This means that maximizing the lower bound is roughly equal to minimizing the loss function of a deep neural network (see Figure 1 in supplementary), except for different regularizers.

the prior in VAE only imposes a regularizer in encoder or generative model, while L 2 penalty for all parameters is always considered in deep neural nets.


The catching agent has to trade-off success (i.e., catching the ball) with effort.

he aims at maximizing the probability of catching the ball with minimal effort.


Hyperkernels achieve this by weighting the integrand of ( 5) in relation to what is "on the other side" of the hyperkernel.

we define convolution hyperkernels by setting g z (x, x ) = r(x, z) r(x , z) in (4) for some r : X × X → R. By (3), the resulting hyperkernel always satisfies the conditions of Definition 1.


Nevertheless, if a sufficient number of edges are missing in the graph, identification sometimes is possible even if R variables are dependent, and monotonicity and MAR do not hold.

techniques from causal inference have been using to derive complex identification results in this setting [7,17].


Interestingly, the above-mentioned "nestedness property" is also fulfilled by minimizers of a parametric submodular minimization problem [12].

it holds for the monotonic max-flow method [25], which is also widely used for obtaining diverse labelings in practice [7,20,19].


This variational family is indexed by parameters η, which are fit to minimize the KL divergence between the variational and true posteriors.

we let {q η } be the family of fully factorized distributions where we parameterize the variational posterior with η = {(x u , σ x ), (z u , σ zu ), (ã, σ a ), ( b, σ b )}.


Our analysis can be modified to give an improved dependence on the errors under this stronger assumption.

the terms in √ ε i disappear from the expressions of A k , A k and A k appearing in the propositions, leading to the optimal convergence rate with a slower decay of ε i .


Let n > m, M ⊂ R N be a Riemannian n-dimensional submanifold.

then for any (dimensionality reduction) map f : M → R m and any open set W ⊂ M, f cannot achieve both perfect precision and perfect recall on W . In this section, we provide a quantitative analysis for the imperfection of f . we prove an upper bound for the precision of a continuous DR map f (thus f achieves perfect recall).


Similar to AdaBoost, DirectBoost works by sequentially running an iterative greedy coordinate descent algorithm, each time directly minimizing true empirical classification error (1) instead of a weighted empirical classification error in AdaBoost.

for each iteration, only the parameter of a weak classifier that leads to the most significant true classification error reduction is updated, while the weights of all other weak classifiers are kept unchanged.


Second, both our reasoning module and visual scene representation are light-weighted, requiring minimal computational and memory cost.

our compact structural image representation requires much less storage during reasoning, reducing the memory cost by 99% compared with other state-of-the-art algorithms.


In this paper, we study a variant of the standard multi-armed bandit problem where the player is temporarily blinded each time he switches arms.

if the player's current choice is different than his choice on the previous round then we say that he has switched arms, he incurs the loss as before, but he does not observe this loss, or any other feedback.


If m = 2, we can obtain explicit expressions for the reward envelopes and for the algorithm.

we have two classifiers, and we assume that the outputs of these classifiers lie in the set ≥ λ such that γ 1 > γ, γ 2 < γ, and λ ≥ 0.


These results suggest that human participants may rely on visual features of intermediate complexity and that the complexity of visual representations afforded by modern deep network models may exceed those used by human participants during rapid categorization.

the top layers (final convolutional and fully connected), while showing an improvement in accuracy, no longer maximize correlation with human data.


The resulting algorithm slightly differs from the Adaptive Natural Gradient algorithm (Amari, Park, and Fukumizu, 1998).

there is little need to adjust a learning rate parameter in the Gauss-Newton approach.


3 An Accelerated Solution to the Fenchel Game via Optimism We are going to analyze more closely the use of Algorithm 1, with the help of Theorem 1, to establish a fast method to compute an approximate equilibrium of the Fenchel Game.

we will establish an approximation factor of O(1/T 2 ) after T iterations, and we recall that this leads to a O(1/T 2 ) algorithm for our primary goal of solving min x∈K f (x).


Under the stick-breaking construction, two hidden variables for each Markov chain are introduced, representing the transition probabilities between the active and inactive states.

we define a m = p(s tm = 1|s (t−1)m = 0) as the transition probability from inactive to active, and b m = p(s tm = 1|s (t−1)m = 1) as the selftransition probability of the active state of the m-th chain.


In this paper, we present quasi-Newton methods for MCMC that are based on approximations to the Hessian from first-order information.

we present a Hamiltonian Monte Carlo algorithm in which the variance of the momentum variables is based on a BFGS approximation.


N (0, 1)), we obtain which tends to be much simpler to solve than (2) as the first two terms are separable in the components of x.

for the choices of P mentioned above, we obtain closed form solutions: where + denotes the positive part and |η (s) | is the sth largest entry in η in absolute magnitude and γ * = min{γ ≥ 0 : the estimators are hardrespectively soft-thresholded versions of η j = A ⊤ j y/m which are essentially equal to the univariate (or marginal) regression coefficients θ j = A ⊤ j y/ A j 2 2 in the sense that η j = θ j (1 + O P (m −1 )), j ∈ [n], hence the term "marginal regression".


More recently, Cook (2007) and Artemiou and Li (2009) argued that principal component regression has potential to play a more important role.

letting u j be the j-th leading eigenvector of the sample covariance matrix Σ of x 1 , .


For example, span{v Proj U y is defined as the projection of y onto subspace U .


In general, this neighborhood contains points from M l as well as other manifolds.

we assume that for all i there exists ≥ 0 such that the nonzero entries of the sparsest solution of corresponds to the d l + 1 neighbors of x i from M l . among all affine subspaces spanned by subsets of the points {x j } j∈Ni and passing near x i up to error, the one of lowest dimension has dimension d l and it is spanned by the d l + 1 neighbors of x i from M l . In the limiting case of densely sampled data, this affine subspace coincides with the d l -dimensional tangent space of M l at x i .


We therefore revert to approximate techniques for computing the marginal probabilities and the partition function Z.

we will undertake a divergence minimization approach, which will yield both an estimate of log Z and approximate marginals.


It mirrors Algorithm 1, with only difference being at the lifted conditioning step.

in lifted conditioning step, instead of going over all possible branches corresponding to all possible counting assignments, the algorithm uses a representative branch which has a variable associated for the corresponding counting assignment.


After constructing a convolutional deep belief network, we perform (approximate) inference of the whole network in a feedforward (bottom-up) manner.

letting I(h k ij ) b k + ( W k * v) ij , we can infer the pooling unit activations as a softmax function: Given a set of poorly aligned face images, our goal is to iteratively transform each image to reduce the total entropy over the pooling layer outputs of a CDBN applied to each of the images.


At a high level, our algorithm works by first approximating the response of an LDS by an autoregressive model of order (𝜏, 𝜏 ), then refining the approximation using wave-filters with a phase component.

the blocks of 𝑀 and 𝑁 corresponding to filter index ℎ and phase index 𝑝 specify the linear dependence of 𝑦 𝑡 on a certain convolution of the input time series, whose kernel is the pointwise product of 𝜑 ℎ and a sinusoid with period 𝑊/𝑝.


The variable x represents model parameters and the role of f is to ensure that the estimated parameters fit some observed data.

f is often a large sum of functions and each term f i (x) measures the fit between x and a data point indexed by i.


Furthermore, we also present a unified view of several well-known policy evaluation methods.

we show that the same Galerkin method can be used to derive Least-Squares Temporal Difference learning, Kernelized Temporal Difference learning, and a discrete-state Dynamic Programming solution, as well as our proposed method.


In the first example, most of the components of this vector are zero, but the first and the last few elements follow a discrete convex trend.


In [6,21], the authors model user clicks by a regularized logistic regression model over observed context features and make decisions by applying Thompson sampling over the learnt model.

no click is treated as negative feedback.


In fact, the additive noise model assumes that all categories of the variables are placed in the "right" order; furthermore, if X → Y holds according to the additive noise model, then for any observation (x, y), there exists a function g(X) such that the noise E = y − g(x) is independent of X, i.e., it has the same distribution for different values of x.

the conditional distribution P (Y |X = x) always has the same shape for different values of x, after being properly shifted according to g(x).


We can rewrite the objective While the combinatorial search over V can now be explicitly computed and optimized using the kernel matrix K n , it still remains highly inefficient to do so.

simply constructing and storing K n takes O(n 2 ) time and space and does not scale to large datasets.


We show that the behavior of AMP is qualitatively similar to the one of naive power iteration.


Next, it estimates the inverse Hadamard transform of the compressed and noisy signal of each user in I ,j .

for each i ∈ I ,j , it computes y i W ri,c which can be described as a multiplication between y i e ri (where e ri is the indicator vector with 1 at the r i -th position) and the scaled Hadamard matrix W, followed by selecting the c-th entry of the resulting vector.


In the remainder of this section, we highlight some major properties of the optimization problem (2).

we show that its solution policy may be purely randomized, we compute a bound on the performance loss of its solution policy w.r.t.


To address these challenges, we adopt an alternating minimization scheme that iteratively fixes one of d and X and minimizes with respect to the other.

we propose an extremely efficient optimization algorithm by effectively exploring the sparse structure of the tensor P and low-rank structure of the matrix X.


We now provide a novel algorithm based on the LP of the previous section.

we show how we can solve this LP exactly in closed form, without explicitly enumerating the exponentially many constraints.


In general, the estimation problem with two different distributions Pr(x, y) and Pr ′ (x, y) is unsolvable, as the two terms could be arbitrarily far apart.

for arbitrary Pr(y|x) and Pr ′ (y|x), there is no way we could infer a good estimator based on the training sample.


The key point is that our algorithm provides provable upper and lower bound on log Z for any graph; the approximation guarantee and computation time depends on the property of DECOMP.


Clearly, the accuracy of TrackIPTree is a tremendous improvement on both sets of apriori fixed clusters -for any choice of coverage on legitimate IPs, the accuracy of spam IPs by TrackIPTree is far higher than the apriori fixed clusters, even by as much as a factor of 2.5.

note that when the coverage required on legitimate IPs is 95%, TrackIPTree achieves 95% accuracy in classifying spam on both data sets, compared to the 35 − 45% achieved by the other clusters.


