 and p = 2.


We are now ready to formulate our estimator.

we assume an initial, 'ideal' estimator taking the form we want to find a model that minimises the average prediction loss in the training set plus a quadratic regulariser that penalises complex solutions (the parameter λ determines the tradeoff between data fitting and good generalisation).


It is obvious that hybrid images of the primary-framework are almost the same with original images, which indicates that the swapped codes contain redundant angle information.

the disentanglement of the primary-framework is defective.


To this end, we establish formal connections between the PR framework with a broad set of algorithms in the control and reinforcement learning (RL) domains, and, based on the connections, transfer well-developed RL techniques for constraint learning in PR.

though the PR framework and the RL are apparently distinct paradigms applied in different context, we show mathematical correspondence between the model and constraints in PR with the policy and reward in entropyregularized policy optimization [43,45,1], respectively.


Since the friction changes smoothly along the track (domain knowledge), we use the adjacency matrix of a nearest-neighbor graph as the MRF kernel K in Equation 2.

we set 1 shows the average reward per time-step of the learned policies as a function of the learning time.


The Generalized Mallows model (GMM) of Fligner and Verducci (1986) extends the Mallows model by introducing a parameter for each of the elements in E and decomposes the inversion distance into a per element dis-tance 1 .

we define v j (π, π 0 ) to be the number of inversions for element j in π with respect to π 0 is v j (π, π 0 ) = i>π 0 j D ij (π, π 0 ).


Note that on this data, the method of [5] maxes out at a p-% rule of 77.6%.

the method is limited in what p-% rules may be achieved.


The proposed neural network is motivated by the low-rank property of pseudo-inverse kernels.

we first compute a generalized low-rank approximation to a large number of blur kernels, and then use separable filters to initialize the convolutional parameters in the network.


We show that there is a trade off between computational complexity and query complexity; We prove that for the case of k-means clustering (i.e., when the expert conforms to a solution of k-means), having access to relatively few such queries allows efficient solutions to otherwise NP hard problems.

we provide a probabilistic polynomial-time (BPP) algorithm for clustering in this setting that asks O k 2 log k + k log n) same-cluster queries and runs with time complexity O kn log n) (where k is the number of clusters and n is the number of instances).


We present a novel algorithm for this problem, misoKG, that is tolerant to both noise and bias and improves substantially over the state of the art.

our contributions are: • The algorithm uses a novel acquisition function that maximizes the incremental gain per unit cost.


We could thus build sparse probabilistic classifiers, providing an accurate estimation of posterior probabilities on a (limited) predefined range of posterior probabilities.

we could derive decision-oriented criteria for multi-class probabilistic classifiers.


The standard way to obtain an interval is to invert a pivotal quantity [5].

since In fact, F is monotone decreasing in x, so to find its endpoints, one need only solve for the root of a smooth one-dimensional function.


Equivalently, we may drop this effort() term from (2) altogether while requiring the search space D to consist of teaching sets of size exactly n. In this paper, we consider test distributions p(x, y) whose marginal on x has a special form.

we assume that p(x) is a uniform distribution over m distinct test stimuli z 1 , .


1 1 To interleave search with the execution of the best plan so far we perform 4D search backward.

the start of the search, sstart, is the actual goal state of the robot, while the goal of the search, s goal , is the current state of the robot.


The second result in [1] is for the regime where the connectivity matrix scales as ln(n)Q/n, Q independent of n, where it is shown that exact recovery has a sharp threshold characterized by the divergence function named the CH-divergence in [1].

if all pairs of columns in diag(p)Q are at D + -distance at least 1 from each other, then exact recovery is solvable in the general SBM.


Our first approach exploits an interesting structure in the problem corresponding to the computation of the rank of the j-th negative sample.

we show that when j > |P|, the rank of the j-th negative sample is obtained by maximizing a discrete unimodal function.


