Hence, we view the rows of D as samples from k distributions D i with distribution D i corresponding to cluster U i .

for each cluster U i , we have partitioning the rows of D with a mixture model is exactly the same as recovering the clusters U i of G. It is left to check that the conditions of Lemma 2 are satisfied.


This is of particular interest in the context of latent collaborative retrieval [24]; unlike standard ranking task, here the number of items to rank is very large and explicit feature vectors and scores are not given.

we develop an efficient parallel stochastic optimization algorithm for this problem.


Then, the probability of deleting the solution with the best f value in Q when implementing lines 8-15 is at most Taking the logarithm, we get where the inequality is since log j−2δ j+1 is increasing with j, and the last equality is since the derivative of log (j−2δ) j−2δ (j+1) j+1 with respect to j is log j−2δ j+1 .

we have where the last inequality is by 0 < 1 − 2δ ≤ 1 and (1 − 2δ) 1−2δ ≥ e −1/e .


Several important methods such as support vector machines (SVM), boosting, and neural networks follow the ERM paradigm [34].

the algorithmic aspects of ERM have received a vast amount of attention over the past decades.


It turns out that each w (k) (η) forms a univariate polynomial whose degree is at most k. Furthermore, since f η (w) are L-smooth µ-strongly convex for any η ∈ [µ, L], standard convergence analysis for GD (e.g., [13], Theorem 2.1.14) guarantees that |w , where κ denotes the condition number.

substituting Equation (6) for w (k)  we see that the faster the convergence rate of a given optimization algorithm is, the better the induced sequence of polynomials (w (k) (η)) k≥0 approximate 1/η w.r.t.


At the setting for some non-negative constant function c(n).

the α i = 1 extremum is locally concave and is a maximum.


For the minimization problem in Theorem 1, the gradient descent algorithm with random initialization can converge to a global optimum only if the step size satisfies where ⇢(R) is the largest singular value of R. Remark 2.

corollary 1 shows that, unlike the optimization of the ordinary least squares problem, the step size required for the convergence of the algorithm depends on the parameter to be estimated, R. estimating linear mappings with larger singular values requires the use of a smaller step size.


Then, enforcing that B obeys the graph structure is equivalent to enforcing that Z ij B ij = 0 for all (i, j).

finding the closest set of parameters B is equivalent to solving min We find it convenient to solve this minimization by performing some manipulations, and deriving a dual.


the stimulus specific part of the response is small compared to non-specific background fluctuations.

statistical pre-processing of the data is required in order to extract topographic maps from the raw experimental data.


There, the occupancy of regulatory binding sites corresponds to the state of neurons, the weights are set by the cooperative interaction among transcription factors, and the thresholds are the effective dissociation constants at a binding site.

implementing a general N -unit neural network requires only O(N ) biochemical species, but up to O(N 2 ) significant binding interactions must be encoded in the molecular sequences.


We demon-strate that our synapse can learn either the conditional probability or the correlation between input and feedback signals.

we can implement a wide range of synaptic learning networks with our circuit.


The reason is that when Algorithm 1 finds a suitable f (r) for players in G r , it completely removes them from consideration for future rounds; subsequent functions may perform poorly with respect to the distributions associated with those players.

this approach may lead to a global classifier with large error on some player distributions.


for either the unperturbed (α = 0) or perturbed (α = 1) process.

where where xc On a digital computer, the above computation can be efficiently implemented in the Fourier domain, despite the huge size of K α and Q α .


The average numbers of agents appearing simultaneously in different steps positioned along the directions k + ±k − = constant and k ± = constant are, respectively, ∆/8 and ∆/4, and 0 for other directions.

the average number of agents common in the pairs of steps {PQ, QQ 1 }, {QQ k , QP}, {QP, QR}, {PQ, QP} are ∆/8, ∆/8, ∆/8 and ∆/4 respectively.


Also, a high cost O(m 2 ) is entailed to compute the Fisher loss in a mini-batch with m samples by computing all the pairwise distances.

we propose a faster-to-compute loss function that approximately maximizes the margin between the positive and negative similarity distributions without making any assumption about the distribution's shape or pattern.


If ρ and V 0 are convex functions over ∆, then the value function V n of the belief MDP is convex over ∆ at any time step n. [Proof in [17,Appendix]] This last theorem is based on ρ(b, a) being a convex function over b, which is a natural property for uncertainty (or information) measures, because the objective is to avoid belief distributions that do not give much information on which state the system is in, and to assign higher rewards to those beliefs that give higher probabilities of being in a specific state.

a reward function meant to reduce the uncertainty must provide high payloads near the corners of the simplex, and low payloads near its center.


This gives the contradiction P X (A 0 (λ)) ≥ ε.

the increasing function λ → m(λ) := inf{δ λ : λ ≥ λ} satisfies m(λ) > 0 for all λ > 0.


These authors further demonstrated that ICA, sparse autoencoders, and sparse coding are mathematically equivalent under mild conditions.

autoencoders may flexibly project the neuroimaging data onto the main directions of variation.


from equation (14b), we have in order to establish that | θ s,λ i | > 0 for all i ∈ S, and moreover that sign( θ s,λ S ) = sign(θ * S ), it suffices to show that Using our eigenvalue bounds, we have In fact, the righthand side tends to zero from our earlier results on W and R, and the assumption that λ n d → 0.


The Perceptron algorithm will serve as our starting point.

recall that whenever the Perceptron makes a prediction mistake, it updates its hypothesis by adding the element t to I t . on any given round, the size of its active set equals the number of prediction mistakes it has made so far.


From (1) we can see that after seeing each new observation w n , the number of terms in the posterior is increased by a factor of K, resulting in an exponential complexity for exact Bayesian update.

we will instead approximate P n by a different distribution, whose parameters will be estimated by moment matching.


More generally, one can have multiple sequences of images from different cars, or medical image sequences from multiple patients.

the VAE prior should be able to capture multiple levels of correlations at once, including time, object identities, etc.


for regularly spaced data with n = 2 J , wavelet bases provide an efficient nonparametric estimator.


The spatial size of the final output, outputting the heatmaps, is set to 16×16.

due to downsampling, for a network with n − 3, n ≥ 4 blocks, the resolution of the input image is H×W = 2 n ×2 n , resulting in 16×16×(32 • 2 n−3 ) tensor.


In the discussion, we also emphasize how our algorithm is non-trivial, not just a simple combinatorial massaging of standard models.

the techniques introduced in this paper offer broader algorithmic takeaways and are worth further studying in the future.


So evaluation of the globally optimal solutions is difficult.

our approach will be to use greedy minimization to compute a locally optimal solution to (11).


These uncertainties from three perspectives might result in an unreliable estimation of ATT.

we present two strategies to tackle the above issue.


Note that only "pure shifts" in combination with "constant depths" lead to a constant PSF across the image, which is the case most methods for camera unshaking are proposed for.

extending blind deconvolution to smoothly space-varying PSFs can increases the range of possible applications.


By property 1, we must have r s ′′ (s, T ) ≤ λ i+1 .

by inequality 2, we have: By property 5, there are at most O(D 3 log n) such samples at level i + 1, i.e.


Consider a product a i for i  k. In order for a i to be chosen at position s after a k+1 is inserted, one of the following events has to occur: i) a i was already chosen at position s before a k+1 is inserted, and a k+1 is inserted at a position `> s, or ii) a i was chosen at position s 1, and a k+1 is inserted at a position ` s 1.

consequently, we have for all i  k Consider a product a i with i  k. This product is chosen at position s only if it was already chosen at position s and a k+1 is inserted at a position `> s. for all i  k, ⇡(i, s, k + 1) = ( 1k+1,s ) • ⇡(i, s, k).


The first K − 1 stick-breaking proportions are drawn from a Beta distribution with parameters (1, α), but we assume that the final stick-breaking proportion is unity (thus implying β * is non-zero only from 1 . . . K).

we only optimize the first K − 1 positions and implicitly take This constrained optimization is performed using the barrier method [17].


The rescaled variance σ 2 and the scale invariant parameter α is determined numerically, where we use the following notations.

it is straightforward to evaluate (4) with ( 5) for given parameters, p and R. For a given finite value of R, we see what happens to the quality of the estimate when the noise level p varies.


In our experiments, we try to linearly blend the 237 models using information from the test data set.

our variable to optimize is a vector x ∈ R N as coefficients of the predicted ratings for each model.


11 A general-purpose QP package (such as CVX) does not explicitly take the sparsity of the solutions into account.

its solution tends to have many very small nonzero coefficients; as a result, the objective values obtained from CVX were always slightly worse than those obtained from feature-sign search or LARS.


Since these features are highly dispersed, these zero entries (and also the non-zero entries) are approximately evenly distributed among all the features.

every feature must have a significant number of zero entries and be lifetime sparse.


The decision maker does not know the coherence in each single trial.

the initial belief state should be a uniform distribution over all initial states (all coherences, not only coherence of that trial).


But due to its weak power in distribution matching, FM will inevitably generate samples outside of the manifold, especially when the data complexity increases.

the generator density p G is usually lower than the true data density p within the manifold and higher outside.


This extension can be defined for any set function, however it is convex if and only if the set function is submodular [17].

we call a permutation ⇡ over [n] consistent 1 1 the Lovász extension can also be written as where ⇡ is a permutation consistent with x and F (;) = 0 by assumption.


A navigating agent not only needs an internal representation that is general, to support cognitive processes such as scene understanding, but also needs to organise and remember the features and structures that are unique to a place.

to support both types of learning, we focus on neural architectures with multiple pathways.


Given quadratic models of V (x k+1 , k + 1), F (x k , u k ) and r(x k , u k ), we can approximate the unmaximised value function, or Q-function, as a quadratic model around the present state-action pair (x k , u k ): Where the coefficients Q are computed by equating coefficients of similar powers in the secondorder expansion of ( 2) Once the local model of Q is obtained, the maximizing δu is solved for and plugged back into (3) to obtain a quadratic approximation of V k : This quadratic model can now serve to propagate the approximation to V k−1 .

equations ( 4), ( 5) and ( 6) iterate in the backward sweep, computing a local model of the Value function along with a modification to the policy in the form of an open-loop term −Q −1 uu Q u and a feedback term −Q −1 uu Q ux δx, essentially solving a local linear-quadratic problem in each step.