For any Each observed variable x ∈ V obs is modeled as random vector in R d , and each hidden variable h ∈ V hid as a random vector in R k . The joint distribution over all the variables V T := V obs ∪ V hid is assumed satisfy conditional independence properties specified by the tree structure over the variables.

for any disjoint subsets The class of models considered are specified by the following structural and distributional assumptions.


This allows us to exploit ideas both from [24] and from [27].

we show that the "topic" modeling problem exhibits a rather simple algebraic solution, where only two SVDs suffice for parameter estimation.


With ω t ∼ N (0, H t ) as an estimate of the system noise, C can be decomposed as C t = A t Σ t A T t + H t . Note that while the transition dynamics in our generative model operates on the inferred latent space, it takes untransformed controls into account.

we aim to learn a latent space such that the transition dynamics in z linearizes the non-linear observed dynamics in x and is locally linear in the applied controls u. Reconstruction of an image from z t is performed by passing the sample through multiple hidden layers of a decoding neural network which computes the mean p t of the generative Bernoulli distribution 1 P θ (X|Z) as where h dec θ (z t ) ∈ R n d is the response of the last hidden layer in the decoding network.


While non-degenerate covariance functions correspond to infinite bases, they nevertheless can correspond to strong assumptions regarding likely functions.

the automatic relevance determination (ARD) squared exponential kernel is often a default choice for Gaussian process regression.


To realize this idea, we advocate a novel input layer for deep neural networks that takes a topological signature (in our case, a persistence diagram), and computes a parametrized projection that can be learned during network training.

this layer is designed such that its output is stable with respect to the 1-Wasserstein distance (similar to [27] or [1]).


We now propose a variant that takes this uncertainty into account by choosing different scaling factors for each perturbation.

we will use the simple choice β t,i = C 2 N t,i with some constant C > 0 that will be specified later.


Define the contamination link function Given a matrix X, define the corresponding contaminated matrix Y with entries Now observe that each of the primitive modes above corresponds to a different choice of random variables A and B, as shown in Table 1.

each of the primitive modes is described by a different assignment to A and B.


This is strictly worse than the complexity O( max{L 0 , M 2 /(2 )} 1/ ) of our approach.

we have managed to remove the secondary term in the complexity bound of S-APG.


Remark A potentially more efficient approximate algorithm could be derived by noting the relations between volume sampling and DVS.

we add a small perturbation to DVS as in Equation 2.1 to transform it into a volume sampling problem, and apply random projection for more efficient volume sampling as in [17].


The key idea of our construction is to sample the independent set of random variables by instantiating this heap from root to leaves.

we will first sample the root node, which is the global max and argmax, then we will recurse, sampling the root's two children conditional upon the root.


In the proof of Theorem 3 we use the fact that nV n (h) satisfies the self-bounding property [14].

for any λ > 0: (see, for example, [9, Theorem 10]).


In informal terms, a learning algorithm is called algorithmically stable if the impact of a single training example on the probability distribution of the final hypothesis always vanishes at the limit of large training sets.

the inference process never depends heavily on any single training example.


6 since it involves taking expectations with respect to ψ, rather than θ: since ψ is enforced to be fast-mixing, these expectations can be approximated by sampling.

suppose that one has generated a set of samples x 1 , ..., x K using the current parameters ψ.


In online isotonic regression, the natural goal is to predict the incoming data points as well as the best isotonic function in hindsight.

for time steps t = 1, . . . , T , the learner observes an instance x i ∈ R, makes a prediction y i of the true label y i , which is assumed to lie in [0, 1].


For each arm, a sample at the time t corresponds to the number of network activity events for 4 consecutive seconds.

the network events are the starting times of packet flows.


To translate the evaluation into an expert response we borrow from previous work [6].

we assume the expert response is given by the indicator I(f * (ξ i , ξ j , θ * ) > ) where ∼ N (0, σ 2 r ).


Our local randomizer can thought of as a transformed, compressed (via sampling), and randomized version of the count sketch [8].


• When all the functions are self-concordant, we establish a similar guarantee if both the gradient and Hessian of the function can be queried multiple times.

we propose to apply the damped Newton method multiple times in each round, and prove an O(min(P * T , S * T )) bound of the dynamic regret under appropriate conditions.


The main contributions of this paper lie in the theoretical analysis of the proposed method.

we show: 1) the proposed method can improve the estimation bound of the standard Dantzig selector approximately from Cs 1/p √ log mσ to C(s − N ) 1/p √ log mσ where the value N depends on the number of large entries in β * ; 2) when N = s, the proposed algorithm can achieve the oracle solution with a high probability; 3) with a high probability, the proposed method can select the same number of correct features under a milder condition than the standard Dantzig selector method.


Kernel-based (and other nonparametric) methods detect nonlinear dependence in random variables without assuming specific relationships among them.

the resulting independence measures attain minimum values when random variables are independent.


The objective of an algorithm in this setting is to maintain a set S 2 I which approximately maximizes f , and to do so with as little memory as possible.

we are interested in algorithms whose memory requirement does not depend on the size of the ground set N , which means that they cannot keep in their memory all the elements that have arrived so far.


This is the main idea that has been explored in [19,25,18].

under the assumption that the messages belong to a linear subspace, [19,25] propose associative memories that can store exponential number of messages while tolerating at most constant number of errors.


Given parameter n ∈ N and sample access to a distribution over N, there exists an algorithm (Algorithm 1) for T(PBD n , ) which takes 2 samples, and runs in time Note that the sample complexity of our algorithm is n 1/4 • Õ(1/ 2 ), matching the informationtheoretic lower bound up to a logarithmic factor in 1/ .

our algorithm does not incur the extraneous Ω(1/ 6 ) term of [1].


The key construction is a variational representation of M-estimation that can express a number of standard robust (and non-robust) methods in a common framework.

consider the following adaptive form of loss function where r is a residual value, is a closed convex base loss, η is an adaptive weight on the base loss, and ψ is a convex auxiliary function.


Essentially, we are searching for latent factors so that, conditioned on these factors, the correlations in the data are minimized (as measured by multivariate mutual information).

we look for the simplest explanation that accounts for the most correlations in the data.


• We numerically demonstrate the effectiveness and robustness of our algorithms in recovering the minimizer of ridge logistic regression on several real datasets (Figures 1 and 2).

our algorithms are at least twice as fast as Newton's methods in finding a high-precision solution while other methods converge slowly.


As the true label signature can be a tiny fraction of a given sequential data point, we break the data point itself into multiple overlapping instances such that at least one of the instance cover the true label signature (see Figure 1).

given a sequential data point X i , we construct a bag represents the τ -th instance (or sub-window) of data points.


Ideally, we would expect the decision boundaries of our models to be close to the true distribution, i.e., the unit ball.

training data 1 Hidden Layer 2 Hidden Layers 3 Hidden Layers 5 Hidden Layers  the non-trivial level set of a narrow fully connected network is always unbounded.


The "correctness" of the policy depends on the extent to which the traffic violation endangers nearby humans and whether any element of the "correctness" specification explicitly forbids such actions.

"correctness" as a binary output is a concept defined by its exceptions, many elements of which are subject to individual valuations [10].


Then, as We can now derive the dual of ESP-design (we consider only which admits as dual We easily show that H ⪰ 0 and that g reaches its maximum on S ++ m for A such that ∇g = 0.

h and A are co-diagonalizable, with Λ Diag(e ℓ−1 (Λ (i) ))Λ = Diag(h 1 , .


Similar to [22], we connect the image and question by calculating the similarity between image and question features at all pairs of image-locations and question-locations.

given an image feature map V ∈ R d×N , and the question representation where W b ∈ R d×d contains the weights.


We show that achieving the guarantee of (1) significantly more efficiently than the work of [MM16,MW17] is likely very difficult.

we prove that for a wide class of kernels, the kernel low-rank approximation problem is as hard as multiplying the input A 2 R n⇥d by an arbitrary C 2 R d⇥k .


for each k-bit string y := y 1 • • • y k , let ⇢ y be an nqubit mixed state such that for each i 2 {1, 2, . . . , k}, there is a two-outcome measurement E 0 that depends only on i and the prefix v := y 1 y 2 • • • y i 1 , and has the following properties (iii) if y i = 0 then Tr(E 0 ⇢ y )  a v ", and (iv) if y i = 1 then Tr(E 0 ⇢ y ) a v + ", where " 2 (0, 1/2] and a v 2 [0, 1] is a "pivot point" associated with the prefix v. Then k = O n/" 2 .


This example is adapted from [12] and [18], which considers the problem of power dispatch.

at each time step t, we try to minimize the power generation cost c i (x t,i ) for each generator i while maintaining the power balance n i=1 x t,i = d t , where d t is the power demand at time t. Also, each power generator produces an emission level E i (x t,i ).


However, since the joint probability of (X, Y ) is unknown but we are provided with an independent and identically distributed (iid) sample of observations {(x i , y i )} n i=1 , we resort to minimizing the empirical risk: , within a class H ⊂ (R) X of functions, calibrated in order to overcome the shift from the true risk to the empirical one.

when H has the form: R being a convex function and c > 0 a constant, Takeuchi et al.


We are interested in maximizing some reward function r : We will look at the Expected Reward Maximization algorithm on S, with some fixed tie-breaking rule.

if we let then ERM is defined as: where ties are broken based on some pre-defined manner.


In this paper, we analyze the multiple source domain adaptation problem and propose an adversarial learning strategy based on our theoretical results.

we give new generalization bounds for both classification and regression problems under domain adaptation when there are multiple source domains with labeled instances and one target domain with unlabeled instances.


Our experiments suggest that both steps of Autowarp -first, learning latent representations using sequence-to-sequence autoencoders, and second, finding a warping distance that agrees with the latent representation -are important to learning a good similarity metric.

we carried out experiments with deeper autoencoders to determine if increasing the capacity of the autoencoders would allow the autoencoder alone to learn a similarity metric.


Steps (2) and (3) of the method update x, decreasing the energy (1) by first taking a gradient descent step with respect to the inner product term in (1) and then taking a "backward" or proximal step involving f . In steps (4) and ( 5), the energy (1) is increased by first marching up the gradient of the inner product term with respect to y, and then a backward step is taken with respect to g. PDHG has been analyzed in the case of constant stepsizes, ⌧ k = ⌧ and k = .

it is known to converge as long as ⌧ < 1/⇢(A T A) [4,5,11].


