The first domain is a blocks world with dynamics based on stochastic STRIPS operators, a −1 step cost, and a goal of stacking the blocks.

the environment state is described as a set of grounded relations (e.g.


We show that for a wide variety of observation models, the Bayes least squares (BLS) estimator may be formulated without explicit reference to the prior.

we derive a direct expression for the estimator, and a related expression for the mean squared estimation error, both in terms of the density of the observed measurements.


We believe that this unification provides a better understanding of the intrinsic difficulties of the best arm(s) identification problem.

our analysis showed that the same complexity term characterizes the hardness of both settings.


Since all histories in an information set belonging to player i are indistinguishable, the strategies in each of them must be identical.

for all h ∈ I, σ i (h) = σ i (I) and σ i (h, a) = σ i (I, a).


we assume that they are both multivariate regularly varying, which correspond to a large non-parametric class of (heavy-tailed) distributions, widely used in applications where the impact of extreme observations should be enhanced, or at least not neglected.


We also explored novel feature selection methods, which exploit the spatial characteristics of brain activity.

we incorporate a bias in favor of jointly selecting nearby voxels.


The comparison problem is intrinsically symmetrical: comparing A to B should give us the same decision as comparing B to A.


PGRD cheaply computes the gradient of the objective utility with respect to the R i parameters through UCT planning.

it takes a simulation model of the CMP and an objective reward function and uses UCT to simultaneously plan actions with respect to the current internal reward function as well as to update the internal reward function in the direction of the gradient of the objective utility for use in the next planning step.


In this paper, we attempt to address two challenges: 1) to break away from the standardized norm of feature-based deep transfer learning 1 , and 2) to learn versatile structures in the data with a data-driven approach.

we are interested in learning transferable latent relational graphs, where the nodes of a latent graph are the input units, e.g., all the words in a sentence.


Given that the discriminator class is sufficiently large, [9] interpreted the GAN training as finding a generator such that the generated distribution ν is as close as possible to the target true distribution µ, measured by the Jensen-Shannon distance d JS (µ, ν), as shown below: Inspired by such an idea, a large body of GAN models were then proposed based on various distance metrics between a pair of distributions, in order to improve the training stability and performance, e.g., [2,3,13,15].

among them, the integral probability metric (IPM) [19] arises as an important class of distance metrics for training GANs, which takes the following form different choices of the function class F in (2) result in different distance metrics.


Thus, the meaning of this near optimality of g * ι,θ (s, a; ω) is interpreted as a near minimization of the Euclidean distance between r and its approximator R ρ * (s, a; ω), so that ρ * works to reduce the distance of the regressand r and the subspace of the regressor ∇ θ ln{d θ (s) ι π(a|s;θ)} of the M-estimator ω.

r(s, a) is almost in this subspace at the point ω = ω * .


As in the realizable case, the algorithm is conservative.

if the loss on example z t is zero then w t+1 = w t . In case the loss is positive the update rule is w t+1 = w t + τ t v t where v t is the same as in the realizable case.


Optimizing M k . First we fix w j k to optimize M k . To avoid over-fitting, as in support vector machine (SVM), we minimize the overall C2B distances from X i 's associated classes to itself and the total amount of slack.

we solve the following convex optimization problem: where C is a trade-off parameter, acting same as in SVM.


W e show in this paper that mainstream inference algorithms based on jointrees [14,8] are a special-case of the arithmetic-circuit approach proposed in [2].

we show that each jointree is an implicit representation of an arithmetic circuit; that the inward-pass in jointree propagation ev aluates this circuit; and that the outwardpass differentiates the circuit.


We follow the three steps in Section 4 to develop MLFre.

we first present an accurate estimation of the dual optimum in Section 5.1, then we solve for the supreme values in (R1 * ) and (R2 * ) in Section 5.2, and finally we present the proposed MLFre in Section 5.3.


These undirected edges provide useful structural knowledge about correlation between the vertex instances.

we allow edges to be of two types -"positive" or "negative" depending on whether the associated adjacent vertices are positively or negatively correlated, respectively.


The idea of MMAX is to consider a modular lower bound tight at the set corresponding to each block of a partition.

at iteration t + 1, for each block i, we approximate f i with its modular lower bound tight at A π t i and solve a modular version of Problem 1 (Line 7), which admits efficient approximation algorithms [2].


In machine learning, the properties of k-means have been studied, for fixed k, by considering the excess reconstruction error E ρ (S n,k ) − E ρ (S k ).

this quantity has been studied for X = R d , and shown to be, with high probability, of order kd/n, up-to logarithmic factors [31].


An additional property of our results is that they are algorithmic in nature.

if we have sufficiently strong efficient PAC-learning algorithms for the target function on each feature set, we can use them to achieve efficient PAC-style guarantees for co-training as well.


Thus, let E be drawn uniformly from I(x), so that P is extended to be a joint distribution over (x, y, E).

we say that a forecaster averaged over all x, y and events of interest E ∈ I(x), whenever the forecaster outputs probability p, then the event E actually holds with probability p. Note that this definition corresponds to perfect binary calibration (1) for the transformed pair of variables y = I[y ∈ E], x = (x, E).


A Central Limit argument appeals to the number of independent eigenfunctions that contribute to this residual, and, the orthogonality of eigenfunctions implies that as more of them meaningfully contribute to each entry in the residual, the more independent those entries become.

if this long tail begins at a low magnitude and decays slowly, then we can treat it as a Gaussian i.i.d.


This is an important and novel contribution when compared to a Gaussian formulation like a classical Gabor.

the bandwidth is then constant around the speed plane or the orientation line with respect to spatial frequency.


Uniform generalization is stronger than the original notion of generalization in Definition 2.

if a learning algorithm generalizes uniformly, then it generalizes according to Definition 2 as well.


In spite of their enormous practical success, unlike more traditional methods such as maximum likelihood inference, GANs are theoretically rather poorly-understood.

two very basic questions on how well they can approximate the target distribution , even in the presence of a very large number of samples and perfect optimization, remain largely unanswered.


The generalized Binomial rule states that all MLNs obtained by conditioning on a singleton atom S with exactly k of its groundings set to true are equivalent to each other.

in order to compute the distribution represented by the MLN conditioned on S, we only need to know how many groundings of S are set to true.


To learn Q automatically, we propose an attention model which guides the random surfer on "where to attend to" as a function of distance from the source node.

we define a Graph Attention Model as a process which models a node's context distribution Q as the output of softmax: where the variables q k are trained via backpropagation, jointly while learning node embeddings.


. . , (x t , y t )}, denote the current version space by V (S t ) = {h ∈ H| S | ∀(x, y) ∈ S t , h(x) = y}.

the next query selected by the algorithm is the algorithm selects the query that in the worst-case over the possible hypotheses, would remove the most hypotheses from the version spaces, when normalizing by the outcome-dependent cost of the query.


In this paper, we instantiate the loss using the structured hinge loss [25].

we define L to be L(x i , y i ; w, A) = max Here, ∆ is the Hamming distance.


that with minimum variance), see [vdV98].

when non degenerate, it is asymptotically normal with limiting variance 4•Var(K (1) (X)) (refer to Chapter 5 in [Ser80] for an account of asymptotic analysis of U -statistics).


We visualize the distribution of all 200 selected b 's using the kernel density estimator.

we adopt the Gaussian kernel, and the kernel bandwidth is selected based on the 10fold cross validation.  . We see that the optimal regularization parameters of OMR significantly vary with different max .


In our initial experiments, we found that the use of a single state to represent all the regions of baseline in the ECG waveform resulted in poor performance when the model was used to infer the underlying state sequence of new unseen waveforms.

a single baseline state allowed for the possibility of the model returning to the P wave state, following a P wave -Baseline sequence.


The new posterior approximation takes the same form, but with the new pseudo-points and new hyperparameters: Similar to the batch case, this approximate inference problem can be turned into an optimisation problem using variational inference.

consider Since the KL divergence is non-negative, the second term in the expression above is the negative approximate lower bound of the online log marginal likelihood (as Z 2 /Z 1 ≈ p(y new |y old )), or the variational free energy F(q new (f ), θ new ).


One exception to this trend is play within the family of S-bots.

consider S1399 and S1239, which are very similar programs, using the same technique for equilibrium computation with the same abstract game.


Our hypothesis is that adversarial sample attacks are deeply entangled with interpretability of DNN classification outputs.

in a fully connected network, all neurons are directly/transitively influencing the final output.


We will be however working in the large document limit -and this will simplify the updates.

in the E-step, in the large document limit, the first term in the update equation for γ has a vanishing contribution.


However, when applied to non-convex problems several drawbacks are revealed.

sGD is widely used for deep learning [2], one of the most interesting fields where stochastic non-convex optimization problems arise.


Garivier and Kaufmann [8] recently provided a general lower bound on the number of samples required in the fixed confidence setting.

they show that for any normal bandit model, under any sampling rule and stopping time τ δ that guarantees a probability of error no more than δ, Recall that M β , defined in (3), is the first time after which the empirical proportions are within of their asymptotic limits.


We show that swapping out locally enforced orthogonality constraints with a global reconstruction cost solves this issue.

we examine the local receptive field network proposed by Le et al.


The prediction of the model is that this would reduce the strength of the input, A, which in turn reduces the level of activity in the head direction cells, α.

if σ z is the jitter of the unreliable landmark, then we expect α to scale as 1/σ 2 z .


We assume each user provides a partial ranking according to his/her ordered partitions.

we assume that the set of offerings S j , the number of subsets ( ˜ j + 1), and their respective sizes ( mj,1 , .


While classical subgradient methods only achieve an error level on the objective function of O(1/ √ k) after k iterations, proximal-gradient methods have an error of O(1/k) while accelerated proximal-gradient methods futher reduce this to O(1/k 2 ) [1,2].

accelerated proximalgradient methods for non-smooth convex optimization achieve the same optimal convergence rate that accelerated gradient methods achieve for smooth optimization.


Unfortunately, inference and learning in this network (and in general in coupled Markov chains) requires very expensive computations, and cannot be performed by simply propagating messages over time and among the variable nodes.

the state of a child variable x k t depends on x k t−dt , s k t , x 1 t and the state of all other children at the previous time step, [x j t−dt ] 2<j<n,j =i .