To the reader not familiar with this model, it should be considered as nothing more but an idealization of the well-known fixed design regression problem Y i = s(i/n) + ε i for i = 1, . . . , n, where ε i ∼ N (0, 1), where the goal is to recover the regression function s. (The white noise model is actually simpler to study from a mathematical point of view).

the least square criterion is defined as Given a Mercer kernel k on [0, 1]×[0, 1], the regularization least square procedure proposes to minimize where (ζ n ) is a conveniently chosen sequence and H k denotes the RKHS induced by k. This procedure can indeed be viewed as a model selection procedure since minimizing regularization aims at selecting the "best" RKHS ball {f, f ≤ R} to represent our data.


Relation to working set algorithm Theorem 3.4 is closely related to Algorithm 1.

our screening test can be viewed as a working set algorithm that converges in one iteration.


This space has a key role in our analysis.

we will assume that Consider the square loss, (x) = (1 − x) 2 .


We use tags that were generated in Vahdat and Mori [28] for the TRECVID MED 11 dataset.

this dataset includes "judgment files" that contain a short one-sentence description for each video.


Our main technical contribution is a characterization of loss functions for manifold structured prediction satisfying such a structural assumption.

we consider the case where the Riemannian metric is chosen as a loss function.


These assumptions imply that the adversary can specify his entire strategy before the game begins.

the adversary can perform all of the calculations needed to specify, in advance, how he plans to react on each round to any sequence of actions chosen by the player.


To deliver robust features, we implement two kinds of high-level pooling frameworks: Bag-of-Systems (BoS) [11] and Spatial-Temporal-Pyramid-Matching (STPM) [16] 3 .

1) BoS is performed with the clustering methods, i.e., MDS, Align and PGD.


We used the logistic loss for a model matrix W ∈ R D×C .

for each training example x i with label y i ∈ {1, .., C}, we defined an individual loss L i (W ) as where for any class c, Then L(W ) is defined as the average of L i (W ) over the whole training set.


We show that a small scale parameter -often interpreted as "equivalent sample size" or "prior strength" -leads to a strong regularization of the model structure (sparse graph) given a sufficiently large data set.

the empty graph is obtained in the limit of a vanishing scale parameter.


The main contributions of this paper are highlighted as follows: • Our result for ASYSG-CON generalizes and improves earlier analysis of ASYSG-CON for convex optimization in [Agarwal and Duchi, 2011].

we improve the upper bound of the maximal number of workers to ensure the linear speedup from O(K 1/4 M −3/4 ) to O(K 1/2 M −1/2 ) by a factor K 1/4 M 1/4 ; • The proposed ASYSG-INCON algorithm provides a more accurate description than HOGWILD!


The time structure of the withinpopulation dynamics appears to differ from that of the between-population interaction.

the latents for each area depend more strongly on its own previous latents as the time delay increases up to 15 ms (Fig.


Kernel ridge regression considers a feature mapping Φ : X → F before performing regularized linear regression.

according to Φ, the kernel function k(x, x ) = Φ(x) T Φ(x ) is defined as the inner product in the space F. When applying kernel ridge regression with a regularization parameter λ to map from X to ZV, if Φ(x) can be explicitly computed, it is known that the closed-form solution is [22]  where Φ is the matrix containing Φ(x n ) T as rows, and K is the matrix with K = ΦΦ T and is called the kernel matrix of X.


Building on this observation, it is reasonable to devise algorithms that can detect and separate out the generalized gradient components corresponding to mistakenly estimated signs a T i zt |a T i zt| along the iterates {z t }.

if z t and x lie in different sides of the hyperplane a T i z = 0, then the sign of a T i z t will be different than that of a T i x; that is, where h := z − x.


[19] proved EEB of g pu is tighter than EEB of g pn when π p / (3) and is Lipschitz continuous; (b) the Rademacher complexity of G decays in O(1/ √ n) for data of size n drawn from p(x), p p (x) or p n (x).

4 under mild conditions, PU learning is likely to outperform PN learning when π p / √ n p + 1/ √ n u < π n / √ n n . This phenomenon has been observed in experiments [19] and is illustrated in Figure 1(a).


In this section, we present numerical results on both synthetic and real datasets to verify the theoretical properties of our algorithm, and compare it with the state-of-the-art methods.

we compare our method, denoted by AltGD, with two convex relaxation based methods for estimating LVGGM: (1) LogdetPPA [9,32] for solving log-determinant semidefinite programs, denoted by PPA, and (2) the alternating direction method of multipliers in [22,24], denoted by ADMM.