On the other hand, we often want to design agents that make decisions adaptively based on the data they collect.

we would like the number of samples itself to be a random variable.


Core to our work is a new and arguably more natural analysis of the Sinkhorn iteration algorithm, which we show converges in a number of iterations independent of the dimension n of the matrix to balance.

this analysis directly suggests a greedy variant of Sinkhorn iteration that also provably runs in near-linear time and significantly outperforms the classical algorithm in practice.


The above refinement is obtained by mimicking the analysis of Hedge forecasters for small losses (see, e.g., [3,Section 2.4]).

one should substitute Lemma 1 with the following lemma in the analysis carried out in Section 5; its proof follows from the mere replacement of Hoeffding's inequality by [3,Lemma A.3], which states that for all η ∈ R and for all random variable X taking values in [0, 1], one has ln E[e −ηX ] (e −η − 1)EX.


At each iteration l, our best approximation of π is given by the pushforward density p l = (T l • • • • • T 1 ) * p 0 .

the SVGD algorithm then seeks a transport map T l+1 = I + Q that further decreases the KL divergence between (T l+1 ) * p l and π, for an appropriate choice of Q ∈ H d . the SVGD algorithm seeks a map Q ∈ H d such that where 0(x) = 0 denotes the zero map.


We would like to point out that our modeling assumptions are violated in this experiment.

w t (e) are correlated across items e because the users do not rate movies independently.


Using the notion of restrictions, we can give names to several types of adversaries that we will study in this paper.

(1) A worst-case adversary is defined by vacuous restrictions P t (x 1:t−1 ) = P. any mixed strategy is available to the adversary, including any deterministic point distribution.


Thus, there is some chance of popping a lower-priority action, to find out if it is useful and should be given higher-priority.

we use Boltzmann exploration to construct a stochastic policy with a Gibbs distribution.


Let D be an isotropic s-concave distribution over R n . For any w * and r > 0, the disagreement coefficient is ) .


In contrast to these fully-supervised works, we address the task without the guidance of temporal segments during training.

instead of detecting all event using the one-to-many-mapping event detector [23,22], we try to localize them one by one using our sentence localizer and caption generator.


In this section, we provide guarantees for CSC 1 (β, x) to return no false positives, in terms of the value of β.

we guarantee this by examining a lower bound β L for β in CSC 1 (β, x).


The optimal solution to the LP: max 3 The Known Objective Problem In this section, we focus on the Known Objective Problem where the coefficients of the objective function c are fixed and known to the learner but the feasible region P (t) on day t is unknown and changing.

p (t) is the intersection of a fixed and unknown polytope .


By combining Theorem 2.1 with previous results [3,4,5], we immediately obtain bounds on the accuracy of the matrix Dantzig selector (4) and the matrix Lasso (5).

for the first time we can show universal recovery of low-rank matrices via Pauli measurements, and near-optimal bounds on the accuracy of the reconstruction when the data is noisy [5].


The edges in our θ-MRF not only connect spatial neighbors but also semantic neighbors.

if two tasks are highly correlated, their parameters given to the same image context should be similar.


When the training process went on, we gradually increased the percentage of monolingual sentences in the mini batch, until no bilingual data were used at all.

we tested two settings in our experiments: • In the first setting (referred to Large), we used all the 12M bilingual sentences pairs during the soft-landing process.


Since SNIPER is trained on 512x512 size chips, it can reap the benefits of a large batch size and training with batch-normalization on a single GPU node.

we can use a batch size of 20 per GPU (leading to a total batch size of 160), even with a ResNet-101 based Faster-RCNN detector.


Separate motion systems have been proposed to explain human perception in scenarios like this example.

braddick [1] proposed that there is a short-range motion system which is responsible for perceiving movements with relatively small displacements (e.g., the player's movement), and a long-range motion system which perceives motion with large displacements (e.g., the flying baseball), which is sometimes called apparent motion.


Then the output can be expressed in the form θT ψ(x[∪s i ]).

the lattice preserves the functional form of its inputs, changing only the values of the coefficients θ and the linear interpolation weights ψ.


In this paper, we provide a solution to this problem and develop an algorithm that enables agents to safely and autonomously explore unknown environments.

we consider the problem of exploring a Markov decision process (MDP), where it is a priori unknown which state-action pairs are safe.


A large part of reinforcement learning is also aimed at studying sequential decision making problems.

one can define a Markov Decision Process (MDP) by a set of states equipped with transition distributions, a set of actions and a set of reward or loss distributions associated with each state action pair.


This problem is challenging as the difference of P and Q is encoded at a much smaller length scale compared to the global structure (Gretton et al., 2012b).

the eigenvalue ratio for the covariance of each Gaussian distribution is 2.0 in P , and 1.0 in Q.


Before we had, for every word type, a variational distribution over the hidden variables corresponding to that word type: We now introduce a variational distribution including new hidden variables s i for unknown words.

for each possible ambiguity class s i of an unknown word w i we introduce variational parameters specific to that ambiguity class.


The relation between walks and the geometric series (10) is that the entries of R l correspond to walk-sums over l-length walks from i to j in the graph, i.e., (R l ) i,j = ρ(W i l →j ).

hence, the variance σ 2 i ≡ P i,i of variable i is the walk-sum taken over the set W i→i of self-return walks that begin and end at i (defined so that (i) ∈ W i→i ).


As noted by [16], people are capable of recognizing that individual stimuli possess an arbitrarily large number of characteristics, but in any particular context will make judgments using only a finite, usually small number of properties that form part of our current mental representation.

by moving to a Bayesian nonparametric form, we are able to bring the ADCLUS model closer to the kinds of assumptions that are made by psychological theories.


The model, though, extends naturally to directed, acyclic graphs as well.

our graphs C n,p with parameters n and p are chosen according to the following distribution: (1) Pick a random permutation σ of n vertices.


In the Appendix, we prove that this leads to unconditional winner-take-all behavior in the W layer.

no more than one W-neuron can be active at a stable steady state.


. . , z n ) denotes the Cartesian coordinates on R n and d n z = dz 1 . . . dz n denotes the volume form on R n . In particular, Before proving this theorem, we prove some results on the relation between u p (q) and d p (q).

there exist δ up > 0 and M up > 0 such that for all q with d p (q) ≤ δ up , we have, lim q→p up(q) dp(q) = 1.


. . , f d ) be a member of the transelliptical graphical model P(G).

if X is also nonparanormal, the graph G encodes the conditional independence relationship of X (the distribution of X is Markov to G).


Our framework allows natural incorporation of partial labels for semi-supervised learning.

we instantiate the framework to unsupervised, multi-class kernelized logistic regression.


The point of this experiment is to show that the Lévy process with Laplacian basis functions forms a natural prior over spectral densities.

samples from this prior will typically look like the types of spectra that occur in practice.


The first layer has a skip connection into the third layer which is stacked onto the input.

if the first and second layer have k 1 and k 2 output channels, the third layer has k 1 + k 2 input channels.


Among the priors that enforce sparsity, the horseshoe has some attractive properties that are very convenient for the scenario described [3].

this prior has heavy tails, to model coefficients that significantly differ from zero, and an infinitely tall spike at the origin, to favor coefficients that take negligible values.


Similarly, KOPLS provides optimal projections for linear multi-regression in feature space.

the solution to (2) also minimizes the sum of squares of the residuals of the approximation of the label matrix: where • F denotes the Frobenius norm of a matrix and B is the optimal regression matrix.


The distributed coding scheme has important consequences for interpreting neural responses: since knowledge about the underlying distribution is spread across the population, the activity of single cells does not reflect the underlying computation in any obvious way.

although the network did reproduce various properties of single neuron and pairs of neuron responses seen experimentally, we found that their modulation with uncertainty provides relatively limited information about the underlying probabilistic computation.


Previous work demonstrated an equivalence between linear value function approximation and linear model approximation [20,21,4], as well as the relationship between errors in the linear model and the Bellman error for the linear fixed point [4].

low error in the linear model could imply low Bellman error in the linear fixed point approximation.


Therefore, even though L[x, y] itself is learned through knowledge of co-occurence of chromaticities in natural images, estimation of the illuminant during inference is still achieved through a simple aggregation of per-pixel distributions.

we set the entries of L[x, y] to minimize a cost function C over a set of training images: where mt is the true illuminant chromaticity of the t th training image, and p t i is computed from the observed colors v t (n) using (4).


Qualitatively quite different from both epitomes and previous stel models, the stel epitome is a model flexible enough to be applied to a very diverse set of images.

we are interested in datasets that might represent well a human's total visual input over a longer period of time, and so we captured two weeks worth of SenseCam images, taken at a frequency of roughly one image every 20 seconds during all waking hours of a human subject over a period of two weeks (www.research.microsoft.com/∼jojic/aihs).


In the completeness case, we claim that these steps will reject with at least 99% probability unless F r (p) = O(C (n 1−r + m −r )).

if F r (p) ≥ KC (n 1−r + m −r ), then m p r ≥ 1, and therefore, by Claim 2.4 we have that E[ γ r ] = F r (p) and Var( γ r ) = O(F r (p) 2 /K 2 ).


In this paper, we report our results using the general bound ) where ↵ can be any constant form the interval [0.5, 1].

the observation that the optimal values of the expected loss and empirical loss are within a V n distance of each other implies that there is no gain in improving the optimization error of minimizing L n beyond the constant V n . if we find an approximate solution w n such that the optimization error is bounded by , then finding a more accurate solution to reduce the optimization error is not beneficial since the overall error, i.e., the sum of estimation and optimization errors, does not become smaller than V n . Throughout the paper we say that w n solves the ERM problem in (2) to within its statistical accuracy if it satisfies We can further leverage the estimation error to add a regularization term of the form (cV n /2)kwk 2 to the empirical loss to ensure that the problem is strongly convex.


Finally, R t (w t ) is given by w A + b α t , hence it satisfies the same update relations.

the fact that w Aα = λ w 2 means that the only quantity we need to cache is b α t as an auxiliary variable r t in order to compute γ t efficiently.


However, a number of experimental results reveal suboptimal behavior and may be explained by assuming that once an observer has committed to a particular hypothesis, subsequent evaluation is based on that hypothesis alone.

observers sacrifice optimality in order to ensure self-consistency.