In this section, we present the empirical performance of DECO via extensive numerical experiments.

we compare DECO after 2 stage fitting (DECO-2) and DECO after 3 stage fitting (DECO-3) with the full data lasso (lasso-full), the full data lasso with ridge refinement (lasso-refine) and lasso with a naive feature partition without decorrelation (lasso-naive).


"Superfast" Numerical Linear Algebra: The structure in these matrices can be exploited for faster linear algebraic operations such as matrix-vector multiplication, inversion and factorization.

the matrix-vector product can be computed in time O(n log n) for Toeplitz and Hankel matrices, and in time O(n log 2 n) for Vandermonde and Cauchy matrices.


As spectral clustering relies on the eigenvectors corresponding to the k smallest eigenvalues (see Algorithm 1) we derive here necessary and sufficient conditions such that in expectation the eigenvectors χ i , i = 1, . . . , k correspond to the k smallest eigenvalues of the normalized Laplacians introduced so far.

we observe that condition E G affects the ordering of the eigenvalues of the normalized geometric mean Laplacian.


We then vary values of hyperparameters and test whether this result is robust.

we sampled 4 different values respectively for temperature spacing {0.5, 3, 7, 10}, number of parallel chains {2, 4, 6, 10}, resulting in 16 combinations of hyperparameters.


In addition, we show examples of classes of functions amenable to our approach.

for simple functions, i.e., those for which minimizing F (S) − a(S) is easy for all vectors 1 a ∈ R n , the problem in Eq.


The law of G * t is given by a gamma process, while the masses are conditionally gamma, The idea of [13] is to define the conditional law of G t+1 given G t and C t to coincide with the conditional law of G t given C t as in Proposition 3.

define where G * t+1 ∼ Γ(α, τ + φ t , H) and w t+1,k ∼ Gamma(c tk , τ + φ t ) are mutually independent.


In this case we say that H(z, r) is a separable Hamiltonian.

we use the term standard HMC to refer to HMC using the identity matrix as G. Although HMC methods often outperform other popular MCMC methods, they may mix slowly if there are strong correlations between variables in the target distribution.


While it leads to good results, our learning procedure is only an approximation.

it optimizes the weights to maximize only the single-step performance.


We propose a selective labeling method by analyzing the out-of-sample error of Laplacian regularized Least Squares (LapRLS).

we derive a deterministic out-of-sample error bound for LapRLS trained on subsampled data, and propose to select a subset of data points to label by minimizing this upper bound.


The agent's behavior during training is -greedy.

the agent selects a random action from the whole action set with probability , and selects a random action from the two best actions in the two action groups (i.e.


However, the total cost could actually be worse than quadratic since the cost of updating each connection could depend on N . Fortunately, the complexity of our collapsed Gibbs sampling algorithm only depends on the number of incident connections, d, or equivalently, the sparsity ρ = d/N .

we must solve a linear system of size d, which incurs a cubic cost, as seen in Fig.


We believe game theory and economic models provide an appropriate modeling framework for understanding interacting regulatory processes.

the problem of understanding coordinate binding of regulatory proteins has many game theoretic properties.


This paper follows up on an experimental discovery: different recommendation systems perform better than others for some users and items but not for others.

the relative strengths of two distinct CF models f 1 (u, i), f 2 (u, i) ∈ F depend on the user u and the item i whose rating is being predicted.


The main problem with generalizing multi-metric learning is that it is based on assumptions that make the feature space both non-smooth and non-metric.

it is often assumed that straight lines form geodesic curves and that the metric tensor stays constant along these lines.


If for all y ∈ Y, d ∈ D we have p Y,D (y, d) > 0, J p Ŷ |D (y|d), p Y T (y) ≤ , where J is given in (3), and Proposition 2 guarantees that, as long as n is sufficiently large, the utility and discrimination control guarantees will approximately hold when p X, Ŷ |Y,X,D is applied to fresh samples drawn from q D,X,Y .

the utility and discrimination guarantees will converge to the ones used as parameters in the optimization at a rate that is at least 1 n log n. The distortion control guarantees (4) are a property of the mapping p X, Ŷ |Y,X,D , and do not depend on the distribution of the data.


4 Where µ is a vector of probabilities: This work proposes to instead explicitly parameterize a distribution over distributions on a simplex, p(µ|x ⇤ , ✓), using a DNN referred to as a Prior Network and train it to behave like the implicit distribution in the Bayesian approach.

when it is confident in its prediction a Prior Network should yield a sharp distribution centered on one of the corners of the simplex (fig.


Let > 0, we in turn have This last probability can be bounded by applying McDiarmid's inequality: changing any Y i value changes ψ(Y x,k ) by at most w i,k • t Y (δ 0 ) when we condition on the event Y δ0 .

this, followed by a union-bound yields Combining with the above we get let δ 0 = δ/2n, with probability at least 1 − δ, for all x ∈ X and k ∈ [n] where the second inequality is an application of Jensen's.


Through such decomposition, the samples are decoupled in the optimization and can be processed separately.

the optimization consists of two iterative updating components.


While lots of online matrix completion algorithms have been proposed recently, they either lack of solid theoretical guarantee [17], or require strong assumptions for the streaming data [19,21,13,18].


Throughout the paper, we use symbol 3 to denote empirical quantities based on the & observed training data ' .

we use 3 # % (' ) to denote the empirical expectation with respect to the training samples, and Assume that input ¡ belongs to a set @ .


We consider tractable representations of probability distributions and the polytime operations they support.

we consider a recently proposed arithmetic circuit representation, the Probabilistic Sentential Decision Diagram (PSDD).


High dimensional state space and control is a recurring theme in these applications.

with the modern social networks customers are classified in a highly granular way, potentially with each customer representing his own class.


For example in Figure 2, o 1 and o 2 affect the waypoint y 3 because of proximity.

we connect an object o k to a trajectory waypoint if the minimum distance to collision is less than a threshold or if o k lies below ō.


Our Contributions: This paper solves online convex optimization with stochastic constraints.

we propose a new learning algorithm that is proven to achieve O( p T ) expected regret and constraint violations and O( p T log(T )) high probability regret and constraint violations.


However, the expressiveness of these models is hampered by practical considerations that almost always lead to the the reliance on simple parametric forms.

non-parametric variants of BNs (e.g., [9,27]) typically involve elaborate training setups with a running time that grows unfavorably with the number of samples and local graph connectivity.


We will now relate the matching condition to the optimality of the divergence.

define w £ R a f X V P ¢; E £ v a E v g ; We will give sufficients conditions for members of w £ to be in OPT ; £ .


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

with probability 1 , the generalization risk is at most " away from the training risk.


Although large entries in a distance matrix A cannot hide, we show it is still impossible to achieve the relative error guarantee in less than mn time for distance matrices.

we show for the ∞ distance metric, that there are instances of distance matrices A with unequal P and Q for which even for k = 2 and any constant accuracy , must read Ω(mn) entries of A.


Importantly, we do not have to make a linearisation approximation for this new sample.

the GP posterior over log ℓ a can be fully exploited when performing active sampling.


This ensures the preservation of conditional dependency relations in G inv .


The distance function is constructed to ensure that objects that are approximately symmetrical to each other (from an inference perspective) are placed in a common cluster.

formally, given a MLN M, let D denote the set of all domains in M. D ∈ D is a set of objects that belong to the same domain.


The entries of H and A were sampled from exponential distributions Exp( ) (For a random variable y 0, its probability density function is e y , and for y < 0, its density is 0).

each entry of H was sampled from Exp(1), then the matrix H was made to be symmetric.


The first i,1 mistakes are made at the first step, [ i,2 − i,1 ] + mistakes are made at the second step, + mistakes are made at the third step, and so on.

the number of mistakes at step k is [ i,k − max k <k i,k ] + and the associated loss is G k . Our main claim follows from combining the upper bounds in Equations 18 and 19.


We prove that the proposed algorithm converges to the optimal solution at a global geometric rate.

such an algorithm is built upon a primal-dual reformulation of the mean squared projected Bellman error minimization problem, which gives rise to a decentralized convex-concave saddle-point problem.


The inference algorithm outlined in the next section combines the body model described above with a probabilistic image likelihood model.

we define φ i (X i ) to be the likelihood of observing the image measurements conditioned on the pose of limb i.


At the same time, models of this class are very demanding in terms of computational resources.

a large amount of memory is required by commonly used fully-connected layers, making it hard to use the models on low-end devices and stopping the further increase of the model size.


Our mean result compares favorably with other methods using medians of mixtures [1,2].

our result is similar to that of [1] at input ≤ 30.


6) and exhibit high agreement in independent simulations.

the goal is to recover the moments of two statistics: the admixture distance, a measure of two individuals' dissimilarity in their ancestry, and the admixture level where 0 means an individual's alleles all come from a single population, and 1 means its ancestry is shared equally among the K populations.


To evaluate INCRIS, we constructed a simple MDP that exemplifies to properties of domains for which we expect INCRIS to be useful.

we were motivated by the applications of reinforcement learning methods to type 1 diabetes treatments [1,17] and digital marketing applications [15].


Second, we use the framework to study population codes of angular variables.

we assess the impact of different noise correlations structures on coding accuracy in long versus short decoding time windows.


The OR contains a 60 × 80 array of light-sensitive "neurons" that each represent local light intensity as a spike rate.

pixels that receive a lot of light spike more frequently than those that receive a little light.


Each new goal g is drawn in an -greedy fashion (Algorithms 1 & 2) with the exploration probability 2 annealed as learning proceeds (from a starting value of 1).

in the controller, at every time step, an action is drawn with a goal using the exploration probability 1,g , which depends on the current empirical success rate of reaching g. if the success rate for goal g is > 90%, we set 1,g = 0.1, else 1.


3 Known time horizon The two-target algorithm consists in exploring new arms until two successive targets 1 and 2 are reached, in which case the current arm is exploited until the time horizon n. The first target aims at discarding "bad" arms while the second aims at selecting a "good" arm.

using the names of the variables indicated in the pseudo-code below, if the length L of the first run of the current arm I is less than 1 , this arm is discarded and a new arm is selected; otherwise, arm I is pulled for m − 1 additional runs and exploited until time n if the total length L of the m runs is at least 2 , where m ≥ 2 is a fixed parameter of the algorithm.


