If the next hypothesis p t+1 of a ROME algorithm is on H t , then by Lemma 2, it and all other members of H t satisfy D(p t+1 ||u) = D(p t+1 ||p t ) + D(p t ||u).

in this case, p t+1 also minimizes D(q||p t ) from among the members q of H t ∩ J t . Lemma 11 implies that p t+1,i /p t,i is the same for all i with x i = 1, and the same for all i with x i = 0.


(Step 1: β): In Metropolis-Hastings, convergence typically improves as the proposal distribution is closer to the target distribution.

it would be ideal to propose β from p(β|ẑ).


The following functional with respect to r is called the free energy: In the last equation, the first term is the Kullback-Leibler (KL) divergence from the trial distribution to the Bayes posterior, and the second term is a constant.

minimizing the free energy (4) amounts to finding a distribution closest to the Bayes posterior in the sense of the KL divergence.


(i) Gaussian q(w n ) First, a Gaussian q(w n ) implies that the KL over the latent variables is closedform, as both the prior and posterior are Gaussian.

we are left with the calculation of the first term in (4) E q(wn) L wn .


Our approach is to compute the target probability using statistics obtained from recent experience performing the task.

we refer to our model as experience-guided search or EGS.


We can't get the results of SVD, LLE and LS within three days on RCV1 and ILSVRC2012 data sets.


As κ → ∞, the right-hand side of ( 6) decreases.

the minimax rate for F(γ, κ) can be no faster than n −1/(1+ρ) , which is the lower bound for F(γ, 1).


The relative efficiency of the block 1,∞ program (6) compared to the Lasso is given by R(θ 1,∞ , θ Las ) = 4−3α .

for sublinear sparsity s/p → 0, the block scheme has greater statistical efficiency for all overlaps α ∈ (2/3, 1], but lower statistical efficiency for overlaps α ∈ [0, 2/3).


(See also [10] for Nyström applied to spectral clustering).

lMDS operates on a number of rows proportional to the embedding dimensionality, d. Dijkstra gets called O(d) times.


This might lead to misinterpreting bandpower changes as changes in connectivity.

a better understanding of connectivity patterns during MI of different limbs as measured by EEG is required to guide the design of new feature extraction algorithms for BCIs.


3 is guaranteed to find a junction tree (with all cliques connected to the same separator).

we can restrict binary search to range δ ∈ [0, log R].


We next focus on the properties of an optimal s. Suppose that two classes are generated from (3) with cross-spectral densities of A 0 and A 1 for classes 0 and 1, respectively.

the signals are drawn from CN (0, A y ⌦ (⌧ )) for y = {0, 1}.


Although we are confident that our algorithm will eventually perform at least as well as the baseline strategy, it should be able to remain alive (not terminated by the manager) long enough for this to happen.

we should make it more conservative (less exploratory) in a way not to violate the manager's safety constraint.


Also, since the standard deviation is of a smaller order than the expectation, the random variables concentrate.

we can pick a threshold t n such that SCAN(A 1 , q, t n ) yields just the nodes in the same cluster as q with high probability.


Moreover, since the A, B were randomly sampled, we have seen that the true optimal solution occurred at the boundary.

relaxing the constraint to linear forced the solution to occur outside of the feasible set, as seen from the results in Table 1 as well as from Lemma 3.


However, although graph convolution in the spatial domain is conceivable, it faces the challenge of matching local neighborhoods, as pointed out in [4].

there is no unique mathematical definition of translation on graphs from a spatial perspective.


We will have to take the dependence between different samples x (j) i and x (k) i into account, but between variable dependencies within a sample do not pose problems.

estimate the confidence of any decision variable z i is straight forward from Equation (5), applied separately to the binary sample set Note that all quantities defined above for the single variable case need to be computed separately for each decision.


To address this problem, we introduce the standard supervised loss (i.e., cross-entropy loss) to C, R L = E (x,y)∼p(x,y) [− log p c (y|x)], which is equivalent to the KL-divergence between p c (x, y) and p(x, y).

we define the game as: It will be proven that the game with utilities Ũ has the unique global optimum for C and G. • Update D by ascending along its stochastic gradient: • Compute the unbiased estimators RL and RP of R L and R P respectively.


Because the similarity structure can be estimated from imaging data even without building an encoding model, RSA allows not only for model testing (by comparing the similarity matrix of neural data with the similarity matrix of the feature vectors when stimuli are represented with an encoding model) but also for exploratory study (e.g., by projecting the similarity structure to a low-dimensional space to visualize its structure, [11]).

originally as a tool for studying visual representations [2,16,10], RSA has recently attracted neuroscientists to explore the neural representational structure in many higher level cognitive areas [23,18].


Let π k be the policy of UBEV in the kth episode.

then with probability at least 1 − δ for all ε > 0 jointly the number of episodes k where the expected return from the start state is not ε-optimal (that is with probability at least 1 − δ UBEV converges to optimal policies and for all episodes T has regret Here polylog(x . . . ) is a function that can be bounded by a polynomial of logarithm, that is, ∃k, C : polylog(x . . . ) ≤ ln(x . . . ) k +C.


In order to impose sparsity we use regularization.

we solve a sequence where the regularization parameter λ > 0 and the constraint sets Ω change from epoch to epoch.


As deep networks are highly nonconvex, even with the same network and optimization method, different initializations can lead to different local optimal.

following [26], we also take two networks with the same architecture but different initializations as two classifiers.


The saddle point equations with respect to probability distributions provide a set of relations between π(x) and π(x): By using the result obtained for the free energy, we can easily perform further straightforward calculations to find all the other observable thermodynamical quantities, including internal energy: which records reproduction errors.

in terms of the considered replica symmetric ansatz, a complete solution of the problem seems to be easily obtainable; unfortunately, it is not.


Given a pool of test points we can rank these test points according to their disbelief index, and points with low index should be rejected first.

this ranking provides the means for constructing a risk-coverage trade-off curve.


The Markov chains in both algorithms are ergodic, but may not be in detailed balance because we rely on approximation.

we guarantee that both algorithms converge for general MRFs, but they may not exactly converge to the target distribution.


The key trick is that after symmetrization, we employ | max{0, z} − max{0, z }| ≤ |z − z |, making three differences of partial risks point-wise (see (18) in the proof).

we have to use a different Rademacher complexity with the absolute value inside the supremum [35,36], whose contraction makes the coefficients of ( 15) doubled compared with Lemma 8 of [19]; moreover, we have to assume G is closed under negation to change back to the standard Rademacher complexity without the absolute value [28].


The training procedure for the new task is as follows, keeping W a t−1 fixed and only back-propagating the newly added parameters of W t \W a t−1 .

the optimization formula for the new task is, min We use stochastic gradient descent to learn the newly added filters with η as the learning rate, The expanded child network will be trained until the required number of epochs or convergence are reached.


In the pseudocode, P W refers to the projection on to the ball W, which amounts to rescaling w to have norm at most D. The accelerated gradient method (e.g., [5]) is summarized as Algorithm 2.

in terms of existing results, for the SGD algorithm we have [4, Section 5.1] , whereas for an accelerated gradient algorithm, we have [5] as long as b = o( √ m), both methods allow us to use a large mini-batch size b without significantly degrading the performance of either method.


However, given very similar score values at the time points of t0 and t3, any of the aforementioned summarized values may not be sufficient to identify the group difference.

if longitudinal cognitive outcomes are available, it would be beneficial to use the complete information for the identification of relevant imaging markers [9,10].


. . , P i n are independent of each other, using the above lower bound for the expectation of their sum and Chernoff bounds we get Pr[P i 1 + . . . .

for a sufficiently large constant c(γ), the probability that the vertex u i is uninfected after two rounds Pr[P 1 + .


Our motivation is the the following: if θ and ψ define the same distribution, then if a certain set of edges are removed from both, they should continue to define the same distribution 1 .

given some graph T , we define the "projection" θ(T ) onto the tree such by setting all edge parameters to zero if not part of T . Then, given a set of graphs T , the piecewise KL-divergence is Computing the derivative of this divergence is not hard-one simply computes the KL-divergence for each graph, and uses the gradient as in Eq.


Furthermore, if a parallel labeling system is available, e.g., multiple annotators working on different labeling workstations at the same time on a network, a single instance selection system can make wasteful use of the resource.

a batch-mode active learning strategy that selects multiple instances each time is more appropriate under these circumstances.


Notice that, using (1), we can write each y i as where ȳi is the i-th data point whose missing entries are filled with zeros and x i is the vector containing missing entries of y i .

substituting (4) in the optimization (3), we would like to solve Notice that matrices ȳi are given and known while vectors In fact, the optimization (5) has two sources of non-convexity: the 0 -norm in the objective function and the product of unknown variables {c ij } and {x i } in the constraint.


The function f M (π e ) can also be made more general to encode how likely it is that the expert uses the policy π e given world model M . e are the observations and rewards.

the conditional distribution over world models given data D e and D a is: The model in Fig.


