In this way, more flexible transformations are allowed than mean-centering and variancescaling.

we build a Gaussian Process to model the prediction on the transformed X, then the prediction is matched with a few observed labels Y (also properly transformed) available in the target domain such that both transformations on X and on Y can be learned.


We choose the coefficients a, b ≥ 0 such that s 0 a 2 + b 2 = 1, whence E{X 2 i,p } = 1 and hence the p-th covariate is normalized as the first (p − 1) ones.


In addition, we show that in the stochastic setting SODA achieves an O a:∆a>0 Kε 2 ∆a pseudo-regret bound that holds simultaneously with the adversarial regret guarantee.

sODA is safe against an unrestricted oblivious adversary and provides improved regret guarantees for at least two different types of "easiness" simultaneously.


Without loss of generality, we assume the coordinates of y are sorted in the order of the cluster indices.


we show how loss in accuracy increases with the degree of risk aversion, ↵, and as the lower bound on social welfare, ⌧ , becomes more demanding.


( 7) (12) Now the membrane potential Vi(t) only depends on the time of the last own spike ti and the refractory contribution amplitude modulation factor at the last spike ' TJi .

we have transferred the effect of all spikes previous to the last one into 'TJi.


Any report of abnormal returns using historical markets should be suspected of "data snooping".

when a dataset is excessively mined by testing many strategies there is a substantial chance that one of the strategies will be successful by simple overfitting.


Indeed, in this paper we are inspired by this geometric insight to propose a new ensembling procedure that can efficiently discover multiple high-performing but diverse deep neural networks.

our contributions include: • The discovery that the local optima for modern deep neural networks are connected by very simple curves, such as a polygonal chain with only one bend.


For every d > 0, m 2 N, distribution D on Z, The dependence on the data distribution is captured through t, which is ideally chosen as small as possible, subject to Q(S) yielding small empirical risk.

(One can use a union bound to tune t based on S.) The fact that the KL bound does not depend on D, other than through t, implies that the bound must be loose for all t such that there exists a distribution D that causes Q to overfit with high probability on size m datasets S ⇠ D m . for fixed t, the bound is no longer distribution dependent.


To further drive intuition behind the E-step complication, let us make the sum-to-one constraint for the variational distributions q n (•) explicit.

k q n (k) = 1 for all data points n, which according to the above block constraint and using Property 1 can be reformulated as the |L| constraints Notice that since q B k can be associated with an internal node in T it may be the case that q B k(l) represent the same q B k across different constraints in (5).


4 Fast Stagewise Unit-Rank Tensor Factorization (SURF) For simplicity, we drop the index r and write the generic form of the problem in (6) as ) , where σ ≥ 0, w (n) 1 = 1, and the factors Before diving into the stagewise/boosting algorithm, we first consider an alternating convex search (ACS) approach [Chen et al., 2012, Minasian et al., 2014 which appears to be natural for solving (7) with any fixed tuning parameter.

we alternately optimize with respect to a block of variables (σ, w (n) ) with others fixed.


Our rate-distortion analysis of the Poisson channel differs from that comparison in a twofold way: First, we do not change-the noise model and second, the MMSE is often more appropriate to account for the coding efficiency than the channel capacity [10].

the assumption of a real random variable to be encoded with minimal mean squared error loss appears to introduce a bias for analog coding rather than for binary coding.


The Bayesian quadrature (BQ) [7,8,9,10] approach to estimating model evidence is inherently model based.

it involves specifying a prior distribution over likelihood functions in the form of a Gaussian process (GP) [11].


We interpret k to be a latent variable that signals the hidden cause of the underlying input distribution, with distribution P k . Critically, we assume that the hidden variable k changes slowly compared to the inter-spike period of the neuron.

we need at least 3 samples from each P k . This corresponds to the multi-view assumption of [2].


Overall, our results are largely negative.

we find that none of the tested algorithms are capable of effectively scaling up to training large networks on ImageNet.


Examples are financial data reported on the Internet, weather data observed by a set of sensors, etc.

in many data mining applications we are interested in learning a global model from such data, like a probability distribution or a clustering of the data, without first transferring all the data to a central repository.


The sum-product algorithm (belief propagation) is an dynamic programming technique for performing marginalization in a factor graph.

it computes sums of the form For tree-structured factor graphs, the sum-product algorithm efficiently recovers the exact marginals.


We used the conditional deep convolutional generative adversarial networks (DCGAN) framework [16,18] to train both a generative model and a discriminator.

the generative network has 4 transposed convolutional layers.


them being the output of a particular hash function.

minimizing (4) without the "z n = h(x n )" constraints: The resulting free codes try to achieve good precision/recall independently of whether a hash function can actually produce such codes.


Since the objective only depends on the distribution of the maximum coordinate of ĉt , we can continue the upper bound with a maximum over any distribution of random vectors whose coordinates are 0 with probability at least 1 − K/L and otherwise are −L or L with equal probability.

let t be a Rademacher random vector, we continue with Now observe that if we denote with a = Pr[Z t = L], the above is equal to We now argue that this maximum is achieved by setting a = K/L.


Since these quasi-indicator functions essentially take values in the discrete set {0, 1} rather than the continuous interval [0, 1], solving (P-rlx) is almost equivalent to solving either (P) or (P').


Section 3 for a detailed problem formulation.

we construct a sequence of convex approximations (upper bounds) to the expected cost, that can be optimized via semidefinite programing [50].


At the same time, we note that more aggressive quantization can hurt accuracy.

4-bit QSGD with 8192 bucket size (not shown) loses 0.57% for top-5 accuracy, and 0.68% for top-1, versus full precision on AlexNet when trained for the same number of epochs.


In this paper we consider unsupervised learning problems by optimal scoring, which was originally proposed to perform LDA by regression [6].

we devise a novel unsupervised framework by using the optimal scoring and the ridge penalty.


The transition probabilities for this Markov chain will be denoted using Q.

we will use some initial distribution for the state of the chain, Q(x (0) ), and will simulate the chain according to the transition probabilities Q(x (i) | x (i−1) ).


In Section 3, we present lower bounds on the sample complexity of sequential tests for minima.

we compute the optimal allocations for this problem and discuss the limitation of naive benchmarks to attain them.


This implies that any solution to (6) has the same cost as in (5) up to a (1 + ")-factor.

by solving (6) we obtain a (1 + ")-approximation to (5).


Instead, we suggest using an analogue of the augmented R-W model, which uses the above noisy-ORlAND-NOT prediction instead of the linear prediction implicit in the augmented R-W model.

we define the following algorithm (with all parameters as defined before), using the notational device that the C/ s are preventive and the C j ' s are generative: Unlike the R-W and augmented R-W models, there is no known characterization of the long-run behavior of this iterative algorithm.


It is intriguing (A) to see this quantity appear as analogous to the score function in the ICA likelihood model, and, (B) to speculate that experiments could show that this' voltage slope at threshold' is a hidden factor in STDP data, explaining some of the scatter in Figure 3A.

an STDP datapoint should lie on a 2-surface in a 3D space of {∆w, ∆t, uk }.


Theorem 1 For R 3way c-NN one can construct a data structure with O(n ρ log 1/cR0 n) query time and O(n 1+ρ ) space, where ρ = 1 The argument for 2-way resemblance can be naturally extended to k-way resemblance.

given three sets S 1 , S 2 , S 3 ⊆ Ω and an independent random permutation π : Ω → Ω, we have: Eq.


We then add event T 1 = ÎT0 to the list of target events, so that when the agent first enters ÎT0 , it creates a new option o T1 whose goal is to trigger T 1 .

the new option's termination function is set to the indicator function ÎT0 , and its reward function becomes the task's reward function plus an option completion reward for triggering T 1 .


We say the variable w n solves the ERM problem in (2) to within its statistical accuracy.

this implies that adding a regularization of order V n to (2) yields a problem that is essentially equivalent.


We observe that the distributions of the gradient g and Hessian M on the RHS of equation 3.5 are fairly complicated.

we need to deal with the interactions of α i 's (the components along x) and b i 's (the components in the orthogonal subspace of x).


For notational convenience we denote by S the symmetrized version of a tensor where care is taken to ensure that existing multiplicities are satisfied.

for a generic third order tensor we set to obtain a symmetric tensor.


In this experiment, we analyze the behavior of all algorithms when given different amounts of trajectories collected under different reference policies.

we consider five reference policies, where each chooses to apply the control action with a fixed probability p ∈ {0.1, 0.2, 0.3, 0.4, 0.5}.


This gives an improved bound on m and lets us partially relax the magnitude bound on parameters so that they can be arbitrarily negative: , then no RBM network of size m, whose weights are upper bounded in value by C, can compute a function which represents n-dimensional IP with margin δ.

for C and 1/δ bounded by polynomials in n, for n sufficiently large, this condition is satisfied whenever m < 2 (1/4− )n for some > 0.


PCFGs make some fairly strong independence assumptions, disregarding substantial amounts of structural information.

it does not appear reasonable to assume that the rules applied at level ¡ in the parse tree are unrelated to those applied at level ¡ £¢ ¥¤ .


However, the strong guarantees these methods enjoy rely on strong assumptions.

it is assumed that the hidden confounder is either categorical with known number of categories, or that the model is linear-Gaussian.


Because of identifiability issues which are subject of ongoing work, we prefer to compare various methods for sparse PCA in a denoising experiment.

we assume that the data were generated from sparse components plus some noise and we compare the various sparse PCA on the denoising task, i.e., on the task of recovering the original data.


The spectral properties of such an sampled matrix have been well studied in the literature [7,1,6,11,14].

it is known that the original matrix is close in spectral norm to the sampled one where the missing entries are filled in with zeros and properly rescaled under certain incoherence assumptions.


Thanks to its simplicity, the sampling procedure can be easily performed.

we provide each For each x ∈ B, generate a y; and set B ′ to contain all the generated (x, y); Update w using the gradient with proper regularizer.


In this paper, we present a RL algorithm for continuous state-action spaces that provides these kind of high-probability safety guarantees for policies.

we show how, starting from an initial, safe policy we can expand our estimate of the region of attraction by collecting data inside the safe region and adapt the policy to both increase the region of attraction and improve control performance.


Despite the fact that the assumption of the Kronecker product structure of covariance makes the statistical model much more parsimonious, it poses significant challenges.

the penalized negative log-likelihood function is non-convex with respect to the unknown sparse precision matrices.