We derive an approximate predictive posterior for GPPVAE that enables out-of-sample predictions of high-dimensional samples.

given training samples Y , object feature vectors X, and view feature vectors W , the predictive posterior for image representation y of object p in view q is given by encode training data dz dZ( 12) where x and w are object and feature vectors of object p and view q respectively, and we dropped the dependency on parameters for notational compactness.


Next, we characterize "easy" instances of KPCA problems under which we are guaranteed a fast rate.

we show that if the decay of the spectrum of the fourth order moment, C , of z ω , is exponential, then the dominating factor, κ(B k , k, m) is in O(1/m).


Now we allow the underlying walk to terminate after finite steps because in real-world applications the observed censored lists are often truncated.

the underlying random walk can be stopped before exhausting every state the walk could visit.


Motivated by the ideas of [21] and [8], we can develop a sequential version of DPP rules.

if we are given a sequence of parameter values λ 1 > λ 2 > .


In this case, the activity rate at stage m is current stage (13) Define x m := λ m−1 (τ m ) − u m−1 − µ (and x 0 = 0 by convention) then the intensity due to events of all previous m stages can be written as τm 0 Ae −ω(t−s) dN (s) = x m e −ω(t−τm) .

x m is sufficient to encode the information of activity in the past m stages that is relevant to future.


Second, we address the complexity of computing a gradient for the empirical risk of a neural network.

we give evidence that computing (or even approximating, up to polynomially large factors) the norm of the gradient of the top layer in a neural network takes time that is "rectangular".


Generalized least squares (GLS) is a method for selecting β when the y i are not necessarily independent and may have different variances.

if we use a linear model with (possibly correlated) mean-zero noise to model the data, i.e., y = xβ + , where β ∈ R is unknown, is a random vector, E[ ] = 0, and Var( |x) = Ω, then the GLS estimator is the best linear unbiased estimator (BLUE) for β [6]-the linear unbiased estimator with the lowest possible variance.


In many problems, the estimate of 1/P (v) will be an underestimate with high probability.

the method will overestimate P (v) and often give no indication that it has done so.


We show that this term can easily be removed, and that doing so gives even lower-variance gradient estimates in many circumstances.

as the variational posterior approaches the true posterior, this gradient estimator approaches zero variance faster, making stochastic gradient-based optimization converge and "stick" to the true variational parameters, as seen in figure 1.


Choosing x t = −v t would incur a loss of − T 2 , while the best fixed action in hindsight, the point 2 T . The player's regret would therefore be Intuitively, the flaw of this naïve strategy is that the hint does not give the player any information about the (d − 1)-dimensional subspace orthogonal to v t . Our solution is to use standard online learning machinery to learn how to act in this orthogonal subspace.

on round t, we use v t to define the following virtual loss function: In words, we consider the 1-dimensional subspace spanned by v t and its (d − 1)-dimensional orthogonal subspace separately.


This can be attributed to the length of the label chain and sparse targets in the chain since RNN b is trained to make correct predictions over all 90 labels, most of them being zero.

the length of target sequences of RNN b is 90 and fixed regardless of the content of training documents.


Our first contribution is to provide a new perspective on IWVI by highlighting a precise connection between IWVI and self-normalized importance sampling (NIS) [17], which instructs us how to use IWVI for "pure inference" applications.


Since y i ∈ A, we have z i ∈ A M,R/4 , and hence the ball B M (z i , R/4) lies completely inside A M,R/2 ⊆ A M,σ−r .

the density inside the ball is at least λ everywhere, and hence the mass inside it is at least Observe that R ≥ 4r and so this condition is satisfied as a consequence of satisfying Equation 1.


Perhaps the simplest useful assumption is that the observations all lie near some low-dimensional subspace.

if we stack all the observations as column vectors of a matrix M ∈ R m×n , the matrix should be (approximately) low rank.


. . ≤ x n . For any pair of jobs (i, j), define d(i, j) as the amount of job i that has been executed before the completion time of job j.

d(i, j) is the amount of time by which i delays j.


For special cases studied in the past, our learning bounds match or improve upon the previously best bounds (see Section 3.3).

our bounds improve upon those of Taskar et al.


The structural predictability model puts us in a position to ask whether empirically observed patterns of syntactic reduction give evidence for speakers' use of some types of cues but not others.

there is a question of whether predictability based on surface cues alone (the NGRAM features of Section 5.2) provides a complete description of information-density effects on speakers' choices in syntactic reduction.


The variational E-step (see [20] for details) calculates the variational parameters z ij , φ i,j (ρ 1:τ |β 1:τ ) = φ i,j β1 (ρ 1 ) τ t=2 φ i,j βt (ρ t |ρ t−1 ), and η (i,β),(j,ρ) for the lower bounds in (9) (13) (10).

given the nesting of the lower bounds, we proceed by first maximizing the GMM lower bound L (i,βt),(j,ρt) GM M for each (i, j, β t , ρ t ).


We include a factor sp, tp ∈ E if sw 1 − d ≤ tp ≤ sw 2 +d.

letting π sp be the actual words in sp, the potentials θ sp,tp (x sp , x tp ) determine the preference of the phrase translation π sp , x tp using a phrase table feature function τ : π, π → R k . θ sp,tp (x sp , x tp ) = γ p τ ( π sp , x tp ) if x sp = tp and a large negative value otherwise, where γ p is the weight vector for the Moses phrase table feature vector.


Intuitively, U-Net-1 to U-Net-k all have the same input size, while U-Net-1 views the smallest region with the highest resolution and U-Net-k views the largest region with the lowest resolution.

for any Second, we need to propagate the higher level information extracted by U-Net-t (2 ≤ t ≤ k) to the next submodule, i.e., U-Net-(t − 1), so that clues from a coarser scale can assist the work in a finer scale.


In this work, we study the multi-factor NMF problem based on the generalized KL divergence.

given an m × n nonnegative matrix V , we seek This simple formulation has a drawback as it is invariant to relative scalings between the factors: for any γ > 0, we have In other words, there exist infinite number of equivalent solutions, which gives rise to an intrinsic ill-posed nature of the mfNMF problem.


Φ ij ∼ N (0, 1/m), satisfy them with high probability.

compression ratios on the order (k log p)/p are achievable for k-sparse signals using a random Φ chosen this way.


It also proves essential to combine knowledge-based feature design with learning methods.

we exploit the fact that in speech we can generate "training examples" by artificially superposing two separately-recorded signals.


To achieve E[f (x a ) − f (x * )] ≤ for the special problem, the IFO complexity of the proposed algorithm is It is interesting to compare Theorem 5 and Corollary 4 with those existing ones for SGD.

it was shown by Shamir [13] that given in [13].


In contrast, SphereNet not only generalizes the hyperspherical constraint to every layer, but also to different nonlinearity functions of input angles.

we propose three instances of SphereConv operators: linear, cosine and sigmoid.


Although the resulting model was therefore simplified, it nevertheless still had a challenging learning task to perform.

the stripe context layers had to learn to encode and maintain the current input value properly, and the Hidden layer had to be able to decode the context layer information as a function of the time input value.


Accordingly, we have chosen this best-performing method to compare to the novel kernel.


The increasing availability of data in EHRs has stimulated a great deal of research efforts that used machine learning to conduct clinical risk prognosis and survival analysis.

various recent works have proposed novel methods for survival analysis based on Gaussian processes [9], "temporal" logistic regression [10], ranking [11], and deep neural networks [12].


On a separate note, experiments [16,17,24] suggest that fast convergence to approximate local minima may be sufficient for training neural nets, while convergence to stationary points (i.e., points that may be saddle points) is not.


the Euclidean distance on the unit sphere.

their algorithm has a query time of O(n ρ ) and space complexity of O(n 1+ρ ) for ρ = 1 2c 2 −1 .


The second algorithm is much faster and takes O(kn) time, where k is the number of configurations of the hyper-parameters alpha and beta that are searched over.

for a 784-1000-784 autoencoder model with mini-batches of size 100 and 50,000 training cases on a GTX 580 GPU, learning takes 1.66 seconds per epoch for standard dropout and 1.73 seconds for our second algorithm.


The motivation behind our method is that projections of previous predictions help to move apart the manifolds that belong to each class in a recursive fashion, in order to achieve better linear separability (Figure 1 shows a vision example separating different image patches).

consider that we have a two class problem which is non-linearly separable.


Even so, we can still make use of the similarities between the manifolds in geometry structures and intrinsic representations to build the alignment.

we have three intuitive observations to explore.


it subtracts from each of these input signals an estimate x obtained by decoding the output spike trains (fig 1A).

xi = D ij r j , where D = (D ij ) are the decoding weights and r = (r j ) are the filtered spike trains which obey ṙj = −λr j + o j with o j (t) = k δ(t − t k j ) being the spike train of neuron j and t k j are the times of its spikes.


Each set Z i is constructed based on S i .

z i = ∪ j∈[3m] B i,j , where B i,j is a subset of {x i,j , x i,j , y i,j , y i,j } and is constructed as follows: x i,j ∈ B i,j iff j ∈ S i , and x i,j ∈ B i,j iff j ∈ S i .


As with PageRank, QD-PageRank is determined by iterative evaluation of equation 3 from some initial distribution, and is equivalent to the primary eigenvector of the transition matrix Z q , where .

although P q (i→j) and P q '(j) are arbitrary distributions, we will focus on the case where both probability distributions are derived from R q (j), a measure of relevance of page j to query q: when choosing among multiple out-links from a page, the directed surfer tends to follow those which lead to pages whose content has been deemed relevant to the query (according to R q ).


The parameters π are generated by a stick-breaking process where ν i ∼ Beta(α, 1).

this process results in a decreasing sequence of probabilities π k . given a finite dataset, the probability of seeing feature k decreases exponentially with k. We consider the multi-way classification, where each training data is provided with a categorical label y, where For binary classification and regression, similar procedure can be applied to impose large-margin constraints on posterior distributions.