This gives |BB T NN T | i,j  p ) i,j to the nearest multiple of w, we obtain the entry exactly.

given N , we can exactly recover AC in O(nk !


Lines 14 -18 in Algorithm 1 are added only for the data analysis in Section 5.3, in which p is massive compared to n in that log(p) is comparable to n, and the algorithm may not scale down the size of p sufficiently for even obtaining a ridge regression estimator afterwards.

a further sparsification step is recommended.


As suggested by Theorem 3 in almost all cases the normalisation improved the performance of the support vector machine solution at a statistically significant level.

we recommend: When training an SVM, always normalise your data in feature space.


In either cases, the functions −p log z + z Given that the function values are non-increasing along the iterates of Algorithms 1, 2 and 3, the sequence of Ω (k) satisfy Ω (k) ∈ C 0 for k = 1, 2, .....

the lower bounds on the diagonal elements of Ω (k) provides the Lipschitz continuity using (17) using the mean-value theorem, the gradient ∇h 1 satisfies ∇h with the Lipschitz continuity constant L = a −2 + S 2 .


For Gaussian graphical models, the number of parameters, the number of edges in the structure and the number of non-zero elements in the inverse covariance or precision matrix are equivalent measures of complexity.

several techniques focus on enforcing sparsity of the precision matrix.


( 12) reveals that the high-β solutions to MF-IIM focus on the nodes for which d in i + b i + h i is smallest.

if b is uniform, the MF magnetization is maximized by focusing the external field on the nodes of smallest in-degree in the network.


Reminding is normally triggered by contextual features present at the moment of retrieval which match distinctive features of the memory being recalled (Berntsen et al., 2013;Wharton et al., 1996), and can occur more often following unexpected events (Read & Ian, 1991).

an individual's current state of understanding can trigger reminding of a past state.


However, solving (3) with this Y is generally NP-hard.

we approximate Y by means of a common multicut LP relaxation [18] with the following two constraints: (1) cycle inequality and (2) odd-wheel inequality.


For multi-electrode recording data of commonly-encountered size, and using our implementation of EM, inference of parameters under these time-constraints would be infeasible.

an ideal parameter initialisation method will not only improve the robustness of the identified parameters, but also reduce the computational time needed for EM convergence.


In natural languages processing, the text sequences are evaluated as the discrete tokens whose values are non-differentiable.

the optimization of GANs is challenging.


The quadratic surrogate allows us to compute the new search distribution analytically where the KL divergence of the new and old distribution is bounded.

we only exploit the surrogate model locally which prevents the algorithm to be misled by inaccurate optima introduced by an inaccurate surrogate model.


3.1) shows that the M-step of MVP can be computed by using only one sample, because the uniform prior typically leads to sparse weights during training.

the EM process develops into the conventional back-propagation.


The extra dependency in log(T ) could be reduced to log(K) or even optimally to 1 by using confidence intervals more carefully tailored, for instance by replacing the log(t) term appearing in the definition of the estimated gradients by log(T /T i (t)) or log(T /KT i (t)) if the horizon T is known in advance as in the algorithms MOSS or ETC (see [4,29,30]), but at the cost of a more involved analysis.

multi-armed bandits provide a lower bound for the approximation error E[L(p T )] − L(p ) of order K/T for smooth convex functions.


As can be seen from Figure 1, each cycle merely includes two feature spaces.

we only need to focus on one cycle and it is easy to extend to the case with multiple cycles.


For example curvature [18] is believed to be connected to uncertainty.

given a problem with known uncertainty at training points, one could use Sobolev Training to match the second order signal to the provided uncertainty signal.


That is, an individual will decide the ranking of the candidates based on the opinions of candidates on these issues.

irrespective of the number of candidates, the number of distinct rankings that prevail in the population are likely to be very few.


V*(so) = E[2:::o r(st, /-t(St))] for every So E S. These are exactly the samples obtained by the TD(l) algorithm [9].

the TD(l) solution is equivalent to the optimal approximate solution.


These leverage scores characterize the importance of the corresponding observations and are key to efficient subsampling with optimal approximation guarantees.

leverage scores emerged as a fundamental tool for matrix sketching and column sampling [22,21,13,36], and play an important role in low rank matrix approximation [11,6], regression [2,28,20], random feature learning [29] and quadrature [7].


Thus, we only need to store X ∈ R k . When a stream element a t = (i t , I t ) arrives, one updates the entries of X: ( By property of stable distributions, the samples x j , j = 1 to k, are also i.i.d.

stable the task boils down to estimating the scale parameter from k i.i.d.


Then MEM is equivalent to maximize ĥ,û l 1 ⊤ĥ subject to X l ûl + Y l µ + W l x l ≥ ĥ, βl ≥ ĥ, Z l ûl ≤ z l  All the three tasks involve convex (and linear) objective function with linear constraints which impose a convex feasible set.

one can use the rich and well-developed literature on convex optimization and linear programming to find the optimum intervention.


Due to the evident reconstructive properties [9,18], the available efficient pursuit methods and the sparsity of representation, we choose the sparse representation as the basic framework for the SRSC and incorporate a measure of discrimination power into the objective function.

the sparse representation obtained by the proposed SRSC contains both crucial information for reconstruction and discriminative information for classification, which enable a reasonable classification performance in the case of corrupted signals.


It follows that the cost of this path is necessarily negative, for otherwise it would have been optimal to apply it to the original flow, improving its score.

we can find the highest-scoring path by simply negating all edge costs and finding the shortest path in the graph.


In order to deal with (p, t)-separable topics the algorithm only needs documents of length t + 1.

in order to reconstruct p-separable topics (t = 1) it only needs bigrams, and in the matrix case (t = 2) only trigrams!


Since, the last column of B is the all one vector, we can write where X is a n × 3 matrix.

aB T = P X + t1 T and the observation error can be expressed as e l = (M − P X) l − t i for Ω(l) = (i, j).


As is common in regression problems, it is convenient to collect all the stimulus vectors and observed responses into matrices.

we described the input by a matrix , the £ th column of which is the input lag-vector ¡ .


By studying simplified versions of the problem, our analysis suggests how precise formulations of assumptions M and SSS can be made and exploited to lead to improved estimators.

the data are For convenience, assume that data are labeled so that R i = 1 for i = 1, . . . , n and R i = 0 for i = n + 1, . . . , N . the labeled sample size is n, and the unlabeled sample size is u = N − n. Let p(x) be the density of X and let m(x) = E(Y | X = x) denote the regression function.


Scores within each category are added and normalized by the maximum possible score.

each data point consists in four normalized scores, which can be seen as a point on the manifold M =]0, 1[ 4 .


Because the optimal policy is not known, it needs to be built by solving a sequence of nested problems.

information propagates both forward and backward.


However, evaluating the posterior mean from equation 7 with x * ∈ X, results in an analytically unsolvable differential equation: f is a complicated function of ∆ f , its own derivative.

we define a two-step approach: first we evaluate a standard GP with the training data, using our initial hyperparameter settings and ignoring the input noise.


However, note that the theoretically optimal choice of the regularization constant does not depend on these quantities.

we have theoretically and empirically shown that for noisy tensor decomposition, the latent approach is more likely to perform better than the overlapped approach.


In practice, we will observe the margins after running the GCC model.

we must bound the probabilities uniformly over all of the possible margins that can arise to obtain a practical bound.


Very small values for β result in the second term in L 1 int in (5) to be neglected.

the domain-specific code, v x1 , will be irrelevant in the loss minimization and the learned domain specific code could be meaningless.


Nonetheless, since φ(s, a) requires access to h(x, z) then we must run message-passing every time we compute a new state s in order to compute the next action.

we run message passing once and then perform less expensive local updates using saved messages from the previous iteration.


Every time we update τ (start of an epoch), we decrease it by a factor of 1 − (except maybe the last round for which τ = 1).

there are at most 1 + log 1 1− (M ) ≤ 1 + log(M ) log(1/(1− )) ≤ 1 + log(M ) epochs.


One can show that, for x ∈ R D at which p is continuous, if k → ∞ and k/n → 0 as n → ∞, then pk (x) → p(x) in probability ( [28], Theorem 3.1).

a natural approach for estimating F (P ) is the plug-in estimator Since pk → p in probability pointwise as k, n → ∞ and f is smooth, one can show FP I is consistent, and in fact derive finite sample convergence rates (depending on how k → ∞).


The Hessian regularizer, the squared norm of the second covariant derivative, ∇ a ∇ b f 2 , corresponds to the Frobenius norm of the Hessian of f in normal coordinates, see Equation 1.

given normal coordinates x r at X i we would like to have an operator H which given the function values f (X j ) on N k (X i ) estimates the Hessian of f at X i , This can be done by fitting a second-order polynomial p(x) in normal coordinates to {f (X j )} k j=1 , where the zeroth-order term is fixed at f (X i ).