Any two nodes i and j are connected (independent of other edges) with probability p if they belong to the same cluster and with probability q otherwise.

pr(ll|ll) = p, Pr(lm|ll) = 1 − p, Pr(ll|lm) = q and Pr(lm|lm) = 1 − q.


Note that each node y i will be either P 1 -active or P * -active.

y i is P * active iff y i is in some φ ∈ Φ * .


Because these three parameters largely influence the assessment, we consider their proposal more as a conceptual framework than as a concrete model order estimation procedure.

the predictor can be chosen independent of the clustering algorithm which can lead to unreliable results (see section 3).


For almost all problems in Bayesian analysis (and certainly the one considered in this paper), the densities are only known up to a normalizing constant.

only f (x) and f (x) are known pointwise, where π(x) = f (x)/Z and π (x) = f (x)/Z .


Our strategy is to partition all the induced trees based on whether they contain the tree edge (k, j) or not.

define T F corresponds to the set of trees that do not contain edge (k, j) and T T corresponds to the set of trees that contain edge (k, j).


Our method outperforms the baseline and ODIN for all tested cases.

our method improves the TNR, i.e., the fraction of detected LSUN samples, compared to ODIN: 41.2% !


Note that leveraging variance reduction ideas in sEM is not straightforward, since sEM is not a stochastic gradient descent algorithm but rather a stochastic approximation [21] algorithm.

the proof techniques we utilize are different than those in stochastic gradient descent algorithms.


To prove convergence bounds for stochastic-gradient based methods, we usually require b g(w) to be an unbiased estimator of the full-batch gradient, and to have small variance Ekb g(w)k 2 , as this controls the speed of convergence.

to see this, suppose w ⇤ is a critical point of f , then we have the progress made by the algorithm at a single step is, in expectation, controlled by the term E[kb g(w k )k] 2 2 ; the smaller it is, the bigger the progress.


We assume Y is bounded, and admits an efficient polar operator.

for any vector d ∈ R h , min y∈Y d y is efficiently solvable.


Because Q( θ k+1 ; θ k ) involves terms log A k i,j and log π k i , in practice we bound A k i,j and π k i away from zero to ensure that Q does not blow up.

we assume a small positive constant θ min so that A k i,j ≥ θ min and π k i ≥ θ min .


Our main result in Theorem 1 extends directly to the case where H is an infinite-dimensional Hilbert space.


From ( 6), it is easy to see that the (E,E)-block of the Hessian is a diagonal matrix given by ( Using this diagonal block, we erase (V,E)-block and (E,V)-block of the Hessian.

we choose a square matrix X such that det X = 1 and After the computation given in the supplementary material, we see that From , it is easy to check that I N + D − Â = Y W , where Â and D is defined in (9) and W is a diagonal matrix defined by For the left equality, theorem 2 is used.


In this section, we show that the variational objective provides a new perspective on some wellknown peculiarities of human probabilistic judgment.

the ostensibly irrational nonlinear weighting of probabilities in risky choice emerges naturally from optimization of the variational objective under a natural assumption about the ecological distribution of rewards.


We use a rational model of preference learning, drawing on ideas from economics and computer science, to explain the behavior of children in several recent experiments.

we show how a simple econometric model can be extended to capture two-to four-year-olds' use of statistical information in inferring preferences, and their generalization of these preferences.


The functional embedding framework provides a new perspective on the pooling functions commonly used in convnets.

it provides a principled framework for learning the pooling operators at each stage of representation based on the underlying geometry of the data, rather than being imposed in a 2D topology a priori as was done previously to learn linearized representations from video [23].


Next, we will search amongst these models to identify those that also have large prior p(M ).

to limit the number of user studies required, we will use a model-based optimization approach [27] to identify which models M to evaluate.


The parameters of ADM and LADM are the same as those in [12] and [20], respectively.

for LADM the penalty is fixed at β = 2.5/ min(m, n), where m × n is the size of X.


When d = 1 the computational complexity of the partition sum is O(n).

the computational cost of the 3rd and 4th moments for a second order test statistic is O(n 3 ).


Our goal is to compare the fidelity of different samples approximating a common target distribution.

we seek to quantify the discrepancy between E Q and E P in a manner that (i) detects when a sequence of samples is converging to the target, (ii) detects when a sequence of samples is not converging to the target, and (iii) is computationally feasible.


Denote by A its adjacency matrix, which is an n by n symmetric matrix with binary entries.

there is an observed edge between nodes i and j, and A ij = 0 otherwise.


However, the model of ( 1) is, by itself, descriptive rather than prescriptive.

the parameters modelled (those determining the responses in the sensory cortex, rather than in associative or motor areas) do not directly control actions or policies of action.


In this paper, we assume that the states of items are independent of the context in which the items are chosen.

the state φ[i] of each item i is drawn i.i.d.


In the ACNN, the convolutional layers focus only on those features most suitable for the current context specified by the side information, as compared to traditional CNNs that use a fixed set of filters over all contexts.

the feature extractors are tuned for each context.


We note that unlike the previous models in [9][10][11][12][13][14][15][16], with the exception of [17], a policy in our model is context-dependent.

since the state is (t, µ t , X(P π t )) and not just the time-belief tuple (t, µ t ), a policy π can recommend different actions for the same belief and at the same time but for a different context.


Our main results are illustrated in Figure 1.

we identify the impossible, intractable, and efficient regimes for the statistical-computational phase transitions under certain regularity conditions.


Algorithm 3 summarizes our second approach for FESL named as FESL-s(election).

the steps in Algorithm 3 on rounds 1, . . . , T 1 is the same as that in Algorithm 2.


There has been sparse research in the direction of nonuniform sampling, e.g., [18,[25][26][27]31].

negahban and Wainwright [26] studied the case of weighted entrywise sampling, which is more general than the setup of uniform sampling but still a special form of random sampling.


However, the assumptions in [19] are much stronger than the assumptions of Theorem 2.

they have strong noise assumptions and a group RIP like assumption (Assumption 4.1-4.3 in their paper).


An important advantage of this transformation is that since the set of all such ψs is S ∞ , the unit sphere in the Hilbert space L 2 .

the square-root representation simplifies the complicated geometry of Γ to the unit sphere.


Our goal was then to see if we could examine the influence a given speaker had on the turn-taking behavior of her conversational partners.

we wanted to see if we could better explain the turn-taking transitions observed in a given conversation between subjects i and} by combining the transitions typical to i and those typical toj.


We propose a simple baseline which selects harder examples with higher probability, as done by Loshchilov and Hutter [29].

we let is the history of prediction probability which stores all p(y i |x i ) when x i is selected to train the network before the current iteration t, H = i H t−1 i , pH t−1 i (y i |x i ) is the average probability of classifying sample i into its correct class y i over all the stored p(y i |x i ) in H t−1 i , and D is a smoothness constant.


It has been shown in a variety of literatures [27,11,22,12,24] that the PCA conducted on elliptical distributions shares a number of good properties enjoyed by the PCA conducted on the Gaussian distribution.

[11] show that with regard to a range of hypothesis relevant to PCA, tests based on a multivariate Gaussian assumption have the identical power for all elliptical distributions even without second moments.


It shows that both methods seek a linear decision rule by maximizing the margin after transforming input data to an intermediate space.

kernel-SVM (K-SVM) [39] first selects a transformation (induced by the kernel) that maps data points into an intermediate space, and then learns a set of linear decision boundaries that maximize the margin.


Next we consider a further generalization, where the unknowns are not only the time stamps of the observations, but also the initial state j.

we only know each set was generated from the same initial state, but do not know the actual initial state.


We adapt the procedure described in [16] to generate the input data.

for each pair of movies, we average the relative ratings from the same users within the same month.


The data, eighty sequences in all, were synthesized from the generating profile sequence of length fifty shown on the top line of the figure.

each data sequence was created by extracting one to four (mean two) patches from the generating sequence of length three to thirty (mean sixteen), sampling from these patches to produce corresponding patches of amino acids in the data sequence, and then filling in the gaps in the data sequence with amino acids sampled from a uniform distribution over amino acids.


We simulate two data sets, SIM-1 and SIM-2, from ground-truth regression coefficients.

we simulate a network formation process starting from time unit 0 until time 1200, where nodes arrive in the network at a constant rate λ 0 = 10 (i.e., on average, 10 nodes join the network at each time unit); the resulting simulated networks have 11,997 nodes.


Notice that in many, and perhaps even most, practical scenarios, being a better predictor on more instances is more important than avoiding such sporadic "embarrassing predictions".

our approach fits any commerce and advertising setting where the agent offers predictions to users on the value of different goods or services, aiming at maximizing the number of users that will find her predictions more accurate than the one provided by the expert.


Our approach interprets expected utility computation as a probabilistic inference problem on an induced Bayesian Network.

our algorithm runs in polynomial time for the important case of independent type distributions.


Standard modeling assumptions yield more concrete guarantees.

for example, if b = a, x ⋆ + ε where ε is independent mean-zero noise with the asynchronous iterates satisfy which is the (minimax optimal) asymptotic covariance of the ordinary least squares estimate of x ⋆ .


In Section 3, we show that the hypothesis learned with the Empirical Risk Minimization (ERM) procedure based on the local minimax risk closely achieves the optimal local minimax risk.

we provide a data-dependent bound on the generalization error, which behaves like the bound for ordinary ERM in the no-ambiguity regime (Theorem 1), and excess risk bounds under uniform smoothness assumptions on F (Theorem 2) and a less restrictive assumption that F contains at least one smooth hypothesis (Theorem 3).


The mergeon encodes the height at which clusters A , B, and C merge.

the fact that M = λ 2 everywhere on A × B represents the merging of A and B at level λ 2 in W. It is clear that in general there is no unique mergeon representing a graphon cluster tree, however, the above definition implies that two mergeons representing the same cluster tree are equal almost everywhere.


In the following analysis we characterize distributions with low variance of sample variance.

we are interested in distributions p ρ such that the quantity E[(σ 2 − σ 2 s ) 2 ] is minimized under the constraint that the variance is fixed, i.e., σ 2 = α.


A natural choice is to consider the average of the rank of the tensor's matricizations.

we consider the combinatorial regularizer Finding a convex relaxation of this regularizer has been the subject of recent works [9,17,23].