As it turns out, the latter approach can be naturally formulated as a constrained matrix completion problem.

when applying the spectral method, the (approximate) values of the target on B are arranged in a matrix H. the main difference between the two settings can be restated as follows: when learning a weighted automaton representing a distribution, unknown entries of H can be filled in with zeros, while in the general setting there is a priori no straightforward method to fill in the missing values.


Third, probabilistic models can easily be combined in a principled way, and their use in the machine-learning field continues to grow [9].

our model could readily and correctly share information with other probabilistic models or be incorporated into broader ones.


The total cost T (X L , X U ) associated with the data is the total misclassification risk, plus the cost of obtaining all labeled data thus far: where the function C(•) returns the cost of obtaining an annotation for its input, and will be defined in more detail below.

to measure the expected utility of obtaining any particular new annotation, we want to predict the change in total cost that would result from its addition to X L . the value of obtaining an annotation for input z is: where z (t) denotes that the input z has been merged into the labeled set with its true label t, and X U z denotes that it has been removed from the set of unlabeled data.


We attribute it to the fact that region embeddings learn predictiveness of co-presence and absence of words in a region; a region embedding can be more expressive than averaging of word vectors.

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


Thus the the usefulness of unlabeled data is tied to how much information about joint distribution can be extracted from the marginal distribution.

in order to make unlabeled data useful, an assumption on the connection between these distributions needs to be made.


Proof: w = 0, ξ = 1, ζ = 0 is a feasible solution for problem (3).

the objective function of ( 3) is upper bounded by C, and should be lower bounded by 0.


A selective predictor is useful if its RC curve is "non trivial" in the sense that progressively smaller risk can be obtained with progressively smaller coverage.

when constructing a selective classification or a prediction model it is imperative to examine its RC curve.


The risk function in Equation ( 2) is difficult to estimate, since the sizes of Θ C and Θ Xt can be exponential in general.

we have to use approximation for estimating the risk function for efficiency.


In many machine learning problems, however, we are given only a limited amount of labeled data, which can cause an overfitting problem.

we also examine the strategy of self-taught learning [30] (related to semisupervised learning [31]).


Multi-agent, Primal-dual, Finite-sum Optimization Recall that under the multi-agent MDP, the agents are able to observe the states and the joint actions, but can only observe their local rewards.

each agent is able to compute Â and Ĉ defined in (6), but is unable to obtain b.


Although the discreteness of transmission can be overcome by averaging over time, this comes at the cost of longer perceptual delays, or lower transmission rates, as longer integration takes place.

we characterize transmission efficiency by reconstruction error as a function of time, Fig.


For that, we assume that each ρ ρ is a fixed hyperparameter.

we replace the EFE objective function in Eq.


The basic inequality (1) for that update becomes: As in the proof of Theorem 2 of this paper, the Generalized Pythagorean theorem applies and dropping one term we get the following inequality: Adding this to the previous inequality we get: By summing over t, multiplying by n − k, and dividing by 1 − exp(−η), the bound follows.

, then this is further bounded by k log n k . the r.h.s.


While such more advanced schemes could reduce variance, they require prior knowledge (or on-the-fly learning) about how classes overlap with one another.

in Section 4 we shall experiment only with the simple random subsampling approach and leave the above advanced schemes for future work.


Sequential Algorithm Theorem 2 states that solutions of (12) are nested.

from y m−1 v = 1 it follows that y m v = 1 for labelings y m−1 and y m obtained according to (12).


A sum of independent Poisson random variables is itself a Poisson with rate equal to the sum of the rates.

these new latent variables preserve the marginal distribution of the observations, w dv and r ud .


This drastically reduces sample complexity and vastly increases generalization capabilities of the classifier since one only need to observe one test sample to achieve invariance Lemma 2.6 also helps us in saving computation, allowing us to apply the computationally expensive Ψ (group integration) operation only once on he classifier and not the test sample.

the kernel in the Invariant SVM formulation can be replaced by the form k Ψ (x, y) = φ(x), Ψ H φ(y) .


Note that the UCB estimates û * j (t)'s may be non-decreasing in j.

the solution of LP τ,b based on û * j (t) depends on the actual ordering of û * j (t)'s and may be different from Eq.


• Y has necessarily one of the following form: • for any y ∈ Y, from the exp-concavity assumption, the function y : y → (y, y ) is convex on the interval on which it is finite 1 .

the risk R is also a convex function (on the convex set of prediction functions for which it is finite).


But it holds promise only for 0-1 and squared losses.

if empirical risk minimization over noisy samples has to work, we necessarily have to change the loss used to calculate the empirical risk.


For example, in biology, interacting proteins are more likely to have the same biological functions than those without interaction.

pCA and PPCA, or more generally most existing DR methods based on the i.i.d.


space with dual B ′ := L q (X , µ) where .

the semi-inner- Having introduced s.i.p.


The first term represents the amount invested in security by player , and is either 0 (if The second term is the expected cost to due to internal or direct risk of the bad event, and is either Ô Ä (which is the expected cost of internally generated bad events in the case ¼ ), or is 0 (in the case of investment, ½ ).

there is a natural tension between the first two terms: players can either invest in security, which costs money but reduces risk, or gamble by not investing.


Note that 2(e ε + 1) 2 ≤ 32 for ε ≤ 1, and ε ≤ 1 is a common regime of interest in differential privacy.

we can always use the simple binary mechanism in this regime and the resulting divergence is at most a constant factor away from the optimal one.


We then prove that the BAVGM algorithm has mean-squared error decaying as O(m −1 n −1 + n −3 ).

as long as m < n 2 , the bootstrap method matches the centralized gold standard up to higher order terms.


Taking the previous conditions into consideration, we want to find a mask M that is smooth and performs well at both SSR and SDR; examples of such masks can be seen in figure 1.

more formally, given class c of interest, and an input image X, to find a saliency map M for class c, our objective function L is given by: (4) where f c is a softmax probability of the class c of the black box image classifier and TV(M ) is the total variation of the mask defined simply as: AV(M ) is the average of the mask elements, taking value between 0 and 1, and i are regularisers.


Just like spherically symmetric models can be nonlinearly transformed into isotropic Gaussians, L p -spherically symmetric models can be mapped into a unique class of factorial distributions, called p-generalized Normal distributions [11].

we are able to quantify the influence of orientation selective filters and contrast gain control on the redundancy reduction of natural images in a joint model.


Importantly, the objective in (3) is jointly convex in Z and K, since tr(ZK † Z) is a perspective function [29].

although the single layer model is not jointly convex in the input features Φ and model parameters U , it is convex in the equivalent reparameterization (K, Z) given Θ.


We must also point that, in this case, preference learning data sets are formed by pairs of objects (v, u), and each object in the pair has the same set of features.

we must modify the ranking methods so they can deal with the duplicated features.


A transfer function P S VU , however, is meaningless because it violates temporal causality.

the denominator can at most add phase-shifts to the systems behaviour.


In the multivariate Gaussian case, the entropy can be computed in closed form: For the contextual case, our regret bound comes also in terms of the quantity γ T , redefined so that the information gain I(y A ; f ) now depends on the observations y A = [y(x)] x∈A of the joint contextaction pairs x = (s, z), and f : S × Z → R is the payoff function over the context-action space.

the kernel matrix x ∈A is defined over context-action pairs.


We first fix a value z \i for Z \i , and a value of R such that R 1 , .

. . , R k all lie in Σ, and consider the ratio of probabilities: Observe that this ratio of probabilities is equal to: which is in turn equal to: Observe that from the stability condition, the ratio of the probabilities is at most the ratio Pr( , which is at most e α2 by properties of the exponential distribution.


The stochastic SLQC optimization Setup: Consider the problem of minimizing a function f : R d → R, and assume there exists a distribution over functions D, such that: We assume that we may access f by randomly sampling minibatches of size b, and querying the gradients of these minibatches.

upon querying a point x t ∈ R d , a random minibatch {ψ i } b i=1 ∼ D b is sampled, and we receive ∇f t (x t ), where f t (x) = 1 b b i=1 ψ i (x).


In any practical deconvolution setting the blur kernel is never perfectly known.

the kernel passed to the algorithms was a minor perturbation of the true kernel, to mimic kernel estimation errors.


But one could imagine requiring something less: perhaps changing distances this way should be allowed to create additional sub-structure, leading to a new partition in which each cluster is a subset of one of the original clusters.

we can define Refinement-Consistency, a relaxation of Consistency, to require that if d is an f (d)-transformation of d, then f (d ) should be a refinement of f (d).


To see this, for users u and v with revealed ratings v , and let •, • Ωuv be the dot product restricted to entries in , which is the cosine similarity of revealed rating vectors restricted to the overlap of their supports.

users u and v are neighbors if and only if their cosine similarity is at least θ.


The event oracle is an axis-parallel rectangle anchored at the origin, where points inside the box are negative and points outside the box are positive.

if there are two features, say query volume and query abandonment rate, an event occurs if and only if both the volume and abandonment rate exceed certain thresholds.


For example, male users tend to rate science fiction and action movies higher than female, which can be informative when predicting their ratings.

a linear model considering no interactive terms can be oversimple and have low predictive power for missing entries.


The obtained results are in Table 3 of supplementary material, where ˆ has the order of magnitude of 10 2 in all the datasets.

our method is able to obtain a good approximation of the DEO.


Beside the reduction of training time (i.e., matrix inversion), the prediction time is also reduced significantly compared to GPR and ν-SVR due to the fact that only a small amount of local models in the vicinity of the query point are needed during prediction for LGP.

the prediction time can be controlled by the number of local models.


Assume m ≥ 12 and n t, we have, with a probability at least 1 − 2e −t , Remark Theorem 3 implies that if the empirical loss L (w * , S) ≤ O(log m/m), for most positive instance x + (i.e., 1 − O(log m/m)), the percentage of negative instances ranked above x + is upper bounded by O( log m/n).

we observe that m and n play different roles in the bound; that is, because the empirical loss compares the positive instances to the negative instance with the largest score, it usually grows significantly slower with increasing n. For instance, the largest absolute value of Gaussian random samples grows in log n. we believe that the main effect of increasing n in our bound is to reduce δ (decrease at the rate of 1/ √ n), especially when n is large.