We observe only the realized graph A and not K, though it holds that EA = K/C(n).

we can only use singular vectors of C(n)A to approximate ŨK S1/2 K . We have: ), the problem is analyzed in [25].


The region of strong dependence shifts to a region with small mass.


The latent structure in [21] is learned by maximizing the lower bound of mutual information between the latent code and the corresponding demonstrations.

existing imitation learning methods which can handle the multi-modal behavior have common features in that they are developed based on the maximum entropy framework and capture the multi-modality of demonstrations by learning the mapping from demonstrations to the latent space.


However, this equality implies and so CUR could be regarded as implicitly performing sparse PCA in the sense that (a) V CUR is sparse; and (b) by Theorem 1 (with high probability), ERR(V CUR ) ≤ (1 + ǫ)ERR(V k ).

the existence of such a V CUR would cast CUR directly as a randomized approximation algorithm for SPCA.


Moreover, if b = 1, then Acc-Prox-SVRG has the same complexity as that of Prox-SVRG, while if b = n then the complexity of this method is equal to that of APG.

with an appropriate mini-batch size, Acc-Prox-SVRG may outperform both Prox-SVRG and APG; even if the mini-batch is not appropriate, then Acc-Prox-SVRG is still comparable to Prox-SVRG or APG.


20 repetitions of this experiment yielded after 42 minutes of training the following results: spike correlation 0.88 ± 0.036, angular error 27.9 ± 3.7 degrees, for U + = 0.0012, U − /U + = 1.055.

apparently the output spike train is less sensitive to changes in the values of U than to changes in w. since only the behavior of a neuron with vector U * but not the vector U * is made available to the neuron during training, the resulting correlation between target-and actual output spike trains is quite high, whereas angular error between U * and U(t), as well as the average deviation in U , remain rather large.


. . , (x m , x 0 m ) 2 (X ⇥X ) m , we will use instead an empirical estimate b q h of q h : b .

we define the empirical error of h for a labeled sample as follows: We first assume that X is a subset of R N for some N 1 and consider a hypothesis set H of linear functions h : x 7 !


When setting a j = α γ t j , we readily see the LOG and EXP penalties.

theorem 5 illustrates a very interesting connection between the LOG and EXP penalties.


The output of this layer, however, will all be positive, which makes discarding the signs no longer an effective way of exposing higher-order dependence.

to extend to another layer, new activation functions and new energy function must be derived.


A more realistic model might incorporate selecting various items with different distributions: for example, the Netflix dataset demonstrates skews in the sampling distribution for different films [16].

given this model our goal is to recover the weight vector w given such pairwise comparisons.


Based on the Koopman operator, the analysis of nonlinear dynamical systems can be lifted to a linear (but infinite-dimensional) regime.

we can consider modal decomposition, with which the global characteristics of nonlinear dynamics can be inspected [4], [5].


We prove the lower bound of single-vertex problem and reduce it to k-MCV.


The major contribution of the present work is to show that a very natural and computationally easy-to-obtain concave extension of DSFs is a nearly tight relaxation of the DSF's multilinear extension.

given this extension, we can use projected gradient ascent (Algorithm.


As opposed to this, RankCentrality is an offline algorithm without any performance guarantee if the sample size in not sufficiently large (see Remark 5).

it is not surprising that, asymptotically, its empirical sample complexity shows a better behavior than the complexity of our online learner.


We suspect that the class studied by Tsybakov [7], used in our minimax proof, is more restrictive than our class.


In fact this is quite common, and identifiability provides geometric structure that we make use of in our lower bounds.

most of our results use this refined notion of elicitation complexity.


If no such α exists, then we choose α(M T ) uniformly at random from V. From Lemma 1, there can exist only one such α ∈ V that satisfies this inequality.

using Markov's inequality, we have P φ [ α(M T ) = α * ] ≤ P φ * (M T , g α * , S, φ) ≥ ψ(δ)/3 ≤ 1 3 .


A source prior is especially important in the single microphone case, which estimates N clean samples plus model coefficients from N noisy samples.

the absence of a strong speech model degrades reconstruction quality.


Under sparse coding schemes, unless the covariance learning is employed, the cross-talk noise does diverge in the large N limit.


. . , x m ), the following upper bounds hold for the empirical factor graph complexity of H 1 and H where s is a sparsity factor defined by s = max Plugging in these factor graph complexity upper bounds into Theorem 1 immediately yields explicit data-dependent structured prediction learning guarantees for linear hypotheses with general loss functions and arbitrary factor graphs (see Corollary 10).

observe that, in the worst case, the sparsity factor can be bounded as follows: where the factor graph Rademacher complexities of linear hypotheses in ).


We usually do not know task and feature structures in advance.


More importantly, the generative models considered in these studies were tailor-made to the specific pscychophysical task presented in the experiment.

it remains to be shown whether more flexible, 'open-ended' generative models are used and learned by humans during perception.


Theorem: Out of all k-dimensional approximations of the form for the probability distribution at time t, the one that minimizes the mean squared error E x { p(t, y|x) − p(t, y|x) 2 w } where averaging over initial points x is with respect to the stationary density φ 0 (x), is given by w j (y) = φ j (y) and a j (t, x) = λ t j ψ j (x).

the optimal k-dimensional approximation is given by the truncated sum Proof: The proof is a consequence of a weighted principal component analysis applied to the matrix M , taking into account the biorthogonality of the left and right eigenvectors.


By the Chebychev's inequality, this latter probability is at most δ.

algorithm 1 is an ( I max , δ)−additive approximation algorithm.


If |R| = 1, then R \ C = ∅ and the product equals 1 by convention.

assume now that f (S, R) is defined when |R| < m and consider the case by the induction assumption each f (S i , R i ) is defined and therefore the product is defined.


Evidence also suggests that feedback signals may operate between hierarchically adjacent areas along the ventral stream [30,5,43] to enable local recurrent processing for object recognition [65,2], especially given ambiguous or degraded visual input [64,51].

feedback processes may be an integral part of both global and local recurrent processing underlying top-down attention in a slower time scale and visual recognition in a faster time scale.


Proof Assume that θ * is a fixed point of the above algorithm.

then the first order optimality conditions require ∂ θ log pi (θ * ) = 0 for all i and the Laplace approximation yields up to second order, the derivatives of p, pi , and p agree at θ * , which implies that θ * is a local optimum.


In contrast, although Double Q-learning can underestimate the values of some actions, it is easy to set the initial action values high enough to ensure optimism for actions that have experienced limited updates.

the use of the technique of optimism in the face of uncertainty can be thought of as an orthogonal concept to the over-and underestimation that is the topic of this paper.


However, little has been known for the optimization landscape of general nonconvex problems.

as suggested by many theoreticians, a strict saddle optimization problem such as streaming PCA could be a first and yet significant step towards understanding the algorithms.


While we focus on feature-rich graphs (e.g., citation data with text attributes, biological data with functional/molecular markers), our approach can also make use of structural features that are present in all graphs (e.g., node degrees).

our algorithm can also be applied to graphs without node features.


A natural question to ask is: what happens when using a matrix P other than the identity or the line vector p −1/2 1 , and what are good choices of such matrices?

we introduce the following family of penalty functions: Definition 1.


In a learning context, where each function f i is the loss associated to a single data point, L is equal to the largest value of the loss second derivative ξ (1 for the square loss, 1/4 for the logistic loss) times R 2 , where R is a the uniform bound on the norm of each data point.

the constraint µ L 8 n is satisfied when λ 8ξR n . In low-dimensional settings, the optimal regularization parameter is of the form C/n [29] where C is a scalar constant, and may thus violate the constraint.


For an m×n matrix, our method has a running time of O(r 2 mn log(1/ )), where r is the rank of the low rank component.

our method has a linear convergence rate, i.e.


We assume that the number of latent variables is smaller than that of the observed.


Note that ( 14) is almost the same as (7) except that it has a proximal-point augmented term.

one can follow the same procedure as in Algorithm 1 to maintain the vector µ and computes µ (t,s) before solving each block subproblem (14).


However, these filler objects are obviously irrelevant (if they affected the attributes of some named term, they would have been named as functions of that term.)

we never have to create them, or worry about their attribute values.


With a spatial algorithm data points corresponding to different time slices can be shuffled without any change in the results.


This dataset is twice as large as the PTB dataset (2000k training, 217k validation and 245k test words) and also features a larger vocabulary (33k words).

it is well suited to evaluate model performance on longer term dependencies and reduces the likelihood of overfitting.


If we fix a realization h, this utility is monotone submodular [11].

f (S, h) is pointwise monotone submodular.


. . , f n ) represent the prediction for the labeled and the unlabeled examples, respectively.

according to the inverse lemma of the block matrix, we can write K −1 as follows: where the term f K −1 f is computed as When the unlabeled data are loosely correlated to the labeled data, namely when most of the elements within K u,l are small, this leads to M u ≈ K u .