Adaptive sampler for MGP The above Gibbs sampler needs a predefined truncation level K c . In [3,26] the authors proposed an adaptive sampler, tuning K c as the sampler progresses, with convergence of the chain guaranteed [16].

the adaptation procedure is triggered with probability p(t) = exp(z 0 + z 1 t) at the tth iteration, with z 0 , z 1 chosen so that adaptation occurs frequently at the beginning of the chain but decreases exponentially fast.


Therefore, the cost function (3) will heavily penalize the classifier if it predicts an output that is believed to be less probable by the prior distribution p LM (•), and it will not penalize the classifier when it does not predict an output that p LM (•) believes to be probable.

the classifier is encouraged to predict a single output mode with high probability in p LM (•), a behavior called "mode-seeking" in [25].


We selected i ∈ {6, 14} for grid and pedigree, and i ∈ {2, 4} for protein.

i = 6 (grids, pedigrees) and i = 2 (proteins) were the smallest i-bounds for which SPRBFAOO could solve at least two thirds of instances within the 2 hour time limit, while i = 14 (grids, pedigrees) and i = 4 (proteins) were the largest possible i-bounds for which we could compile the heuristics without running out of memory on all instances.


In Corollary 2 and Theorem 4, we showed that the step size required for convergence to a specific solution depends on the solution itself.

we showed that there is a direct connection between the smoothness parameter of the training loss function and the Lipschitz constant of the function estimated by the network.


Another possible objective function is to maximize the scatter of the responses.

we would like to choose the next stimulus such that the response vector r is far away from the previously-seen response vectors r 1 , .


A typical choice of prior is a Gaussian distribution with diagonal covariance: Here β k ∈ R D denotes the classifier parameters for class k. Note that this prior assumes that each β k is independent of all other β i 's.

a-priori, the weights for label k are not related to any other label's weights.


The function F ( ) is said to satisfy the local error bound with parameter 2 (0, 1] if 9 > 0 such that for any 2 S , dist( , where C is a positive constant possibly depending on .

when = 1/2, F ( ) is said to be locally quadratic and when = 1, it is said to be locally linear.


In biological systems, for example, creating a taxonomy of the instances is often one of the first steps in understanding the system.

much of the work on analyzing gene expression data [3] has focused on creating gene hierarchies.


consider a stationary Markov decision process (MDP).


We theoretically analyze generalization bounds for DA under the setting of multiple source domains with labeled instances and one target domain with unlabeled instances.

we propose average case generalization bounds for both classification and regression problems.


Next, we rewrite the density into the form required by (1) by substituting σ(ψ k ) for π k : Choosing a k (x) = x k and b k (x) = N k for each k = 1, 2, . . . , K − 1, we can then introduce Pólyagamma auxiliary variables ω k corresponding to each coordinate ψ k ; dropping terms that do not depend on ψ and completing the square yields where Ω ≡ diag(ω) and κ(x) ≡ x − N (x)/2.

conditioned on ω, the likelihood of ψ under the augmented multinomial model is proportional to a diagonal Gaussian distribution.


The learning process of student model then switches to: Such a sequential procedure of obtaining f ω * (i.e., f ω T ) is the learning process of the student model with training data D train and loss function provided via the teacher model µ θ , and we use an abstract operator F to denote it: Just as the training and testing setup in typical machine learning scenarios, the teacher model here similarly follows the two phases setup.

in the training process of teacher model, similar to qualified human teachers are good at improving the quality of exams, the teacher model in L2T-DLF refines the loss function it sets up via optimizing its own θ.


It is now easy to dominate P ∈ H f (ρ; c, k) ∪ H f (ρ; c, k) far away from B 0 .

choose any p 0 ∈ B 0 ∩ P , which was guaranteed to exist in the preceding point; since min p∈P B f (x, p) ≤ B f (x, p 0 ) holds for all x, it suffices to dominate p 0 .


There exists γ = O(θ 2 /n) such that for all > 0, there exist M = O( √ n log(n/( θ)) and δ = Ω( 3 θ/ √ n) and a family C 0 of affine functions g : R n → R of bounded operator norm with the following property.

suppose φ : R → Furthermore, the functions of C are identically distributed as random variables over D n . we have statistical dimension bounds (and hence query complexity bounds) for functions that are sufficiently close to periodic.


We then ask how much information the spike train conveys about (a) which of the random segments Si(t) and (b) which of the amplitudes O"j was used.

the experiment consists of a series of trials of length 2 s where the fast component is one of the sequences {Si}, and after 1 s, the amplitude switches from 0"1 to 0"2 or vice versa.


The Pareto Optimization method for Subset Selection (POSS) [24] reformulates the original problem Eq.

( 1) as a bi-objective maximization problem: POSS maximizes the original objective and minimizes the subset size simultaneously.


This paper proposed a Bayes-Sard cubature method, which provides an explicit connection between classical cubatures and the Bayesian inferential framework.

we obtained polynomially exact generalisations of standard BC in Thm.


We devise a novel neural network architecture for this task which we train end-toend.

we introduce Latent Attention, which computes multiplicative weights for the words in the description in a two-stage process with the goal of better leveraging the natural language structures that indicate the relevant parts for predicting program elements.


If one can construct a dual witness Y ≥ 0 such that λ 1 (X + A T YA) = v, X v for any estimator v, then this estimator is the maximum likelihood estimator.

using the power iteration estimator v = v t , such a dual witness can provide a certificate of convergence of the power iteration (11).


Based on the shared-latent space assumption discussed in Section 2, we enforce a weight-sharing constraint to relate the two VAEs.

we share the weights of the last few layers of E 1 and E 2 that are responsible for extracting high-level representations of the input images in the two domains.


The upper bound in Theorem 2 holds for any arbitrary sample path of information arrivals that is captured by the matrix H, and matches the lower bound in Theorem 1 with respect to dependence on the time horizon T , as well as the sample path of information arrivals η k,t 's, the number of arms K, and the minimum expected reward difference ∆.

this establishes a minimax regret rate of order 1 ∆ K k log T t=0 exp −c • t s=1 η k,s for the MAB problem with auxiliary information that is formulated here, where c is a constant that may depend on problem parameters such as K, ∆, and σ. Theorem 2 also implies that the adaptive exploration policy guarantees the best achievable regret (up to some multiplicative constant) under any arbitrary sample path of auxiliary information (and in particular, under the subclasses discussed in §3.1 for any values of λ and κ).


We give policy-language biases that enable solution of very large relational Markov decision processes (MDPs) that no previous technique can solve.

we induce high-quality domain-specific planners for classical planning domains (both deterministic and stochastic variants) by solving such domains as extremely large MDPs.


In this section we turn our attention to an object that perhaps connects δ R and δ d most intimately: the operator geometric mean (GM), which is given by the midpoint of the geodesic (3), denoted as  Table 1: Some of the similarities between δ R and δ d . All matrices are assumed to be in P n . The scalars t, s, u satisfy The GM (21) has numerous attractive properties-see for instance [30]-among these, the following variational characterization is very important [31,32], especially because it generalizes the matrix geometric mean to more than two matrices.

this "natural" generalization is the Karcher mean (Fréchet mean) [31,32,11]: This multivariable generalization is in fact a well-studied difficult problem-see e.g., [33] for information on state-of-the-art.


We then demonstrate how statistical inference may be performed on causal parameters identified by this algorithm.

we consider cases where only a single sample is available for parts of the model due to full interference, i.e., all units are pathwise dependent and neighbors' treatments affect each others' outcomes [24].


As noted in Section 3, looser bounds would be obtained if classical Bernstein inequalities in Hilbert spaces [14] were used instead.

lemma 7.2 would result in a range for t of qn −r/(r+1) ≤ t ≤ C ∞ , implying k * = O(n 1/(r+1) ) rather than O(n 1/r ), and thus Theorem 3.2 would become (for k ≥ k * ) d α,p (S ρ , S k n ) = O(n −αr/(r+1)+1/(p(r+1)) ) (compared with the sharper O(n −α+1/rp ) of Theorem 3.2).


Converting a verification problem into this canonical representation does not make its resolution simpler but it does provide a formalism advantage.

it allows us to prove complex properties, containing several OR clauses, with a single procedure rather than having to decompose the desired property into separate queries as was done in previous work [11].


The regret is characterized in terms of the independence number α of the undirected observability graph.

they prove that √ T α log K is the optimal regret (up to logarithmic factors) and show that a variant of Exp3, called ELP, achieves this bound when the graph is known ahead of time, where α ∈ {1� .


We denote the size of this subset as B H to avoid confusion with the training batch size.

we report results for B H = 1 and B H = 320.


Since the occupancy record of this DP, m K+1 := {m K+1 } ∪ {m K+1,k : k = 1, . . . , K + 1}, is not yet defined at the onset, with probability 1 we turn to the top-level DP when departing from state K + 1 for the first time.

we define p(•|c t = K + 1) according to the occupancy record of ancestors in the stock urn.


We argue that all inputs that fool the classifier without confusing humans can pose potential security threats.

we show that previous defense methods, including the certified ones [16,17], are not effective against this more general attack.


All δ values from Theorem 9 are above 1, and not shown.

3 for the SBM model class, the C cannot be proved stable even for the LFR data.


In order to model the initial distribution without cluttering the notation, we assume there is a special state θ beg always present at the beginning of the sequence, and only at the beginning.

we always condition on (θ 0 = θ beg ) and (θ n = θ beg , n > 0), and drop these conditioning events from the notation.


To this end, we start by description of a weighted independent set problem as the MRF model.

consider a binary MRF on an n 1 × n 2 grid G = (V, E): Here, the parameters are chosen as follows.


One of the long-studied properties of biological neural circuits which has yet to fully impact the machine learning community is the nature of local competition.

a common finding across brain regions is that neurons exhibit on-center, off-surround organization [1,2,3], and this organization has been argued to give rise to a number of interesting properties across networks of neurons, such as winner-take-all dynamics, automatic gain control, and noise suppression [4].


much larger decision tree policies than necessary.

dAGGER cannot leverage the fact that our oracle provides not just the optimal action to take in a given state, but also the cumulative reward of every state-action pair (either directly as a Q-function or indirectly as a distribution over possible actions).