In fact, we show that the values z(t, j), t = 1, . . . , k are distributed identically to points on a normalized Brownian motion.

let W (s) be a standard linear Brownian motion and let S(s) be the normalized Brownian motion We then show that, for every j, there exists times s 1 , .


However, safety concerns have been raised in regard to both this specific application and other biomedical applications of policy search algorithms.

the existing state-of-the-art gradient-based algorithms, including the current natural actor-critic algorithms, are unconstrained and could potentially select dangerous policies.


To resolve this problem, we enforce that samples appear at most once in running averages.

when a sample is revisited, its old assignment is first removed before it is reassigned.


It is based on modifying max-product by drawing upon a dual co-ordinate descent and barrier method.

the algorithm is as follows: (1) For small enough parameters ε, δ, run subroutine DESCENT(ǫ, δ) (close to) convergence.


Thus, provided we can compute S j,k for a single pair of features, we can avoid storing the entire matrix S. To avoid performing the whitening step for all of the input features, we can instead perform pair-wise whitening between features.

to compute the squared correlation of x j and x k , we whiten the jth and kth columns of X together (independently of all other columns), then compute the square correlation between the whitened values xj and xk .


We proved that our confidence estimate has low coverage error under mild assumptions on the learning algorithm.

we did so even for a changing policy that uses the confidence estimates.


Define λ to be sufficiently positive that the MAP configuration is either (−1, −1, −1, −1) or (1, 1, 1, 1), abbreviated by −1 and 1, respectively.

the solution is −1 when α + β < 0 and 1 otherwise.


A robust MDP considers the case where the transition probability is determined in an adversarial way.

when action a is taken at state s, the transition probability p s,a (•) can be an arbitrary element of the uncertainty set U(s, a).


The only other policy π 0 with which the learner could match the feature expectations of π 0 in her worldview (restricting again to deterministic policies) is the one that moves to the central cell as quickly as possible and then alternates between states 2 and 3. and hence π 0 is a better teaching policy than π * regarding the performance that a learner matching feature expectations in her worldview achieves in the worst case.

this example shows that providing demonstrations from the truly optimal policy π * does not guarantee that the learner's policy achieves good performance in general.


To learn the parameter vector a t ∈ IR d from data {(x ti , y ti )} m i=1 we would like to minimize the empirical error m i=1 L(y ti , a t , U x ti ) subject to an upper bound on the number of nonzero components of a t , where L : IR × IR → IR + is a prescribed loss function which we assume to be convex in the second argument.

this problem is intractable and is often relaxed by requiring an upper bound on the 1-norm of a t . we consider the problem min where γ > 0 is the regularization parameter.


The choice of the exponential rate νρ h is made to cover a very large class of functions, as well as to relate to results from prior work.

for a standard partitioning on R p and any α, β > 0, any function f such that f (x) ∼ x→x β||x − x || α fits this assumption.


4 A single neuron activation vector is one multidimensional variate, and a layer of neurons gives us a set of multidimensional variates.

we can consider two layers, L 1 , L 2 of a neural network as two sets of observations, to which we can then apply CCA, to determine the similarity between two layers.


However, there is a crucial sense in which this result "just barely" holds, rendering it arguably less interesting to us here.

let Γ * n denote the partition of S = {1, 2, . . . , n} in which each individual element forms its own cluster.


In this paper, we develop a sparse variant of Hinton's deep belief network algorithm, and measure the degree to which it faithfully mimics biological measurements of V2.

we take Ito & Komatsu [7]'s characterization of V2 in terms of its responses to a large class of angled bar stimuli, and quantitatively measure the degree to which the deep belief network algorithm generates similar responses.


We test whether the same DNN prior (G 6 ) that was trained on inverting the feature representations of ImageNet images generalizes to enable visualizing DNNs trained on different datasets.

we target the output neurons of two DNNs downloaded from Caffe Model Zoo [20]): (1) An AlexNet DNN that was trained on the 2.5-million-image MIT Places dataset to classify 205 types of places with 50.1% accuracy [26].


We study the reward of our algorithms, and their regret against a fixed comparator x: Comparison of Regret Bounds The primary contribution of this paper is to establish matching upper and lower bounds for unconstrained online convex optimization problems, using algorithms that require no prior information about the comparator point x.

we present an algorithm that, for any To obtain this guarantee, we show that it is sufficient (and necessary) that reward is Ω(exp(|g 1:T |/ √ T )) (see Theorem 1).


This approach not only eases computational overhead but also helps achieve low regret by being able to transfer information between actions.

bPM-TS runs orders of magnitude faster than any existing algorithms, opening the path for the model of partial monitoring to be applied on realistic settings involving large numbers of actions and outcomes.


Statistical learning methods are used to derive Bayesian estimates of the conditional probability of firing for each cell given the kinematic variables (we consider only hand velocity here).

we use non-parametric models of the conditional firing, learned using regularization (smoothing) techniques with cross-validation.


The second challenge introduced by using nested mini-batches is determining the size of M t . On the one hand, if M t grows too slowly, then one may suffer from premature fine-tuning.

when updating centroids using M t ⊂ {1, . . . , N }, one is using energy estimated on samples indexed by M t as a proxy for energy over all N training samples.


The learner's likelihood function is The teacher wants the learner to arrive at a posterior p(θ | D) peaked at θ * by designing a small D. As discussed above, this can be formulated as (2) with the KL loss() and the cardinality effort() functions: namely uniform over the version space consistent with D. The optimal teaching problem becomes min n,x1,y1,...,xn,yn − log One solution is the limiting case with a teaching set of size two D = {(θ * − /2, −1), (θ * + /2, 1)} as → 0, since the Teaching Impedance T I = log( ) + 2c approaches −∞.

the teacher teaches by two examples arbitrarily close to, but on the opposite sides of, the decision boundary as in Figure 1(right).


In a suitable metric it has been shown that this condition becomes sufficient and necessary [10].

if Θ is constant, F s is negative definite if and only if (MF) s is negative definite.


It is defined as the ground distance (i.e., cost function) between every two perceptual features, extending the notion of a distance between single elements to a distance between sets of elements.

consider two probability distribution x ∼ µ and y ∼ ν; EMD can be then defined as: where Π(µ, ν) denotes the set of all joint distributions γ(x, y) with marginals µ(x) and ν(y), and c(x, y) is the cost function (e.g., Euclidean or cosine distance).


The cluster assumption states that a cluster (or a high density area) contains only (or mostly) points belonging to the same class.

if x 1 and x 2 belong to the same cluster, the corresponding labels y 1 , y 2 should be the same.


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

because Shannon information does not make any assumptions about the code, it is not always apparent how information should be best extracted from the responses: just because information is 'there' does not mean we know how to get at it.


Similarly, the optimization problem of Sparse coding [10] is: From these formulations, it is clear there are links between ICA, RICA, sparse autoencoders and sparse coding.

most methods use the L 1 sparsity penalty and, except for ICA, most use reconstruction costs as a non-degeneracy control.


We will first show that, x (k) is sub-Gaussian with mean m k and parameter σ 2 k for a regime of λ we are interested in.

we will show that for |λ| ≤ 1/(2m k−1 r), By definition, due to distributional independence, we have E[e λx (k) ] = E[e λx (k) ] (l/ l) .


As shown in Figure 1c, in numerical experiments higher-order terms improve the Hogwild covariance on the block diagonal relative to the A = 0 approximation, and the improvements increase with local mixing rates.

the off-block-diagonal first-order term in the Hogwild covariance is nonzero and it depends on the local mixing performed by S. if global synchronization happens infrequently relative to the speed of local sampler mixing (e.g.


Asymptotically, the solution has the form of an almost everywhere constant function, with highly localized spikes near the labeled points, and so no learning is performed.

an interpretation in terms of a density-based kernel K p , as in the onedimensional case, is not possible.


To further test the effectiveness of our active selection scheme, we compare with another baseline that sequentially adds one feature at a time from a filtered pool of features.

we first rank the individual features based on their classification accuracy, and only consider the top 80 features (using 80 features achieves essentially the same accuracy as using 184 features).


As a by-product, we can prove that it is also optimal in terms of some other performance metrics.

we consider the following 3 metrics considered in prior works, and formally establish the optimality of polynomial code for each of them.


In this paper, we propose an alternative approach to LDS parameter estimation under exponential family observation noise.

we reformulate the LDS as a two-view generative model, which allows us to approximate the estimation task as a form of matrix factorization, and apply recent global optimization techniques for such models (Zhang et al., 2012;Yu et al., 2014).


Stronger models might enable us to identify a wider class of blurring kernels rather than just box filters.

they could provide a better strategy for identifying the blur direction.


When the instance is non-anomalous and all its views are consistent, all of the views are generated from a single latent vector.

the latent vector assignments for all views are the same, When it is an anomaly and some views are inconsistent, different views are generated from different latent vectors, and some latent vector assignments are different, i.e.


To fully describe Algorithm 1 we need to specify how to compute the confidence-set bounds (β t ) t . On the one hand, these bounds have to be accurate -the confidence set E t should contain the unknown θ * ; on the other hand, the larger they are, the larger the regret bounds we obtain.

the β t should be as small as possible subject to being accurate.


It consists of three parts: (1) the encoder maps input frames to the physical state of the objects (i.e., position, velocity, etc.).

we take in a sequence of 3 RGB frames from the simulation.


We prove recovery of the sparse and low rank components under a set of requirements which are tight and match those for the convex techniques (up to constant factors).

under the deterministic sparsity model, where each row and each column of the sparse matrix S * has at most α fraction of non-zeros, we require that α = O 1/(µ 2 r) , where µ is the incoherence factor (see Section 3).


We can also define the minimax loss, which is defined by choosing the w(•) which minimizes WorstCaseLoss(•).

minimaxLoss(M, cost, k) := min In the next section, we describe the optimal algorithm for a restricted class of M . That is, we obtain the mapping w which optimizes WorstCaseLoss(w; M, cost, k).


We propose SafetyNets, a framework that enables an untrusted server (the cloud) to provide a client with a short mathematical proof of the correctness of inference tasks that they perform on behalf of the client.

safetyNets develops and implements a specialized interactive proof (IP) protocol for verifiable execution of a class of deep neural networks, i.e., those that can be represented as arithmetic circuits.