This leads to the sensitivity to the class noise as the example lasso chooses may be mislabeled.

the sparse representation classification (SRC), a lasso based classification schema [21], is not suitable for applications in the presence of class noise.


For the relevant range of b (.2 to .8), the shift shift in starting point is approximately linear in b (Figure 3e inset), so that the new distance to the boundary is approximately z + kb.

the new mean decision time is z+kb A tanh Az+Akb c 2 .


words) where the transition probability indicates the strength of association in memory, and with the further constraint that node labels are only generated when the node is first visited.

repeated visits are censored in the output.


In fact, for Lpwq " 1 m ∞ m k"1 L k pwq where L k pwq " `pf w px k q, y k q, for any single weight W phq ri, js in the network we can compute d dt pW phq ri, jsq 2 " 2W phq ri, js¨d W phq ri,js dt BW phq ri,js , using the sharp chain rule of differential inclusions for tame functions [Drusvyatskiy et al., 2015, Davis et al., 2018.

if we can prove the theorem for every individual loss L k , we can prove the theorem for L by taking average over k P rms.


But, it is known that f is strongly convex since A is positive definite [27].


To obtain non-trivial and feasible guarantees, we need to relax the coverage constraint in SC.

we instead solve the Streaming Bicriteria Submodular Cover (SBSC) defined as follows: Definition 1.


In the result of the leA, the coefficients have normalized variance.

those components, that have large Euklidian norm, occur as features with high entropy in the source patches.


Section 3 investigates the close relationship with Rademacher complexities, Section 4 develops the maximal average margin principle for ordinal regression, and Section 5 reports experimental results of application of the MAM to classification and ordinal regression tasks.

let the class of hypotheses be defined as the signed distance of a sample (X, Y ) to the hyper-plane w T x = 0, or the margin M (w) ∈ R, can be defined as SVMs maximize the worst-case margin.


Note also that Equation 4provides necessary conditions for an additive bias term to achieve perfect correction, since any such correction must achieve parity in the sign.

it might make it simpler to find a better bias term since Equation 4 (and therefore, set of possible α values) is less constrained.


Under our formulation, each segment (node) in the sketch needs to be assigned to one of the candidate groups (labels).

our candidate selection problem becomes a segment labeling problem, where the set of possible labels for a given segment is the set of candidate groups that contain that segment.


The forms of E and E s are exactly the same for posterior kernel embeddings and conditional kernel embeddings.

the following theorem in [13] still hold.


Thus, we can show that a previously proposed parameterization [10] cannot represent all unitary matrices larger than 7 × 7.

such a parameterization results in what we refer to as a restricted-capacity unitary recurrence matrix.


However, in this learning problem, what matters is not f but the way f behaves with respect to ν.

trying to capture the properties of f via the distribution-free ||f || ∞ bound is not satisfactory.


For simplicity, we focus on linear features, that is, h i (x) = u i , x , where u i ∈ IR d . In addition, we assume that the vectors u i are orthonormal.

if U denotes the d × d matrix with columns the vectors u i , then U ∈ O d . The functions f t are linear as well, that is f t (x) = w t , x , where w t = i a it u i . Extensions to nonlinear functions may be done, for example, by using kernels along the lines in [8,15].


They rely on upper bounds of a MAP solution, under partial assignments, in order to prune the search space.

any method capable of providing upper bounds tighter than those of a relaxation, can potentially have an impact in the performance of a branch-and-bound MAP solver.


Let θ i be the P × |o t,i | matrix of parameters for intersection i.

we additionally define U(p) as the unit vector with a 1 in row p. assuming exp(x t,i ) is element-wise exponentiation , Pr(a t,i = p|o t,i , θ i ) = ãt,i (p); We implemented two baseline controllers for comparison: (1) a uniform controller giving equal length to all phases; (2) A SCATS inspired adaptive controller called SAT that tries to achieve a saturation of 90% (thought to be used by SCATS) for all phases.


That is, given the matrices A and C, there are two distinct vectors θ 1 and θ 2 in ∂P such that A T θ 1 = A T θ 2 .

if eventually θ 1 is the true parameter vector satisfying C 1 and C 3 , then θ 2 satisfies these conditions as well.


It is worth noting that this dataset has a fixed number of discrete positions.

3(b) shows a reasonable coverage of the manifold with some abrupt changes.


iWVI can be seen as an instance of auxiliary variational inference [1] where a joint divergence upper-bounds the divergence of interest.


How frequently this walk visits a particular node in the long run, or equivalently the stationary distribution, is the score of the corresponding item.

effectively this algorithm captures preference of the given item versus all of the others, not just immediate neighbors: the global effect induced by transitivity of comparisons is captured through the stationary distribution.


For kernel methods, our bounds hold for algorithms approximating the empirical risk up to a factor of 1+ε, for log(1/ε) = ω(log 2 n)).

they provide conditional quadratic lower bounds for algorithms with, say, a log 1/ε runtime dependence on the approximation error ε.


In the case of PubMed, our algorithm only needs to work on covariance matrices of order at most n = 1000, instead of the full order (n = 141, 043) covariance matrix.

at values of the penalty parameter λ that target cardinality of 5 commands, we observe a dramatic reduction in problem sizes, about 150 ∼ 200 times smaller than the original sizes respectively.


Then the weighted prediction will be almost equal to the prediction of that hypothesis (i.e., close to +1 or −1 for all queries), and therefore the responses to all queries are relatively certain and non-informative.

the convergence of the algorithm could become quite slow in such conditions.


We must design the system so that we can include the latest research on inference while reusing pre-existing analyses and optimizations.

we use an intermediate representation (IR) for probability distributions that serves as a target for modeling languages and as a basis for inference algorithms, allowing us to easily extend the system.


By Lemma 2 the probability of obtaining at most R repetitions is upper-bounded by δ.

it is sufficient that the bound holds whenever the number of repetitions is greater than R. For any such r > R, we have ∆(n, r, pn ) > δ.


For this approach, I i , i = 1, . . . , m are the total number of training data, so the solution to (4) is not affected by n i .

we remove it from (8), so r i + r ′ i = 1.


Our empirical results on data-sets show that the resulting bound is often not essentially weaker than the original one.

the conventional tools turn out not to be so 'ill-suited' after all.


Note that every conditional model for R i is fit only using rows where L \ {L i } are observed.

the fitting procedure fails in datasets with few enough samples that for some R i , no such rows exist.


Last, the induced binary problems are treated as separate problems and are learned independently.

there might be strong statistical correlations between the resulting classifiers, especially when the induced binary problems are similar.


In addition, the factor functions and the technology for exactly inferring the states described in this paper can be extended to other applications of factor graphs in fields such as medical diagnoses, social network analysis, and preemptive attack detection.

we assert that further investigation is necessary to understand the different usecases of this model.


Define the kernel power K with k i,j = g p i,j .

the off-diagonal kernel entries k i,j i 6 = j monotonically decrease with respect to increasing p. 9 p ⇤ such that Corollary 3 is satisfied for Beyond the examples outlined here, similar conditions can be enumerated for a wide range of parametrized kernel functions, and are easily checked for model-based kernels e.g.


The high dimension also prohibits MH AuxVar.

we only run the Gibbs SBA algorithm on this large grid-structure MRF.


Then where c 0 is an absolute constant and Remark 2 If ↵, ✓ 0 , ✓ 1 are unknown, any estimate-then-explore strategy (or the strategy described in Corollary 1) would be unable to choose an m that depended on these parameters, so we can treat it as a constant.

for the case when ✓ 0 and ✓ 1 are bounded away from {0, 1} (e.g.


However, we cannot know whether those who are buying the product have been exposed to the advertisement or not.

every individual always has one missing label.


To see this notice that Y = Xβ * + W and Y N , X N being "close" to Y, X imply 2 N QY N = 2 N X N ( Qβ * ) + "extra noise terms" + 2 N QW.

assuming the control of the magnitude of the extra noise terms, by using the Q-rationality assumption and that Q is estimated to be a multiple of Q, the new ground truth vector becomes Qβ * which is integer-valued.


Unfortunately, inference in large, complex graphical models is computationally intractable.

approximate inference methods are needed.


In contrast, DFA appeared to be quite sensitive to the choice of architecture and our architecture search was guided by the performance of TP methods.

the numbers achieved by DFA in our experiments should be regarded only as a rough approximation of the attainable performance for the algorithm.


( 9) is fully specified by the player's choices on rounds t, t − 1, t − 2.

the Ω(T 2/3 ) lower bound for switching costs and bandit feedback extends to the full-information setting with a memory of size at least 2.


The set of latent variables is Y = {Z 1:D 1:T , F 1:D 1:T −1 }.

the function Q can be represented as shown below.


The smoothing algorithm described in the previous section computes the optimal posterior process by providing us with the stationary solution functions A(t) and b(t).

when subsequently optimising the parameters we only need to compute their explicit derivatives; the implicit ones vanish since ∇ A L θ,Σ = 0 and ∇ b L θ,Σ = 0.