First, we define ¦ R to be the ¢ § -by-¢ § matrix whose entries correspond to the probabilities that one state-action pair follows another when the agent behaves according to 1 .

the element on the $ &© (' %) P ¨ § row and $ 5© D ' @) P ¦ uR can be viewed as the stochastic transition matrix of a Markov chain over state-action pairs.


Although information is lost due to hash collisions, it is nevertheless possible to construct a marginal likelihood that exactly accounts for this censoring: the finite projection property of the Dirichlet process specifies what the distribution should be of the hashed counts in each of the buckets.

the hash function forms a partition of V, and Equation (1) specifies the resulting marginal Dirichlet distribution.


Inference: We proof-sketch in the supplementary material that inference can be performed efficiently in parallel as we can decompose the problem of optimizing the orderings induced by the different queries to optimizing each ordering separately.

for a query i of class c the computation of the y i kj 's, ∀k ∈ P c \ i, j ∈ N c can happen independently of the computation of the y i k j 's for some other query i = i.


Further experiments are being done to demonstrate coupling effects like the ones described in Section 5.

the use of external optical signals directly onto the detector to implement optical coupling has been demonstrated.


We thus see that this algorithm is equivalent to the MAP algorithm derived in §4.1.1 for Gaussian scale mixtures.

for direct MAP estimation of latent variable x, the EM Gaussian scale mixture method and the variational bounding method yield the same algorithm.


We will show that the characterization of γ-Lipschitz functions via the norm of the differential can be extended from the 2 setting in (4) to arbitrary Banach spaces by considering the gradient as an element in the dual of B.

for any Banach space B with norm • B , we will derive the loss function where λ, γ ∈ R are regularization parameters, and show that a minimizer of this this is an approximation to the Wasserstein distance on B.


The task then is to identify the circles to which each alter v i belongs, as in Figure 1.

the goal is to find nested as well as overlapping communities/clusters in u's ego-network.


Conversely, all stationary points of the function are stable points of the algorithm.

since the problem is convex in W and H individually, but not jointly in both, the algorithm can be thought of as taking exact steps toward minimizing over W (as H is held fixed) and then minimizing over H (as W is held fixed), whence the alternating-minimization name.


Optimizing a continuous synthetic function In this problem, the objective is to optimize a blackbox function sampled from a GP, whose domain is R 2 , given a set of evaluations of different functions from the same GP.

we consider a GP with a squared exponential kernel function.


However, the problem becomes significantly more challenging with multiple items for sale.

conitzer and Sandholm proved that the problem of finding a revenue-maximizing deterministic CA is NP-complete [Conitzer and Sandholm, 2004].


The model provides a unified explanation for experimental data previously explained by two competing models, namely, the additive offset model and the dynamic weighting model for incorporating prior knowledge.

the model predicts psychometric and chronometric data for the random dots motion discrimination task [13] as well as Carpenter and Williams' saccadic eye movement task [10].


In addition, the model contains a proxy whose association is 0.40 and influence is 14.5.

the variance of the proxy is 14.5 times greater than that of the model; this arises because other associated variables cancel most of this variance in the full model.


As illustrated in Figure 1, our RSDN architecture consists of two subnets, and each subnet is a fully connected multi-layered perceptron of 2K+1 layers, i.e., an input layer, 2K-1 hidden layers and a visible layer at the top.

if we stipulate that layer 0 is input layer, there are the same number of neurons in layers k and 2K-k for k = 0, 1, • • • , K. the Kth hidden layer is used as code layer, and neurons in this layer are further divided into two subsets.


Thus, we can infer the causal direction by comparing the description length variability of conditional distributions across different values of the conditioning variable with the causal direction being the less variable.

we propose three decision rules for causal inference based on this criterion.


Our main results highlight the respective roles of these parameters and show how can they be chosen so that the corresponding solutions achieve optimal learning errors.

we show for the first time that multi-pass SGM with early stopping and a universal step-size choice can achieve optimal learning rates, matching those of ridge regression [23,4].


In the following we focus on measures with discrete support.

we consider distributions µ, ν ∈ P(X) that can be written as linear combinations µ = n i=1 a i δ xi and ν = m j=1 b j δ yj of Dirac's deltas centred at a finite number n and m of points (x i ) n i=1 and (y j ) m j=1 in X.


This is to be contrasted with the following property of the occupancy measures generated by O-REPS (proved in the supplementary material): Lemma 1.

for any t > 0, D (q t q t+1 ) ≤ η 2 2 q t , ˆ 2 t . if the losses are estimated by bounded sample averages as done by Peters et al.


Here λ > 0, l i (λ) = (K nn (K nn + λnI) −1 ) ii are the leverage scores and q ≥ 1 controls the quality of the approximation.

given λ, the Nyström points are sampled independently from the dataset with probability p i ∝ l i (λ).


The attractive facet of this modification is that it leverages the sparsity of the solution and intermediate iterates in a manner that falls within a block coordinate descent framework.

suppose as detailed above at any outer loop Newton iteration, we partition the variables into the fixed and free set, and then first perform a Newton update restricted to the fixed block, followed by a Newton update on the free block.


Similarly to previous works [26], we do not report the computational time saved by screening because it can be easily computed from the rejection ratio.

if the Lasso solver is linear in terms of the size of the data matrix X, a K% rejection of the data can save K% computational time.


One of the most desirable characteristics is the fact that the partitioning of the learners knowledge is made explicit.

the model learns a much more differentiated and context bound representation when γ is small, and a more context general and less differentiated representation when γ is large.


Therefore, in the present work, we suggest replacing this question with a more nuanced one: how elicitable are various properties?

heeding the suggestion of Gneiting [7], we adapt to our setting the notion of elicitation complexity introduced by Lambert et al.


The minimax optimal predictions are defined as follows: for all j ∈ [n], Theorem 2 suggests a statistical learning algorithm for aggregating the p ensemble classifiers' predictions: estimate b from the training (labeled) set, optimize the convex slack function γ(σ) to find σ * , and finally predict with g j (σ * ) on each example j in the test set.

the resulting predictions are guaranteed to have low error, as measured by V . it is easy to prove [9] that V is at least max i b i , the performance of the best classifier.


The payoff function for the induced MDP is defined to reflect the goal of maximizing the total expected reward across all agents.

payoff , denote the payoff obtained from all agents at time t. Given a (nonstationary) policy π = {π 1 , π 2 , .


Our coherence measure can be used in conjunction with any clustering algorithm.

let G = (V, W ) be a weighted graph of points and let V = S ∪ (V \ S) be a possible partition (computed by some clustering algorithm).


We show that Σ = XT X/ √ nm can be estimated in a consistent, accurate manner on the basis of public users alone.

we express the error Σ − Σ F as a function of the total number of observations.


The primary contribution of this paper is theoretical.

we present a new algorithmic framework: Nonconvex Inexact Proximal Splitting (NIPS).


Each component of ⇠ i 2 {0, (1 ) 1 } d is an independent draw from a scaled Bernoulli( 1) random variable.

dropout noise corresponds to setting xij to 0 with probability and to x ij /(1 ) else.


We choose coordinate descent for solving (1) as the baseline algorithm.

denote the solution of i } are given by the following recursion: We use the same initial starting point as TranSync.


Our fully convolutional architecture enjoys the benefits of the network modifications that are widely used by FCNs for semantic segmentation [16,2].

we reduce ResNet-101's effective stride from 32 pixels to 16 pixels, increasing the score map resolution.


In a recent work, [10] proposes an algorithm for computing the top eigenvector of A T A by running intermediate SVRG steps to approximate the shift-and-invert iteration.

at each step SVRG is used to solve: min , where a i is the i-th column of A.


[18] showed that there exists cross-subject consistency in the cortical regions useful for prediction in this task.

experts partitioned each dataset into 24 non overlapping regions of interest (ROIs), then reduced the data by discarding all but 7 ROIs and, for each subject, averaging the BOLD response across voxels within each ROI and showed that a classifier trained on data from 5 subjects generalized when applied to data from a 6th.


While the above examples show that there cannot be a bound on the sample size as a function of the expectation of the value, it turns out that there is a very tight connection between the first moment and uniform convergence: Theorem 2.3.

the following dichotomy holds for a distribution µ on R + : the empirical revenues uniformly converge to the true revenues if and only if E µ [V ] < ∞.


This LOCALSPECTRAL optimization problem was introduced in [14], where it was shown that the solution to LOCALSPECTRAL may be interpreted as a locally-biased version of the second eigenvector of the Laplacian.

although LOCAL-SPECTRAL is not convex, it's solution can be computed efficiently as the solution to a set of linear equations that generalize the popular Personalized PageRank procedure; in addition, by performing a sweep cut and appealing to a variant of Cheeger's inequality, this locally-biased eigenvector can be used to perform locally-biased spectral graph partitioning [14].


[8] tackle the problem of selecting features in a budgeted learning scenario.

they solve a problem that can be viewed as the inverse of traditional active learning; given class labels, they seek to determine the best features to compute for each instance such that a good predictive model can be trained under a budget.


In our variational approach we introduce a new distribution q and make the following approximation 4 p(z 1:N , h 1:N 1:T , Θ 1:K |v 1:N 1:T , Θ1:K , γ) ≈ q(h 1:N 1:T |z 1:N )q(z 1:N )q(Θ 1:K ).

( we approximate the posterior distribution of the hidden variables of the model by one in which the hidden states are decoupled from the parameters given the indicator variables and in which the indicators are decoupled from the parameters.


Nesterov [12] proposed a simple first-order method to exploit this type of structure, and our focus is a slight variant of this procedure.

given some initialization θ 0 ∈ Ω, consider the update where γ u > 0 is a parameter related to the smoothness of the loss function, and is the ball of radius ρ in the norm defined by the regularizer.


Boundedness necessarily implies that the function values converge to finite values at negative and positive infinity, and hence asymptote to horizontal lines on both sides.

the derivative of the activation converges to zero as the input goes to both −∞ and +∞.


Suppose that our subdomains are Φ 1 = {y} and Φ d = Θ d for d ≥ 2.

we consider the restriction of the Mondrian to a slice of the space where the first coordinate takes on value y.


They may be much heavier-tailed, biasing large signals less drastically while shrinking noiselike signals heavily towards zero.

the priors by [6,7], along with the Strawderman-Berger prior [7], have a very interesting and intuitive representation later given in ( 2), yet, are not formed in a conjugate manner potentially leading to analytical and computational complexity.