For any ⌘, ⌘ ′ > 0, there exists z ∈ , with z k ≠ 0 for all k ∈ [p], such that the following holds for the distribution-weighted combining rule h Lemma 6 indicates that for the solution z, h ⌘ z has essentially the same loss on all source domains.

our problem consists of finding a parameter z verifying this property.


Of course, we would not know ahead of time how to bin the data to avoid losing accuracy.

we suggest an adaptive quantization scheme, pictured in Figure 2(c), to make the accuracy loss (if any) as little as possible.


Roughly speaking, the PMAC-learning framework guarantees that the learned function is a constantfactor approximation of the true submodular function with high probability.

after learning a submodular function, one obtains an approximately submodular function.


Moreover, some inequalities used in Section 5 are very loose; most of the rounding operations in fact produce much less error than the upper bound K • u .

we can set the values of K • u much larger than the theoretical values without compromising the quality of approximation.


Until the ball is pushed far from the target location, DDPG agents realize they are pushing at the wrong direction and switch to the opposite together.

the ball is pushed back and force and hardly stabilized at the target location.


We will use the following notation: Note that q t , v t satisfy the Bellman equations underlying π t , P and r t . For reasons to be made clear later in the paper, we shall need the state distribution at time step t given that we start from the state-action pair (x, a) at time t − N , conditioned on the policies used between time steps t − N and t: It will be useful to view µ N t as a matrix of dimensions |X × A| × |X |.

µ N t,x,a (•) will be viewed as one row of this matrix.


This result is surprising since as illustrated in Figure 1, the learning target functions with AE ¼ and AE ¼ ¼ are visually quite similar.

it intuitively seems that the result of AE ¼ ¼ is not much different from that of AE ¼.


By applying clockwise torque to the main rotor to make it rotate, our helicopter experiences an anti-torque that tends to cause the main chassis to spin anti-clockwise.

in the invention of the helicopter, it was necessary to add a tail rotor, which blows air sideways/rightwards to generate an appropriate moment to counteract the spin.


Neural responses from all 10 or 20 trials were histogrammed in 20 ms bins aligned with stimulus pulse durations.

in the regression framework, the instantaneous input vector § comprised the sound amplitudes at each possible frequency at time £ , and the output was the number of spikes per trial collected into the £ th bin.


Indeed, since all of the squared MMD estimates calculated on the training data using each of the base kernels are negative, it is unlikely the statistic computed on the data used for the test will exceed the (always positive) threshold.

when no entries in η are positive, we (arbitrarily) select a single base kernel k u with largest ηu /σ u,λ .


Given a point (x, y) = (µ 1 (H), µ 2 (H)) on the graph in Figure 2, where µ i (H) is the probability by which player i plays Heads, we know that our expected reward is  We wish to show that We consider each part separately.

in the losing section, we let g(t) = x = t and h(t Similarly, we can show that we receive 1/4 reward over C w . C R 1 (x, y)dt = 1/3 > 0, and we have shown that we receive a payoff greater than the Nash equilibrium payoff of zero over every cycle.


In particular, we define the set of all local neighborhoods as N B {(i, j)| i,j = 1, ρ(x i , x j ) ≤ γ} .

we minimize the following objective function Note that we do not need to have the same metric in both the input and the feature space, they do not even have in general the same dimension.


These techniques have been extensively applied to CDMA systems.

most of last generation digital communication systems such as Global Positioning System (GPS), wireless 802.11b, Universal Mobile Telecommunication System (UMTS), etc, may take advantage of any improvement on this topic.


Since in general normalized cuts are thought to be superior to mincuts, the transductive inference approach that we used in the later experiments is built on the above spectral hypergraph clustering method.

as shown in [20], with the least square loss function, the classification function is finally given by f = (I − ξΘ) −1 y, where the elements of y denote the initial labels, and ξ is a parameter in (0, 1).


For the second inequality, note that the middle term is a maximizer for the right-hand side.

p 1 must be able to do (weakly) better if it knows µ 2 compared to if it just assumes that µ 2 = µ 1 .


S ij evaluates the similarity between the i-th and j-th nodes.

the importance of a feature can be thought of as the degree it respects the graph structure.


since λ n (B/m) < 1, we can apply the Neumann series representation [Suhubi, 2003, page 390] to this expression: we can re-write f (S) as an infinite sum of traces of matrix powers: Note that B i S here means (B S ) i and not (B i ) S . Submodularity.


[Mohri et al., 2012]) that if every function f in a class F can be written as the summation of two functions g and h from classes G and H, respectively, then H).

we split each revenue function into n + 1 components such that the sum of these components equals the revenue function.


A key insight towards efficient weighted-voting is that this path visits all leaves (containing x) of any subtree of T0.

weighted voting might be implemented by keeping a weight w(A) at any node A along the path, where w(A) aggregates the weights Q(T ) of every subtree T that has A as a leaf.


Nevertheless, its application to continuous distributions requires discretization of these distributions, leading to computational intractability when one desires good accuracy and, hence, has to use fine discretization with large n, which leads to the necessity of solving an optimization problem of large dimension.

this approach is not directly applicable in our setting of continuous distributions, and it is not clear whether it is applicable in the decentralized distributed setting with arbitrary networks.


Hyper-parameters (such as σ defining the bandwidth of the Gaussian kernel) are selected with cross-validation or heuristics [12,6,8].

the basis is not directly adapted to the data, loss, and task at hand.


Unfortunately, in a sequence correction task such as our problem, one domain contains all correct sequences, while the other contains everything not in the correct domain.

the mapping from correct to incorrect is not one-to-one, it is one to many.


By moving the means of the Gaussians arbitrarily far apart, L(D | θ C ) can be made arbitrarily worse than L(D | θ D ), where θ C and θ D are MLEs on C and D respectively.

even for two well-separated Gaussians, uniform sampling can perform arbitrarily poorly.


In channel coding, the factors of the graph are parity-checks and the correlations are high but can change from positive to negative by the flip of a single variable.

the pair-wise mutual information is zero for any two variables (unless the factor only contains two variables) and we could not define a prefixed tree structure with the algorithm in [9].


We now provide an unbiased estimator of the expected generalization error of θn , which generalized the Akaike information criterion [22] (for a proof, see [23]): Proposition 2 In addition to the assumptions of Proposition 1, we assume that where The criterion G is a sum of two terms: the second term corresponds to a variance term and will converge to zero in probability at rate O(n −1 ); the first term, however, which corresponds to a selection bias induced by a specific choice of sampling distributions, will not always converge to the minimum possible value L u (θ 0 ).

in order to ensure that our active learning method are consistent, we have to ensure that this first term is going to its minimum value.


Note that TS features require calculating and storing only one number per one category.

using TS as new numerical features seems to be the most efficient method of handling categorical features with minimum information loss.


Note that the bound (10) consists of two main terms: the first term captures the geometric convergence of the population EM operator from Theorem 1, while the second term characterizes the radius of convergence in terms of sample complexity, which is O( 1/n).

with probability at least 1 − δ, we have where c, c are universal constants.


Although transitions are rarely observed, the emission means are set to be distinct so that this example is likelihood-dominated and highly identifiable.

fixing a computational budget, we expect large M to be preferable to large L, covering more of the observation sequence and avoiding poor local modes arising from redundant information.


For a suitably chosen prior and posterior, [5] have also shown that R S (G Q ) is small when the corresponding Majority Vote classifier B Q achieves a large separating margin on the training data.

the PAC-Bayes theorem yields a tight risk bound for large margin classifiers.


. . , x n } drawn i.i.d from P (X), respectively: Quantum Mean Maps We still take the expectation of the features of the data, except we require that the feature maps (•) produce valid density matrices representing pure states (i.e., rank 1).

quantum mean maps have the nice property of having probabilities along the diagonal.


The dependence is only on the size of global observable set σ, which is guaranteed to be not more than dimension of adversary's action space.

though we have adopted the CPM model [1], our algorithms achieve meaningful regret bounds for countably infinite or even continuous learner's action space.


The proposed algorithm provides a modal decomposition of the dynamics into a finite number of modes, and thus it could be considered as a feature extraction procedure for a nonlinear dynamical system.

we consider applications using extracted features from our analysis such as state prediction, sequential change-point detection, and dynamics recognition.


It is wellknown that a Markov process on variables x will converge to a desired equilibrium distribution 7r(x) if the transition probabilities obey the detailed balance conditions, w(x'lx)7r(x) = w(xlx')7r(x'), (10) where w(x'lx) is the probability of generating x' from x.

any search algorithm searching on a flat space should have dynamics which obeys, and PEIL does not obey this.


It follows from a result of Cesa-Bianchi and Lugosi [1] that for some constant B.

by Markov's inequality we conclude that If |||M ||| * = ∥D∥ * = o (n/(q + log(pq))) 1/4 , then returning to (6.1), this gives us a bound on R c (M )− Rc (M ) that is o P (1).


In particular, because of the equivalence H ≡ S m and the data processing inequality, the algorithmic stability of a lazy learner provides a lower bound to the stability of any possible learning algorithm.

we can relate algorithmic stability (uniform generalization) to the size of the observation space by quantifying the algorithmic stability of lazy learners.


In fact, we find that these features are better face detectors than a linear filter trained from labeled data, achieving up to 86% AUC compared to 77% on labeled validation data.

our results emphasize that not only can unsupervised learning algorithms discover object-selective features with no labeled data, but that such features can potentially perform better than basic supervised detectors due to their deep architecture.


Compared with NABird dataset, CUB dataset is a well-known dataset for ZSL.

we first conduct experiments on CUB dataset for comparing with the state-of-the-art approaches.


Further, on the duration problem in the CAP data sets, PBCMC also achieved the reported accuracy using λ = 5000 and φ = 10 −2 , and on the location problem it acheived negligibly better accuracy using λ = 5000 and φ = 10 −3 or φ = 10 −4 .

simply using an even more stable parameterization of PBCMC did not improve performance on these problems either.


We use a probabilistic argument to switch from the randomization over F 0 to the randomization over S ⇠ D and show that there exists a deterministic F ✓ F such that f i (S) = f j (S) for all f i , f j 2 F w.h.p.

over S ⇠ D. By a union bound this holds for all samples S. for such a family of functions F = {f 1 , .