Let g : [a, b] → R be the following linear function, Then, for all t ∈ [t, b], where the first inequality follows from the concavity of f (•, x), and the second inequality follows from f (t, x) − p(t) ≤ ε/γ and from p(a) ≤ h(a) ≤ f (a, x).

x is an ε-approximation for all t ∈ [t, b] that satisfy g(t We finish the proof by considering three cases.


Evaluation of this function and its gradient is in general intractable (#P-complete).

it is common to optimize an approximate objective, such as the pseudo-log-likelihood.


Therefore, to minimise squared error, the postsynaptic membrane potential should represent the mean of the posterior over possible output function values it should be computing based on the input spike trains: Biophysically, to a first approximation, the somatic membrane potential of the postsynaptic neuron can be described as some function(al), f , of the local dendritic membrane potentials, This is interesting because Pfister et al.

[11,12] have recently suggested that short-term synaptic plasticity arranges for each local dendritic postsynaptic potential, v d i , to (approximately) represent the posterior mean of the corresponding presynaptic membrane potential: it would be tempting to say that in order to achieve the computational goal of Eq.


First, mutation must be large enough to counteract the effects of drift towards random corners of the hypercube.

the fixed point of the average distance to 1/2, (D(t + 1)) defined in equation ( 2) , must be sufficiently close to zero.


The standard algorithm [3] which is exponential in the treewidth will make O(2 k ) calls to this oracle.

when the cost of computing the KL-divergence is large, this algorithm becomes even more attractive as it results in exponential speedup over the standard algorithm.


We can now describe the expected state evolution.

at iteration t = 0, one has: Using recursion, we can write: in order to prove the convergence of Algorithm 1, one needs to show that lim t→+∞ We state this precisely in the next theorem.


Calculating the gradient with respect to each d r and equating it to zero, we obtain Swapping the sums over m and i with the sum over j, using the auxiliary variables, and noting that d j does not depend neither on m nor on i, we obtain Similarly to the way we solved for α r , we now define the vector u r = v r − j =r ν j,r d j to get the following iterate for d r : where, as above, we incorporated the case d r = 0, by applying the operator [•] + to the term 1 − γ/ u r . The form of the solution implies that we can eliminate d r , as it becomes 0, whenever the norm of the residual vector u r is smaller than γ.

the dictionary learning procedure naturally facilitates the ability to remove dictionary words whose predictive power falls below the regularization parameter.


However, a function with true error e 3 could produce a lower observed error than one with true error e 1 or e 2 (since e 3 −B(3, k) < e 2 + B(2, k) and e 3 − B(3, k) < e 1 + B(1, k)), and thus could be chosen as ĥk .

the smallest bound we can place on the true error of ĥk in this example is e 3 = 0.2.


⇠ i = 1) or determining that ↵ is strictly greater than 0, is at least as hard as detecting that any of the distributions up to distribution N is heavy.

a lower bound on the total expected number of samples of all considered distributions for this strictly easier detection problem is also a lower bound for the estimate-then-explore strategy for the most biased coin identification problem.


In contrast, if π d,1 is close to 0 and therefore π d,2 = 1 − π d,1 is close to 1, the article can be considered to be more related to a subject on English or English-speaking countries.

a pivot is assigned considering the language biases of the articles.


Remark 2.5 For brevity's sake our major aim was to show the best possible rates using our techniques.

theorem 2.3 states rates for the SVM under the assumption that (λ n ) is optimally chosen.


Further, weak Lipschitzness, unlike Lipschitzness, does not constraint the local decrease of the loss at any point.

weak-Lipschitzness is a property that lies somewhere between a growth condition on the loss around optimal arms and (one-sided) Lipschitzness.


Our result extends significantly those in [7; 26; 10; 18; 8] in various aspects, but most saliently from the restrictive full binary tree [7; 26] to any rooted tree.

our results imply that a generic HMM (whose underlying tree is "completely" unbalanced) cannot be exactly represented by any polynomially-sized shallow SPN, which, to our best knowledge, has not been shown before.


This imposes limitations, including how, because the tensor arose based on some underlying graph that the partitioning metric was designed explicitly for a graph.

the applications are limited in scope and cannot model, for example, the airplane-airplane-airport multiplex network we analyze in Section 3.2.


( 8) take the form of softmax functions.

they can be used as soft approximations of the hard max operation in Eq.


Our main theoretical contribution is the following theorem, which establishes that under TTEI sampling proportions converge to the proportions w β derived above.

while the sampling proportion of the optimal arm is controlled by the tuning parameter β, the remaining 1 − β fraction of measurement is optimally distributed among the remaining k − 1 arms.


Conditioned on the Bernoulli parameter π k , the expectation of the sparsity is The conditional distribution of the topic β k given the vocabulary subset b k is Dirichlet(γb k ).

topic k is represented by those terms with non-zero b kv s, and the smoothing is only enforced over these terms through hyperparameter γ. Sparsity, which is determined by the pattern of ones in b k , is controlled by the Bernoulli parameter.


Efficient computability is a major concern in our paper, and recursively defined kernels on combinatorial data structures, can be very expensive.

in Section 4 we describe a strategy based on a combination of linearizing each level of the kernel (relative to a given dataset) and a randomized low rank projection step, which reduces every stage of the kernel computation to simple operations involving small matrices, leading to a very fast algorithm.


The details of the proof are in Appendix A of the supplementary material.

perturbations smaller than M F,X / √ 2L F cannot deceive the network F for a data point X.


The f i marginals of the variational posterior (6) depend only on the corresponding inputs x i .

we can write the ith marginal of q(f |m, S; X, Z) as Using our variational posterior (5) the lower bound (4) simplifies considerably since (a) the conditionals p(f |u; X, Z) inside the logarithm cancel and (b) the likelihood expectation requires only the variational marginals.


. . , i N ), the cross entropy in (3) approaches zero and the cross entropy in (4) goes to +1.

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


Then, with probability 1 − 4δ ln(T ), we have for any T ≥ 3.

assuming R = 1, we have, for λ small enough, with probability at least 1 − δ, Proof.


In the kth iteration, let γ (k) i denote the optimal domain warping from q i to µ (k) and let q(k) i = (q i , γ ) ], [q i ]) 2 .

the cost function decreases iteratively and as zero is a lower bound, n i=1 d([µ (k) ], [q i ]) 2 will always converge.


These equations have 2n + 1 roots, so the total number of intervals in one round is O(n 3 ).

over T rounds, the number of intervals is O(n 3T ).


expert 2 was the better one on block i), in which case ) + , where (x) + = x if x ≥ 0 and 0 otherwise.


In addition, the smallest non-zero coefficient has to be lower bounded to ensure a certain level of signal-to-noise ratio.

we augment our previous definition of B(s, ρ) to have a signal strength control Then we can obtain the following modified Theorem.


The smallest bounding box of X is defined as B (X) := B (a, b) where the lower bound a is given by the pointwise infimum of X and the upper bound b is given by the pointwise supremum of X, that is a α := inf{x α : x ∈ X} and b α := sup{x α : x ∈ X} for all α = 1, . . . , d. Note that B (X) = B (conv (X)).

if X is convex, the smallest bounding box for X depends only on the extreme points ext (X), i.e., B (X) = B (ext (X)); this bounding box can be easily calculated if the number of extreme points is not too large.


This suggests that R V (ŷ bc ) = ∞, which in fact may be confirmed by direct calculation.

when n = 2, ŷbc is weakly consistent, but not consistent.


Mounting evidence suggests that CVD and cancer share a number of risk factors, and possess various biological similarities and (possible) interactions; in addition, many of the existing cancer therapies increase a patient's risk for CVD [2,29].

it is important that patients who are at risk of both cancer and CVD be provided with a joint prognosis of mortality due to the two competing diseases in order to properly manage therapeutic interventions.


Maximum A-Posteriori (MAP) Models: a variation on Likelihood models involves Maximum A Posteriori (MAP) estimates.

if there exist a prior information regrading the uncertainty on the i-th row of P a , which can be described via a Dirichlet distribution [4] with parameter α a i , the resulting MAP estimation problem takes the form max the MAP uncertainty model is equivalent to a Likelihood model, with the sample distribution f a i replaced by f a i + α a i − 1, where α a i is the prior corresponding to state i and action a.


num: 512 sh:1 sw:1 sh:2 sw:2 sh:1 sw:1 sh:2 sw:2 sh:1 sw:1 sh:2 sw:1 sh:1 sw:1 sh:2 sw:1 sh:1 sw:1 ph:1 pw:1 ph:0 pw:0 ph:1 pw:1 ph:0 pw:0 ph:1 pw:1 ph:0 pw:1 ph:1 pw:1 ph:0 pw:1 ph:0 pw:0 Specifically, the feature map in the last layer is sliced from left to right by column to form a feature sequence.

the i-th feature vector is formed by concatenating the i-th columns of all of the maps.