Algorithm ROBUST-LRA (see Algorithm 1) is (", )-differentially private.

furthermore, given a matrix A 2 R n⇥d , it runs in poly(k, n, d) time, e O(k(n + d)) space, and outputs a rank k matrix M such that, with probability 9/10 over the randomness of the algorithm, where OPT k (A) := min rank(X)k kA Xk p . for p = 1, we get Remark 1.


3 With our theoretical understanding in hand, we proceed with an experimental evaluation of the efficacy of multilinear functions applied to hypothesis testing.

given a binary vector, we attempt to determine whether or not it was generated by an Ising model.


The image view consists of 120-dimensional visual features extracted from representative frames selected from videos, and the textual features are 100-dimensional.

we give the competing algorithms, both CCALin and ALS CCA, the advantage of the knowledge of the eigengap at k. we estimate the spectrum of the matrix b T for the Mediamill dataset and set the gap-dependent parameters in CCALin and ALS CCA accordingly.


The connection enables us to make contributions by borrowing from each body of literature to the other.

it helps us highlight the role of weighted sampling for SGD and obtain a tighter guarantee on the linear convergence regime of SGD.


Our approach maintains the simplicity and flexibility of fairness through awareness, but provably only requires a small number of random samples from the underlying metric, even though we make no structural assumptions about the metric.

our approach works even if the metric provably cannot be learned.


Our paper belongs to this line of research.

we propose a novel general case PSRL algorithm, referred to as DS-PSRL, that exploits model parametrization (generalization).


In the last decade, policy gradient methods have significantly grown in popularity in the reinforcement-learning field.

they have been largely employed in motor control and robotic applications, thanks to their ability to cope with continuous state and action domains and partial observable problems.


If we introduce an abrupt change into the sequence, for example by reversing the second half of the sequence, s i is found to be ≈ 1 for the frame following the change.

the system has detected not to use the previous frame at this point, but to revert to using the latent appearance image f . We have proposed a method for localising an object undergoing affine transform whilst simultaneously learning its shape and appearance.


(C) If F ≥ Fmax , then Fh ≥ Fhmax ∀h ∈ H; this enforces that all hypotheses must be satisfied.

truncated version of each hypothesis submodular utility and distance function: F h,αn,αj is truncated from below at α j and from above at α n (it is assumed that α n > α j ), and is offset by −α j so that F h,αn,αj (∅) = 0.


Variational inference is coordinate ascent in the bound on the probability of a single document.

we alternate between the following two equations until the objective converges: (5) where \]i is the first derivative of the log r function.


Apart from the aforementioned, there have been a lot more works that incorporate social graph information in recommender systems; however, few works have been devoted to the theoretical understanding of this problem.

it is widely open as to by how much one can improve the performance with the aid of graph side information.


Moreover, this simple ALP algorithm achieves very good performance within a constant distance from the optimum, i.e., O(1) regret, except for certain boundary cases.

for 1 ≤ j ≤ J, let q j be the cumulative probability defined as q j = j j =1 π j with the convention that q 0 = 0.


Corollary 1 and 2 provide answers to our motivative questions.

for loss functions which are (8, T )-smooth, we can obtain a lower sample complexity; if they are (∞, T )-smooth, there is an -LDP algorithm for the empirical and population excess risks achieving error α with sample complexity which is independent from the dimensionality p in the term α.


The operations of shape recipes (pyramid decomposition, convolution, and image reconstruction) are all linear operations, and so they can be combined into a single linear convolution.

we can think of shape recipes as inferring the high resolution range data z high via a single convolution where I is the Fourier transform of the intensity image i.


To learn a mapping of (context x) Ñ (program a), we optimize the parameters of a conditional distribution ⇡ ✓ pa | xq that assigns a probability to each program given the context.

⇡ ✓ is a distribution over the countable set of all possible programs, denoted A.


To approximate the Q-functions, we use a linear combination of 20 radial basis functions.

for each action, we consider 9 Gaussians with means uniformly spread in the interval [−20, 20] and variance equal to 16, plus a constant feature.


Rather, it is possible to incorporate influences of its own spiking history on the neural response.

the firing rate is then determined by a combination of both the external stimulus and the spiking-history of the neuron.


The condition (RMAP) can be therefore used to decide whether or not the MAP instantiation for P Φ is the unique MAP instantiation for every P ∈ P Ψ .

we have an algorithm to test the robustness of P Φ with respect to the perturbation P Ψ .


Therefore, we use the prior information on the size of the clusters to select α.

we define expected size ratio, R, as R = min(s1,s2) max(s1,s2) where s 1 and s 2 are the expected size of the two clusters (known a priori).


Instead of assuming no interdependency between Y i as in [19], our major goal here is to model the structure in the latent layer Y .

we assume a conditional model for the first layer based on an exponential family p(y|x) = q 0 (y) exp(−y U x − Ω(U x)), where q 0 (y) = y ∈ Y .


Changing variables from r to r' = UTr and using I det(U)1 = 1, this can be rewritten as Becausepe(e) is a function ofeTe only, Pe (Ue) = pe(e), and therefore Im(r, 8; UWm ) = Im(r, 8, W m ).

because we have assumed a model in which the noise is described by independent Gaussians, or generally the distribution of the noise e is a function of e T e only, the mutual information is invariant to unitary transformations of the output.


In each iteration, we search the best gain pair < x, ŷ′ > by traversing the value of y ′ in f x→y ′ .

change the value f (x) to ŷ′ in order to achieve the highest score increase.


In the case of kernel-based reinforcement learning, the so-called "self-approximating" property allows for a much more efficient implementation in vector format (see also [Rus97]).

because our definition of r m,ah in (5) only depends on the values of h at the states in S , the AACOE (3) can be solved in two steps: In other words , we first determine the values of hm at the points in S using ( 7) and then compute the values at new locations x in a second step using (8).


[27] theoretically assessed backpropagation based methods and found that Guided BackProp and DeconvNet, under certain conditions, are invariant to network reparamaterizations, particularly random Gaussian initialization.

they show that Guided BackProp and DeconvNet both seem to be performing partial input recovery.


We view {z n } N n=1 and matrix Γ as latent variables, and optimise the parameters θ = {µ, Φ, Λ, Ψ} by EM.

we view the weight matrix Λ as a matrix of parameter and estimate the entries by maximum a posteriori (MAP) learning.


Thus if all features are reliably tracked in all the frames, the problem can be solved trivially using SVD [11].

performing an SVD on the measurement matrix of the rotating cylinder stimulus recovers the correct structure even if the measurement matrix is contaminated with significant amounts of noise and if the number of frames is relatively small.


Co-training [2] is a method that has had substantial success in scenarios in which examples can be thought of as containing two distinct yet sufficient feature sets.

a labeled example takes the form ( x 1 , x 2 , ), where x 1 ∈ X 1 and x 2 ∈ X 2 are the two parts of the example, and is the label.


We have tested several candidates and finally found the absolute value of the input to be the best choice, as claimed in [9].

the parameters with relatively small magnitude are temporarily pruned, while the others with large magnitude are kept or spliced in each iteration of Algorithm 1.


In this work, we aim to study the advantages and limitations of both topologies and further enrich the path design by proposing a dual path architecture.

we first provide a new understanding of the densely connected networks from the lens of a higher order recurrent neural network (HORNN) [19], and explore the relations between densely connected networks and residual networks.


Despite the fact that, as we show, this does not correspond to a convex-concave optimization problem (even for simple linear generator and discriminator representations), we show that: Under suitable conditions on the representational powers of the discriminator and the generator, the resulting GAN dynamical system is locally exponentially stable.

for some region around an equilibrium point of the updates, the gradient updates will converge to this equilibrium point at an exponential rate.


We have shown that sampling-based inference in a simple generative model of V1 can be interpreted in multiple ways, some previously discussed as mutually exclusive.

the neural responses can be interpreted both as samples from the probabilistic model that the brain has learnt for its inputs and as parameters of the posterior distribution over any experimenter-defined variables that are only implicitly encoded, like orientation.


This approach aims to provide insight into odor encoding in the brain by identifying an olfactory space that relates smoothly to changes in large-scale neural firing patterns.

we develop a Gaussian process based latent variable model (GPLVM) [13] for identifying latent structure underlying population activity in the olfactory cortex.


In order to guarantee convergence, previous works impose stringent assumptions that require significant effort for practitioners.

they require a custom implementation of the greedy step (called the LMO) for every probabilistic model with respect to an unnatural variational family of truncated distributions.


A kernel q V (V|W) is a mapping from values in W to normalized densities over V [5].

kernels act like conditional distributions in the sense that v∈V q V (v|w) = 1, ∀w ∈ W. Conditioning and marginalization in kernels are defined in the usual way.


If L r ⊂ V denotes a set of data points that are a priori known to belong to class r then we simply enforce L r ⊂ A r in the definition of an allowable partition of the vertex set.

any allowable disjoint partition also respect the given set of labels.


The authors considered the USPS handwritten digit database and reported the test error rates achieved by the linear SVM trained on the data pre-processed with KPCA: the conclusion was that the larger the number of principal components, the better the performance.

the KPCA step was useless or even counterproductive.


One of the main results is the Fisher-Tippett-Gnedenko theorem [11,12] that characterizes the limiting distribution of this maximum as n converges to infinity.

it proves that a rescaled version of this maximum converges to one of the three possible distributions: Gumbel, Fréchet, or Weibull.


Tracing its origins to [17] for offline optimization problems, OMD proceeds by taking a gradient step in the dual (gradient) space and projecting it back to the primal (decision) space via a mirror map generated by a strongly convex regularizer function (with different regularizers giving rise to different algorithms).

oMD includes as special cases several seminal learning algorithms, such as Zinkevich's online gradient descent (OGD) scheme [29], and the multiplicative/exponential weights (EW) algorithm [1,13].


A set of hidden variables {ω ij } represents which part (i) produced feature f j .

these variables ω ij then take values {0, 1} restricted to ω ij = 1 means feature j was produced by part i; each part can produce multiple features, each feature is produced by only one part.


Furthermore, we achieved a sparse solution for the point estimation method.

we obtained less than 5 non-zero sources out of 116 for most of the tasks for the RCV1 collection.