Then the infimum depends only on the value of P 1 , P 2 on Ω 0 , the subset of Ω on which P 1 and P 2 are absolutely continuous with respect to each other: This is intuitive as we can always find a detector φ such that its value is arbitrarily close to zero on Ω \ Ω 0 .

if P 1 and P 2 have measure-zero overlap, then inf φ Φ(φ; P 1 , P 2 ) equals zero, that is, the optimal test for the simple test (P 1 , P 2 ) has zero risk.


In this paper, we instead propose a novel combination of unsupervised joint alignment with unsupervised feature learning.

we incorporate deep learning into the congealing alignment framework.


Let f : X → (−∞, +∞] be proper, lower semicontinuous and uniformly essentially strongly convex with modulus γ ∈ Γ U .

then (5) If, in addition, γ(r) := γ(r)/r is strictly increasing, then Df * is uniformly continuous with modulus of continuity χ(r) = γ−1 (r/2).


After this Markov chain has converged (i.e., finished the burn-in stage), we draw J samples {Z (j) } and estimate the expectation statistics To make prediction on unlabeled testing data using the prediction rule (5), we take the approach that has been adopted for the variational MedLDA, which uses a point estimate of topics Φ from training data and makes prediction based on them.

we use the MAP estimate Φ to replace the probability distribution p(Φ).


A standard way to approach optimization with inconsistent noise is to estimate the value of each set used by the algorithm to an accuracy ε via independent randomized sampling, where ε is chosen small enough so as to obtain approximation guarantees.

assuming that the algorithm only makes polynomially many value queries and that the function f is such that F (S) ∈ [b, B] for any set S, then a classical application of the Chernoff bound combined with a union bound implies that if the value of each set is estimated by averaging the value of m samples with m = Ω B log n bε 2 , then with high probability the estimated value F (S) of each set used by the algorithm is such that In other words, randomized sampling is used to construct a function which is εapproximately submodular with high probability.


Given an algorithm that identifies stationary points of the likelihood function of Bayesian networks (e.g., EM), we can now identify all stationary points of a component likelihood.

we just apply this algorithm to the sub-network of each component S, and then extract the parameter estimates of variables in S while ignoring the parameters of boundary variables.


Intuitively, aj represents the number of "imaginary samples" observed prior to observing any data.

if X is distributed multinomial with parameters 0 = (01, ... , Or), and p(O) is Dirichlet, then the probability that our next observation is Xj is aj/a*.


. . , y (n) } be an n-dimensional random vector following an n-variate Gaussian distribution N (µ, Σ), and let G = (V, E) be a Markov network representing the conditional independence structure of N (µ, Σ).

the set of vertices V = {1, . . . , n} corresponds to the set of variables in Y , and the edge set E contains an edge (i, j) if and only if y (i) is conditionally dependent on y (j) given all remaining variables; i.e., the lack of an edge between i and j denotes the conditional independence of y (i) and y (j) , which corresponds to a zero entry in the inverse covariance matrix Σ −1 ( [1]).


Similar to NS-ES and ES, NSR-ES operates on entire episodes and can thus evaluate reward and novelty simultaneously for any sampled parameter vector: ✓ i,m t = ✓ m t + ✏ i .

we compute f (✓ i,m t ) and N (✓ i,m t , A), average the two values, and set the average as the weight for the corresponding ✏ i .


We use these insights to propose an improved ensembling procedure in Section 5, which we empirically validate in Section 6.

we test VGG-16 [ For each model and dataset we train two networks with different random initializations to find two modes.


Note that since each edge was chosen randomly, we obtain the same distribution of causal graphs if we simply fix σ to be the identity permutation.

c n,p is just a random undirected graph G n,p in which we've directed all edges from lower to higher indexed vertices.


Recently, sparse learning via 1 regularization [20] and its various extensions has received increasing attention in many areas including machine learning, signal processing, and statistics.

the group Lasso [1,16,22] utilizes the group information of the features, and yields a solution with grouped sparsity.


We naturally define the reward as the negative acquisition cost.

in the episode (s 0 , a 0 , r 1 , s 1 , .


This section shows that the choice of a good parameterisation is not obvious, and that the use of the Riemannian geometry of the simplex [Ama95, GC11] is important in designing Langevin MCMC algorithms.

we show that an unnormalized parameterisation, using a mirroring trick to remove boundaries, coupled with a natural gradient update, achieves the best mixing performance.


Fortunately it turns out not hard to tackle this issue by using semi-definite programming (SDP) relaxation which linearizes the quadratic terms.

let I be the identity matrix, and define Then θ U x can be replaced by θ M u x and the quadratic term in (15) can be expanded as Since given (π, θ, γ, v) the objective function becomes linear in M , so after maximizing out these variables the overall objective is convex in M . Although this change of variable turns the objective into convex, it indeed shifts the intractability into the feasible region of M : Here M 0 means M is real symmetric and positive semi-definite.


The key aspect of our resulting formulation is that, unlike the standard sampler for the infinite-state HMM, dynamic programming can be used.

we describe a simple algorithm that monotonically decreases the underlying objective function.


By using either Oja's algorithm or Neon2 online , there exists an algorithm, denoted by ApproxNC-Stochastic, which uses stochastic gradient evaluations or stochastic Hessian-vector product evaluations to find the negative curvature direction for general stochastic nonconvex optimization problem (1.1).

approxNC-Stochastic returns a unit vector b Based on ApproxNC-Stochastic, we present our negative curvature descent algorithm in Algorithm 1.


In this paper, we provide worst-case regret bounds on Bayesian non-parametric methods, which show how these algorithms can have low regret.

we examine the case where the prior (over functions) is a Gaussian process -thereby extending the work of Kakade and Ng [2004] to infinite-dimensional spaces of experts.


4 can improve significantly (in terms of the constants) depending on the regularity of the loss function: smoother loss functions will result in "simpler" learning problems and vice-versa.

when corresponds to the squared geodesic distance, the more "regular" is the manifold M, the learning problem will be.


Since our underlying distribution p β (•) is Gaussian, it is natural to consider messages which are Gaussian distributions.

let Then, ( 6) is equivalent to synchronous consensus propagation iterations for K t and µ t . The sequence of densities is meant to converge to an approximation of the marginal conditional distribution of x j .


For deterministic experiments, we compare AdaNCG, AdaNCG + with their non-adaptive counterparts.

the non-adaptive counterpart of AdaNCG named NCG uses NCS(f, x, 2 /2, δ).


The isometric nature of the mapping φ 0 ensures that the manifolds M 0 remain Riemannian and that the curves γ 0 remain geodesic.

each γ 0 remains parametrizable (Gallot et al., 2004).


This question was recently considered in [21] and our results offer new insights.

recalling the results in [16], we see that in the slow rate setting there is essentially no difference between random features and Nyström approaches, neither from a statistical nor from a computational point of view.


Namely, they both require |θ 0,i | ≥ c max{σs 0 log p/ n, σ/ √ n}, which is √ s 0 larger than the ideal detection level [10].

in order for the coefficient θ 0,i to be detectable with appreciable probability, it needs to be larger than the overall 2 error, rather than the 2 error per coordinate.


In this section, we present simulations that verify our results.

we generate a matrix M ( ) ∈ R 1000×100 , and a fixed s * = 10-sparse vector x by using the construction given in (4.2) and (4.1) where ∈ (0, 1).


For each order, a derivation operation π Op of a given type Op specifies a sequence of symbols Y 1 . . . Y k (possibly the empty sequence ǫ), and can be applied to a partial derivation: with ⊕ being list concatenation.

a derivation operation involves popping the top item off the stack, choosing a derivation operation of the appropriate type, applying it to add some symbols to D yielding D ′ , and pushing a list of new items A back on the stack.


Intuitively, larger δ and smaller ε indicate more severe mode collapse.

if a large portion of the target P (S) ≥ δ in some set S in the domain X is missing in the generator Q(S) ≤ ε, we declare (ε, δ)-mode collapse.


Importantly, however, the unconfoundedness assumption is not assumed to hold for the observational data, which may be subject to unmeasured confounding.

both selection into the observational study and the selection of the treatment may be confounded with the potential outcomes of any one treatment.


However, the optimal inference network q(z|x 1 , ..., x N ) would be the true posterior p(z|x 1 , ..., x N ).

the conditional independence assumptions in the generative model imply a relation among joint-and single-modality posteriors: the joint posterior is a product of individual posteriors, with an additional quotient by the prior.


3.2-3.3 we discuss their dependence structure and explain how to estimate them from data respectively.

using the notions reviewed above, we define the Fiedler delta function ∆λ 2 in the following way: ∆λ 2 (u, v, G) is the variation in the Fiedler eigenvalue of the graph Laplacian that would result from flipping the value of X uv in G. Concerning the range of the Fiedler delta function, we can easily prove the following proposition: Proposition 2.


We propose the following model to capture the lingering of gradients.

for every x ∈ R d and index i ∈ [n], let δ(x, i) ≥ 0 be the lingering radius of ∇f i (x), meaning that 2 as long as we travel within distance δ(x, i) from x, the gradient ∇f i (x) can be reused to represent ∇f i (y).


The next result shows that no DR map f , continuous or not, can achieve perfect recall and perfect precision simultaneously -a widely observed but unproved phenomenon in practice.

it rigorously justifies the intuition that perfectly maintaining the local neighbourhood structure is impossible for a DR map.


For each of the candidate windows, we use a feature representation that is extracted from a trained Convolutional Neural Network (CNN).

we pass the image as input to the CNN and use the activation vector of the penultimate layer of the CNN as the feature vector.


We first examine how the L p,q WN-DNN architecture influences their generalization properties.

for L p,q normalization where q ≤ p * and 1/p + 1/p * = 1, we obtain a complexity bound that is independent of width and only has a square root dependence on the depth.


We chose to work with sequences of images, so the results are easy to analyse; yet BRUNO does not make any image-specific assumptions, and our conclusions can generalise to other types of data.

for non-image data, one can use a general-purpose Real NVP coupling layer as proposed by Papamakarios et al.


It can be seen that reshaped-WF outperforms all other three algorithms in computational time cost.

it is two times faster than truncated-WF and six times faster than WF in terms of both the number of iterations and computational time cost.


We show how to build hierarchical, reduced-rank representation for large stochastic matrices and use this representation to design an efficient algorithm for computing the largest eigenvalues, and the corresponding eigenvectors.

the eigen problem is first solved at the coarsest level of the representation.


Traditional approaches are not necessarily appropriate for probabilistic inference because there are tradeoffs between faster sampling times and convergence which are not easy to characterize.

different inference methods may affect the amount of parallelism that can be exploited in a model.