The same analysis applies to the factor f v too.

z τ f u (Z τ )f v (Z τ ) exhibits the whole inverted S of the Q-Q plot.


This observation is based on an important property of Newton-like methods, namely that there is much more freedom in choosing a Hessian approximation than in computing the gradient [7,3].

a smaller sample O k can be employed for updating the inverse Hessian approximation H k than for computing the batch gradient g S k k in the search direction −H k g S k k . In summary, by ensuring that unresponsive nodes do not constitute the vast majority of all working nodes in a fault-tolerant parallel implementation, or by exerting a small degree of control over the creation of the samples S k in the multi-batch method, one can design a robust method that naturally builds upon the fundamental properties of BFGS updating.


We used the following conjugate priors (Dirichlet distributions) over θ l and α l,m as: Here, ξ and ζ are hyperparameters and in this paper we set ξ = 2 and ζ = 2, each of which is equivalent to Laplace smoothing for θ l,i and α l,m , respectively.

the objective function to find Θmap is given by Of course, the third term on the RHS of Eq.


In the typical RL task formulation, an acting agent does not have access to the complete environment specification, or, even if it does, the environment is non-differentiable.

in our setup, an agent that was trained by any REINFORCE-like algorithm would underuse available information about the environment.


This scale change is given by the product of the square roots of the eigenvalues of X T X.

the relationship between the probability distribution in the input and output spaces includes the proportionality factor, y'det(xTx), as formally derived in Appendix 1.


However, these special conditions may fail to hold for generic complex objectives involved in practical applications.

it is desired to further understand the convergence behaviors of CR for optimizing nonconvex functions over a broader spectrum of geometrical conditions.


Since the predictor might lose information with respect to these features in order to yield the predictions, only the features from the encoder go through the decoder to be merged back into the main path (as illustrated in Figure 2 and formalized in Equation ( 1)), so that more information is kept for use in the primary task.

all parameters learned in the predictor are to be thrown away after training, i.e.


1 CF-2g closely resembles the serial Seq-2g, but the elements e 2 V are no longer processed in a fixed order.

the sets A, B are replaced by potentially stale "bounds" Â, B, where Â is a subset of the "true" A and B is a superset Intuitively, the bounds hold because A e , B e are stale versions of A ι(e)−1 , B ι(e)−1 , which are monotonically non-decreasing and non-increasing sets.


The next lemma shows that each subcluster added to active is correctly grown to the true cluster: (1) every vertex added to such a cluster is correct, and (2) no two clusters in active can be merged.

clusters obtained from active are the true clusters.


Since a large percentage of connections are pruned away, the network structure should become much simpler and probably even much "thinner" by utilizing our method.

the loss function derivatives are likely to be very small, especially when the reference model is very deep.


While we were able to improve their performance by turning off dropout, these unregularized networks didn't match the training set performance of the corresponding network configurations that had pooling units (see Section 3 of the supplementary material).

our final network contained two layers of 50 hidden units and pooling units.


But for a new data point, x test , the only features that are nonzero in both x test and w ada are the first three.

in particular, we have the AdaGrad solution will label all unseen data as a positive example!


Suppose now that the induction holds for some t 1.

since by definition of v t , subtracting ⌘t v t from x t can only decrease positive entries in x t (see proof of Observation 2), and both v t , v + t are vertices of P (and thus in {0, 1} n ), and ⌘t = 2 t , it follows that each entry i in x t+1 is given by: x t+1 can also be written in the form 2 t st+1 for some st+1 2 N n . By definition of t and the monotonicity of {⌘ t } t 1 , we have that 2 t 2 t+1 is a positive integer.


Unlike expensive setups such as Sawyer or Baxter, our setup has higher calibration errors and lower accuracy due to in-accuracte kinematics and hardware execution errors.

unlike existing self-supervised datasets; our dataset is diverse and huge but the labels are noisy.


This has the effect of producing more promising candidate solutions for the next generation, which is coevolution in a broad sense.

eSGD not only integrates EA and SGD as complementary optimization strategies but also makes use of complementary optimizers under this coevolution mechanism.


In addition to p i -denoting the fact that wise man i wears a red hat -to model each agent's individual knowledge, we need to use a modality K j , j ∈ {1, 2, 3}, which represents the relative notion of knowledge at each time point t 1 , t 2 , t 3 .

k j p i denotes the fact that agent j knows that agent i wears a red hat.


We start from an existing probabilistic logic programming language, ProbLog [De Raedt et al., 2007], and extend it with the capability to process neural predicates.

the idea is simple: in a probabilistic logic, atomic expressions of the form q(t 1 , ..., t n ) (aka tuples in a relational database) have a probability p. the output of neural network components can be encapsulated in the form of "neural" predicates as long as the output of the neural network on an atomic expression can be interpreted as a probability.


. . , x M are conditionally independent given y, then their covariance matrix is diagonal, i.e., cov(x, x|y) = diag(y).

by the law of total covariance, the covariance in ( 6) has the form where the last equality follows by the multilinearity property of cumulants (see Appendix C.1).


Specifically, if we have K observations of x, denoted x (k) , k = 1, ..., K, then the estimate of µ i , denoted μi , is given by We can still use the maximum entropy formulation described above; the only difference is that we replace µ by μ.

the maximum entropy distribution is given by q(x| μ) (Eq.


As can be seen from the proof, this constant has no special meaning and can be replaced by any constant strictly larger than 1 at the cost of increasing the constant C. Note that the O P (1/ √ n) rate obtained by Biau and Bleakley (2006) has a factor 1 in front of the bias term.

theorem 1 provides a significant improvement unless the bias overly dominates the second term of the bound (i.e., the complexity of G is too small).


In addition, GNC-AGD and Natasha2 needs additional Hessian Lipschitz condition.

we only plot the gradient complexity of SVRG, SCSG and our proposed SNVRG in Figure 1.


It can be shown that the Hessian matrix of F is positive semidefinite, and strictly positive definite for α > 0.

the optimization problem (3) is convex, respectively strictly convex (see e.g.


The left group shows clear bias towards horizontal orientation tuning, the middle towards vertical.

subspaces can be learned directly from partitioning on the sparse REFIRE network offering a complementary approach to learning structured models directly from the data [26,27].


Note that each random function H i is deterministically defined by the random variables ((X j , Y j )) i j=1 .

the risk of H i is also a deterministic function of ((X j , Y j )) i j=1 .


Benchmark dataset: Except for synthesizing natural looking profile view face images, the proposed DA-GAN also aims to generate identity preserving face images for accurate face-centric analysis with state-of-the-art deep learning models.

we evaluate the possibility of "recognition via generation" of DA-GAN on the most challenging unconstrained face recognition benchmark dataset IJB-A (15).


The number of such singular values can scale as fast as O(min{m, n}).

when we place nulls (zeros) in f (λ), it is desirable to ensure that these nulls are pronounced after we approximate f (λ) by f L (λ).


A deviation of 0 indicates that we have achieved perfect modularity and 1 indicates that this dimension has equal mutual information with every factor.

we use 1 − δ i as a modularity score for code dimension i and the mean of 1 − δ i over i as the modularity score for the overall code.


The nodes in the first set are instantiated to the corresponding values in the test instance for which Z t is to be predicted.

the first set of nodes represents the antecedent of the LBR rule and the second set of nodes represents the consequent.


Since at least half of the Q measure of the classifiers err on an example incorrectly classified by the majority vote, it follows that the error rate of the majority vote is at most twice the error rate of the Gibbs classifier.

given enough training data, the PAC-Bayes theorem will give a small risk bound on the majority vote classifier only when the risk of the Gibbs classifier is small.


The higher an action is ranked by the learner's current policy, the more it is preferred by the learner, thus easier to learn.

similar to [6], we define a hope action that combines the task loss and the score of the learner's current policy.


Given the property of MDS codes, the master can decode an entire row after obtaining any m results in that row; likewise for the columns.

the master can recover the final output using a peeling algorithm, iteratively decoding the MDS codes on rows and columns until the output C is completely available.


The method does not use explicitly that the graph is directed.

the result is equally valid for Markov graphs.


To express such negative costs in a graph cut, we point out an equivalent formulation with positive weights: since m(V) is constant, minimizing m(S) = i∈S m(i) is equivalent to minimizing the shifted function m(S) − m(V) = −m(V \ S).

we instead minimize the sum of positive weights on the complement of the solution.


Our basis is constructed by extending the standard coupled basis with an additional component to model the mean residues.

it extends the standard coupled basis [32,12] and achieves better performance.


Therefore, the optimal class y can be obtained by In the calculation of ( 8), the integral over Gaussian means and covariances for a frame can be solved analytically to be Student distributions.

we can compute a Bayesian predictive score for a frame, and then can compute a phoneme sequence score by using the Viterbi algorithm.


We further restrict the setting by permitting the payoff of each player to depend only on their own action and the actions of their neighbors (a subset of the other players).

we may view our setting as a graphical game [10; 11].


Clearly, F is defined up to a scale factor: if F 33 = 0, one can assume that F 33 = 1.

each pair x ↔ x of matching points in images I and I yields a linear constraint on the eight remaining coefficients of F. Because of the quantification and the presence of noise in images, these linear relations are satisfied up to some error.


Furthermore, the version of the reward-modulated Hebbian learning rule that we propose does not require extraneous information about what is noise and what is signal.

we show in this study that reward-modulated learning is a possible explaination for experimental results about neuronal tuning changes in monkey pre-motor cortex.


On the other hand, reducing the skewness discussed in this paper may improve learning performance even when such techniques are used.

developing a concrete method of balancing positive and negative rewards (as discussed in Section 4.2) can be very helpful, and will be investigated in the future.


Since such beliefs are modeled as Gaussians, this probability has a unique global maximum at µ b = µ c and Σ N → 0 + .

a final cost J final = w 0 µ b − µ c 2 2 + w 1 tr Σ N can approximate the negated log-probability of successfully catching the ball while rendering the optimal control problem solvable.


Second, RAPCF only filters forward and does not smooth backward.

there will be impoverishment in distant ancestors v t−L , since these states are not regenerated.