Example 1 (Linear model with nonlinear Bayes rule): In this scenario, we have two normally distributed classes with different covariance matrices.

denote X|Y = j ∼ N (µ j , Σ j ) for j = ±1, then µ −1 = (−2, 1) T , µ 1 = (1, 0) T , and .


. . , b K that can be nonnegatively (or conically) combined to approximate the input vectors a i .

where the combining coefficients c kn are restricted to be nonnegative.


( 5) can be written as a feed-forward pass in a recurrent neural network by stacking multiple computation blocks.

the first step is a convolution layer and the third step can be considered as a deconvolution layer sharing weights with the first step.


The remaining question pertains to the correct way to group the initial points into clusters that capture the information about the position on the graph after t-steps.

can we replace the initial point with an initial cluster, that enables prediction of the location on the graph at time t, with similar accuracy?


EM's simple message passing scheme also facilitates a very efficient parallel implementation.

all the δ messages for the current iteration in Alg.


. . S 4 .Details are provided in the appendix.

we need a chaining result (Lemma 4) that allows us to compute bounds for products of terms efficiently.


However, negative equivalence constraints violate this assumption, as dependencies between the hidden variables are introduced.

assume we have a group Ω = {(a 1 i , a 2 i )} P i=1 of index pairs corresponding to P pairs of points that are negatively constrained, and define the event E Ω = {Y complies with the constraints}.


Here we show in a rigorous mathematical treatment how homeostatic processes, which have previously received little attention in this context, can overcome common theoretical limitations and facilitate the neural implementation and performance of existing models.

we show that homeostatic plasticity can be understood as the enforcement of a 'balancing' posterior constraint during probabilistic inference and learning with Expectation Maximization.


Considering the highly complexity of the mapping function between the LR feature vectors and the desirable sparse codes, we propose to learn a set of mapping functions for each possible local image structures.

the K-means clustering algorithm is used to cluster the LR/HR patches into K clusters.


The MMSB provides better fits to real network data than single community models [23,27], but cannot account for node popularities.

we extend the assortative MMSB [9] to incorporate per-community node popularity.


Nearest-neighbor graphs have first been proposed for Shannon entropy estimation by Kozachenko and Leonenko (1987).

in the mentioned work only the case of N N S graphs with S = {1} was considered.


Notice that we could differently weight the encoding and the reconstruction energies in the loss function.

assigning a very large weight to the encoding energy corresponds to turning the penalty on the encoding prediction into a hard constraint.


In that case, let a * 1:T denote the revenue-minimizing buyer-optimal sequence.

once a buyer decides to start accepting the offered price at a certain time step, she will keep accepting from that point on.


Rewards: the reward function r(S, v) at state S is defined as the change in the cost function after taking action v and transitioning to a new state S 0 := (S, v).


However, identifying which models of neural dynamics can (or cannot) reproduce electrophysiological or imaging measurements of neural activity has been a major challenge [1].

many models of interest -such as multi-compartment biophysical models [2], networks of spiking neurons [3] or detailed simulations of brain activity [4] -have intractable or computationally expensive likelihoods, and statistical inference has only been possible in selected cases and using model-specific algorithms [5,6,7].


For that case, we can obtain a weaker lower bound by applying Reverse Generalized Holder inequality [15], obtaining from a different perspective a bound previously derived in [16].

let s 1 , • • • , s n−1 < 0 and s n such that 1 si = 1.


The quantity 1/ mc(U ) is then the maximum "min-max" margin with respect to all possible embeddings.

the rows of matrix P represent the "weights" of the binary classifiers and the rows of matrix Q the "input vectors" associated with the objects.


Note that the σ 2 and κ 2 parameters of LCA, though important to estimate, are not ultimately used for kNN classification.

after a model is trained, we simply Training error is computed using leave-one-out cross-validation.


Also, we note that (17) is tantamount to selecting the variational approximation with maximal Bayesian evidence [6].

we are selecting the Ĥ, out of a class of variational approximations to H, that most probably explains the training data t, marginalized over the weights.


Like for PCA [3], independent components analysis [1], and mixtures of factor analysers [5], the variational Bayesian algorithm for state-space models can be used to learn the structure of the model as well as average over parameters.

using F it is possible to compare models with different state-space sizes and optimise the dimensionality of the state-space, as we demonstrate in the following section.


[9] exploit group norm sparsity priors to learn dictionaries of natural images by first clustering the training image patches, and then learning a dictionary where members of the same cluster are encouraged to share sparsity patterns.

they use the 2,1 norm defined as ψ(H) = k ||h k || 2 , where h k are the elements of H that are members of the k-th group.


Our approach outperforms all the competitors significantly across all categories.

we achieve 12.19%, 6.32% and 10.22% improvement in AP for Cars, Pedestrians, and Cyclists, in the moderate setting.


These inputs are not usually available to humans during driving.

[41] applies GAIL to the task of modeling human driving behavior on highways.


However, it was recently discovered that by restricting attention to a sub-class of tensors such as low-CP rank orthogonal tensors [1] or low-CP rank incoherent 1 tensors [2], one can efficiently find a provably approximate factorization.

exact recovery of the factorization is possible for a tensor with a low-rank orthogonal CP decomposition [1].


Contrarily, in this section, we derive and study algorithms that perform hard policy-updates.

we generalize the prominent Approximate Policy Iteration (API) [13,7,11] and Policy Search by Dynamic Programming (PSDP) [1,19].


As a result, if max θ∈A λ | θ, xj | < mλ, we can also conclude that [β * λ ] j = 0 by (R1).

(R1) can be relaxed as In this paper, (R1 ) serves as the foundation for constructing our screening rules, Slores.


Let M be a (p, g)-black box for query class F. Then for any database stream X and stream of linear queries F over F, BBSCHEDULER(X, F, M, ✏, , , n, p, g) is (✏, )-differentially private for ✏ < 1 and (↵, )-accurate for sufficiently large constant C and For concreteness, we instantiate this general result with SMALLDB [BLR08], a differentially private algorithm for generating a synthetic database y that closely approximates a true database x on a every query from some fixed set F of k linear queries.

sMALLDB outputs some y :  In some applications it is more natural for accuracy bounds to improve as the database grows.


2, the two algorithms share similar theoretical guarantees with potential advantages of one on the other depending on the specific setting.

fLIPFLOP performs slightly better in settings 2, 3, and 4, whereas (A, B)-PROD obtains smaller regret in setting 1, where the constants in the FLIPFLOP bound show their teeth.


We assume that each trajectory in the behaviour data is generated by an agent with a fixed reward function.

we assume that the reward function does not change within a trajectory.


This applies to a large variety of complex networks (e.g., social and biological networks) as well as to data sets engineered as networks via similarly graphs, where one often attempts to get a first impression on the data by trying to identify groups with similar behavior.

finding communities allows one to find like-minded people in social networks, to improve recommendation systems, to segment or classify images, to detect protein complexes, to find genetically related sub-populations, or discover new tumor subclasses.


We now investigate to which extent the statistics of natural images with dichotomized pixel intensities can be characterized by pairwise correlations only.

we would like to know how the role of pairwise correlations opposed to higher-order correlations changes depending on the dimensionality.


Even if an especially clever agent were to realize the trap, they still could not unilaterally alter the outcome by restraining their own behavior.

cPR appropriation problems have socially-deficient Nash equilibria.


The Higgins' metric [7] is defined as the accuracy that a low VC-dimension linear classifier can achieve at identifying a fixed ground truth factor.

for a set of ground truth factors {v k } K k=1 , each training data point is an aggregation over L samples: l |, where random vectors z l are drawn i.i.d.


Our results show comparable (or improved) performance as compared to other models.

we observe that the global kernels out-perform the local kernels 2 by a small margin.


Because n t is usually quite large in practice, the number of control items is usually very small.

as n t → ∞, we have k * = 1, that is, there is no need for control items beyond fixing the unidentifiability issue of the biases.


In addition, we prove that the related RS, termed the Shapley mediator, does satisfy the stability requirement.

we show that the Shapley mediator possesses a potential function [27], and therefore any better-response learning dynamics converge to an equilibrium (see, e.g., [13,17]).


In this paper, we present a learning algorithm that explicitly considers safety, defined in terms of stability guarantees.

we extend control-theoretic results on Lyapunov stability verification and show how to use statistical models of the dynamics to obtain high-performance control policies with provable stability certificates.


Specifically, the perspective we are promoting here is that probability distributions defined by graphical models should be viewed as multi-linear functions, and the construction of jointrees should be viewed as a process of constructing arithmetic circuits that compute these functions.

the fundamental object being factored is a multi-linear function, and the fundamental result of the factorization is an arithmetic circuit.


As in [6], we formulate the problem of learning a fair set of predictions as a convex program.

our objective is to minimize the expected loss E x,y⇠D [L(f (x), y)], subject to the multifairness constraints defined by C. 5 we show that a simple variant of stochastic gradient descent due to [20] learns such linear families efficiently.


The presence of introns substantially complicates the identification of the transcribed and coding regions.

it is usually insufficient to identify regions depleted of stop codons to determine the encoded protein sequence.


Note that in social networks such as Twitter, the graph is directed.


More general notions of FDR with corresponding online procedures have recently been developed by Ramdas et al [14].

they incorporate the notion of memory and a priori importance of each hypothesis.


Optimal parameters were determined separately for each of the 22 cells, based solely on the training data.

in the cross-validation testing we used a validation paradigm: Using the 31 training stimuli, we removed an additional datapoint and trained our algorithm on the remaining 30 points.


The individual weights are chosen independently and randomly from a Gaussian distribution with mean and variance given by [J ij ] J = 0 and J 2 ij J = g 2 /N , where square brackets are ensemble averages [9 -11,13].

the control parameter g which scales as the variance of the synaptic weights, is particularly important in determining whether or not the network produces spontaneous activity with non-trivial dynamics (g = 0 corresponds to a completely uncoupled network and a network with g = 1 generates non-trivial spontaneous activity [4,9,10]).


Define See Figure 2  We conjecture that the difference Right γ n,k − Left γ n,k can be used as before to construct a density estimate.

if γ n,k is a shortest path from the anchor point X 0 to X s , we believe that under similar conditions on k and n as before, is a consistent estimator of the quantity log p(X s ) − log p(X 0 ).