Finally, the optimal solution A * of ( 11) is composed of α * jgi = S( ), where S is the shrinkage operator defined as follows: Given the smooth approximation of Ω(W) in (11) and the corresponding gradient presented in Theorem 1, we could apply gradient descent method to solve the problem.

we replace the overlapping-group-lasso penalty in (8) with its smooth approximation f µ (W) to obtain the following optimization problem min where is the augmented logistic regression loss function.


The key object in this paper is the Gaussian Mixture Model (GMM), whose probability density is and where p N is a (multivariate) Gaussian with mean µ ∈ R d and covariance Σ 0.

0} K j=1 and weights α ∈ ∆ K , the K-dimensional probability simplex.


Discriminatively training the disaggregation bases B1:k is naturally framed as a structured prediction task: the input is X, the multi-variate desired output is A ⋆ 1:k , the model parameters are B1:k , and the discriminant function is F ( X, B1:k , A 1:k ).

1 we seek bases B1:k such that (ideally) While there are many potential methods for optimizing such a prediction task, we use a simple method based on the structured perceptron algorithm [5].


It is clear, however, that if we evaluate the performance of our batch policies in terms of experimental time, then there is a very significant advantage over Sequential.

the amount of experimental time for a policy is approximately equal to the number of requested batches, assuming that the batch size is selected to allow for all selected experiments to be run in parallel.


In [5] Tsybakov showed that if P has a noise exponent q then ERM-type classifiers can obtain rates in (1) which are of the form n − q+1 q+pq+2 , where 0 < p < 1 measures the complexity of the hypothesis class.

rates faster than n −1/2 are possible whenever q > 0 and p < 1.


The second term κ i (p) measures the effect of interactions between edges corresponding to the same parameter (edges from the same E i ) on the same path from input to output.

if for any path from an input unit to an output unit, no two edges along the path share the same parameter, then κ (2) (p) = 0.


Since we do not have an exact form for p(•), nor p(x), we apply Gibbs sampling to draw samples from p(x).

at sweep round t, we update each variable x i by drawing samples from Such conditional probabilities are straightforward to approximate.


Next, we define the following modification of f , which could be the function returned by an -erroneous oracle.

note that f (x) and f (x) differ only in the region where | i∈A x i − j∈B x j | ≤ 1 2 n. the value of f (x) in this region is within ], while f (x) = 1 2 , so an -erroneous oracle for f (x) (both in the relative and absolute sense) could very well return f (x) instead.


However, threshold potentials with nonuniform weights are strictly more general than concave cardinality potentials.

there exists w and y such that © w;y @AA cannot be expressed as j j @jR j AjA for any collection of concave j and sets R j .


Stochastic quantization performs well in CNNs [13,23].

for the ternary case, the quantized value is sampled from a Bernoulli distribution.


In many important applications, the algorithm designer is faced with multiple objectives to optimize.

in online advertising it is fairly common to optimize multiple metrics, such as clicks, conversions, and impressions, as well as other metrics which may be largely uncorrelated such as 'share of voice', and 'buyer surplus'.


This interpretation makes it easy to formulate a cost: given any x ∼ P G (x), we want P G (c|x) to have a small entropy.

the information in the latent code c should not be lost in the generation process.


Despite its consistency, the algorithm has some significant caveats.

theorem 3 implies that we may need k > rank(M ) to guarantee consistency, which increases the state dimensionality.


Convexity has played an increasingly important role in machine learning in recent years, echoing its growing prominence throughout applied mathematics (Boyd and Vandenberghe, 2003).

a wide variety of two-class classification methods choose a real-valued classifier f based on the minimization of a convex surrogate φ(yf (x)) in the place of an intractable loss function 1(sign(f (x)) = y).


We finally remark on the uniform convergence assumption, which holds when the constraint functions are linear [25], but unfortunately does not hold for general convex Lipschitz functions.

one can simply show examples where there is no uniform convergence for stochastic convex Lipchitz functions in infinite dimensional spaces [21].


To address such a challenge, we develop a general inferential theory of the EM algorithm for parameter estimation and uncertainty assessment of high dimensional latent variable models.

we make two contributions in this paper: • For high dimensional parameter estimation, we propose a novel high dimensional EM algorithm by attaching a truncation step to the expectation step (E-step) and maximization step (M-step).


Note that the shape of W i,j (t) is remarkably reminiscent of the experimentally found measurements for STDP [8].

the shape of the STDP curve depends on the type of neuron and is different for connections from excitatory to excitatory than from excitatory to inhibitory or inhibitory to inhibitory neurons (Figure 3).


We now outline an approximation procedure that restricts the candidate neighborhood assignments.

for a local optimization at any node i, we may restrict the possible local neighborhoods at any iteration t to only those configurations that are at most h Hamming distance away from the best local configuration for i in iteration t-1.


More recent work (Yuille in preparation) clarifies the class of maximum likelihood inference problems that can be "solved" by GLRW and by non-linear GLRW.

we show that a non-linear RW can perform ML estimation for the non-generic case studied by Cheng.


• We propose SchNet: a neural network specifically designed to respect essential quantumchemical constraints.

we use the proposed cfconv layers in R 3 to model interactions of atoms at arbitrary positions in the molecule.


This is done because (i) it puts λ n and the regularization term γ together, and (ii) it allows more freedom in the choice of the prior for β. Additionally, it has an interesting interpretation, in that the SVM loss function behaves like a globallocal shrinkage distribution [9].

γ −1 corresponds to a "global" scaling of the variance, and λ n represents the "local" scaling for component n. The {λ n } define the relative variances for each of the N data, and γ −1 provides a global scaling.


The ESS and TN-ESS are comparable between HMC and the Gibbs sampler.

for 100 inducing points and the RBF covariance, ESS and TN-ESS for HMC are 11 and 1.0 • 10 −3 and for the Gibbs sampler are 53 and 5.1 • 10 −3 .


We circumvent this hardness by positing an assumption on the structure of rewards.

we assume that the slate-level reward is a sum of unobserved action-level rewards that depend on the context, the action, and the position on the slate, but not on the other actions on the slate.


We have studied the task of efficiently identifying NN models with low accuracy by exploring possibilities within the theoretical framework of sublinear algorithms and evaluated our approach by moving to experiments.

we have proved that there is a one-sided error property tester with complexity O( √ nk 2 / 2 ), i.e., a sublinear (randomized) algorithm that decides whether an input graph G is a k-NN graph or requires many edge modifications to become a k-NN graph (i.e.,  precision 1 and recall 2/3 when taking -far graphs as relevant).


The different tests did not require any special tuning of parameters.

the step-size in [13] (their ↵ variable) is always 0.1.


In our deep generative replay framework, the model retains previously acquired knowledge by the concurrent replay of generated pseudo-data.

we train a deep generative model in the generative adversarial networks (GANs) framework [10] to mimic past data.


[6] and Ruozzi and Tatikonda [7], we will use graph covers in order to better understand the behavior of the Bethe approximation with respect to the partition function.

we will show that if a graphical model is strictly positive and contains even one frustrated cycle, then there exists a choice of external field and a 2-cover without frustrated cycles whose partition function provides a strict upper bound on the partition function of the original model.


The most accurate model for a given target cluster t minimizes the loss with respect to the conditional probability p(x, y|t) referred to as the target distribution.

in [2] it is shown that: where: by using sample-specific weights r t (x, y) that match the training distribution c p(c)p(x, y|c) to the target distribution p(x, y|t) we can minimize the expected loss with respect to the target distribution by minimizing the expected loss with respect to the training distribution.


Compute V k and Σ k , the top k eigenvectors and eigenvalues of (∑ We now present empirical evaluation of our kernel learning framework and our semi-supervised kernel dimensionality approach when applied in conjunction with k-nearest neighbor classification.

using different regularization functions, we show that our framework can be used to obtain significantly better kernels than the baseline kernels for k-NN classification.


The difference among the three methods lies in the correlation between the second and third tasks.

in SNR 1 and MTRL, those correlations are close to 0, indicating that these two tasks are nearly uncorrelated, and hence the knowledge shared among the three tasks is very limited for SNR 1 and MTRL.


It follows from Theorem 1 in [11] and Theorem 1 in [10] and Lkk γ . Theorem 1 states that the quality of the bound depends on both Lkk and the maximum of { Lkm } m∈[K],m =k .

the worstcase regret will be lower if the k th true hypothesis w * k can predict the labels for training examples in both the k th task itself as well as those in all the other related tasks in high confidence.


We consider heavy-tailed payoffs with finite (1 + ϵ)-th raw moments in the analysis.


2 More efficient representations are possible if the goal is to preserve only the distances in a certain range.

o( −2 log n) bits are sufficient to distinguish between distances smaller than 1 and greater than 1 + , independently of the precision parameter [10] (see also [16] for kernel generalizations).


This idea exploits the observation that CNN filters have a very characteristic structure that reappears across data sets and problem domains.

cNN weights can typically be captured with a small number of degrees of freedom.


Network output error measures the difference between N e (I e ) and N p (I s ).

we report the root-mean-square error (RMSE) over all pixels and channels.


This leads to the Stochastic Gradient Method (SGM).

starting from an initial estimate x 0 ∈ K obeying the constraints, SGM iteratively applies the following updates at every iteration t, the current iterate x t is updated by adding µ t g t , where g t is an unbiased estimate of the gradient ∇F (x t ) and µ t is the learning rate.


For all activation functions σ i , i = 1, 2, ..., d − 1, there exist positive constants q(i) and Q σ (i) such that for any 0 ≤ Note that Assumptions 1 and 2 hold for a variety of commonly used activation functions including ReLU, sigmoid, softPlus and tanh.

in Assumption 2, the existence of the constants q(i) and Q σ (i) are more important than the particular values they take, which affect only the constant terms in our bounds presented later.


The offline approximation algorithm for ranking with submodular valuations will be a crucial tool in our analysis of online submodular set cover.

this offline algorithm can viewed as constructing the best single permutation S for a sequence of objectives F 1 , F 2 . . . F T in hindsight (i.e., after all the objectives are known).


The conditions of Theorem 2 hold for many cases in practice, in particular in the case of the exponential loss (Φ = exp) or the logistic loss (Φ(−x) = log 2 (1 + e −x )).

linear convergence is guaranteed in those cases since both the exponential and logistic losses are strongly convex over a compact set containing the converging sequence of α t s. MDEEPBOOSTSUM(S = ((x 1 , y 1 ), .