The main advantage of NCE is that it allows us to fit models that are not explicitly normalized making the training time effectively independent of the vocabulary size.

we will be able to drop the normalizing factor from Eq.


Hence ⇣ qr,⌘ (t) increases quickly away from zero.

2 q,⌘ (t) ⌘⇣ 2 qr,⌘ (t) also increases quickly, which drives the algorithm away from the saddle point.


Let X be a measurable space and H be an RKHS of functions on X with k : X × X → R as the bounded reproducing kernel.

by the reproducing property, it is easy to verify that P f = ⟨f, m P ⟩ H and P n f = ⟨f, m Pn ⟩ H for all f ∈ H, where ⟨•, •⟩ H denotes the inner-product of H, and the worst case error (2) can be written as the difference between m P and m Pn : where 3), it is easy to see that for every f ∈ H, the integration error |P n f − P f | is bounded by the worst case error: We refer the reader to [21,11] for details of these derivations.


The probability of a spike event at ¢ , a priori of knowing the stimulus history, is flat: ¢ ¥ spk @ ¢ © ¤ ¦¥ § £ !

r R . the mutual information between the stimulus and the single spike events is [20]:


In the simulations of the right column, param-  eters are such that part-whole relationships are not enforced.


On the other hand, in the presence of noisy labels, as the classifier improves the effect of the noise increases -the classifier may give correct predictions, but will still have to update due to wrong labeling.

in an advanced stage of the training process the majority of the updates may actually be due to wrongly labeled examples, and therefore will not allow the classifier to further improve.


Nonetheless, our approach greatly outperforms a version of their model that was not trained with these extra signals, and even a version of their model trained using 9K ground-truth programs.

rNs can achieve very competitive, and even super-human results under much weaker and more natural assumptions, and even in situations when functional programs are unavailable.


Since, for all combinations of a and 0, the columns of Taoa,ou are linearly dependent on the columns of U, we can write Taoa,ou == UW T for some q x q matrix of weights W. If t is a test that is linearly dependent on Q, then anyone-step extension of t, aot, is linearly dependent on Q.

this is because we can write the outcome vector for t as u(t) == (UwT)T for some (1 x q) weight vector w and the outcome vector for aot as u(aot) == (Taoa,ou(t)T)T == (Taoa,oUwT)T == (UWTwT)T. aot is linearly dependent on Q.


These assumptions lead to decoupling of the marginal and conditional distributions.

the distribution q λ k (η k ) can now be factorized into two GW distributions and a distribution of W. To avoid over-cluttering symbols, we omit the subscript k of the specific cluster k. Posterior distribution of µ Y , Λ Y : The update rules follow the standard posterior distribution of Gaussian-Wishart conjugate priors.


Moreover, much of the theoretical analysis developed in former work is based on normalized vectors [16], and it has been shown that utilizing norm information improves the estimates of random projections [17].

a natural question to ask is, if we know a priori that data is 2 -normalized, can we come up with a better nonlinear map?


Our second example (shown in Figure 2b) exhibits a setting where A and B suffer very different reconstruction error when projected onto one dimension: A has high reconstruction error for every projection while B has a perfect representation in the horizontal direction.

asking for a projection which minimizes the maximum reconstruction error for groups A and B might require incurring additional error for B while not improving the error for A.


1a), in which light intensity values were randomly selected every 30 ms from a Gaussian distribution having a mean of 4 mW/mm 2 and an RMS contrast of 18%.

the photoreceptors were presented with exactly the same visual stimulus, and the movie is many correlation times in duration, so we can replace averages over stimuli by averages over time (ergodicity).


When considering the remaining budget, the ALP algorithm can be viewed as "sampling without replacement".

we can show that b τ follows the hypergeometric distribution [23] and has the following properties:  Then, we prove Theorem 1 based on Lemma 2.


For sending small messages, e.g., the d-dimensional model θ or aggregated gradient, this latency dominates the message size-dependent transmission latency.

it is important to reduce the number of communication rounds, even more so than the bits per round.


Since the state of the system at time 0 is perfectly observed to be 10, given any time-discretization, the maximum value of s i at step i of the Markov chain is (10 + i).

message dimensions are always finite, and we can directly apply the forwardbackward algorithm.


It follows from the definitions of generalized false-positive/false-negative rates and calibration that all trivial classifiers h lie on the diagonal defined by c f p (h) + c f n (h) = 1 (Figure 1a).

all classifiers that are "better than random" must lie below this diagonal in false-positive/false-negative space (the gray triangle in the figure).


Those 3 instructions (accumulation, popcount, xnor) take 1 + 4 + 1 = 6 clock cycles on recent Nvidia GPUs (and if they were to become a fused instruction, it would only take a single clock cycle).

we obtain a theoretical Nvidia GPU speed-up of factor of 32/6 ≈ 5.3.


The second measure relates behavioral and fMRI data by exploiting the informational association between the predicted neural activity u t and the state of RNN, h t . First, note that, at each time t, h t is a Markov state for the RNN, in that given h t , RNN outputs after time t are independent of their past.

we can decompose: as the effect changing r t1 has on the predicted RNN state h k t at time t, times the effect that a change in h k t has on the action probability ⇡ t2 at t 2 .


Observers update the probability of co-occurrence, defined in our case rigidly as 1-back occurrence, inductively as the items on the list are presented.

we model this as the observer's sequential Bayesian updates of the probability p(x), stored at every time step as a discrete memory engram m. in this encoding model, information about the displayed list of items is available in distributed form in memory as p(x i , x l |m), with each engram m storing one instance of co-occurrence.


From this reasoning, the distribution of all possible path lengths through a residual network follows a Binomial distribution.

we know that the path lengths are closely centered around the mean of n/2.


Performing the multiplications across all time steps results in a total complexity of O(N T d).

the total complexity of the proposed E-step estimation method is O(N T L 2 + N T d) per loopy BP iteration.


For the image denoising and inpainting tasks, however, the choices of clean and noisy input are natural: they are set to be the desired image after denoising or inpainting and the observed noisy image respectively.

we propose a new training scheme that trains the DA to reconstruct the clean image from the corresponding noisy observation.


However, this gradient step cannot be followed directly because we need to maintain the constraints.

the constraint w 1 = 1 can be maintained by first projecting the gradient onto it, obtaining δw = (I − 1 k 11 )∇ w J H . the weight vector can be updated by where α is a step-size parameter.


We stop iterating when the function value at the new point is less than ε away from the function value at θ * , and return the corresponding optimal exponential utility policy.

our algorithm evaluates f at a subset of the points {θ 1 (θ 2 /θ 1 ) i/n : i = 0, • • • , n} where n is a power of 2.


But if we apply Binomial rule [9] on S, we get a new theory where {X, Z} becomes a single occurrence equivalence class and we can resort to domain independent inference.

7 application of Binomial rule before single occurrence would lead to larger savings.


In fact, the quantity 1/2 is nothing special here and we can set any value close to one with a compensation that the number of samples n will increase.

we put 1/2 for the simplicity of the proof.


The contrapositive implies that if two quantities are dependent, there must exist a causal path between them.

when the joint distribution of a causally sufficient system changes, for any effect variable, the distribution of its direct causes, P (causes), must change independently from P (effect | causes); otherwise, the parameter sets (or variables, from a Bayesian perspective) involved in the two modules change dependently, and there must exist a causal path connecting the parameter sets, which implies the existence of confounders and contradicts the causal sufficiency assumption.


In the real world, there will always be perturbations -stochasticity in the environment, modeling errors, or wear and tear.

the specific task design and notion of success used for the simulated characters are not adequate.


It is worth pointing out to a familiar reader that what we call ρ 1 is more appropriately denoted by ρ (n−1,1) in the literature; but we will stick to ρ 1 for brevity.

q is related to the Fourier Transforms of the irreducible representations ρ 0 and ρ 1 .


In terms of texts, it is not reasonable to augment the data using signal transformations as done in image or speech recognition, because the exact order of characters may form rigorous syntactic and semantic meaning.

the best way to do data augmentation would have been using human rephrases of sentences, but this is unrealistic and expensive due the large volume of samples in our datasets.


Similarly, for left turns, $ is biased to the left.

the activity offset between the two rings leads to an anticipation time % for ADN neurons, see Figure 2.


However, it can be shown that if two SRVFs belong to different orbits, then the distance between them is non-zero.

this distance d is a proper distance (i.e.


Some simple algebra (omitted for the sake of brevity) shows that the weight matrix of the product graph can be written as Performing a random walk on the product graph G × is equivalent to performing a simultaneous random walk on the graphs G and G [4].

the (in + j, i n + j )-th entry of A k × represents the probability of simultaneous k length random walks on G (starting from vertex v i and ending in vertex v j ) and G (starting from vertex v i and ending in vertex v j ).


However, log-likelihood is non-convex in the entries of the kernel matrix, and this learning problem is conjectured to be NP-hard [1].

previous work has instead focused on more restricted convex learning settings: learning only a single weight for each row of the kernel matrix [2], or learning weights for a linear combination of DPPs with fixed kernel matrices [3].