Finger flexion states are estimated from the behavioral flexion traces (e.g., Figure 1 (a)).

samples on the extension parts of the traces are labeled with state "extension," samples on the flexion parts of the traces are labeled with state "flexion," and samples during rest are labeled with state "rest."


It is a subclass of weighted coverage functions [20].

we have the following chain relationship between these classes of functions: FL ⊂ WC ⊂ WCT ⊂ SCMM ⊂ DSF ⊂ All-Submodular-Functions [3].


Lemma 3 The function f Θ : R M → R M defined above is permutation equivariant iff all the offdiagonal elements of Θ are tied together and all the diagonal elements are equal as well.

this result can be easily extended to higher dimensions, i.e., X = R d when λ, γ can be matrices.


We first consider synthetic experiments using Gaussian measurements on Gaussian signals.

we considered a two layer network given by G(x) = relu(W 2 relu(W 1 x)) where each W i has i.i.d.


We show that this approach provides a natural trajectory, in which simple, cheap classifiers are applied initially, and used to provide guidance on which of our more expensive classifiers is likely to be more informative.

we show that we can get comparable (or even better) performance to a method that uses a large range of expensive classifiers, at a fraction of the computational cost.


Not all probability distributions can be efficiently sampled.

high-dimensional distributions such as the uniform distribution on the cube [0, 1] d are especially susceptible to sampling difficulties due to the curse of dimensionality, where in some cases it is necessary to take exponentially many (in the dimension of the space) sample points in order to obtain a satisfactory estimate.


Inference Atomization The problem of high sensitivity to small input differences considered previously is but one of the consequences of the inference atomization.

observe that the analysis of the trial outcome data is separated from the blinding assessment.


To overcome this limitation, the authors in [12] extended the analysis in [20] to treat the specific constrained problem (3).

[20,12] consider block-coordinate-descent methods where in each step t a subset J t of variables is updated.


To compute ARE, we pursue a similar optimization scheme as used for RESCAL which has been shown to scale to large datasets [22].

we solve the regularized optimization problem min via alternating least-squares, which is a block-coordinate optimization method in which blocks of variables are updated alternatingly until convergence.


If language users are rational, they might choose to structure their utterances so as to optimize communicative properties.

information-theoretic and psycholinguistic considerations suggest that this may include maximizing the uniformity of information density in an utterance.


We apply this convex duality framework to the f-divergence and Wasserstein distance families, providing interpretation for f-GAN, including vanilla GAN minimizing the JS-divergence, and Wasserstein GAN.

we generalize [1]'s interpretation of the vanilla GAN problem (1), which only holds for an unconstrained discriminator set, to the more general case with linear space discriminator sets.


However, it changes the distribution of paths seen during training.


One can imagine that in regions where a process is changing its output value rapidly, corrupted input measurements will have a much greater effect than in regions where the output is almost constant.

the effect of the input noise is related to the gradient of the function mapping input to output.


The true error bound for this stochastization of the large-margin classifier is not dependent on the dimensionality except via the margin.

since the Gaussian tail decreases exponentially, the value of P-l(f) is not very large for any reasonable value of f. at P(3), we have f :::; 0.01.


Again, we can observe that SURF gives better predictions at a lower computational cost, as well as good sparsity.

the paired t-tests showed that for all five real datasets, the RMSE and SC of our approach are significantly lower and higher than those of Remurs and GLTRM methods, respectively.


In this paper, we advance our understanding of the computational power of subsampling gradients of the f i for the problems of top eigenvector computation and regression.

we show that under assumptions of numerical sparsity of the input matrix we can achieve provably faster algorithms and new nearly linear time algorithms for a broad range of parameters.


Assume that we have a ground-truth distribution over the subspace spanned by the first d dimension of the Euclidean space R D . We will assume that this distribution is a single Gaussian N (0, λ 2 I d ).

suppose this distribution is corrupted with Gaussian noise along the orthogonal subspace of dimension D − d. for any "true" point x drawn from N (0, λ 2 I d ), its observation x e is drawn from N (x, σ 2 I D−d ).


The key insight is that the estimate Fi converges to F i point-wise at the same rate as the Hoeffding-Asumza inequality.

one does not pay an additional M factor from applying a union bound.


We have shown that Exploration and Hybrid achieve a sample complexity of O(1/ε) with much better dependency on the problem dimension than the ones in existing work.

if the smallest magnitude of the non-zero components of the optimal solution is not too small, the rate of Hybrid can be boosted to near the minimax optimal sample complexity of full information algorithms.


The elements of L are called loss functions.

the performance of the learner is measured in terms of its regret, The simplest possible case, which will be the focus of this paper, is when the losses are linear, i.e., when t (w) = f t , w for some f t ∈ F ⊂ R d . In fact, the linear case is not only simple, but is also fundamental since the case of nonlinear loss functions can be reduced to it: Indeed, even if the losses are nonlinear, defining f t ∈ ∂ t (w t ) to be a subgradient 1 of t at w t and letting ˜ t (u) = f t , u , by the definition of subgradients, t (w if an algorithm keeps the regret small no matter how the linear losses are selected (even when allowing the environment to pick losses based on the choices of the learner), the algorithm can also be used to keep the regret small in the nonlinear case.


Similar to the previous works, we want to evaluate our embedding using supervised learning tasks.

we want to evaluate whether the proposed embedding yields better results for a (1) learning-to-rank (2) classification and regression (3) link prediction tasks.


The theory of boosting is well-developed for the case of binary classification.

the exact requirements on the weak classifiers in this setting are known: any algorithm that predicts better than random on any distribution over the training set is said to satisfy the weak learning assumption.


In this section, we provide an analysis of NormSpecSync under a generalized Erdős-Rényi noise model.

the noise model we consider is given by two parameters m and p. we assume the observation graph G is fixed.


From (5) and using the above assumption, it is obvious that and thus R 0 ) is a valid upper bound to Θ ; furthermore it can be tightened as we progress in the MCGD method.

observe that ( α, Θ, R) with R = Θ is an optimal solution to P(R 0 UB ), we have In other words, for all feasible (α, Θ, R) to ) is an upper bound to R and Θ .


Observe that the distance between two uniformly chosen points on a sphere of radius r is close to r √ 2.

the median distance between two such points is exactly r √ 2, and with high probability for large d, the distance will be within some small radius ε of r √ 2.


Local convergence of the iterates is studied for the inference phase assuming that the true image is a fixed point for a proximal (lies on a manifold represented by proximal).

contraction of the learned proximal is empirically analyzed to ensure the RNN iterates converge to the true solution.


For each image-sentence sequence )}, we augment each sentence T l n with multiple sentences for training.

when we perform the LSA-based text summarization, we select top-κ highest ranked summary sentences, among which the top-ranked one becomes the summary sentence for the associated image, and all the top-κ ones are used for training in our model.


because each f (i) is monotone and submodular, f is monotone and submodular as well, so the greedy schedule's approximation guarantees apply to f . by Theorem 3, we have R benefit ≤ T t=1 t . Thus by Lemma 2, where R = T t=1 r t . To bound E [R benefit ], it remains to justify the assumption that each meta-action ãt always takes unit time.


On the other hand, the information contained in the variance is not removed because the variance contains a discontinuous spatial component, S " ¡ (' .

the variance of a correlation profile which has only high Fourier modes can still preserve its slowly varying components.


However, it also forms the basis of our projected gradient descent strategy for computing the projection in Eq.

5 under more general divergences D. we will do this by iterating for some step-size λ.


Most previous work on feature selection in supervised learning with functional data is quite recent and focuses on regression problems; for instance, on the analysis of fMRI images (Grosenick et al., 2008;Ryali et al., 2010) and NIR spectra (Xiaobo et al., 2010).

adaptations of lasso and other embedded methods have been proposed to this end (see, e.g., Kneip and Sarda (2011); Zhou et al.


Let the state space S and Player 2's action space A 2 remain the same as in M . Let A 1+ , p + (•|•, •, •), r + (•, •, •) be Player 1's action set, the transition kernel, and the reward function of M + , such that for any a 1 ∈ A 1 and a 2 ∈ A 2 and an admissible transition probability p(•|s, a 1 , a 2 ) ∈ P k (s, a 1 , a 2 ), there is an action a 1+ ∈ A 1+ such that p + (•|s, a 1+ , a 2 ) = p(•|s, a 1 , a 2 ) and r + (s, a 1+ , a 2 ) = r(s, a 1 , a 2 ).

player 1 selecting an action in A 1+ is equivalent to selecting an action in A 1 and simultaneously selecting an admissible transition probability in the confidence region P k (•, •).


The parameter θ ∈ R + is a quantity that may be set to obtain sharper upper bound on the violation of constraints and may be chosen arbitrarily.

a larger value for θ imposes larger penalty on the violation of the constraints and results in a smaller violation for the objectives.


An efficient way to counteract this lag will be to apply forces anticipating changes in the desired velocity.

anticipation can be beneficial even when one can act instantaneously on the plant.


One basic fact is that 1 distance is not embeddable in 2 [17].


In particular, simply fitting a multidimensional rectangle distribution to the data (by using PCA to align the axes, and then assuming a uniform distribution on each axis) works surprisingly well for a wide range of distributions.

using the analytic eigenfunctions of a uniform distribution on data sampled from a Gaussian, works as well as using the numerically calculated eigenvectors and far better than boosting or RBMs trained on the Gaussian distribution.


Despite only requiring weak feedback, our TPP learning algorithm has provable regret bounds and empirically performs well.

we propose a set of trajectory features for which the TPP generalizes well on tasks which the robot has not seen before.


To this end, we develop a fully nonparametric causal discovery method that relies only on observational data.

our method does not a priori postulate a particular functional model for the interactions between the variables or a particular noise structure.


As an example of a real world data set, we used the NGRIP ice core data (provided by Niels-Bohr institute in Copenhagen, http://www.iceandclimate.nbi.ku.dk/data/), which provides an undisturbed ice core record containing climatic information stretching back into the last glacial.

this data set as shown in figure 3 contains 4918 observations of oxygen isotope concentration δ 18 O over a time period from the present to roughly 1.23 • 10 5 years into the past.


the qualitative aspects of the dynamics reflected in the absolute values and imaginary parts of the eigenvalues are well characterised.