(X, Y ) = ( X, Ŷ ) whenever ν k are picked so that ) and .

choice of ν 3 = (3ζ(U,V )) p (µ(Ω (3) )) 1−p and )) 1−p for any p ∈ [0, 1] in these respective intervals guarantees exact recovery.


In this paper, we address the finer and more challenging question of determining, under the general LSBM, the minimal number of misclassified items given the parameters of the model.

for any given s = o(n), our goal is to identify the set of parameters such that it is possible to devise a clustering algorithm with at most s misclassified items.


Note that R pu ( g pu ) keeps decreasing and goes negative.

this should be fixed since R(g) ≥ 0 for any g. it holds that is not always true, which is a potential reason for uPU to overfit.


From a mathematical point of view, the relevance of Fourier theoretic ideas in all these cases is a direct consequence of equivariance, specifically, of the fact that the {T s g } g∈G operators form a representation of the underlying group, in the algebraic sense of the word [20].

it has been shown that whenever there is a compact group G acting on the inputs of a neural network, there is a natural notion of Fourier transformation with respect to G, yielding a sequence of Fourier matrices {F s } at each layer, and the linear operation at layer s will be equivariant to G if and only if it is equivalent to multiplying each of these matrices from the right by some (learnable) filter matrix H s [7].


We apply our results to the K-MAX problem and expected utility maximization problems.

for K-MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first Õ( √ T ) bound on the (1− )-approximation regret of its online problem, for any > 0.


However, such a strategy entails additional storage costs.

for implementing Algorithm 2, we must store the gradients {rf i (↵ t i )} n i=1 , which in general can cost O(nd) in storage.


Branching: The branching operation in our method is carried out using the property of a simplex.

since, in a n-simplex, any r + 1 vertices are not on a r − 1-dimensional hyperplane for r ≤ n, any n-simplex can be divided as S = ∪ p i=1 S i , where p ≥ 2 and S i are n-simplices such that each pair of simplices S i , S j (i ̸ = j) intersects at most in common boundary points (the way of constructing such partition is explained in Section 4.2).


In this work we re-examine the phenomena of statistical adaptation in the motion sensitive neurons of the fly visual system.

we are interested in adaptation to the variance or dynamic range of the velocity distribution [10].


The Cue-based model, in contrast, is unsure: it assigns roughly equal probabilities of helping or hindering to these cases (bar heights near 0.5).

the Cue-based model is effective at inferring simple object goals of maze-world agents, but is generally unable to distinguish between the more complex goals of helping and hindering.


• We propose new losses that effectively address the aforementioned challenges.

we propose a weighted cross entropy loss for classification that accounts for the imbalance in the impact of misclassification for background class as opposed to object classes (Section 3.2), a teacher bounded regression loss for knowledge distillation (Section 3.3) and adaptation layers for hint learning that allows the student to better learn from the distribution of neurons in intermediate layers of the teacher (Section 3.4).


We also note that bounded vertex cover Bayesian networks have a close connection to naive Bayes classifiers.

variables outside a vertex cover are conditionally independent of each other given the vertex cover.


Our approach again outperforms all approaches by a large margin.

our approach achieves ∼12% higher scores than 3DVP [12] on Cars in moderate and hard data.


The advantages of such a black-box framework are two fold: theoretically, it highlights a structural connection between the two problems; practically, it allows one to simply plug in any of the vast majority of solvers available for SLR and directly get an algorithm for SPCA with provable guarantees.

• For hypothesis testing: we match state of the art provable guarantee for computationally efficient algorithms; our algorithm successfully distinguishes between isotropic and spiked Gaussian distributions at signal strength ✓ & q k 2 log d n . This matches the phase transition of diagonal thresholding [22] and Minimal Dual Perturbation [4] up to constant factors.


Similarly, we would expect that if the first two parts of condition 5 hold (about X and Y ) then the last one about ω would also hold as it is predicting outcomes Y .

the last part of condition 5 is essentially a requirement on our ω-leaner base method that it's not doing anything strange like adding unnecessary noise to Y thereby making it have fewer moments.


Incomplete datasets may still exhibit a certain structure.

certain variables may always be observed in the dataset, while others may always be unobserved (hidden).


However, ignoring the uncertainty in the hyperparameters can lead to pathologies.

the reliability of the variance for Z depends crucially upon marginalising over all unknown quantities.


We alleviate this problem by imposing a weight w i to each observation in the constraint.

this weight is chosen to be , where f is a reasonable guess of the final minimizer f . Our goal is to weight the Hinge loss in the constraint, w i H −ε (y i f (x i )), so that it approximates the indicator function 1{y i f (x i ) < −ε}.


In a survey question that asks to compare Proto and Criticism Condition and Raw Condition, a subject added that "[Proto and Criticism Condition resulted in] less confusion from trying to discover hidden patterns in a ton of images, more clues indicating what features are important".

in a question that asks to compare Proto and Criticism Condition and Proto Only Condition, a subject said that "The addition of criticisms made it easier to locate the defining features of the cluster within the prototypical images".


Model Discrimination: The above results indicated that we had identified several high-performing networks in quite distinct architecture families.

the strong performance constraint allows us to identify several specific candidate model networks for the biological system, reducing a much larger set of mostly non-performing neural networks into a "shortlist".


. . , T T ), training/validation data with labels, λ   Even though we can solve (IP) via its LP relaxation, the resulting LP can be too large in practical applications for any general-purpose LP solver.

the number of variables and constraints is roughly O(T × |T max | + N × T × K max ), where T is the number of trees; |T max | is the maximum number of nodes in a tree; N is the number of examples; K max is the maximum number of features an example uses in a tree.


This extends the applicability of Riemannian Manifold MCMC to a much wider class of statistical models than at present.

we do so by estimating the covariance structure of the tangent vectors at a point on the Riemannian manifold induced by the statistical model.


and the optimal λ is given by Using this corollary we can make a simple modification to Algorithm 1 in order to use it for variable selection.

we modify the computation of the matrix D (penultimate line in Algorithm 1) as D = Diag(λ), where the vector λ = (λ 1 , .


In this work, we present a resampling-free analysis for the alternating minimization algorithm applied to the multi-response regression.

we focus on the high-dimensional setting of multi-response linear models with structured coefficient parameter, and the statistical error of the parameter can be expressed by the complexity measure, Gaussian width, which is related to the assumed structure.


The nodes of DDTs correspond to hyperrectangles (cells) in [0, 1] d (see Figure 1).

given a hyperrectangle A = d r=1 [a r , b r ], let A s,1 and A s,2 denote the hyperrectangles formed by splitting A at its midpoint along coordinate s. define A s,1 = {x ∈ A | x s ≤ (a s + b s )/2} and A s,2 = A\A s,1 .


If we define C ⇤ := max(kx ⇤ xk, ku ⇤ ūk), then the unique solution x( ) of ( 5) satisfies kAx( If in addition the parameter 10C⇤ kdk min( P , D ) , then we have In practice, we solve (5) approximately, using an algorithm whose complexity depends on the threshold ✏ for which the objective is accurate to within ✏.

we seek x such that where the left-hand inequality follows from the fact that f is strongly convex with modulus 1 .


This means that we end up minimising an upper bound on the loss, instead of the loss itself.

solving (6) requires only that we are able, for any value of θ, to find for each value of θ, we are able to identify the mapping which is consistent with the model (eq.


Here we train a Recurrent Neural Network (RNN) on a language modeling task.

we conduct word-level prediction experiments on the Penn Tree Bank (PTB) dataset [Marcus et al., 1993] using the 929k training words and its 73k validation words.


We also provide it online on our project page.

we take a snapshot of our model and generate captions for a subset of MS-COCO images [20] using greedy decoding.


This pooling-over-position arrangement yields complex cells with larger receptive fields.

the operation transforms a position sensitive input to a (somewhat) translation invariant output.


This notion of covering is related to the Ψ function discussed in the previous section.

if a set L α-covers a graph than it is necessarily the case that Ψ(L) ≥ α.


We now apply a version of Fano's lemma [11] to the set of distributions P T θ for θ ∈ Θ(δ).

using the proof of Lemma 3 in [11]  Each term KL(P θi (X t )||P θ i (X t )) is at most the KL divergence g(δ) between Bernoulli variates with parameters 1/2 + δ and 1/2 − δ.


The k-segment mean problem optimally fits a given discrete time signal of n points by a set of k linear segments over time, where k ≥ 1 is a given integer.

we wish to partition the signal into k consecutive time intervals such that the points in each time interval are lying on a single line; see Fig.


Later, another thresholding algorithm for the fixed budget setting addressed the FWER and FWPD metrics [7].

if their procedure is given a budget exceeding (n ) then the FWER is driven to zero, and the FWPD is driven to one.


Using the parametrization described in Subsection 2.1, we hope to cooperatively optimize GGN and LON, simultaneously estimating 3D camera pose, 3D room layout, and 3D object bounding boxes, in the sense that the two networks enhance each other and cooperate to make the definitive estimation during the learning process.

we propose three cooperative losses which jointly provide supervisions and fuse 2D/3D information into a physically plausible estimation.


As a basic performance measure, we evaluated the ability of our method to predict new protein-protein interactions, showing that it achieves better performance than several other models.

our results validate our assumption that we can explain interactions via the interactions of active sequence motifs.


The GKR proof protocol operates backwards from the circuit outputs to its inputs.

gKR uses sum-check to reduce the prover's assertion about the circuit output into an assertion about the inputs of the output layer.


We split z ∈ R d into local copies of x i ∈ R d , while enforcing the equality constraints x i = z for all i.

we consider the following reformulation of (1.1) where h(z) Our algorithm uses the Lagrangian relaxation of the equality constraints, and at each iteration a (possibly non-uniformly) randomly selected primal variable is optimized, followed by an approximate dual ascent step.


We can introduce a Poisson point process to control the duration of non-recombinant inheritance.

given that C i,t = k, then with probability e −dr +(1−e −dr )π kk , where d is the physical distance between two loci, r reflects the rate of recombination per unit distance, and π kk is the self-transition probability of ancestor k defined by HMDM, we have C i,t+1 = C i,t ; otherwise, the source state (i.e., ancestor chromosome k) pairs with a target state (e.g., ancestor chromosome k ) between loci t and t+1, with probability (1−e −dr )π kk .