If b were scaling with n, then it would be really easy to differentiate scores of items that are at the two opposite end of the dynamic range; in which case one could focus on differentiating scores of items that have their parameter values near-by.

the interesting and challenging regime is where b is constant and not scaling.


On the other hand, the proposed method overall worked reasonably well and it consistently outperformed the passive learning scheme.

the proposed method would be robust against the misspecification of models and thus it is reliable.


By conditional expectation, Thus, we can see that the key idea of decoupling L(a, w) into Q L dec (a, a ′ , w), is essentially "recoupling" Q L dec (g 1 , g 2 , w) via conditional expectation -the "recoupled" term L can be viewed as an approximation of L(a, w).

notice that by Fact (ii), for any convex function φ, by choosing φ(t) = |t| p , we can control moments of sup w∈S L(a, w) via sup w∈S L(a, w) For tetrahedral polynomials, L = L, so the approximation is exact.


Such a measure could assign absolute numbers to each of the possibilities, but any such static assignment would make it impossible for the propositions u(x 1 , X ) > u(x 2 , X ) and u(x 1 , X ) < u(x 2 , X ) to hold simultaneously, as is desired of a context-sensitive utility measure.

we see that it is impossible to design a utility function u such that u : X × C → R. If we wish to incorporate the effects of context variation on the desirability of a particular world possibility, we must abandon a foundational premise of existing statistical decision theory -the representational validity of absolute utility.


While this significantly improves the image in the blurred areas, serious artifacts are observed in the background.

in addition to recovering the blurring kernel, we need to segment the image into blurred and unblurred layers.


Even within one of the 10 classes of CIFAR, the intra-group diversity is very high compared to any of the 10 classes of MNIST.

for CIFAR it is inappropriate to assume, as the metrics of the previous subsection do, that each labelled class corresponds to a single mode of the data distribution.


Unfortunately, the disparity range covered by a phasetuned population is limited by phase wraparound.

a single population cannot cover the large range of disparities encountered in natural scenes unless the scale of the receptive fields is chosen to be very large, which results in very low resolution depth estimates.


In a different regime where we introduce increasing correlations between the synaptic inputs [1] we find that the neuronal gain is changed little with increasing input rates but increases substantially with increasing input correlations (Fig 4c) .

the adaptive aSTDP rule can normalize the mean postsynaptic rate even when the input statistics change.


This makes covariance estimation in high dimensions even harder: the data dependence lowers the effective sample size available for constructing the estimator [TZ84].

stronger regularization λ will be needed.


More precisely, there exists π * ∈ Π HR , (p * , r * ) ∈ s∈S Ps such that  This leads to sup π∈Π HR w π, µ * , (s ini ) = w π * , µ * , (s ini ) = inf µ∈CS w π * , µ, (s ini ) .


In case of single-task learning, the number of target samples refers to the samples used at learning time, while for BAT it represents the size S of the auxiliary training set used to estimate the transfer error.

while in single-task learning the performance increases with the target samples, in BAT they just make estimation of E λ more accurate.


Again since both the KL and the U function in P2 are convex, and the constraints in F 2 are linear, P2 is a convex program.

we can employ a technique similar to that used to solve MaxEnDNet to solve the PoMEN problem.


The results shown in figure (1) demonstrate that with a linear model the likelihood function will display a peak corresponding to each source location, but with the multiplicative model only a spurious location that is consistent among the kernels remains and information about the two sources is lost.

we use a model in which time difference and level difference information is first combined multiplicatively within frequency channels and is then summed across frequency.


Since dj measures how "connected" point j is to other points in the same cluster, it will be dj = O(n) in this case, so the sum, which is over 0(n 2 ) terms, is in turn divided by djdk = O(n 2 ) .

as long as the individual Ajk's are small, the sum will also be small, and the assumption will hold with small fl.


Then for prediction, the random initialized part in the weights has strong cancellation, while the structured signal part in the weights collectively affects the output.

the latter can be much smaller than the former while the network can still give accurate predictions.


In contrast, we postulate that the apical dendrite has an explicit error representation by simultaneously integrating top-down excitation and lateral inhibition.

we do not need to postulate separate temporal phases, and our network operates continuously while plasticity at all synapses is always turned on.


For the full breaking, adjacent breaking, and any top-k breaking, the three statements in Theorem 2 are equivalent for any data D. For any position-k (with k ≥ 2) and any bottom-k (with k ≤ m − 1), 1 and 2 are not equivalent to 3 for some data D. Ford, Jr. [6] identified a necessary and sufficient condition on data D for the MLE under PL to be unique, which is equivalent to condition 1 in Theorem 2.


Note that some projection indices are suitable for finding super-Gaussian components (heavy-tailed distribution) while others are suited for identifying sub-Gaussian components (light-tailed distribution) [9].

traditional PP algorithms may not work effectively if the data contains, say, both super-and sub-Gaussian components.


If the Jacobian of the update rule at a stable fixed point z has spectral radius less than one, then the fixed point is asymptotically stable.

if a fixed point z is hyperbolic, then linear stability implies asymptotic stability.


Note that the conventional training procedure for regularization by noise such as dropout [34] relies on the objective with S = 1 (Section 4).

we show that using more samples achieves tighter lower-bound and that the optimization by IWSGD has great potential to improve accuracy by proper regularization.


1 (b), implies the following factorization Using d-separation in this DAG, we see that in cases where any values are missing, neither MCAR nor MAR assumptions hold under this model.


Although physiologically meaningful change is often accompanied by the change in rate, there has been evidence that the higher order statistics can change without a corresponding change of rate [4,5].

statistical tools that capture higher order statistics, such as divergences, can improve the state-of-the-art hypothesis testing framework for spike train observations, and may encourage new scientific discoveries.


(iii) Lipschitz smoothness constant of h (✓) increases with the decrease in .

we have to choose smaller step-size, based on Theorem 4.2, which means slower convergence.


Obviously, if we use small enough pieces, then any signal can be composed of any other.

the larger those pieces are, the more similar S 1 is to S 2 .


In particular, our approach adds additional terms to the energy function (i.e., regions to the Hasse diagram) such that the high-order factors can be pre-computed and remain constant during local message passing within each machine.

high-order factors are only accessed once before sending messages across machines.


If an internet user clicks uniformly at random over the outgoing links, then the probability that the user will click on page i given she is currently on page j is where the denominator is the out-degree or number of web pages that page j links to.

m is the transition matrix of a Markov chain and under mild conditions, the probability that a "random surfer" will be on any page regardless of where she starts is given by the vector p that solves p = Mp.


To avoid this problem, we modify the backward computation by employing some algebraic substitutions.

we can compute the derivatives efficiently using dynamic programming, as shown in Algorithm 2 and Algorithm 3.


If H is a graph and p ≥ 1, Ω H,p reduces to the Laplacian regularization Ω G,p .

note that for characteristic functions of sets, the difference between the hypergraph cut and its approximations such as clique and star expansion carries over to Ω H,p and Ω G CE ,p , respectively.


The result applies, in particular, to the SVM objective.

we obtain a rate of 1/n on the convergence of the SVM objective (with fixed regularization parameter) to its infinite data limit.


However, across all problems, no person consistently outperformed the model.

for all aggregation methods, except for the mode, we demonstrated a wisdom of crowds effect, where the average performance of the model was better than the best individual over all problems.


The success rate of planner that uses the obstacle map generated by this classifier from the raw image is 90.3%, showing that obstacle identification from the raw image is indeed challenging.

the success rate of the VIN, which was trained without any obstacle labels, and had to 'figure out' the planning process is quite remarkable.


The processing for this type of system uses acoustic information to obtain sequences of phones for a given conversation and then discards the acoustic waveform.

processing is done at the level of terms (symbols) consisting of, for example, phones or phone N -grams.


Cues are based on the idea of suspicious coincidences: roughly, "things that move together likely belong together".

frequencies which exhibit common onsets, offsets, or upward/downward sweeps are more likely to be grouped into the same stream (figure 2).


However, simulation of the marginal model is three times faster compared to the full model.

it seems beneficial to evaluate and train RBMs based on sampling and annealing of the marginal model p(h) rather than the full model p(v, h).


However, when the recurrent kernel is enlarged to the parts of other characters, such as "p", the context carried by the kernel is unnecessary.

we need to weaken the signal that comes from unrelated context and combine the feed-forward information with recurrent information in a flexible way.


Similarly, the approach based in local and global consistency (LLGC) [6] can be thought of as using the same transformation but on the normalized Laplacian and a Gaussian likelihood.

it is easy to see that most of these algorithms can exploit the proposed evidence maximization framework.


However, as seen in Figure 1, the domain of images and videos is quite different, as it is often the case that images of objects are taken in controlled settings that differ greatly from where they appear in real-world situations, as seen in video.

we cannot typically expect a detector trained on images to work well in videos.


Before seeing the preferences ¢ , the vector £ ¢ ¥ £ ¢¡ forms a joint prior Gaussian derived from the GP.

after incorporating the ¢ information, the posterior mean of £ is where the user preference function for a song s, £ ¡ 0) , is a linear blend of kernels ¡ £ ¢ §¥ £ ¡ 0) § that compare the metadata vector for song ) with the metadata vectors £ £¢ for the songs that the user expressed a preference.


As the optimization proceeds and the classification accuracy increases, the number of incorrectly classified samples reduces on average, and this leads to the reduction in the average effective batch size (more samples generate zero derivatives).

our hypothesis is that there is an optimal value of scaling parameter ↵ for a given combination of dataset, metric and task.


We use the non-parametric distribution of Z by sampling random pairs {X s , Y s } from the classes seen during training (such that X s and Y s belong to the same category) and generating from them Z = E(X s , Y s ) using the trained encoder.


The a priori knowledge we want to encode in this toy problem is local invariance under rotations.

the output of the decision function on a given training point Xi and on its image R(Xi,C:) obtained by a small rotation should be as similar as possible.


Bayes-rationality specializes to economic rationality once we instantiate the underlying intuitions behing the completeness and transitivity assumptions in a context-sensitive preference inference theory.

rational value inference in the form we propose can formally replace static assumptions about preference orderings in microeconomic models that currently exclusively use ordinal utilities [12].