For example, suppose we know that f con is Gaussian with known covariance matrix and unknown mean.

proposition 1 says we cannot design R ε so that it can decontaminate (1−ε)f tar +εf con for all f tar ∈ D. it is impossible to design an algorithm capable of removing Gaussian contamination (for example) from arbitrary target densities.


Then its dual norm is defined as For instance, the dual norm of Further, Lemma 3.

for any vectors x, y and any norm ||.|| A , the following inequality holds: 4 its second derivative (Hessian in higher dimensions) is bounded away from zero by at least β.


We use standard approximate dynamic programming with a quadratic, normalized cost r(x, u) = x T Qx + u T Ru, where Q and R are positive-definite, to compute the cost-to-go J π θ .

we use a piecewiselinear triangulation of the state-space as to approximate J π θ , see [39].


In contrast to methods based on influence functions that require large number of samples due to their asymptotic nature, we empirically show that the developed ALOOCV works well even when the number of samples and features are small and comparable to each other.

aLOOCV is capable of predicting overfitting and hence can be used for model selection and choosing the regularization hyperparameter.


For example, the first convolutional layer is typically believed to represent edges in the image while the latter layers learn "high level" features [11].

it is common to treat the last few layers as representations for classification.


Our study proves that the fast learning rate is possible for RFSVM in both realizable and unrealizable scenarios when the optimized feature map is available.

the number of features required is far less than the sample size, which implies considerably faster training and testing using the random features method.


Nevertheless, as we shall see, optimizing these networks with gradient descent leads to very different solutions.

we show that for fully connected networks with single output, optimizing the exponential loss over linearly separable data using gradient loss again converges to the homogeneous hard margin support vector machine solution.


The optimism in the face of uncertainty paradigm has already lead to several successful results for solving decision making problems.

it has been applied in the following contexts: multi-armed bandit problems [1] (which can be seen as single state MDPs), planning algorithms for deterministic systems and stochastic systems [8,9,17], and global optimization of stochastic functions that are only accessible through sampling.


Instead, we adopt the likelihood p(X| Φ, ŷ) as the criterion to select parameters, which is indirectly related to our evaluation criterion (i.e., accuracy).

we test multiple values of c and , and select the value that produces a model with the maximal likelihood on the given dataset.


While cumulants are essentially unique with the additivity and homogeneity properties [17] when no restrictions are made on the probability space, the preprocessing step of ICA gives additional structure (like orthogonality and centering), providing additional admissible functions.

[20] designs "robust cumulants" which are only minimally effected by sparse noise.


In this paper, we introduced a novel multi-armed bandit setting where the objective is to perform as well as the arm with the best risk-return trade-off.

we relied on the mean-variance model introduced in [10] to measure the performance of the arms and define the regret of a learning algorithm.


In this paper, we assume that the GP over user preference functions is zero mean.

at any particular time, the user does not want to listen to most of the songs in the world, which leads to a mean preference close enough to zero to approximate as zero.


We therefore suspect that there is considerable room for improving our analysis of high dimensional data in the sub-exponential decay case.

removing the exponential dependence on d under reasonable assumptions is an interesting direction for future work.


We now have the machinery to describe the development in [20] of a Winsorized Poisson graphical model.

kaiser and Cressie [20] assert in a Proposition of their paper that there is a valid joint distribution consistent with these Winsorized Poisson node-conditional distributions above.


Tensor power method in Algorithm 1 requires significant storage to be deployed: Ω(d 3 ) memory is required to store a dense d × d × d tensor, which is prohibitively large in many real-world applications as tensor dimension d could be really high.

we show in this section how to compute tensor decomposition in a memory efficient manner, with storage scaling linearly in d. we consider the case when tensor T to be decomposed is a population moment E x∼D [x ⊗3 ] with respect to some unknown underlying data distribution D, and data points x 1 , x 2 , • • • i.i.d.


This simulation is inspired by the experimental study of noise correlations in cortical area MT due to [10], where it was suggested that time-varying correlations between pairs of neurons play a significant role in explaining behavioral variation in smooth-pursuit eye movements.

the findings in [10] and [4] suggest that short-term increases in noise correlations are likely to occur after feedback arrives and neurons within and upstream from MT synchronize.


Each iteration of SNGD is as simple and efficient as SGD, but is much more appropriate for non-convex optimization problems, overcoming some of the pitfalls that SGD may encounter.

we define a family of locally-quasi-convex and locally-Lipschitz functions, and prove that SNGD is suitable for optimizing such objectives.


Finally, we can recover an approximation bound that is analogous to the primal bound, which bounds the approximation error between H + and the best representable approximation to the on-policy fixed point  Finally, using H Π = OH Π and Lemma 1, we obtain To compare the primal and dual results, note that despite the similarity of the bounds, the projection operators do not preserve the tight relationship between primal and dual updates.

even if (1−γ)q = Hr and (1−γ)(Oq) = (OH)r, it is not true in general that (1−γ)(POq) = (POH)r. The most obvious difference comes from the fact that in the dual, the space of H matrices has bounded diameter, whereas in the primal, the space of q vectors has unbounded diameter in the natural norms.


Messages relate to the Θ coefficients by the following definition where the (nonstandard) notation Θ \y x indicates the matrix Θ V x for which V is the set of all the nodes in X that are reachable by x if y were removed from the graph.

θ \y x is summarizing the effect that x has on the entire network except for the part of the network that x can only reach through y. Propagation: The message-passing scheme is organized to recursively compute the Θ matrices using Theorem 3.2.


the long bandwidth kernel in panel (b), real spike trains, 4 and 8 spike trains), outperforming the GP IGIP method requires an optimal kernel choice, which can not be judged from the data alone.

the adaptive kernel method generally performed more poorly than GP IGIP.


In other words, the weights are initialized as z U m (h t ) = 1 M ∀m and are then updated recursively using the likelihood of the dynamics model for that particle of observations as z U m (has ) ∝ z U m (h)P (s |a, s, P m ) = z U m (h) P m (s, a, s ).

one advantage of this definition is that it enforces a correspondence between the history and belief representations in the finite-dimensional space, in the sense that z we can work in history space during planning, alleviating the need for complete belief updates, but via a finite and well-behaved representation of the actual belief -since different histories corresponding to the same belief are mapped to the same representation.


We represent pegs G % ¨ § © by numbers 1, 2, 3, respectively.

given an instance of size , we push onto ds the values ¦ ¡ ¨¥ £ .


Here we focus on a complementary problem-learning generative graphics networks via the idea of "graphics as inverse vision".

we learn our generative model with recognition models that recover 2.5D sketches from generated images.


This in turn gives rise to two approximation algorithms with provable guarantees.

we make the following contributions: -We bound the difference between objective values of stationary points and the global optimum.


The proof of Theorem 4 proceeds by providing a "guess" for the Investor's action, which is a modification of the original Black-Scholes hedging strategy.

when the current price is S and the remaining budget is c, then the Investor invests (S, c) := SU S (S, c).


To motivate our construction, we would like to note that boosting algorithms define weights in two different domains: the vectors w t ∈ R m which assign weights to examples and the weights {α t : t ∈ [T ]} over weak-hypotheses.

in the terminology used throughout this paper, the weights w t ∈ R m are primal vectors while (as we show in the sequel) each weight α t of the hypothesis h t is related to a dual vector λ t . we show that Eq.


It is well known [7,14] that uniformly sampling O(n/γε −2 log(n/δ)) points with replacement is sufficient to obtain w.p.

1 − δ the following guarantees on Φ m We can now rewrite Π n as using uniform sampling we can reduce the size of the search space H m by a 1/γ factor (from n to m n/γ) in exchange for a γ additive error, resulting in a computation/approximation trade-off that is linear in γ. Exploiting the error bound for γ-preserving dictionaries we are now ready for the main result of this paper: showing that we can improve the computational aspect of kernel k-means using Nyström embedding, while maintaining optimal generalization guarantees.


( 4) if, in addition, we ask that v = sign(θ 0 ).

earlier work [14,23,21] required generalized irrepresentability plus sign-consistency in zero noise, and established sign consistency in non-zero noise.


Worse yet, even if the scores are calibrated, the ranking of the scores itself may not be reliable.

a higher confidence score from the model does not necessarily imply higher probability that the classifier is correct, as shown in [6,7,8].


Indeed as the number of possible models k is increased to 14, the number of effective models is also increased, but only moderately, from an average of 2 to an average of 3.5.

only small number of models are used in practice, which avoids severe overfitting.


Even though our setting is far more general than exponential utility and exponential families, the framework we develop can also be used to show that their results can be extended to interactions between traders who have what we call "compatible" risks and beliefs.

for any risk-based trader possessing a risk ρ with dual α, we can think of that trader's "belief" as the least surprising distribution p according to α.


The tracking oracle tracks the optimal proportions at each round.

the tracking oracle samples the arm with the largest ratio its optimal and empirical proportions.


In this paper, we propose a novel homotopy smoothing (HOPS) algorithm for solving the problem in (1) that achieves a lower iteration complexity than O(1/ ).

the iteration complexity of HOPS is given by Õ(1/ 1−θ ), where θ ∈ (0, 1] captures the local sharpness (defined shortly) of the objective function around the optimal solutions.


It is also interesting to note that Theorem 17 appears to be tight in the sense that none of the hypotheses can be removed.

for neurons with general non-montonic activation functions, or for neurons with monotonic activation functions whose output magnitude violates the aforementioned bounds, there are example networks that can efficiently compute any real-valued function.


For example, one might object to our "pessimism bias" interpretation of lower expected reward rates for unchosen alternatives, by attributing it to a confirmation bias [23] or a sampling bias [24].

subjects may report especially low reward estimates for unchosen options, either because they retroactively view discarded options as more undesirable (confirmation bias), or because they failed to choose the unchosen options precisely as they decided these were less valuable to begin with for some reason (regardless of the reason, this scenario introduces a sampling bias of the unchosen arms).


[5] 29.5 ± 0.4% In the second experiment we study the statistical performance of our method as the spectral function changes.

we choose functions giving rise to Schatten L p prenorms, as discussed in Section 4.