Such a self-reconstruction task can help narrow the domain gap (see more explanation below).

assuming that the forward and reverse projections have the same importance for ZSL, our DIPL model solves the following optimization problem: where W ∈ R d×k is a projection matrix from the semantic space to the feature space, and λ, γ are the regularization parameters.


We also examine the speed-up of sp-tree over other algorithms.

the CPU time and the speedup of sp-tree search over LSH is summarized in Table 2.


Hence the algorithm reduces to minimizing ρ a (i.e., choosing a split as balanced as possible) at each internal node a ∈ I.

greedy minimization of ( 2) is equivalent to GBS.


For ¬ ¼, let ´Þ ¬µ consist of the elements of Ù ¾ such that ´Ù Þµ ¬ (that is, the ¬-ball centered at Þ).

then (any «-packing must have at most ´ ¬ «µ elements within distance ¬ of Þ.)


Specifically, assume that the binary response y ∈ {−1, 1} to each query x ∈ X is an independent realization of the random variable Y satisfying , where h * ∈ H is fixed but unknown.


A scoring rule is a function S : P × O → R where, for all P ∈ P, P ∈ argmax Q∈P E X∼P S(Q, X).

if you are paid S(P, X) upon stating belief P ∈ P and outcome X occurring, then you maximize your expected utility by stating your true belief.


3 for the quantification of the quality.

we quantify by how much the social network information can reduce sample complexity.


We do so by "flipping" the edges in P to obtain a matching with higher weight.

let M = M − (P ∩ M ) + (P ∩ M * T ) be the matching containing all edges in M except the ones in P , which are replaced by the edges in P ∩ M * T . It is easy to see that M is a matching in T e (t), and that w(M ) > w(M ).


Our approach addresses it as a search over the possible parameter vectors θ ∈ IR |F | .

rather than optimizing the log-likelihood itself, we introduce a Laplacian parameter prior for each feature f k takes the form We define P (θ) = k P (θ k ).


To further improve the computational efficiency of MKL, we extended the level method [6], which was originally designed for optimizing non-smooth functions, to the optimization of convex-concave problems.

we regard the MKL problem as a saddle point problem.


More precisely, let δ (l)→i be the message sent from subnetwork l to X i and δ i→(l) the opposite message.

then we define the COMPOSE message passing scheme as follows: each subnetwork computes its local pseudo-max-marginals over each of the individual variables, given, as input, the pseudo-max-marginals over the others.


We use π to denote the NP update to distinguish it from the conventional Bayesian posterior π(θ|x noting that in general the nonparametric posterior π(θ | x 1:n ) will be different to the standard Bayesian update as they are conditioning on different states of prior knowledge.

as stated above, π(θ|x 1:n ) assumes artificially that F 0 ∈ F Θ .


Each push at level i uncovers a collection of length-(i + 1) paths terminating at S -for example, in the figure, we have uncovered all length 2 and 3 paths, and several length 4 paths.

the crucial observation is that each uncovered path of length i starting from a node v is accounted for in either q i v or r i v . in Figure 1, all paths starting at solid nodes are stored in the estimates of the corresponding states, while those starting at blurred nodes are stored in the residue.


To study the marginal, our primary tool is the theory of convex geometry.

we will use an extension of the Prékopa-Leindler inequality developed by Brascamp and Lieb [11], which allows for a characterization of the integral of s-concave functions.


It is worth noting that we used a Gaussian model for inverse covariance even though the data was highly non-Gaussian.

neural firing time series data is generated by a nonlinear, mutually-exciting point process.


The observed data vector ¡ is then given by a linear combination of the $© , plus additive isotropic Gaussian noise.

¡ ¡ &% § (' , 1 Here, it must be stressed that in these low-level neural network models, the hidden variables that the neurons represent are not what we would typically consider to be the 'causal' variables of a visual scene.


For infinite capacity Q, R(✓ ; Q) is zero for all ✓ 2 ⇥, and the objective reduces to maximum likelihood.

when Q is the set of Gaussian distributions (as is the case in the standard VAE), then a Gaussian variational family regularizes the true posterior p ✓ (z | x) toward being Gaussian [10].


This suggests a modification of Algorithm 1, in which one-step co-training is carried out on the problem of extracting anchors rather than the problem of labeling web pages.

one might map f 1 's predictions from web pages to anchors, by giving a positive label to anchor a iff a links to a page x such that f 1 (x) = 1; then use WL 2 algorithm A 2 to learn a predicate p 2 ; and finally, map the predictions of p 2 from anchors back to web pages.


Let µ denote a vector of marginal probabilities (also called mean parameters) for each node and edge of the MRF.

µ includes µ i (x i ) ∀i ∈ V and µ ij (x i , x j ) ∀(i, j) ∈ E. The set of µ that arises from some joint distribution p is referred to as the marginal polytope: The MAP problem is then equivalent to solving the following LP: It can be shown that there always exists a maximizing solution µ which is integral and gives the optimal x.


The toolkit for proving our results derives from the deep literature of Euclidean functionals, see, (Steele, 1997;Yukich, 1998).

our strong consistency result uses a theorem due to Redmond and Yukich (1996) that essentially states that any quasi-additive power-weighted Euclidean functional can be used as a strongly consistent estimator of Rényi entropy (see also Hero and Michel 1999).


Of course, C ⋆ is not known, and it is not possible to exhaustively consider all possible values in the domain of C. Instead, we examine only a finite number of possible values of C over a fine discretization of its domain.

let N ǫ/2 (S r−1 ) denote a finite ǫ /2-net of the r-dimensional ℓ 2 -unit sphere; for any point in S r−1 , the net contains a point within an ǫ /2 radius from the former.


Impressive progress has been made in a variety of application domains, such as speech, natural language and image processing, thanks to the development of new learning strategies [15,53,30,45,26,3] and of new architectures [31,44,46,23].

these architectures tend to become ever deeper, with hundreds of layers, each of which containing hundreds or even thousands of units.


we do not have evidence that this reduced freedom in training labels adversely affect the power of the teaching set solution.


3 shows, we could scale to these large domains because, the complexity of sampling the proposal is feasible even when generating the ground MLN is infeasible.

we show the time taken to generate the proposal distribution (C-Time) and the the number of weighted samples generated per minute during inference (I-SRate).


This produces XORs of average length nq.

note that every two complementary XOR constraints involving the same subset of V (e.g., c ⊕ d and c ⊕ d ⊕ 1) are chosen with the same probability irrespective of q.


We can then plot the cumulative regret In the first experiment, the baseline reward is nonlinear.

we generate rewards using r t (s t,at , st , a t ) = θ T s t,at + 2I(|[s t ] 1 | < 0.8) + n t where n t = N (0, 1) and θ ∈ R 8 is a fixed vector listed in supplement section 1.1.


The original model of [18] does that by exploiting an integral form of the RBF kernel.

given two patches x and x , convolutional kernel networks provide two vectors .


Letting |I j | = n, we now simplify the objective function: where the last two equalities follow first by applying our derivation for λ and then the fact that qj sums to one.

this shows that we can choose an arbitrary index i ∈ I j and propagate the value g i + β j log qi + β j log n. for numerical reasons we choose the one that maximizes qi .


In particular, the key intermediate quantity for filtering is the "extended and marginalized" belief In many models (including Hidden Markov Models (HMMs) and Kalman filters), the extended state p t is linearly related to the predictive state q t -a property we exploit for our framework.

for some linear operator W . For example, in a discrete system ψ t can be an indicator vector representing the joint assignment of the next k observations, and ξ t can be an indicator vector for the next k + 1 observations.


In contrast, our result is an improved It is worth emphasizing that the proposed homotopy smoothing technique is different from recently proposed homotopy methods for sparse learning (e.g., 1 regularized least-squares problem [20]), though a homotopy strategy on an involved parameter is also employed to boost the convergence.

the involved parameter in the homotopy methods for sparse learning is the regularization parameter before the 1 regularization, while the parameter in the present work is the introduced smoothing parameter.


from pseudo-inputs , where K mm is a M ×M covariance matrix.

from (5) we have , where u = 1 − Yf , K mn is the cross-covariance matrix between {v m } M m=1 and {x n } N n=1 , and Note that if we drop the diag(•) term in (10) due to the i.i.d.


The error bound condition has been developed for such kind of problems [21].

if F (x) has a polyhedral epigraph, then there exists c > 0 such that dist(x, Ω * ) ≤ c(F (x) − F * ) for any x ∈ R d . It then implies HOPS has an O(log( 0 / )) iteration complexity for solving a non-smooth REL minimization with a polyhedral epigraph.


We call p n (x) the truncated Chebyhshev series of degree n. For analytic functions, the approximation error (in the uniform norm) is known to decay exponentially [26].

if f is analytic with f ( b−a 2 z + b+a 2 ) ≤ U for some U > 0 in the region bounded by the ellipse with foci +1, −1 and sum of major and minor semi-axis lengths equals to ρ > 1, then (2) Given a matrix A ∈ S d×d and a function f : R → R, the spectral-sum of A with respect to f is where tr (•) is the matrix trace and λ 1 , λ 2 , .


The results of all the methods are compared in Table 1, where evaluation measures include both peak signal-to-noise ratio (PSNR) and running time (Time).

compared with the state-of-theart single-image SR methods (e.g., SR-CNN, ANR and K-SVD), our multi-frame-based method can surpass them by 0.28∼0.54 dB, which is mainly attributed to the beneficial mechanism of temporal dependency modelling.


We choose a vertex i and look for a spanning tree which minimizes the inverse centrality R(i) of vertex i, that is we solve the problem Note that R(i) is the contribution of vertex i to the total resistance of T and that, by equations (3.1) and (3.2), R(i) = nT + ii + Rtot n . The above problem can then be interpreted as minimizing a tradeoff between inverse centrality of vertex i and inverse connectivity of the tree.

(4.3) encourages trees which are centered at i and, at the same time have a small diameter.


The first and best-studied instance of this family of kernels are random walk kernels, which count matching walks in two graphs [5,7] to quantify their similarity.

the geometric random walk kernel [5] is often used in applications as a baseline comparison method on graph benchmark datasets when developing new graph kernels.


We established the equivalence of our distributed OCC DPmeans, OFL and BP-means algorithms to their serial counterparts, thus preserving their theoretical properties.

the strong approximation guarantees of serial OFL translate immediately to the distributed algorithm.


If E is uncountable, this σ-algebra is too coarse to resolve events of interest 1 .

it does not contain the singletons (one-point sets), such that the measure P E X is incapable of assigning a probability to an event of the form {X E = x E }.