Then the space of all possible preference vectors is given by i becomes arbitrarily large for all i = i, then with high probability item i is ranked higher than any other item i and there is no way to estimate θ i to any accuracy.

we further put the constraint that θ The parameter b characterizes the dynamic range of the underlying preference.


At the same time, the reconstruction quality rose to about 0.9 (Figure 3a).

a simple local activity stabilization can help alleviating damage to the system.


However, the former requires more positive rewards, while the latter calls for more negative rewards, causing an unsolvable dilemma.

it is more realistic just to balance the likelihood of under/overestimation, so that all actions/policies can compete fairly with each other.


To track to the previous frame (t − 1), a backward dense optical flow field Since the optical flow field is typically real-valued, the sampling position p t+τ becomes fractional.

the corresponding feature x( p t+τ ) is derived via interpolation with a specific sampling kernel G, written as In this paper, we will not go deeper into the usage of different choices of sampling kernels G and use the bilinear interpolation as default.


Furthermore, all X feasible for the SDP have identical trace equal to n, so that the search space of the SDP is compact.

our results stated below apply: For p = √ 2n , for almost all C, even though (Max-Cut BM) is non-convex, any local optimum Y is a global optimum (and so is X = Y Y ), and all saddle points have an escape (the Hessian has a negative eigenvalue).


By the definition of GLIE exploration, each state in the finite state space will be visited infinitely often w.p.1.

it is sufficient to only prove that the OAL algorithm will converge to the optimal policy over individual state games w.p.1.


The existence of interactions or univariate variables being modeled by first layer hidden units is determined by the nonzero weights entering those units [31].

we would like to penalize the number of nonzero elements in rows of the input weight matrix, as a group belonging to a block.


The first objective is often monotone submodular and the latter objectives are monotonically decreasing supermodular functions [LKG + 07, KLG + 08].

the problem is often formulated as an instance of cardinality constrained maximization with a small number of submodular objectives.


The network on the left hand side is equivalent (i.e.

represents the same function) to the network on the right for any nonzero it is helpful to understand what are the feasible node-wise rescalings for RNNs.


In the LTLS trellis architecture, every path has another (closest) path within 2 edge deletions and 2 edge insertions (see Figure 2).

the minimum Hamming distance in the underlying coding matrix is restrictively small: ρ = 4, which might imply a poor error correcting capability.


We believe that context effects can be viewed as a reflection of a trial-to-trial learning, and the cumulative effect of these trial-to-trial modulations corresponds to what we classically consider to be category learning.

any compelling model of category learning should also be capable of explaining context effects.


The surprisingly competitive discriminative baseline shows that it is possible to achieve high onset accuracy on this task without adapting to the test instrument.

it is reasonable to ask how much of the gain our model achieves is due to its ability to learn instrument timbre.


However, in our case z is a Gaussian random variable, which means (f (z; w), y) is now itself a random variable given a label y.

we use the expectation E z∼N (µ,Σ;θ) (f (z; w), y) as the overall loss between the label y and a time series S given its Gaussian representation N (µ, Σ; θ).


A possible explanation is that the activation of neurons is exponentially decayed by input/output gates during the propagation.

setting large bias values for these gates may facilitate the transmission of information through many layers at the beginning of the learning.


(5) h () terms are estimated using parametric approximation to the total and conditional distribution It is assumed that the total distribution of features is a Gaussian distribution with covariance ~.


It should be noted that as the PLCC, Kendall and RMSE values increase/decrease fast and look saturate when the trial number is large, it is difficult to visually distinguish the performances of different methods.

in this paper, we rescale the Kendall and PLCC values by Fisher transformation, i.e.


Instead of comparing absolute running times, we observe that all four solvers are based on the Z3 theorem prover [6] and rely on the counterexamples from Z3 to adjust their search strategy.

we compare the number of queries to Z3, which is usually the performance bottleneck for verification tasks.


On the other hand, about 10% (or 2.8%) of the samples have < 500 (or < 200) nonzeros.

we must deal with empty bins if we do not want to exclude those data points.


The remaining (infinite) states are equivalent from agent's perspective, as, in expectation, these states will exhibit the mean dynamics of the prior.

the only parts of the infinite model that need to be initialised are those corresponding to the states the agent has visited as well as a catch-all state representing all other states.


By standard sample complexity bounds, n should also be order of 1/ 2 .

one may wonder, if the size of the minibatch should be order of n. This is not true, since the required training set size is 1/ 2 times the VC dimension of the hypothesis class.


Thus, in stark contrast to the setting considered here, the size of a τ -divergence ambiguity set has no impact on the corresponding optimal estimator.

moreover, the least favorable prior Q = N d (µ, S ) for a τ -divergence ambiguity set always satisfies in order to harm the statistician, nature only perturbs the second moments of the signal but sets all second moments of the observation as well as all cross moments to their nominal values.


Although recently reformulated in the context of deep learning [2], verifying its relevance in practice requires the computation of mutual informations for high-dimensional variables, a notoriously hard problem.

pioneering works in this direction focused either on small network models with discrete (continuous, eventually binned) activations [3], or on linear networks [4,5].


While, many research works take it for granted and omit the fact that the learned features by Softmax are only separable not discriminative.

the performances of deep models in many recognition tasks are limited.


We restrict each radar to scanning subsectors that are a multiple of 90 • , starting at 0, 90, 180, or 270 • .

with N radars there are 13 N possible actions at each decision epoch.


The experimental setup is similar to that used in [8], except for one important difference: in the current experiments, the positive training sets do not include additional protein sequences extracted from a large, unlabelled database.

the recognition tasks performed here are more difficult than those in [8].


Therefore using equation (3.9) and equation above, we have that Note that since g | α has Gaussian distribution, we have, p g|α (0) = (2π) −d/2 ( α 6 6 ) −d/2 .

thus using two equations above, we can bound E [h(α)] by it suffices to control the RHS of (3.10), which is much easier than the original Kac-Rice formula.


Crucially, there exists controllable trajectories satisfying the joint specification: H¬red ∧ P yellow ∧ H (yellow ∧ P blue) =⇒ (¬blue S brown) .

a specification optimizing agent must jointly perform both tasks.


A key observation from Theorem 2 is that the optimal multilabel classifier can be obtained by thresholding the marginal instance-conditional probability for each label P(Y m = 1|x) and, importantly, that the optimal classifiers for all the labels share the same threshold ⇤ .

the effect of the joint distribution is only in the threshold parameter.


For p ij = 1/2, we need about Θ( log T D(p ij ||1/2) ) time-slots to distinguish it from 1/2 [5].

when choosing i as the first candidate, the regret of comparing it with all other arms can be estimated by R( 1) ij (t)||1/2).


Applying Lemma 2.3, we obtain T ρ + 2 T ρ log(1/δ) = .

solving the roots of this equation, we obtain θ T is ( , δ)-differentially private for the above value of σ 2 .


Also, ||x|| 2 ≤ ||x|| 2 ≤ p, so we obtain the desired bound.

proof sketch of Theorem 2: Under the same notation as the previous proof, notice that Proof sketch of Corollary 1: Let * = (1 − γ) log 2 p. Since i = 2 −i(1−β) ∀i < L − * + 1 and i = 0 otherwise, we have for ≥ * and since L = log 2 p, λ ≥ 2 β(L− * ) 2 β−1 = p βγ 2 β−1 , which is increasing in p. we can pick k = 2 * and since 2 * /p = p −γ , the result follows.


In particular, the tensor factorization algorithm returns a set of vectors {( µ h , w h ) : h = 1, . . . , k}, where each ( µ h , w h ) estimates a particular column of C c (for some µ cl ) and a particular diagonal entry of W (for some w l ).

it is important to note that the tensor factorization algorithm doesn't provide a one-to-one correspondence between the recovered column and the true columns of C c . µ 1 , .


Second, the coarser time granularity of the high level process implies fewer resampling steps and fewer opportunities for particle depletion.

if we can model how much drift is expected to occur over a small section of the robot's trajectory, we can maintain this extra uncertainty longer, and resolve inaccuracies or ambiguities in the map in a natural fashion.


Because of the flexibility this allows in constructing Σ s , and therefore Σ b , some additional notation is required to proceed.

a new decomposition of Σ b is defined as where Also, using commutative properties of the trace operator, L(γ) only depends on the data B through the d b ×d b sample correlation matrix BB T . to reduce the computational burden, we replace B with a matrix B ∈ d b ×rank(B) such that B B T = BB T . This removes any per-iteration dependency on n, which can potentially be large, without altering that actual cost function.


The first layer learns 128 filters of size 1 × 1.

441 input features are compressed to 128 learned features.


In the parameter transfer setting, a number of samples in the source region are often needed to accurately estimate the parameter to be transferred.

it is desirable to be able to use unlabeled samples in the source region.


If (X , H) is neighborly, then GBS requires n 0 = O(log |H|) queries in noiseless conditions [Now08].

under the conditions of Theorem 2, the query complexity of the simple noise-tolerant GBS algorithm is O(log |H| log log δ ), a logarithmic factor worse than the optimal query complexity.


From Figure 1 we also see that SVRG-BB is not sensitive to the choice of η 0 .

sVRG-BB has very promising potential in practice because it generates the best step sizes automatically while running the algorithm.


However, we will bound functions like in equation ( 1), which do not write as a sum of partly dependent variables.

we need a result on more general functions than sums of random variables, but which also takes into account the known dependency structure of the variables.


When the Gaussian likelihood p(y|x) is replaced with a Bernoulli distribution (which leads to the logistic data fit term above), it is no longer possible to compute the marginalization (4) or the posterior distribution p(x|y; γ), which is used both for optimization purposes and to make predictive statements about test data.

a heuristic Laplace approximation is adopted, which requires a second-order Newton inner-loop to fit a Gaussian about the mode of p(x|y; γ).


Given three real numbers γ init 0 , γ escap 0 and γ fin 0 we set down φ 1 .

we can map M 0 onto the intervals ]γ escap 0 , γ init 0 [ and ]γ escap 0 , γ fin 0 [ respectively: if γ0 refers to the sigmoid function, φ 1 0 • γ0 will be a logistic curve, growing from γ escap 0 to γ init 0 .


The designs are obtained from the continuous solutions using the hard constraint, with replacement method of [10] and a simple design truncation as in [22].

these simulations consider univariate measurements (n e = 1).