On the other hand, the specified components are highly informative, producing almost the same results as a classifier directly trained on a discriminative manner.

we observe some overfitting in the NORB dataset.


Clean data and irrelevant or even corrupted information are often mixed together, which motivates us to consider the high-dimensional, big but dirty data problem.

we study the subspace clustering problem in this setting.


Train II") and compare it with our method intensively (see supplementary materials), considering there exists trade-offs between accuracies on benign and adversarial examples.

a combined method is also evaluated to testify our previous claim of orthogonality.


we shall add back such dependency and as well as support the existence of a proximal term.

(to minimize ψ(x) + f (x) where ψ(x) is a proper convex simple function.)


As a result, even though the ultimate objective may be to maximise total reward, we need to use objective functions that are closer in spirit to the likelihoods common in probabilistic unsupervised learning.

we consider functions of the form In this expression, the two log probabilities reflect the accuracy of stimulus representation, and of value prediction, respectively.


When BN is applied after a linear layer, it is well known that the output is invariant to the channel weight vector norm.

denoting a channel weight vector with w and ŵ = w/ w 2 , channel input as x and BN for batch-norm, we have This invariance to the weight vector norm means that a BN applied after a layer renders its norm irrelevant to the inputs of consecutive layers.


Analyzing the above procedures in this alternative model yields several interesting details.

the natural modification of CAL produces a method that (in the realizable case) makes the same number of label requests as before, except that now it makes zero mistakes, since CAL will request a label if there is any uncertainty about its label.


In the most general formulation, the constraints, Q : R IK ×R KJ → R NQ and R : R IK ×R KJ → R NR , are biaffine maps, that define N Q inequality and N R equality constraints jointly in a and b.

each inequality constraint has the form By rearranging terms and combining the N Q constraints in matrix notation, we may write from which it is clear that the constraints are linear in a.


The result says that, subject to widely believed conjectures in computational complexity, no efficient algorithm exists for approximating the partition function of certain hard-core models.

recall that the hard-core model with fugacity is given by (2.1) with Assuming NP 6 = RP, there exists no FPRAS for computing the partition function of the hard-core model with fugacity on regular graphs of degree d. no FPRAS exists when = 1 and d 5.


For the problems on which PBCMC was superior, DWM's best results were not obtained using the most reactive parameterization.

simply using an even more reactive parameterization of DWM did not improve performance on these problems.


Thus, it is interesting to compare the recovered subspace to models of rapid gain control.

victor [15] proposed a retinal gain model in which the gain signal consists of time-delayed copies of the excitatory kernel.


Unfortunately, if we now wish to extend DP mixtures to groups of clustering problems, the assumption that G 0 is smooth conflicts with the goal of sharing clusters among groups.

even if each group shares the same underlying base distribution G 0 , the smoothness of G 0 implies that they will generate distinct cluster parameters (with probability one).


We subsample the raw data from [7] to evaluate the ability to find the trend from smaller samples.


Unsupervised feature learning algorithms have recently attracted much attention, with the promise of letting the data guide the discovery of good representations.

unsupervised feature learning is an important component of many Deep Learning algorithms (Bengio, 2009), such as those based on auto-encoders (Bengio et al., 2007) and Restricted Boltzmann Machines or RBMs (Hinton et al., 2006).


This gives the variational lower bound L(q): where H[•] denotes the Shannon entropy.

from inspecting the VB lower bound ( 12) and ( 6) we arrive at the following induced factorizations without forcing further factorization upon (11): the approximate posterior on assignment matrices factorizes across time; and the approximate posterior on latent states factorizes across tracks.


Consider the support set J t = S t−1 ∪ supp ∇F (x t−1 ), r . We have for any 0 < θ ≤ 1/ρ + , where ν = √ s − r + 2.

picking θ = 1/ρ + gives Note that the lemma also applies to the two-stage thresholding algorithms (e.g., CoSaMP [15]) whose first step is expanding the support set.