As mentioned in the introduction, we avoid such arbitrary hard separation thresholds by considering repulsive priors that smoothly push components apart.

we propose two repulsion functions defined as with A = {(s, j) : s = 1, . . . , k; j < s} and g : + → [0, M ] a strictly monotone differentiable function with g(0) = 0, g(x) > 0 for all x > 0 and M < ∞.


As described previously, we will use a negative exponentiated distance for κ.

κ = αe −βd + ᾱd − β d, where d, d are the OTMANN distance and its normalised version.


We write s := 1 1+ρ and define P := Q s µ.

then for all sequences (f n ) of measurable functions f n : X → R the following are equivalent: for measurable f : X → R we have S P (f ) = 0 if and only if R P (f ) = R P . Proof: For n ∈ N we define E n := {f n > 0} {h > ρ}.


The core of the algorithm is the way knownness is computed and how it's used to make the estimated transition function optimistic.

if we use a uniform fixed grid instead of the knownness-tree, the algorithm starts to act similar to MBIE (Strehl & Littman, 2005).


In the theoretical analysis of the Lin-UCB [7] and its kernelized version Kernel-UCB [18] θ a is replaced by θ, and the goal is to learn one single estimator using data from all the arms.

the data from the different arms are pooled together and viewed as coming from a single task.


We propose a hybrid approach of [2] with BQ, taking the form of a quadrature rule, that (i) carries a full probabilistic interpretation, (ii) is amenable to rigorous theoretical analysis, and (iii) converges orders-of-magnitude faster, empirically, compared with the original approaches in [2].

we prove that super-exponential rates hold for posterior contraction (concentration of the posterior probability mass on the true value of the integral), showing that the posterior distribution provides a sensible and effective quantification of the uncertainty arising from numerical error.


A Markov chain is defined using these affinities by setting the transition probability 3 4 (' ) from vertex 5) to vertex to be proportional to the edge affinity, 0 6 (' ) .

3 4 1' ) 7¥ 98 A@ CB ) 0 1' ) where 8 ) D¥ FE FG IH B 0 1' ) gives the normalizing factor which ensures E G IH B 3 P (' ) P¥ Q § .


This is a less restrictive setting than the setting we consider in this paper.

our construction can be directly used as a label-private learner simply by splitting the training set into two parts and discarding the labels in one of them.


Suppose that the available features can be organized into overlapping subsets according to a notion of similarity, and that the features useful in one task are similar, but not necessarily identical, to those best suited for other tasks.

a feature that is useful for one task suggests that the subset it belongs to may contain the features useful in other tasks (Figure 1).


Like QPEs, EFPEs impose a lower bound on the realization of every sequence.

given a sequence q and an extension with action a (i.e., the sequence qa), the realization r i (qa) has to satisfy the lower bound r i (qa) ≥ q r i (q), where q > 0 is a real constant (different sequences can have a different value for q ).


In addition, if we set γ k = O( 1 k q ) as 1 2 < q < 1 and the noise satisfy This proof of Theorem 2 is different from previous one.

we cannot expect some sort of convergence to f (x * ), where x * ∈ argmin f due to nonconvexity.


Note that, in both cases, our DSC-Net-L2 achieves the lowest error rate.

for COIL20, we obtain an error of 5.14%, which is roughly 1/3 of the error rate of the best-performing baseline AE+EDSC.


We show that it outperforms the DBM at solving a variety of inference tasks including classification, classification with missing inputs, and prediction of randomly selected subsets of variables.

we use the MP-DBM to outperform the classification results reported for the standard DBM by Salakhutdinov and Hinton [18] on both the MNIST handwritten character dataset [14] and the NORB object recognition dataset [13].


However, this approach can result in moving toward the saddle point.

if an eigenvalue is negative, the Newton step moves along the eigenvector in a direction opposite to the gradient descent step, and thus moves in the direction of θ * .


First, we assume a function P which, given a sentence x, identifies a set of candidate phrases, not necessarily coherent, for the sentence, P(x) ⊆ P. Second, we assume a score function which, given a phrase, produces a real-valued prediction indicating the plausability of the phrase for the sentence.

the phrase recognizer is a function which searches a coherent phrase set for a sentence x according to the following optimality criterion: among all the coherent subsets of candidate phrases, the optimal solution is defined as the one whose phrases maximize the summation of phrase scores.


While some mistakes can be spotted by eye, most of the neurites can be identified and simple tracing tools can be used to obtain final skeletons/segmentations [30,31].

the identified clusters exhibit homogeneous colors and dendritic pieces that either form connected components or miss small pieces that do not preclude the use of those tracing tools.


To enable a faster local update, we replace q ijk (x, y, z | φ ijk ) in (1) with a simpler "mixture-of-deltas" variational distribution, where A is a randomly chosen set of triples (a, b, c) with size O(K), and a δ aaa ijk + (a,b,c)∈A δ abc ijk = 1.

we assume the probability mass of the variational posterior q(s i,jk , s j,ik , s k,ij ) falls entirely on the K "diagonal" role combinations (a, a, a) as well as O(K) randomly chosen "off-diagonals" (a, b, c).


We consider two kinds of representative domain adaptation settings: one is domain adaptation with multiple sources and the other is domain adaptation combining source and target data.

we use the integral probability metric to measure the difference between two domains.


But, as we discussed above in the financial markets example, there exist cases where a player should be measured against alternatives that depend on the past and the player should take into account the correlations between actions.

we consider competitor functions of the form Φ t : t=1 denote the class of all such functions.


Partitioning the workers into three disjoint and non-empty groups G 1 , G 2 and G 3 , the outline of this stage is the following: we use the spectral method to estimate the averaged confusion matrices for the three groups, then utilize this intermediate estimate to obtain the confusion matrix of each individual worker.

for g ∈ {1, 2, 3} and j ∈ [n], we calculate the averaged labeling within each group by Denoting the aggregated confusion matrix columns by µ gl := E(Z gj |y j = l) = 1 |Gg| i∈Gg π i µ il , our first step is to estimate C g := [µ g1 , µ g2 , .


In [3], image segmentation is formulated as a MWIS problem.

given an image, they first segment it with different bottom-up segmentation schemes to get an ensemble of distinct superpixels.


Refinement: Temporally Constrained Clustering Spectral clustering methods are designed for an unordered "bag" of feature vectors, but, in our case, the temporal ordering of image is an important source of information.

the stablity of appearance is easily computed by computing the motion energy 3 between two frames.


Different choices for the (mini-)batch size b can lead to different algorithms.

for b = 1, the above algorithm corresponds to a simple SGM, while for b = m, it is a stochastic version of the batch gradient descent.


Then there exists C > 0 depending on f and K such that the following holds for n sufficiently large with probability at least 1 − 1/n.

for each taking → 0 and δ → 0 gives us an exact characterization of the asymptotic behavior of Quick Shift in terms of mode recovery.


Then, the noisy ICA model X = A(S + ξ) + η in which ξ is considered part of the latent source signal S = S + ξ, and the model X = AS + (Aξ + η) in which ξ is part of the noise are indistinguishable.

the latent source S and its covariance are ill-defined.


We begin by specifying a model of decision-making that makes the standard assumptions that decision makers tend to choose things with greater utility and that utilities are additive.

if f j is a binary vector indicating the effects produced by option o j and u is a vector of utilities assigned to each of the m effects, then the total utility associated with option o j can be expressed as U j = f j T u.


However, the representational properties which distinguish networks which memorize from those which generalize remain unclear.

we hypothesize that the representational similarity in a group of generalizing networks (networks trained on the true labels) should differ from the representational similarity of memorizing networks (networks trained on random labels.)


In the second step we "pull" μ back into the feasible regime by taking a convex combination with the uniform distribution u (see [3] for a related approach).

this step solves the simple problem of finding the smallest λ ∈ [0, 1] such that 0 ≤ μ ≤ 1 (where μ = (1 − λ)μ + λu).


In this paper, we introduce SURGE, which is a system that induces surface regularization to depth and normal estimation from a single image.

we formulate the problem as DCRF which embeds surface affinity and depth normal compatibility into the regularization.


If for a given neuron x t−1 k is always positive, then the gradient ( 4) is only strictly positive when h t−1 k = 1 and h t j = 1 and strictly negative when h t−1 k = 1 and h t j = 0.

the connection R jk is strengthened when neuron k appears to cause neuron j to activate and inhibited if neuron k fails to activate neuron j.


We consider -depending on the result -one of the following assumptions on the noise: The second assumption is weaker than the first.

the first assumption implies that not only the noise, but also the target function f * is bounded in supremum norm, while the second assumption does not put any additional restriction on the target function.


The subunit model performs well when compared to the Rust-STC model, but we were interested in obtaining a more absolute measure of performance.

no purely stimulus-driven model can be expected to explain the response variability seen across repeated presentations of the same stimulus.


Example: Online shoppers rate an item with a high probability either if they love the item or if they loathe it.

the probability that a shopper supplies a rating is dependent on the shopper's underlying liking [16].


Next, we estimate the offset parameters , 0, to align multiple motion sequences in a common domain.

we choose 0(1) , 0(2) , ... , o(n) so that the shifted motion sequences minimize the deviation from a common prototype model by analogy to the signal-to-noise-criterion ( 1).


First, instead of feeding a single sampled word as the input to the generator RNN, we use the softmax distribution over words instead.

during the generating process of transferred x 2 from G(y 1 , z 2 ), suppose at time step t the output logit vector is v t . We feed its peaked distribution softmax(v t /γ) as the next input, where γ ∈ (0, 1) is a temperature parameter.


Therefore, during training, we randomly jitter each 'canonical' positive example to increase the robustness of our model.

we consider translation shift (of ±16 pixels), scale deformation (of 2 ±1/4 ), and also horizontal flip.


Similarly, the uncorrelated projections of e can be equivalently represented by respectively n − K, K − 1 and one uncorrelated one-dimensional examples.

for the projection on 1, with a single example, the sample variance is null, resulting in the absence of unbiased variance estimator of λ 3 .


Suppose `t is convex and L-Lipschitz; in particular, for every x 2 R, there is a sub-derivative `0 t (x) such that |`0 t (x)|  L. Then there is an explicit learning strategy that guarantees regret R T = O(L p T n ) for all T . This is so even assuming the measurement E t and loss function `t are chosen adaptively, in response to the learner's previous behavior.

the algorithm applies to L 1 loss and L 2 loss, and achieves regret O( p T n ) for both.