But none of these approaches model the interactions between output variables.

they can be considered a special case of our model, where only unary functions f i are present.


For this posterior family, the non-Gaussian terms p(y n |x T n z) are difficult to handle, while the Gaussian term N (z|µ, Σ) is easy because it is conjugate to q.

we set pe (z|λ) ≡ N (z|µ, Σ)/N (z|m, V) and let the rest of the terms go in pd .


the type of SGD, regularization, and learning rate schedule, the same as those used in individual learning.

collaborative learning can be simply put on top of individual learning.


In the Bayesian point of view, the constraint terms could be viewed as the logarithm of the prior distributions over the states S. Therefore, the objective can be viewed as a log posterior distribution over S. Now the Viterbi algorithm is not applicable directly since at any time t, the state S it depends on all the states at all time steps, because of the regularization terms which are non-Markovian inherently.

in the following section we transform the optimization problem (5) into a convex quadratic program which can be efficiently solved.


With an abuse of notation, we denote the expected reward-to-go under a history as: For π ∈ Π HR and µ ∈ C S (λ), define w(π, µ, h t ) E (p,r)∼µ u s (π, p, r, h(t)) = u(π, p, r, h(t))dµ(p, r).

w(π, µ, (s ini )) = u(π, p, r) dµ(p, r) is the minimax objective.


The numerator samples are generated in the same way as above, but a half of denominator samples are replaced with negative training samples.

while the numerator sample set contains only positive training samples, the denominator sample set includes both positive and negative training samples.


Thus, a product could obtain a higher (lower) rating depending on if it is assessed together with worse (better) products.

information about batches tested by consumers in each rating session is a very important issue.


We now consider Kalman Filter(KF) [14] as the optimal centralized counterpart of (5).

it is wellknown that the steady state KF satisfies a Riccati equation, and when the parameter of interest is scalar, the Riccati equation simplifies to a quadratic with the positive root comparing with the complete graph ( 16), we have and the upper bound can be made tight by choosing α = 1 for |a| < we should choose an α < 1 to preserve unbiasedness as well.


Such noise injection is known to compete with the explanatory power of autoregressive feedback mechanisms and may result in degenerate, near deterministic models [BVV + 15].

there have been a variety of papers that propose deep stochastic state sequence models, which combine stochastic and deterministic dependencies, e.g.


If the posterior distribution focuses mass on the expert policy parameters the expected value of the MAP parameters will converge to the expected value of the expert's policy.

to examine the speed of convergence to the desired expert policy we report the performance of the MAP policy in the MDP task.


Nevertheless, the reward signal, which comes at the moment of the final action of classification, is rather weak to the agent.

the concept of not wasting queries can be slow to learn.


When solving a finite sum problem (defined in 1) with a stochastic oracle, one needs at least n 2 /2 oracle calls in order to obtain an accuracy level of:  where A is a d × d diagonal matrix whose diagonal entries are κ, 1 . . . , 1, and q = (1, 1, 0, . . . , 0) is a d-dimensional vector.

one can easily verify that f ± are smooth and strongly convex functions with condition number κ, and that the minimizer of F σ is (σ/n)q, and using Equation ( 7), we see that δ(n) = κ+1 n 2 .


Although non-linear embeddings can mitigate this problem [8], complex graph patterns can still require a computationally infeasible embedding dimension.

no method yet exists that is able to compute embeddings of large graph-structured data -such as social networks, knowledge graphs or taxonomies -without loss of information.


In order to derive the matrix-valued kernel we note that (4.19) can be written as (u, L, u) where L is the n × n block matrix whose , q block is the d × d matrix I d L q .

we define w = L 1 2 u so that we have u = P L− 1 2 w (here L −1 is the pseudoinverse), where P is a projection matrix from IR dn to IR d . Consequently, the feature map in equation (2.7) is given by Φ = B = L− 1 2 P * and we conclude that K q (x, t) = (x, P L−1 P * q t).


Then the probability of an assignment in the HDP can be written as (after integrating out mixture weights [1,11], and if we only consider terms that would not be constants after we do the asymptotic analysis [5]): For the likelihood, we follow the same assumption as in the parametric case: the observation densities are Gaussians with a shared covariance matrix σ 2 I d . Further, the means are drawn independently from the prior N (0, ρ 2 I d ), where ρ 2 > 0 (this is needed, as the model is fully Bayesian now).

p(µ Now, we can perform the small-variance analysis on the generative model.


Indeed, take d = K, A = {1, . . . , K}, define the vectors {m a } a∈A as the canonical basis {e a } a∈A of R d , and take θ ∈ R d the vector whose component θ a is the expected reward for arm a.

then, M t is a diagonal matrix whose a-th diagonal element is the number N t (a) of times the a-th arm has been played up to time t. the exploration bonus in GLM-UCB is given by β a t = ρ(t)/ N t (a).


Voxels supporting similar representation of sensory input or tasks tend to spatially cluster together.

we used a Gaussian Process to impose a smooth prior on log(s) [17].


This was implemented by optimizing an input image using gradient ascent to maximize the simplified variant of the normality score described in section 5.5 (see, e.g., [39]).

we trained a classifier on the digit '3' from the MNIST dataset, with a few geometric transformations.


Furthermore, for satisfiable sentences the length of the shortest proof is bounded by the number of variables.

in this case, all nodes visited on a branch of the search tree that resolved to unsatisfiable have y ij = −1 and the nodes on the branch leading to satisfiable have y ij = +1.


Recall that, if Z (S) differs from Z (T ) , there are three possibilities: D (S) differs from D (T ) , or g (S) * differs from g (T ) * , or both of them occur.

it is necessary to consider two kinds of distances: the distance between D (S) and D (T ) and the distance between g (S) * and g (T ) * .


Sorting the sets of motifs from all runs at the same time is an NP hard multidimensional assignment problem [45].


In particular, hippocampal recordings in rats indicate that brief sequences of prior experience are replayed both in the awake resting state and during sleep, both of which conditions are linked to memory consolidation and learning (Foster & Wilson, 2006;Davidson et al., 2009;Gupta et al., 2010;Ambrose et al., 2016).

the mental look back into the past seems to occur exactly when credit assignment is to be performed.


First, we show that on the gradient flow (gradient descent with infinitesimal step size) trajectory induced by any differentiable loss function, for a large class of homogeneous models, including fully connected and convolutional neural networks with linear, ReLU and Leaky ReLU activations, the differences between squared norms across layers remain invariant.

as long as at the beginning the differences are small, they remain small at all time.


While d does not satisfy the triangle inequality, it provides a useful measure of dissimilarity normalised by the amount of computation.

our experience suggests that d puts more emphasis on the amount of computation at the layers over structure and vice versa for d. it is prudent to combine both quantities in any downstream application.


Thirdly, D MESA is sensitive to the overall spatial layout of the denisities.

if the difference between F 1 and F 2 is a low-frequency signal, e.g.


Using our method, the state of the GRU tends to remain in the basin of the attraction of h * = 0.

our method can improve robustness against noise, which is an advantage of the GRU [33].


LASH does not impose any such constraint of reducibility on the similarity function.

lASH must truly compute new similarities after each agglomerative step.


We designed the classifier for use in a clinical setting, where physicians have little time to label beats, let alone tune classifier parameters.

it was important that the method should require few cardiologist-labeled heartbeats, and have no user-defined parameters.


In our experience setting the number of projections equal to the depth of the tree is sufficient.

for a tree of depth k, we use only k projection vectors v, as opposed to 2 k with a PCA tree.


However, in sparse KPCA, u = X e i because the projection directions are simply the examples in X.

for sparse KPCA we have: given a kernel matrix K the deflated kernel matrix K can be computed as follows: where τ = K[:, i k ] and i k denotes the latest element in the vector i.


ĉω2 = 0 and ĉω1 = 0, by the groups constructed through Figure 2, ĉTv = (ĉ ω ) v⊆ω = 0 for all v ⊆ ω 1 .

so the partial derivative of the objective (7) with respect to c ω1 at ĉω1 is ∂L ∂c ω1 c ω 1 =ĉ ω 1 the probability of {ĉ ω2 = 0} equals to the probability of { ∂L ∂c ω 1 c ω 1 =ĉ ω 1 = 0}, which is 0.


Denote by xl 1 the minimal nonzero element in x l , and l ) a tree constructed by removing the nodes corresponding to the indices in the set {i : l shares the same root as T l . It follows from Algorithm 1 that i:vi∈V 1 l v i ≤ 0.


The utterance "I ate the..." can be completed by any edible object, but "I ate the red..." only by those objects which are both edible and red.

adjectives which preceed unexpected nouns can be used to "smooth out" what might otherwise be sudden spikes in information density.


If inference problems can be solved in polynomial time, then so can learning (using, e.g., structured perceptron).

when learning is hard, inference must be hard as well.