We first explain the reason the required preview window width W is proportional to the hidden vector size D. Figure 2 shows the singular value distribution of WikiText-2 LM softmax weights.

we observed that the distributions are similar for all three cases when the singular value indices are scaled with D. it is important to preserve the ratio between W and D. The ratio of singular values in a D/8 window over the total sum of singular values for 256, 512, and 1,024 hidden vector dimensions is 0.42, 0.38, and 0.34, respectively.


receptive fields) with more invariance gained by the pooling layers, akin to classical convolutional neural networks.

the multilayer scheme produces a sequence of maps (I k ) k≥0 , where each vector I k (z) encodes a point-say f k (z)-in the linear subspace F k of H k . we implicitly represent an image at layer k as a spatial map As mentioned previously, the mapping to the RKHS is a key to the multilayer construction.


Since the mutual information is a general measure of dependence (linear and non-linear) between two variables, I action and I state measure the ease in inferring the goal from the actions and states, respectively, generated by the policy π.

if Alice wants Bob to do well, she should choose a policy with high information, and vice versa if not.


MCMC itself can be computationally demanding, and the expected estimation error depends directly on the correlation between successive points in the Markov chain.

efficiency can be achieved by taking large steps with high probability.


We generated spike trains by using a multivariate Poisson distribution with only a thirdorder correlation whose Poisson mean depends on periods as: (a)  ), but they only differ in the third-order correlation.

classical Poisson-HMMs that employ an independent Poisson assumption [1,2,7] are not able to segment them into distinct states.


These graphs only represent the conditional uncorrelatedness, but conditional independence, among variables.

by extending the Gaussian to the elliptical family, the gain in modeling flexibility is traded off with a loss in the strength of inference.


Moreover, the expression derived for G EC is not a simple explicit functions of µ µ µ but contains additional optimizations over Lagrange parameters.

we cannot expect that message-passing algorithms similar to loopy belief propagation or expectation propagation (EP) [4] which sequentially update moments and Lagrange parameters are guaranteed to converge.


This statement holds for any acyclic substructure embedded within the full graph with cycles -not just the spanning trees Tl , ... ,T L used to implement TRP.

algorithms such as TRP /BP attempt to reparameterize a distribution on a graph with cycles so that it is consistent with respect to each embedded tree.


Of course, when ∆ = +∞, the problem includes any non-convex optimization, where we cannot hope for an efficient solution for any finite .

the crucial quantity to study is the optimal tradeoff of and ∆: For which , ∆ the problem can be solved in polynomial time, and for which it can not.


improved recalls), however, it seems no use to the final predictions, whose quality we believe is mainly determined by the second stage.

in the paper we mainly study the case of single-stage detectors [30,25,31,23].


By Lemma 3 merging clusters cannot decrease the margin.

seperation(T ′′ ) = seperation(T ′ ) = seperation(T ).


However, their analysis crucially relies on exact minimization of the learning objective, whereas SGD with a finite number of steps only finds an approximate minimizer.

to our knowledge, no prior work applies to this problem.


When λ is small, the Gaussian likelihood is highly restrictive, constraining most of its relative mass to a very localized region of x space.

a tighter prior more closely resembling the 0 quasi-norm can be used without the risk of local minima, which occur when the spines of a sparse prior overlap non-negligible portions of the likelihood (see Figure 6 in [15] for a good 2D visual of a sparse prior with characteristic spines running alone the coordinate axis).


Using ( 13) in (10), we see that where m is the stacked means of the UBM.

the GLDS kernel inner product is Note that Γ in ( 12) is almost the UBM covariance matrix, but is not quite the same because of a squaring of the p(j|z i ) in the diagonal.


Similarly to Phase I, the gradient has a relatively small magnitude, and noise is the key factor to help the algorithm escape from the saddles.

an SDE approximation need to be derived.


Another example is cross-language classification where labeled documents in English are much more than ones in some other languages such as Bangla, which has only 21 Web pages in the ODP.

it would be great if we could learn the knowledge across different feature spaces and to save a substantial labeling effort.


It is easy to see that this is the best P 2 can do, because P 1 can average $0 by always choosing Sell.

choosing Heads with 25% probability and Tails with 75% probability is an optimal strategy for P 2 in the "Play" subgame.


The ordered tree structure is a special case of the non-negative garrote discussed in [24] when the hierarchical relationship is depicted by a tree.

the asymptotic properties established in [24] are applicable to the ordered tree structrue.


However in the anonymised setup the context document is required for answering the query, whereas the original version could also be answered by someone with the requisite background knowledge.

following this procedure, the only remaining strategy for answering questions is to do so by exploiting the context presented with each question.


Note that we do not need to know the label of a training sample in order to enforce this loss.

the proposed loss function is completely unsupervised and can be used along with supervised training as a semi-supervised learning method.


Then, the AUC, A, associated to C is given by: which is the value of the Wilcoxon-Mann-Whitney statistic [10].

the AUC is closely related to the ranking quality of the classification.


If > 1/L, then the adversary can simply draw a point uniformly at random in [0, 1) and be guaranteed that the target will be within /2 with a probability greater than 1/L.

the privacy constraint is automatically violated, and no private learner strategy exists.


Given our generation model p(x i t |z i t ), the true posterior distribution is intractable.

the standard practice is to employ a variational approximation q(z i 1:T |x i 1:T ) to the true posterior [17].


( 24) is actually the thresholding point of estimated σ 2 for the h-th component to be discarded.

h EVB = H * if and only if σ 2 H * +1 < σ 2 < σ 2 H * .


When x tm = 0, the power consumption of device m at time t is zero (P m 0 = 0), and when x tm ∈ {1, . . . , Q − 1} its average power consumption is given by P m xtm .

the total power consumption is given by y t = M+ m=1 P m xtm + n t , where n t ∼ N (0, 0.5) represents the additive Gaussian noise.


Presumably, accumulating more probabilistic evidence of more complex conditional dependencies has a cost, both in terms of storage, and in terms of the computational load of performing subsequent inference.

discarding information after making a decision can help to keep this storage and the computational complexity at a manageable level, freeing computational resources to perform other tasks.


The threshold has to be manually tuned to the data and tasks by trial and error.

a learning method is required to more effectively address the exploding gradient problem in training of GRUs.


However, GRProp observes non-zero gradient for subtasks A, B, and C that are propagated from the parent nodes.

even though the reward is delayed, GRProp can figure out which subtask to execute.


Our key technical contribution is to formulate the estimation of an upper bound of the inner product as a constrained convex optimization problem and show that it admits a closed form solution.

the estimation of the inner product can be computed efficiently.


In the general case with noise , we lose the spectral property that the global transition matrix has K eigenvalues which are 1.

the algorithm described in Section 3.2.1 cannot be applied directly to make ĀP block diagonal.


Based on Lemma 4.3, we conjecture that in these cases, ground truth is the only solution that is stationary regardless of the latent representation.

updating the latent representation once in a while prevents the generative parameters from converging to local minima, i.e.


Note that a network which minimizes this loss function is maximizing its information content, even if the number of neurons, N , far exceeds the input dimension K, so that N K. As becomes clear from inspecting the loss function, the network is trying to code an N + K dimensional signal with only N neurons.

just as in the undercomplete autoencoder, all of its information capacity will be used.


Next, the spatial configuration C is decided by a random permutation of some assignment of the target and distractors to the N 2 cells in A (such that there is exactly one item in each cell).

for a given search array A, the spatial configuration as well as stimulus parameters are fixed.


The main idea promoted by Remix-UCB is to divide the time available in epochs 1, . . . , s max (the outer loop of the algorithm), such that at the end of each epoch s, for all the remaining arms k the following holds, , where τ k identifies the time instants up to current time t when arm k was selected.

using (4), this means that, for all k, with high probability: at the end of epoch s we have, with high probability, a uniform control of the uncertainty with which the empirical rewards µ τ k k approximate their corresponding rewards µ k . Based on this, the algorithm eliminates the arms that appear significantly suboptimal (step 2 of the update of Remix-UCB).


expected number of boxes × expected volume of a box) in X sampled from a RBP is a constant τ The expected length of the interval in u According to the definition of the RBP (Poisson distribution in Step 1), we have an expected number of τ • D d=1 1 + λL (d) bounding boxes in X .

the expected total volume of the bounding boxes for a given budget τ and a domain X is τ • D d=1 L (d) .


Then: If S = A, i.e., using the augmented Lagrangian smoother, γ 0 := √ Lg = 1, and c k := 0, then the (1P2D) update satisfies: for all k ≥ 0.

the worst-case analytical complexity of Algorithm 1 to achieve an ε-solution xk is O( √ ε).


Also, a linear combination of kernels is a kernel.

if the base kernel is valid, then the convolutional kernel is also valid.


Moreover, we have the same constraints as in the name-face problem [6]: a person can be assigned to at most one name-verb pair, and vice-versa.

given a document with P i persons and W i name-verb pairs, the number of possible assignments is j , where j is the number of persons assigned to a name-verb pair instead of null.


We provide an algorithmic framework for computing higher order derivatives of matrix and tensor expressions efficiently, which fully operates on tensors, i.e., all variables are allowed to be tensors of any order, including the output variables.

for the first time higher order derivatives and Jacobians can be computed directly.


Temporal correlations in the high-dimensional neural population are modelled via a lower number of shared latent processes x k (•), k = 1, . . . , K, which linearly relate to a high-dimensional signal h h h(•) ∈ R N in neural space.

inter-trial variability in neural firing that is shared across the population is modelled via the evolution of the latent processes on a given trial.


This fixes the completeness problem, while retaining soundness: no combination of contours in the two images in a) can produce matching shapes.

with rough alignment we can cast the first step of shape matching as context selection: which subset of contours, or context, to use for feature computation.


On the other hand, our experiment also showed an unexpected behavior that has not been reported before, for d ≤ 4, the change of MI caused by the use of DN is negative, i.e., DN increases statistical dependency for such cases.

though effective for high dimensional models, DN is not an efficient coding transform for low dimensional multivariate t models.


Remark I It is important to realize that the problem in ( 5) is non-convex due to the nonconvex constraint w 2 i ≥ 1.

the optimal solution found by the dual problem in ( 7) is not necessarily the optimal solution for the prime problem in (5).