We now show why starting with extreme teaching instances as in curriculum learning, as opposed to the boundary strategy, is optimal under our setting.

we consider the problem of selecting an optimal teaching sequence of length t = 2, one positive and one negative, (x 1 , 1), (x 2 , 0).


Since the discovery by Adrian [1] that action potentials are generated by sensory neurons with a frequency that is substantially determined by the stimulus, the idea of rate coding has become a prevalent paradigm in neuroscience [2].

today the coding properties of many neurons from various areas in the cortex have been characterized by tuning curves, which describe the average firing rate response as a function of certain stimulus parameters.


We think of permutations as complete matchings in a bipartite graph.

we consider an n × n bipartite graph and each permutation corresponds to a complete matching in the graph.


The related applications in biological and medical research can be found in existing literature (Hudson et al., 2009;Bandyopadhyaya et al., 2010;Ideker and Krogan, 2012).

given n We are interested in estimating the difference of the precision matrices of two distributions: We define the empirical covariance matrices as: S  where S X and S Y are the empirical covariance matrices.


Therefore, importantly, n does not necessarily grow to infinity, if for example N, 1 σ grow appropriately with p. That means that Theorem 2.5.A and Theorem 2.5.B imply non-trivial guarantees for arbitrary sample size n. The proposition below shows that if σ is at most exponential in −(1 + ) (p+2n) 2 2n + (2 + p n ) log (RQ) for some > 0, then for appropriately chosen truncation level N the LBR algorithm recovers exactly the vector β * with n samples.

with one sample (n = 1) LBR algorithm tolerates noise level up to exponential in −(1 + ) p 2 /2 + (2 + p) log(QR) for some > 0.


Similar to our algorithm, the VSUGS method proposed takes a sequential updating approach, but relies on a different approximation.

what we approximate is a joint posterior over both data allocation and model parameters, while VSUGS is based on the approximating the posterior of data allocation.


[2017], we then calculate the expectation of visibility and depth along each ray.

given a ray R with N samples R 1 , R 2 , ... , R N along its path, we calculate the visibility (silhouette) as the expectation of the ray hitting the voxels: Similarly, the expected depth can be calculated as , where d j is the depth of the sample R j .


In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set C. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set C is simple for a quadratic objective function.

our results hold if one can find a ⇢-approximate solution of a quadratic program subject to C in polynomial time, where ⇢ < 1 is a positive constant that depends on the structure of the set C. Under this condition, we show that the sequence of iterates generated by the proposed framework reaches an (✏, )-second order stationary point (SOSP) in at most O(max{✏ 2 , ⇢ 3 3 }) iterations.


A stochastic process {L(ω)} ω∈R + is a Lévy process if it has stationary, independent increments and it is continuous in probability.

l must satisfy  Poisson process, a special case of a Lévy process.


A Nash equilibrium is defined by a joint assignment of actions such that no player has any incentive to deviate.

if σ p = θ denotes the action choice for the protagonist and σ a = {a t , b t } the choice for the antagonist, then the joint action σ = (σ p , σ a ) is a Nash equilibrium if U p (σ p , σ a ) ≤ U p (σ p , σ a ) for all σp , and U a (σ p , σa ) ≤ U a (σ p , σ a ) for all σa .


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

the warm start model was learnt based on full bilingual data.


One practical consequence of this result is that we can apply a standard automatic differentiation procedure to L to obtain an unbiased gradient estimator.

we convert the stochastic computation graph into a deterministic computation graph, to which we can apply the backpropagation algorithm.


In the following theorem, we show that the latter is the case.

multiplicative weights has the property of not doing better than the best expert in hindsight.


Thus, we can divide such that in each interval, all c t+1 functions are linear.

there exist constants l j i and m j i such that c λ t+1 (s j ) = By the duality theorem, we have that c λ t (s) equals to the optimal value of the following LP on y and q.


The hyperparameter θ constructs the kernel matrix A(θ) ∈ S d×d of the training data {x i } d i=1 (see [22]).

one can find a good hyperparameter by minimizing the negative log-marginal likelihood with respect to θ: For handling large-scale datasets, [29] proposed the structured kernel interpolation framework assuming θ = [θ i ] ∈ R 3 and where W ∈ R d×r is some sparse matrix and K ∈ R r×r is a dense kernel with r d. in [29], r "inducing" points are selected and entries of W are computed via interpolation with the inducing points.


That is, we examine if the correspondences identify regions that exhibit a relationship with the task (in our experiment, a  language task) even if they are ambiguous or not detected based on the standard single-subject GLM analysis.

can we transfer evidence for the activation of specific regions between subjects, for example between healthy controls and tumor patients?


We also show that some learnability assumption on C is necessary in order to achieve a high-utility (C, d, ⌧)-metric multifair predictions efficiently.

we give a reduction from inverting a boolean concept c 2 C to learning a hypothesis f that is metric multifair on a collection H derived from C, where the metric samples from d encode information about the concept c. Recall, that for any H and d, we can always output a trivial (H, d, 0)-metric multifair hypothesis by outputting a constant hypothesis.


For problems with more than two components, most of the existing approaches [5,6,7,8] rely on the Tensor Methods.

for a D-dimensional linear MR problem, [6] propose a convex optimization formulation using a third-order tensor, which results in a computational cost of O(N D 12 ) and a sample complexity of O(D 6 / 2 ), limiting its application to problems of small dimension.


Neither of these situations can be handled by generic algorithms for finite PM games.

the model considered an action space X for the learner, that has a small subset of actions defining a global observable set (see Assumption 2 in Section 2).


2 Note that we do not assume the samples from GP(m, k) are in H. In the following, without loss of generality, we assume the GP prior considered in regression has µ = 0 and Σ = I.


We compare our proposed techniques to the current state-of-art generative and discriminative models developed in [7].

[7] introduced 3 encoding architectures -Late Fusion (LF), Hierarchical Recurrent Encoder (HRE), Memory Network (MN) -each trained with a generative (-G) and discriminative (-D) decoder.


Deep Gaussian Processes (DGP) [Damianou and Lawrence, 2013] are multilayer predictive models that are highly flexible and can accurately model uncertainty.

they have been shown to perform well on a multitude of supervised regression tasks ranging from small (∼500 datapoints) to large datasets (∼500,000 datapoints) [Salimbeni and Deisenroth, 2017, Bui et al., 2016, Cutajar et al., 2016.


H p (r) is a non-decreasing, non-negative function, and there exist δ Hp > 0 and M Hp ≥ 0 such that, r ≥ H p (r) ≥ r − M Hp r 3 , for all r < δ Hp .


In this paper, we consider the problem of recovering a signal x * ∈ R n from (possibly noisy) magnitude-only linear measurements.

for sampling vector a i ∈ R n , if then the task is to recover x * using the measurements y and the sampling matrix A = [a 1 . . . a m ] .


The intrusions have a considerable variety.

they are: N0 -1 kHz pure tone, N1 -white noise, N2 -noise bursts, N3 -"cocktail party" noise, N4 -rock music, N5 -siren, N6 -trill telephone, N7 -female speech, N8 -male speech, and N9 -female speech.


The most relevant previous work is that for which a classifier is directly implemented via a GP, without an explicit connection to the margin associated with the SVM [7].

gP-based classifiers have been developed by [17].


With the expansion (5) in hand, we can show a number of finite-sample convergence guarantees for the robustly regularized estimator Based on the expansion (5), solutions b ✓ rob n of problem (6) enjoy automatic finite sample optimality certificates: for ⇢ 0, with probability at least 2 are constants (which we specify) that depend on the loss `and domain ⇥.

with high probability the robust solution has risk no worse than the optimal finite sample robust objective up to an O(⇢/n) error term.


sample drawn from a common entity-wise GP with covariance Σ : U × U → R. This provides a case of p(t|θ) in a SRM, where θ = θ Σ determines the GP kernel function Σ. Optimizing the GP kernel Σ via evidence maximization means to learn the dependency of entities in U, summarized over all the entities v ∈ V. There is a drawback if applying HGP to link prediction.

the model only learns a one-side structure, while ignoring the dependency in V. the attributes of entities v cannot be incorporated even if their entity attributes are available.


Sparse-overcomplete representations have a number of theoretical and practical advantages, as demonstrated in a number of recent studies [6,8,3].

they have good robustness to noise, and provide a good tiling of the joint space of location and frequency.


Beyond the traditional visual captioning task [41,28,43,40,18] that produces one single sentence, generating long and topic-coherent stories or reports to describe visual contents (images or videos) has recently attracted increasing research interests [19,35,22], posed as a more challenging and realistic goal towards bridging visual patterns with human linguistic descriptions.

report generation has several challenges to be resolved: 1) The generated report is a long narrative consisting of multiple sentences or paragraphs, which must have a plausible logic and consistent topics; 2) There is a presumed content coverage and specific terminology/phrases, depending on the task at hand.


One important aspect of our model is that, in terms of graphical representation, the distribution p(ω) is independent of x, while the distribution p θ (z|x, ω) is conditional on x.

p(ω) tends to capture uncertainty of model parameters (epistemic uncertainty), while p θ (z|x, ω) reacts sensitively to uncertainty in data, varying across different input points (heteroscedastic uncertainty) [14].


By simple concentration results we have � � A − A� 2 = ν with high probability; however, � A is in general not a good estimator of A in Frobenius norm when A is high-rank.


Thus, we expect ICMC to perform better on models that are closer to real-world graphs and have power-law distributed degrees.

as strongly supported by experiments (see Figure 3), we hypothesize that ICMC solves exact matrix completion from an almost optimal number of entries for sampling graphs drawn from the preferential attachment model.


Our main focus are learning problems that can be formulated as stochastic convex optimization.

these are problems in which the goal is to minimize the expected loss: F P (w) .


This definition of F trades off between exploitation (maximizing the most plausible F h ) and exploration (distinguishing between more and less plausible F h ) by allowing each Fi to reach its maximum value either by having F h reach ↵ i or having G h reach  i .

each of the thresholds can be satisfied with either a sufficiently large utility F h or a sufficiently large distance G h . Figure 2 shows the logical relationships between these components.