The speedups for both methods are the same when the delay is few milliseconds.

if a gradient takes longer than one millisecond to compute, RR is on par with HOGWILD!


The orthogonal joint diagonalization algorithm with the GP/DICA cumulants (JD-GP) demonstrates promising performance.

the GP/DICA cumulants significantly outperform the LDA moments.


A set of solutions C ⊆ S of a solution space S is a cluster if it is a maximal subset such that any two solutions in C can be connected by a sequence from C where consecutive solutions are neighbors.

clusters are connected components of the "solution graph" which has solutions as nodes and an edge between two solutions if they differ in the value of exactly one variable.


A similar challenge arises in a multi-batch implementation of the L-BFGS method in which the entire training set T = {(x i , y i ) n i=1 } is not employed at every iteration, but rather, a subset of the data is used to compute the gradient.

we consider a method in which the dataset is randomly divided into a number of batches -say 10, 50, or 100 -and the minimization is performed with respect to a different batch at every iteration.


Again, though, even this form of adversary appears unnaturally strong in the context of causal graphs.

given the random variables V the causal relations between them are pre-determined.


The empirical success of sparse coding can be justified by theoretical analysis [17], which showed that a modification of sparse coding with added locality constraint, called local coordinate coding (LCC), represents a new class of effective high dimensional non-linear function approximation methods with sound theoretical guarantees.

lCC learns a nonlinear function in high dimension by forming an adaptive set of basis functions on the data manifold, and it has nonlinear approximation power.


Unlike CS, however, PLST takes principal directions v m (to be introduced next) rather than the random ones, and does not need to solve an optimization problem during decoding.

pLST considers only a matrix V with orthogonal rows, and decodes r(x) to the predicted labels by h(x) = round(V T r(x)+ ȳ), which is called round-based decoding.


(3) We are still able to model spatial correlations in a 2-D lattice, despite the fact that this introduces dependency cycles in the model.

our semi-supervised training procedure can be interpreted as a MAP estimator, where the parameter prior for the model on labeled data is governed by the conditional entropy of the model on unlabeled data.


kernel and the r.k. of an RKBS need not be symmetric, the space in (7) is an interesting example of an RKBS, which is induced by a pd kernel.

it can be seen that many RKBSs (B pd p (R d ) for any 1 < p < ∞) have the same r.k (ignoring the scaling factor which can be made one for any p by choosing µ to be a probability measure).


Within the context of statistical inference, Bayesian optimization methods have been developed where a prior distribution over the space of functions is assumed and uncertainty is tracked during the entire optimization process [6,7].

non-parametric Bayesian approaches such as Gaussian Processes have been applied for derivative-free optimization [8,9], also within the context of the continuum-armed bandit problem [10].


As a consequence, the neuron's firing will be memoryless, and close to a Poisson process.

we found Fano factor close to 1 and quasi-exponential ISI distribution (figure 1-E) on the entire range of parameters tested.


Despite recent progress on the frame prediction problem [10,23], it remains quite challenging, in part because the dimensionality of the true-value space is so large.

in practice, it can be substantially easier to solve inverse dynamics prediction, with input (o t b:t+f , a t b:t 1 , a t+1:t+f 1 ) and true-value a t . the agent is trying to "postdict" the action needed to have generated the observed sequence of observations, given knowledge of its past and future actions.


All are fit using stochastic gradient descent (SGD) (Robbins and Monro, 1951).

we compare the following methods: • A global EFE model, which cannot capture group structure.


Note that while all h i -s, g and g are assumed to take values in {−1, 1}, u 1 can take values in R. Then: Now we show that for any α 1 u 1 + α 2 g ∈ MV(u 1 , g) there exists a close hypothesis in MV(u 1 , g).

this hypothesis is α 1 u 1 + α 2 g: Note that for every x on which g and g agree, i.e.


We can also express (5) using vector notations, which will become useful for deriving the Neural DUDE in the next section.

we let ∆ |S| be a probability simplex in R |S| .


They are usually referred to as the so-called categorical data.

each instance in those datasets is described by one or more attributes.


With the above generative process, we can train the network by optimizing the CMMD objective with proper regularization.

let D s XY = {(x d i , y d i )} N i=1 denote the given training dataset.


The GGG allows to avoid the limits of the CHL for modelling topology.

it allows to take into account the noise and to model isolated bumps.


Matrix Classification: We consider a matrix classification problem under the setting described in [21].

we first randomly generate a low-rank matrix classifier X * , which is an n × n symmetric matrix of rank r. Then, we specify a sampling ratio θ ∈ (0, 1] and sample p = ⌊θn 2 ⌋/2 independent n × n symmetric matrices W 1 , .


Therefore, problem (2) can be reformulated as where p x,0 and p x,1 denote the elements of the vector p corresponding to the probability values P(X = x, Y = 0) and P(X = x, Y = 1), respectively.

the simple application of the minimax theorem [9] implies that the saddle point of the above optimization problem exists and moreover, the optimal decision rule is a MAP rule for a certain probability distribution P * ∈ C. there exists a pair (δ * , P * ) for which Although the above observation characterizes the optimal decision rule to some extent, it does not provide a computationally efficient approach for finding the optimal decision rule.


This is the general approach we will follow in this paper.

in section 3, we consider the case where the given set of sample points x 1 , .


In this paper we follow the formulation in [4], where the tasks' structure is summarized by a positive definite matrix D which is linked to the covariance matrix between the tasks, W W . Here, W denotes the d × T matrix whose t-th column is given by the vector w t (we have assumed for simplicity that the mean task is zero).

we learn W and D by minimizing the function where γ is a positive parameter which balances the importance between the error and the penalty.


This memory term, which allows nodes to remember previous values and incorporate them into the next update, directly relates the Min-Sum Splitting algorithm to accelerated multi-step first order methods in convex optimization.

we show that a proper choice of the splitting parameters recovers the same matrix that support the evolution of shift-register methods used in numerical analysis for linear solvers, and, as a consequence, we recover the same accelerated rate of convergence for consensus [45,4,24].


The main objective of this paper is precisely to tackle this issue in the context of image modeling.

our approach is based on convolutional kernel networks, which have been recently introduced in [18].


finding ψ * will not be useful unless we can establish that v * ≥ 0, i.e.


Finally, we are currently investigating the use of DIFFRAC in semi-supervised image segmentation.

early experiments on estimating the number of clusters using variation rates of our discriminative costs are very promising.


Alternatively, maximum belief propagation (max-BP) allows a joint maximum a posteriori (MAP) set of state variables to be determined.

given a song, we generate the observations Y i , i = 1 . . . F , (where F is the number of frames in the song) and seek a set of states X i that maximize the joint product Our smoothness function ψ is the product of tempo and phase smoothness components ψ T and ψ P . For the tempo component, we use a Gaussian on the log of tempo.


In the following sections we will consider several examples of applications of LDOMD to online combinatorial decision problems involving both 0-1 polytopes and general polytopes in R d . Consider an online decision problem in which the decision space C contains (not necessarily all) independent sets in a matroid M = (E, I).

on each trial t, one selects an independent set I t ∈ C, and receives a loss vector t ∈ [0, 1] |E| , with t e specifying the loss for including element e ∈ E; the loss for the independent set I t is given by e∈I t t e . Here it is natural to define a mapping φ : C→{0, 1} |E| that maps each independent set I ∈ C to its characteristic vector, defined as φ e (I) = 1(e ∈ I); the loss on selecting I t ∈ C is then φ(I t ) • t . Thus here d = |E|, and φ(C) ⊆ {0, 1} |E| .


Thus, an effective and compact region embedding cannot be trivially obtained from a word embedding.

effectiveness of the combination of three tv-embeddings ('r: 3 tv-embed.'


As for the comparison to the setting of discounted losses of [9], we note that the latter can be cast as a special case of our setting (since the discounting weights take the special form β t,T = γ t . . . γ T −1 therein, for some sequence γ s of positive numbers).

the fixed-share forecaster can satisfy the bound stated in [9, Theorem 2], for instance, by using the online tuning techniques of Section 7.3.


We conclude this section with a further relevant example of our proposed approach to the setting of applications to relational knowledge, which we plan to investigate in future work.

we consider settings where the output space M corresponds to the Hyperbolic space (or Poincaré disk), which has recently been adopted by the knowledge representation community to learn embeddings of relational data to encode discrete semantic/hierarchical information [34,33].


The earliest theoretical analyses of HMC were the asymptotic "optimal scaling" results of [27], for the special case when the target distribution is a multivariate Gaussian.

they showed that the Metropolis-adjusted implementation of HMC with leapfrog integrator requires a numerical stepsize of O * (d − 1 4 ) to maintain an Ω(1) Metropolis acceptance probability in the limit as the dimension d → ∞.


The form of the optimization problem (Q) is similar to that of the COSSO for smoothing spline ANOVA models (Lin and Zhang, 2006); however, our method differs significantly from the COSSO, as discussed below.

our method is scalable and easy to implement even when p is much larger than n. We now derive a coordinate descent algorithm for fitting a sparse additive model.


To avoid double counting, the high level SLAM process can only weigh the match between the new observations and the existing high level maps.

all of the observations for a single high level motion step (single low level trajectory) must be evaluated against the high level map, before any of those observations are used to update the map.


However, the reshaping process for test datasets has a critical difference from reshaping training datasets.

we should reshape test datasets, conditioning on the identified domains from the training datasets -the goal is to discover latent domains in the test datasets that match the domains in the training datasets as much as possible.


Motivated by the above applications and the increasing need to satisfy multiple objectives, we study a wide class of multi-objective online optimization problems, and present both hardness results and (almost tight) bi-objective approximation algorithms for them.

we study resource allocation problems in which a sequence of items (also referred to as impressions) i from an unknown set I arrive one by one, and we have to allocate each item to one agent (for example, one advertiser) a in a given set of agents A.


Qualitatively, however, generated samples from VEEGAN seem better than other methods.

the samples from VEEGAN +DAE are meaningless.


. . , u r } ⊆ S d , such that We expect this assumption to hold when the target function U is of the form U (x) = ∑ r i=1 f i (u ⊺ i x) for functions f i ∶ R → R with uniformly bounded third derivatives.

this class includes the target functions used in logistic regression.