The weighted cluster indicator matrix L solving the maximization problem in Eq.

( 11) can be computed by solving a kernel K-means problem [5] with the kernel Gram matrix given by DisCluster is equivalent to a kernel K-means problem.


The inner minimization problem is a convex conic program on K. Also, as the pointwise minimum of a family of concave quadratic functions of α, the solution to the inner problem is a concave function of α, and hence the outer optimization problem is also convex (see [15] for further details).

( 3) is a concave maximization problem subject to linear constraints and is therefore a convex problem in α.


For instance, the open country scene is characterized by its upper part as a wide sky area.

to estimate the depth well in that region by sacrificing some regions in the bottom may help an image to be categorized to the correct category.


Denoting the linear regression coefficient of log(v) versus log(κ) by β and the linear regression coefficient of log(α) versus log(κ) by ξ, it can be easily shown that (5) entails: Hence, if log(α) and log(κ) are statistically uncorrelated, the linear regression coefficient between them, which we termed ξ, would be 0, and thus from (6) the linear regression coefficient of log(v) versus log(κ), which we named β, would be exactly − 1 3 .

any trajectory that produces log(α) and log(κ) that are statistically uncorrelated would precisely conform to the power-law in (1) 3 .


Note that it is distinct from the original Poisson distribution (1); in particular its normalization constant involves a summation over finitely many terms.

no restrictions are imposed on the parameters for the normalizability of the distribution.


However, r only the n = 1 arc cosine kernel preserves the norm of its inputs: the n = 0 kernel maps onto a unit hypersphere in feature space, while higher-order (n > 1) kernels may induc spaces with severely distorted dynamic ranges.

we hypothesize that only n = 1 a kernels preserve sufficient information about the magnitude of their inputs to work effe composition with other kernels.


In a large-scale distributed learning setting, large batch size is used in optimization.

in such a setting, it is preferable to use a method that can scale well with batch size.


Use B to denote the set of all balanced strategies and Ḃ to denote the set of all balanced gain distributions.

interestingly, as demonstrated in [9], a balanced adversary π inflicts the same regret on every learner: (See Lemma 10) given an adversary strategy π, we can define the value-to-go V π t0 (s) associated with π from time t 0 in state s, , s t+1 ∼ P (.|s t , π(s t , t), s t0 = s).


First, it does not eliminate from consideration correlations that can be explained by direct edges among the observables.

it forms clusters even in cases where the dependencies can be fully explained by a standard Bayesian network structure.


However, in practice some features could be more informative than the others for learning purpose.

it makes more sense to assign different sampling probabilities for different features and make the features compete with each other.


In this view, ✓ a resembles the initial number of balls of color a in the urn at time t = 1 and N a (t) resembles the total number of balls of color a added into the urn after t draws.

the probability the t th draw is of color a is proportional to N a (t).


For any distribution q(w) and function f (w) we have min w [f (w)] ≤ E q(w) [f (w)].

the minimizer of the Gibbs risk is a point estimate, which with Theorem 6 implies: Corollary 7.


If a graphical constant-sum game admits a fully-mixed NE x * and all players employ MW updates, then Before proving the lemma, we note that V i (t) is indeed the variance of the following random variable, and thus is always non-negative: the random variable realizes the value Proof.

we first expand , and noting that in this interval the function ln(1 + y) + y 2 /4 is concave but the function ln(1 + y) + y 2 is convex, by the Jensen's inequality, we have By following the proof of Theorem 3 in [23], one can show that the above double summation is zero.


Our approach is based on continuous extensions that are linear operators on the class of set functions, namely, linear continuous extensions.

as a specific example, considering the multilinear extension, we can write where F denotes the extension of f . the value of F (x), when ⇠ , is an unbiased estimator for F (x) and unbiased estimates of the (sub)gradients may be obtained analogously.


This high-level behavior coincides with the dynamics of competitive learning; both act to decrease the difference between a stored weight and the applied input.

no explicit computation of the direction or magnitude of weight updates is necessary-the circuit naturally performs these computations for us.


Given the result in Equation ( 12), the proof is trivial for the computation of L ell as we only need to realize that q k (f given in Equation ( 10) has a block-diagonal covariance structure.

q k(n) (f n• ) is a Q-dimensional Gaussian with diagonal covariance.


This implies that at least 0.1t of the "good" classifiers of that player should mislabel it, which amounts to 1/6 of the "good" classifiers.

the error of the majority of the functions for that player is at most 6 = .


Let diag(A) be the matrix A with off-diagonal elements replaced by zero and A 1/2 be the squared root matrix of A.

it is easy to see that Γ has the same nonzero pattern as Σ −1 .


As we discuss below, the model is trained conditionally, and therefore encodes a joint distribution over the rating variables, conditional on all of the voxel measurements.

there will be no free parameters corresponding to the voxel nodes due to the use of a conditional model, while rating nodes R s (j, •) have an associated node potential parameter Q node (s, j).


We investigate the application of the proposed approximation scheme using these subsets.

consider the information projection of an arbitrary probability measure P with density 3 p to the set D = F S i given by: Applying Theorem 3, we can compute that p S = p(x)δ S (x)/Z, where Z is a normalization factor: the normalization factor is a marginal density at x S = c S . We may now compute the restriction explicitly: In other words, the information projection to a sparse support domain is the density of x S conditioned on x S = c S . The resulting gap is: for a given target sparsity k, we solve: , where J(s) = log p(x S = c S ).


For scheme (b), ψ k corresponds to 1 n (α (by ( 128)) ( 25) and by setting β to 0 in (126), we get L 2 2 ≥ n 5 similar to (23).

l 2 1 saves a factor of n compared with L 2 2 .


From Equations (4.1), (4.2) we have 3) It is easy to check that under our assumptions, w ≥ nk 2 and m ≥ k 2 .

(1 − nk/w) k ≥ 1/e and (1 − k/m) −k ≥ 1/2e.


Intuitively, the age (or "oldness") should not be too large to ensure the convergence.

it is a natural and reasonable idea to assume an upper bound for ages.


For example, training on salt-and-pepper noise does not imply robustness on uniform white noise and vice versa.

changes in the noise distribution between training and testing constitutes a crucial challenge to deep learning vision systems that can be systematically addressed in a lifelong machine learning approach.


While there may be many different approaches to define what is a good clustering (and, in general, deciding what is a good clustering is a difficult problem), for the problem of classifying time-series samples there is a natural choice, proposed in [21]: those samples should be put together that were generated by the same distribution.

define target clustering as the partitioning in which those and only those samples that were generated by the same distribution are placed in the same cluster.


3, in which PIB features show similar characteristics for the events related to both SOZ and non-SOZ.

we utilize the factor graph model presented in this paper to further filter the detected abnormal events based on their spatial and temporal patterns and isolate the SOZ-likely events.


However, we also find that the feature matching objective has several drawbacks.

we develop a novel formulation of the discriminator and generator objectives to address these drawbacks.


A cut (S 1 , S c 1 ) defines a 1-local minimum if and only if it has lower energy than all cuts that do not intersect it.

if a 1-local minimum is not a global minimum then the cut (S 1 , S c 1 ) necessarily intersects all of the cuts defined by the global minimizers.


[22] (when all elements of F have cardinality two).

equation ( 13) then becomes and the inequalities ( 12) defining the polytope of concavity are we define By Theorem 2, the set C is the region of concavity for the Bethe entropy ( 14) within [0, 1] |F | .


Furthermore, suppose that K ⇤ = p p d U U T with U 2 R p⇥d is a generic (dense) orthogonal matrix with unit norm columns.

the factor p p d is simply the scaling needed so that the average magnitude of the entries in K ⇤ is a constant, independent of the dimensions p and d. In this case, rank(K ⇤ ) = d and kK ⇤ k F = trace(U T U ) = p. These two facts imply that the tightest bound on the nuclear norm of K ⇤ is kK ⇤ k ⇤  p p d. we take = p p d for the nuclear norm constraint.


By setting the derivative of Q(p, g) with respect to g 1 and g 2 to zero and compare them 3 , we have, After some algebra, we obtain, According to Proposition 5, we have ∂φ(g,y) ∂g1 due to the order preserving of the top-k subgroup probability space.

the above equation cannot hold since at least one of components in the sum is negative according to Definition 3.


However, the algorithms in [9] assume that φ(x) is a Lipschitz continuous function, i.e., the subgradient of φ(x) is bounded.

when the domain X is unbounded, the algorithms in [9] cannot be directly applied.


Conversely, supervised domain adaptation (SDA) algorithms do require labeled target data, and because labeling information is available, for the same quantity of target data, SDA outperforms UDA [38].

if the available target data is scarce, SDA becomes attractive, even if the labeling process is expensive, because only few samples need to be processed.


We plot pseudo-regret realization from each simulation, i.e., E[ ⇤ T ] T , where E[ ⇤ T ] is the expected reward for the Oracle, computed via Monte Carlo simulation, and T is the total reward achieved by the algorithm.

lower pseudo-regret realization implies better performance.


Finally, for SVM-PERF's training and score prediction we used the binary executables downloaded from the URL http://www.cs.cornell.edu/people/tj and used the SVMPERF's settings that are the closest to our optimization formulation.

we used L1-norm for slack variables and allowed the constraint cache and the tolerance for termination criterion to grow in order to control the algorithm's convergence, especially for larger values of the regularization constant.


An equivalent condition is that to each I i we can associate a base permutation σ i so that τ ji = σ j σ −1 i for any (i, j) pair.

the problem of finding a consistent set of τ ji 's reduces to that of finding just m base permutations σ 1 , .


Our main assumptions in the development of a mean field theory are that L;#i bCijvj is a Gaussian random variable, and that bCij and Vj are independent.

where (v 2 ) == N-1 L;i v; is the second moment of the firing rate.


We consider parametrized policies π θ , and hence wish to optimize for the parameters (θ).

we overload notation and use η(π) and η(θ) interchangeably.


We emphasize that all our results hold for any fixed input, which need not be random; we average only over the weights and the biases.

fM1 cannot be directly solved by batch normalization [11], which renormalizes by averaging over inputs to N , rather than averaging over initializations for N . Theorem 1 (FM1 for fully connected networks (informal)).