Following the classical marginalization, it suffices to define a projection L ∈ R (n−1)×n with property L1 n = 0 n−1 .

all biases of the form 1 n w are mapped to the nullspace of L. The Matrix T-distribution under affine transformations [10, Theorem 3.2] reads LX ∼ T (α, β, LM, LΨL , Ω) and in our case (Ω = I d , LM = L1 n w = 0 (n−1)×d ), we have ) Note that due to the statistic LX, the likelihood is constant over all X (or S) mapping to the same D. As we are not interested in any specifics about L other than its nullspace, we replace the image with the kernel of the projection and define matrix Q : which accounts for arbitrary latent feature correlation Σ and all mean matrices M = 1 n w . In hindsight, the combination of Bayesian and classical marginal likelihood might appear arbitrary, but both strategies have their individual strengths.


Thus, the response of the or-node U i is defined as, Collaborative Edge: For any pair of leaf-nodes (L j , L j ′ ) respectively associated with two different or-nodes, we define the collaborative edge between them according to their contextual cooccurrence.

how likely it is that the object contains contours detected via the two leaf-nodes.


This perspective allowed us to re-evaluate the necessity of regularization methods such as weight decay, and to suggest new methods for normalization, targeting the computational, numerical and task-specific deficiencies of current techniques.

we showed that the use of L 1 and L ∞ -based normalization schemes could provide similar results to the standard BN while allowing low-precision computation.


As a consequence, similar experiments might yield equally interesting insights into the functioning of DNNs, especially as a comparison to high-quality measurements of human behaviour.

human data for our experiments were obtained using a controlled lab environment (instead of e.g.


Hence, (10) and (12) with γ = 0 reduce to the 1minimization of SSC.

our framework is a generalization of SSC, which simultaneously finds similarities and missing entries for incomplete data.


Finally, we assume that the Causal Markov Assumption and the Causal Faithfulness Assumption [23] both hold.

the conditional independences in the observational distribution correspond one-to-one with the d-separations in the causal DAG.


Given that we do not have a perfectly tight bound, one important consideration arises: does the minimum of the stochastic bound predict the minimum of the true error rate (as predicted by a large holdout dataset).

can we use the stochastic bound to determine when we should cease training?


S, and then use this to define pairwise regrets as above.

for the three tournament solutions discussed above, we will define the following natural notions of individual regret: ∈ TC(P) In the special case when P admits a Condorcet winner i * , the three individual regrets above all reduce to the Condorcet individual regret, r CW P (i) = ∆ P i * ,i .


This classification can be turned into a ranking of outlierness by considering the signed distance to the separating hyperplane.

the further an object is located in the outlier half space, the more likely it is to be a true outlier.


Our CL encoder architecture is based on what we call a spatially-replicated LSTM: Given that, for a sequence of length T , a convolution outputs a T × H × W × C sized tensor X, a spatially replicated LSTM takes as input each of the X(:, i, j, :) cells, and encodes its output to the same spatial location.

the weights of the LSTM at each spatial location is shared, but the hidden states are not.


With a softmax function, we normalize the parameters for all the devices to obtain a probability distribution of the placement.

the probability of choosing the j-th device for the m-th operation is given by: where D is the total number of devices, and u m denotes the vector of parameters {u m1 , u m2 , • • • , u mD } for the m-th operation.


(iv) We want to be able to use such an algorithm together with the public unlabeled data to construct a differentially private learner that outputs a classifier, which can then be used to answer as many classification queries as we wish.

given the second goal above, the final private learner would be completely agnostic to the intricacies of the underlying non-private learner and its model.


Although it accurately predicts the intrinsic images of the train class, its performance drops when tested on other classes.

the shape predictions suffer the most, as they are the most dissimilar from anything seen in the training set.


Recent years have seen a surge of work at the intersection of social choice and machine learning.

significant attention has been given to the learnability and applications of noisy preference models [16,2,1,3,24].


The contributions of this paper are as follows: • A theoretical result (Lemma 4) that upper bounds the conditional mutual information of arbitrarily large sets of random variables in polynomial time.

we do not assume that an efficiently computable mutual information oracle exists.


Our hypothesis is that the state-of-the-art algorithms such as [11] are seeking features invariant to the domains whereas we seek an explicit similarity metric explaining both differences and similarities of domains.

instead of seeking an invariance, we seek an equivariance.


Setting k = 1 is equivalent to collecting the node labels disregarding the neighbor order and comparing the label distribution between all node pairs.

consider the following smoothing algorithm for the WL kernel.


The latter is due to the fact that, by multiplying the activations by a constant scale larger than one, β becomes more insignificant.

for β = 0 the distance between the peaks of the distribution is non-consequential.


One important property of the Arnoldi relation is that the residuals are orthogonal to the space spanned by the columns of V = Q (t+1) H (t) .

in the following section, we introduce methods to speed up the matrix-vector product v = Aq.


On the other hand, due to the symmetry between X and Z, we can also view CCA as an approach to feature-aware LSDR.

cCA is equivalent to first seeking projection directions W z of Z, and then performing a multi-output linear regression from x n to W z z n , under the constraints W x X T XW T x = I, to obtain W x .


This paper considers a number of related inverse filtering problems for hidden Markov models (HMMs).

given a sequence of state posteriors and the system dynamics; i) estimate the corresponding sequence of observations, ii) estimate the observation likelihoods, and iii) jointly estimate the observation likelihoods and the observation sequence.


We propose to construct and learn CNNs to directly estimate these messages in the message passing procedure, rather than learning the potential functions.

we directly learn factor-to-variable message estimators.


The key to swapbased algorithms is that cluster assignments are never frozen.

when considering the swap of x(4) and x(7), clarans assigns x(2), x(3) and x(4) to the cluster of x(1) before computing the new MSE.


The distribution complexity r(Q) of the planting distribution Q is the smallest integer r ≥ 1 so that there is some S ⊆ [k], |S| = r, so that the discrete Fourier coefficient Q(S) is non-zero.

the distribution complexity of Q is r if Q is an (r − 1)-wise independent distribution on {±1} k but not an r-wise independent distribution.


Therefore, the total number of ⊥'s will not exceed Before moving to the next section, we should mention that Algorithm 2 is not computationally efficient.

if H contains exponentially many functions in the natural parameters of the problem, which is often the case, the running time of Algorithm 2 becomes exponential.


One aspect of our RL algorithm that is relevant to understanding our results is that we use a linear function approximator to store our policy.

for each state feature, we maintain a vector of real-valued weights indexed by the possible actions.


To obtain mean-squared error estimates from MSB, we select our posterior mean as a point-estimate (the comparison algorithms do not generate posterior predictions, only point estimates).

for each simulation scenario, we sampled multiple datasets and compute the matched distribution of r A m . rather than running simulations and reporting the distribution of performance for each algorithm, we compare the algorithms per simulation.


From a theoretical perspective a key question has become to characterize statistical and computational tradeoffs, that is if, or under which conditions, computational gains come at the expense of statistical accuracy.

recent results considering least squares, show that there are large class of problems for which, by combining Nyström or random features approaches [10,11,12,13,14,15] with ridge regression, it is possible to substantially reduce computations, while preserving the same optimal statistical accuracy of exact kernel ridge regression (KRR).


This ensures that we find the optimal point in the final grid G k . We also want I G k i to consist only of intervals of width 1 /k with endpoints contained in G k . This ensures that the overall number of points processed over all iterations is at most O(nk), and by bounding the number of intervals in each I i in each iteration we can achieve significantly better performance.

the scaling algorithm for convex functions by Ahuja and Orlin (2001) can be seen as a particular realization of this framework where the refinement process keeps the size of each I i to exactly one.


Given this transition matrix, one can imagine a random walk starting at every point on the graph.

the probability distribution of the positions of a random walk, starting at Xj after t time steps, is given by the j-th row of the t -th iteration of the I-step transition matrix.


Proximal-gradient methods and accelerated proximal-gradient methods [1,2] are among the most important methods for taking advantage of the structure of many of the non-smooth optimization problems that arise in practice.

these methods address composite optimization problems of the form minimize where g and h are convex functions but only g is smooth.


For example, the standard generalization error bounds for stochastic gradient descent (SGD) on convex Lipschitz functions cannot be obtained by proving uniform convergence for all empirical risk minimizers (ERM) [13,26].

there exist empirical risk minimizing algorithms whose generalization error is √ d times larger than the generalization error of SGD, where d is the dimension of the problem (without the Lipschitzness assumption the gap is infinite even for d = 2) [13].


To this end, we need to introduce certain assumptions on (i) the construction of weighted points {(w i , X i )} n i=1 and on (ii) the smoothness of the integrand f ; we discuss them in Sections 4.1 and 4.2, respectively.

we introduce the notion of powers of RKHSs [22] in Section 4.2, which enables us to characterize the (relative) smoothness of the integrand.


As an isotropic Gaussian vector has mutually independent components, there is no statistical dependency among elements of u.

x/ √ z equals to a transform that completely eliminates all statistical dependencies in x.


For matrices, • p denotes the Schatten p-norm, X p = ( i σ i (X) p ) 1/p , where σ i (X) are the singular values of X.

• * = • 1 is the trace or nuclear norm, • F = • 2 is the Frobenius norm, and • = • ∞ is the operator norm.


In addition, it is worth mentioning that when performing the SVM, each x n was normalized to be V i=1 x n i = 1 because discrimination is much easier in the V − 1-dimensional simplex than in the original V dimensional space.

classification is generally not determined by the number of words on the page; actually, normalization could also significantly improve the performance.


The recent work of [8], concerning generalized weighted variants of k-NN, shows that asymptotic normality holds under the weaker restriction k/n 4/(4+d) → 0 if f is twice differentiable at x. Asymptotic normality as stated above yields some insight into the rate of convergence of k under the stated conditions on k. In fact, [8] shows that such a result can be obtained in expectation for n = n(x) sufficiently large.

their conditions on k allows for a setting of k ≈ n 4/(4+d) (not allowed under the above conditions) yielding a minimax-optimal While consistency results and bounds on expected error are now well understood, we still don't have a clear understanding of the conditions under which high probability bounds on |f k (x) − f (x)| are possible.


Table 1 shows that a three layer GSN clearly outperforms all other models, except for the MNIST random-background dataset.

when comparing the GSN-3 to the radial basis function support vector machine (SVMrbf), i.e.