Because the size of the smallest interval known to contain X ⇤ is halved with each additional query, this yields the query complexity Unfortunately, once the level of privacy L increases above 1, the bisection strategy is almost never private: it is easy to verify that if the adversary sets b X a to be the learner's last query, Q n , then the target is sure to be within a distance of at most ✏.

the bisection strategy is not ( , L)-private for any L > 1, whenever ✏ < /2.


Let y designate the label of a particular example.

then, the first two components of each instance were drawn from a two-dimensional normal distribution, N (µ, ∆ ∆ −1 ) with the following parameters, the label of each examples determined the mean of the distribution from which the first two components were generated.


From (2) we see the high level and low level rewards are a linear combination of the same set of reward weights w. This allows us to combine both types of expert advice presented above to obtain the following unified optimization problem min w,η,ξ This optimization problem is convex, and can be solved efficiently.

even though the optimization problem has an exponentially large number of constraints (one constraint per policy), the optimum can be found efficiently (i.e., in polynomial time) using, for example, the ellipsoid method, since we can efficiently identify a constraint that is violated.


As in boosting or kernel methods, nonlinearity can be introduced by including nonlinear features in x.

we are mainly interested in the scenario that d n. there are many more features than the number of samples.


here, we are given α > 0, and wish to produce c : (1) instances with low density are deemed to be anomalous.


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

our notion will require that a fair classifier treat similar subpopulations of individuals similarly, in a sense we will make formal next.


Our inequalities are directly applicable to a number of sequential decision problems.

we use them to design and analyze new algorithms for sequential hypothesis testing, best arm identification, and sorting.


Standard LSH approaches to the nearest neighbor problem work by scoring points in the set That is, LSH considers only those points in W that are covered by at least one of the same α sets as x. Predictive indexing, in contrast, maps each cover set Q i,J to an ordered list of points sorted by their probability of being a top-10 nearest point to points in Q i,J .

the lists are sorted by h Q i,J (p) = Pr q∼D|Q i,J (p is one of the nearest 10 points to q).


To further underscore the differences we note that skeletons are naturally more compact than networks.

all examples of skeletons in this paper are irreducible, meaning that for each two nodes v, u ∈ V (S), in(v) � = in(u).


Our main findings suggest that the marginal cost of ensuring differential privacy in the context of distribution learning is only moderate.

for a broad class of shape-constrained density estimation problems, we give private estimation algorithms that are nearly as efficient-both in terms of sample size and running time-as a nearly optimal non-private baseline.


There are several applications where we need to model vector-valued outputs.

this involves modelling multiple outputs or tasks.


While our work makes a step towards theoretical understanding of SGD for training neural networs, it is far from being conclusive.

the real data could be separable with respect to different metric than 2 , or even a non-convex distance given by some manifold.


therefore the trace of a square matrix may be seen as the total variance along any set of orthogonal directions: the trace of a square matrix is the sum of its eigenvalues.


The inverse of the ancestor function is the descendent function τ , which maps u ∈ U to the subset {u ′ ∈ U : u ∈ π ⋆ (u ′ )}.

u is a descendent of u ′ if and only if u ′ is an ancestor of u. Graphically, we can depict T as a rooted tree: U defines the tree nodes, ALL is the root, and { u, π(u) : u ∈ U \ ALL} is the set of edges.


2 Where we define , and D KL [q||p] for q, p ∈ [0, 1] is defined as the divergence of Bernoulli distributions with parameters q and p. From now on, following [2], we will define the clustering distance function d D of Eq.

( 1) as: the clustering distance is the probability that two independently drawn instances from D will be in the same cluster under one clustering, and in different clusters under another clustering.


All the experiments in this paper are evaluated through single-crop validation process on the validation dataset of ImageNet-1k.

an image region of size 224 × 224 is cropped from the center of an input image with its shorter side being resized to 256.This 224 × 224 image region is the input of the network.


Our general result is stated in terms of a maximization over a vector parameter `= (`1, . . . , `d) with `t 2 {0, . . . , ⇡ t p} for all t. We will see in the proof that `t represents the number of items of type t that are unknown to the decoder after p⇡ t `t are revealed by a genie.

we define the following: • Given `and , we let S `be a random set of indices in [p] such that for each t 2 [d], the set contains `t indices corresponding to entries where equals t. we define S `to be uniformly distributed over all such sets.


We present an alternative probabilistic formulation, called semi-parametric PCA (SP-PCA), where no assumptions are made about the distribution of the latent random variable θ. Non-parametric latent distribution estimation allows us to approximate data density better than previous schemes and hence gives better low dimensional representations.

multi-modality of the high dimensional density is better preserved in the projected space.


Assuming that the time resolution in the observed time series is sufficiently high, one might also want to encourage the amount of disagreement over time to be Markovian.

if the previous time point disagreed with the other latent traces, then the current point should be more likely to disagree.


Proposition 4.1 (The SVAE objective lower-bounds the mean field objective) The SVAE objective function L SVAE lower-bounds the mean field objective L in the sense that for any parameterized function class {r(y; φ)} φ∈R m . Furthermore, if there is some φ * ∈ R m such that ψ(x; y, φ * ) = E q(γ) log p(y | x, γ), then the bound can be made tight in the sense that Thus by using gradient-based optimization to maximize L SVAE (η θ , η γ , φ) we are maximizing a lower bound on the model log evidence log p(y).

by optimizing over φ we are effectively learning how to condition on observations so as to best approximate the posterior while maintaining conjugacy structure.


However, we show that despite the improvements over standard SGMCMC, the discretization error of SGRLD still causes problems on the simplex.

it leads to asymptotic biases which dominate at the boundary of the space and causes significant inaccuracy.


Those assumptions, including low-rankness, incoherence and uniform sampling, are now standard and widely used in the literatures, e.g., [14,17,22,24,28,33,34,36].

the analyses in [17,33,36] show that, in terms of theoretical completeness, many nonconvex optimization based methods are as powerful as the convex program (2).


In addition to knowing the description of the algorithm, an adaptive adversary is stronger and can record all of the past actions of the algorithm, and use these arbitrarily to decide the future payoff vectors and site allocations.

communication: We do not explicitly account for message sizes, since we are primarily concerned with scaling in terms of T and k. We require that message size not depend k or T , but only on the number of experts n. we assume that n is substantially smaller than T and k. All the messages used in our algorithms contain at most n real numbers.


Our approach exploits the fact that, in MCVI, the randomness in the gradient estimator is completely determined by a known Monte Carlo generating process; this allows us to leverage knowledge about this generative procedure to de-noise the gradient estimator.

we construct a computationally cheap control variate based on an analytical linear approximation to the gradient estimator.


In order to encode the image label x cls , the encoder r cls = φ cls enc (x cls ) takes the vector of C cls binary probabilities x cls ∈ R C cls , one for each of the C cls possible object classes, and broadcasts the corresponding values to all H × W spatial locations (u, v) in h. Formally r cls ∈ R H×W ×C cls and ∀u, v, c : r cls uvc = x cls c . Encoding the object detection label x det is similar, but reflects the geometric information captured by such labels.

each bounding box b m of the M extracted by SSW is associated to a vector of C cls + 1 probabilities (one for each object class plus one more for background) .


However, we observe that such an operation can be further combined using a regular 3-D convolution [10].

the m d f ⇥ d f input feature maps can be viewed as a single 3-D feature map with a size of d f ⇥ d f ⇥ m. The special depth-wise convolution, or equivalently the global average pooling layer, is essentially a 3-D convolution with a kernel size of d f ⇥ d f ⇥ 1, where the weights in the kernel is fixed to 1/d 2 f . Moreover, in this view, the channel-wise convolution is a 3-D convolution with a kernel size of 1 ⇥ 1 ⇥ d c . These two consecutive 3-D convolutions follow a factorized pattern.


Consider the following problem: given two matrices A 2 R d⇥n1 and B 2 R d⇥n2 that are stored in disk, find a rank-r approximation of their product A T B.

we are interested in the setting where both A, B and A T B are too large to fit into memory.


Controlled interventions provide the most direct source of information for learning causal effects.

a dose-response curve can be learned by varying the treatment level and observing the corresponding outcomes.


We estimate the probability distribution obtained by applying matrix P repeated starting from any initial condition.

let p t (i) = P(X t = i) denote the distribution of the random walk at time t with p 0 = (p 0 (i)) ∈ R n + be an arbitrary starting distribution on [n].


We show how this scheme can be used to perform Bayesian inference, and how simple extensions of this basic idea make it possible to combine sources of information and to propagate uncertainty through multiple layers of random variables.

we focus on one Monte Carlo method, namely importance sampling with the prior as a surrogate, and show how recursive importance sampling approximates hierarchical Bayesian inference.


In this section, we outline a practical implementation of this approach and we present some of our theoretical results.

we consider the relative value iteration algorithm for average-cost MDPs that is described , for example, in [Ber95].


( 14) is minimized when the column vectors of Y are orthogonal to each other (refer to Appendix A for details).

minimizing L C (α, Y ) will maximally scatter the points away from each other on G(1, n).


We will often abuse notation and write σ (i) for valuations of entities i ∈ V when the intended meaning is either already defined or is clear from the context.


Unlike in the synthetic experiments training data was either used for building models, or associating models to tasks.

we set |S 1 t | = |S 2 t | = 0.5|S t | for η = 0.5, and used one half of the examples to build models (set the weights of prediction functions), and the remaining half to evaluate each of the k models on the T tasks, and associating models to tasks.


We begin by constructing an appropriate measurement of the quality of this approximation.

p(x) is D-close to S if a random sample from p is no more than a distance D from S in expectation.


Black-box identification approaches are widely used to learn dynamic models from a finite set of input/output data [1].

in this paper we focus on the identification of large scale linear systems that involve a wide amount of variables and find important applications in many different domains such as chemical engineering, economic systems and computer vision [2].


The SMC procedure of Algorithm 1 can used to solve this problem by defining a sequential decomposition of the graphical model.

we consider the decomposition corresponding to first including the node θ and then, subsequently, introducing the nodes z 1 to z M in any order.


A factored, n-agent Dec-MDP is locally fully observable iff an agent's local observation uniquely determines its local state: Local full observability is not equivalent to independence of observations.

a problem may be locally fully observable without being observation independent (since agents may simply observe outcomes of non-independent joint actions).