In the second set of experiments, we evaluate the performance of the coordinate-update mfNMF algorithm based on the multiplicative updating algorithm of the SMS problem (Section 4.1).

we consider the mfNMF problem that approximates a randomly generated target nonnegative matrix V of dimension m×n with the product of three stochastic factors, W 1 (m×m ), W 2 (m ×n ), and W 3 (n × n).


with 1) the optimisation over F φ (X ) ⊕ P as per Section 6.1, and 2) the optimisation over a subspace of F φ (X ) as per Section 6.2.

we use the second method if the data is linearly separable, otherwise we use the first.


As generator at stage I, we adopt a U-Net-like architecture [20], i.e., convolutional autoencoder with skip connections as is shown in Figure 2.

we first use several stacked convolutional layers to integrate I A and P B from small local neighborhoods to larger ones so that appearance information can be integrated and transferred to neighboring body parts.


Despite these specific choices, we expect the model results to be general to the entire class of sparse coding models.

the choice of comparing neural responses with Monte Carlo samples from the model's posterior distribution was taken in agreement with experimental results that report high neural variability.


Despite the apparent complexity of this objective, we find that approximate inference using an alternating minimization algorithm, like in [15], works well in practice.

we create explicit auxiliary variables w i (n) for the coefficients, and solve the following modified optimization problem: Note that the second term above forces coefficients of y(n) to be equal to the corresponding auxiliary variables w i (n), as β → ∞.


Here we depart from the view that LGN aims to minimize receptive clutter by selecting a single output from a population of scale and gain specific contrast filters [12].

in order to determine contrast at an image location, we apply the smallest filter with boosted contrast output above what can be expected to be noise for that particular filter.


The labels of outliers are generated by y o = sign( −β, x o ).

outliers follow the model having opposite sign as inliers, which according to our experiment, is the most adversarial outlier model.


When the algorithm knows the size of each cluster, we modify the definition of ē by replacing the terms |A (t−1) j |'s with the actual sizes |A j |'s.

when the algorithm knows that the clusters are of equal size, we obtain the following refinement rule: we put user i to A (t) j * , where j * = arg max j∈[k] c 1 • e {i}, A (t−1) j + Π i (u (j) ) for some hyperparameter c 1 > 0.


In both settings, we allow our algorithm to be improper, that is, the learned functions need not belong to F. We compare the sample complexity of our algorithms to their PAC counterparts in the realizable setting.

in the traditional realizable PAC setting, m ,δ denotes the number of samples needed for ( , δ)-learning F. m ,δ is the total number of samples drawn from a realizable distribution D, such that, with probability 1 − δ, any classifier f ∈ F that is consistent with the sample set satisfies err D (f ) ≤ .


Assume for instance that θ i = 0 for all i ≥ 3.

it follows from (1) that for any x ∈ {−1, 0, 1} n , with H(x) defined as in the proof of Proposition 1: two parameters θ, θ ′ such that θ 1 θ 2 = θ ′ 1 θ ′ 2 and θ i = θ ′ i = 0 for all i ≥ 3 cannot be distinguished.


As all deep learning training methods are based on stochastic gradient descent, our Co-teaching works in a mini-batch manner.

we maintain two networks f (with parameter w f ) and g (with parameter w g ).


The intrinsical ill-posed nature of the problem is circumvented using Bayesian regularization methods.

working under the framework of Gaussian regression [18], in [16] the system impulse response is modeled as a Gaussian process whose autocovariance is the so called stable spline kernel that includes the BIBO stability constraint.


5 The motivation for the specific choice of the problem formulation (6-8) is that projecting S onto § or § f can be done inexpensively.

the first projection step S ¡ 6 B 3 C ¢¡ §£ 8 ¢ B DB S ¥¤ E S B B f ¦ ¡ S 4¤ @) 5 § ¡ involves minimizing a quadratic objective subject to a single linear constraint; the solution to this is easily found by solving (in V !


In this paper we give algorithmic evidence that this similarity is likely not a coincidence.

we give a simple, general, and efficient procedure for transforming a black-box solver for sparse linear regression to an algorithm for SPCA.


Of course, each of these prediction rules minimizes a different prediction error measure.

median prediction minimizes the mean absolute error and is the prediction rule we use in our experiments.