Since g(z ) in Theorem 2 is assumed to follow a linear form, and L is a squared loss, Equation 2can still be solved using linear regression.

the Shapley values from game theory can be computed using weighted linear regression.


To this end, note that z = (x, y) and z = (x , y ) are samples independently drawn from ρ.

the objective function of (3) can be rewritten as , where the minimization is achieved by Likewise, 2 where the minimization is obtained by letting Moreover, observe that , where the maximization is achieved with 1 The work [6,21] studied the regularized ERM problem, i.e.


But such an approach will not be able to encode robot dynamics which might be hard to measure.

our second solution is to learn an embedding space for robot hardware itself.


Note that in the graph sketching problem, the weight of k is bounded by d due to the special structure of cut function.

in the noiseless setting, we can leverage the sparsity d and use much fewer offsets P n in the spirit of compressed sensing.


Clearly, it suffices to construct unbiased estimates of `t(w) for any w 2 W. We will use a discretization approach.

assume that we are given a finite subset W 0 of W that will be constructed later.


It is also necessary to show that the covariance of the numerator is O(1/M ).

to bound Cov(Y M,1 , Y M,2 ), we require bounds on the quantity Assuming g is sufficiently smooth, a Taylor series expansion of g Lk(l) (Z) around where ξ Z ∈ E Z Fk(l) (Z), Fk(l) (Z) .


( 1) with A ij = y i y j K(x i , x j ) and b i = −1.

the multiplicative updates for computing the maximal margin hyperplane in hard margin SVMs are given by: where A ± are defined as in eq.


The final weights are therefore given by This equation is also solved by LSTD, including the incremental updates of A and b as new samples are acquired [14].

lSTD can be seen as computing the transitions between the basis functions using a Monte Carlo approach.


A better choice is to run Dijkstra's algorithm [6], which finds the minimum distances from a single vertex to all other vertices in the graph.

dijkstra's algorithm must be run N times.


In all the problems considered here, the magnitude of the effect of any reference on a query is inversely proportional to the distance metric ( , ).

the net effect on the query is dominated by references that are "close by".


The problem of learning preferences is related to the problem of learning H n,2 : if we associate each pair (i, j) with the vector in C n,2 whose i'th coordinate is 1 and whose j'th coordinate is −1, it is not hard to see that P n ⊂ H n,2 : for every σ, h σ = h w,0 for the vector w ∈ R n , given by w i = σ(i).

every upper bound for H n,2 implies an upper bound for P n , while every lower bound for P n implies a lower bound for H n,2 .


It is sometimes difficult to assess the significance of an improvement in variation of information or rand score since changes can be dominated by modifications to a few large objects.

we decomposed the variation of information into a score for each object in the ground truth.


Recommendation games with the Shapley mediator are exact potential games.

due to Monderer and Shapley [27], any recommendation game with the Shapley mediator possesses at least one PNE, and the set of pure Nash equilibria corresponds to the set of argmax points of the potential function; therefore, SM satisfies S. Corollary 1.


Both pair-based rankers and regression-based rankers implicitly made this assumption, as they tried to learn a single rank function for all queries using the same set of features.

after we have generated feature vectors by combining the queries and URLs, we can create a "training data matrix" of size N × P , where N = NQ j=1 n j is the total number of "data points" (i.e., Query+URL) and P is the total number of features.


One of the advantages of gradient descentbased proofs is that they continue to work even when some perturbations are introduced in the updates.

we believe that our framework can deal with approximate projections and we hope to pursue this in future work.


This motivates the development of number systems where objects counted frequently (essential resources) are counted with small numbers (for better discriminability).

it is reasonable to assume that, in general, larger numbers will be encountered relatively less frequently than smaller ones in natural environments, and hence, that the functions p(m) and p(c) will be monotone decreasing 3 .


These local histograms will need to be communicated across the nodes, and followed by global merge and sort operations.

each node would need to communicate O(n) sized histograms, which can lead to a significant communication bottleneck.


The code of RTRMC is optimized for the MC problem.

we mainly compare the oracle calls of RTRMC for fair comparison.


In this section, we wish to extend these results to the case where the target function is not fixed in advanced.

we seek a single distribution weighted combining rule that can perform well for any f ∈ F and any mixture D λ .


[13] provide a minimax lower bound for this model, and we match it as well.

we achieve the optimal convergence rate for online matrix decomposition for this model.


When this is not the case, as in BNs, every function ψ j can be multiplied by a constant z j such that the assumption holds, and the result is obtained after dividing by j z j .

here we assume positivity but discuss how this can be relaxed below.


Fast Iterative Algorithm The computation of matrix square root in our parametric SR-SOP depends on EIG, which is limited supported on GPU, slowing down training speed of the whole network.

we employ the recently proposed iterative method [26] to speed up computing matrix square root.


However, in our application this algorithm would have sample complexity of n = Õ( p M • log I), for M autonomous queries in T total queries.

direct use of this mechanism would result in a pricing for non-adaptive users that depends on the number of queries by autonomous users.


While we describe the algorithm in this way for ease of explanation, in practice we do not actually compute the GBMS step for all x dn values, but only for the missing ones, which is all we need.

our algorithm carries out GBMS denoising steps within the missing-data subspace.


Best-boundfirst search can quickly discard unpromising nodes, but should not be used frequently at the top levels of the tree since the bound estimate is not accurate enough yet.

we propose to learn policies adaptive to different problem types and different solving stages.


• For any n-bit x, it is hard for a party knowing only the public key P and the encryption E P (x) to compute x.

4 in such a scheme, anyone knowing the public key of Alice can efficiently compute and send encrypted messages to Alice, but only Alice, who is the sole party knowing her private key, can decrypt those messages.


The basic idea of the algorithm is to replace the MDP M with M ≡ ( S, A, Pa , ra , γ), where Pa = Ka Ḋa and ra = Ka r a (r a ∈ R n a is the vector composed of sample rewards r a i ).

instead of solving an MDP with n states, one solves a model with m states only.


Unlike thresholding individual trajectory lengths, other methods use the entire tensor to solve the tensor biclustering problem.

when k is very small, the accumulated noise can dominate the signal strength.


Performance are reported according to the Gaussian loss G and Hellinger loss H . Unsurprisingly, methods trained with respect to a specific loss perform better than the competitor with respect to such loss.

as a further measure of performance we also introduced the "Recognition" loss R . This loss has to be intended as a measure of how "well" a predictor was able to correctly reconstruct an image for digit recognition purposes.


All units within a feature map share the same weights.

each feature map can be seen as convolving the feature maps of the previous layers with small-size kernels, and passing the sum of those convolutions through sigmoid functions.


It is crucial to note though that, since the experts are compared on a sequence-bysequence basis, the MR criterion ignores the possibility that different experts may induce different sequences of choices by the opponent.

mR makes sense only under the assumption that Nature's choices are independent of the decision maker's choices.


Meanwhile, the Tucker decomposition of a single tensor computes a latent "core" tensor but has no dynamics or noise capabilities.

the MLDS achieves the best of both worlds by uniting the two models in a common framework.


In general, with l different labels, we can define Y ∈ R l×N .

x and Y are composed by stacking up the training and testing data as: X = [X tr X tst ] and Y = [Y tr Y tst ].


In this work, instead of generating audio from scratch, our goal is to augment the input audio channels so as to introduce spatial information.

unlike Wavenet, efficient audio generation can be achieved without sacrificing audio fidelity, by transforming the input audio.


Hence, the proposed approach will provide a principled way of combining labels from any combination of human and previously existing machine-based classifiers.

the labels given by labeler i to image j (which we call the given labels) are denoted as L ij and, under the model, are generated as follows: under the model, the log odds for the obtained labels being correct are a bilinear function function of the difficulty of the label and the expertise of the labeler, i.e., More skilled labelers (higher α i ) have a higher probability of labeling correctly.


The expression can thus be rewritten as The objective is then to show that the above equals p( *).

it proceeds directly from the definition of * that, for all other than *, all but one of the terms in the sum above approach zero, and the remaining term equals p 1 ( *).


In contrast, our network can only implement pairwise interactions, a connection between two spiking neurons implementing the conditional probability linking the two corresponding binary variables.

we need to assume additional conditional independencies between the nodes in the generative model, so that their joint probability can be pairwise factorized: In words, it means that variables bias each other's probabilities, but do not influence each other's dynamics, i.e they do not affect each other's transition probabilities.


6 Then it is easy to see that the second highest probability of collision is achieved for the hash value corresponding to the coordinate with the second largest absolute value, etc.

we consider the indices i ∈ [d] sorted by their absolute value as our probing sequence or "ranking" for a single cross-polytope.


One can further show that the marginal distribution of all b m is a t-distribution with t = b m |α m /σ bm|αm and 2â α degrees of freedom, which allows a principled way of determining whether a regression coefficient was excluded by means of standard hypothesis testing.

variational Bayesian Least Squares (VBLS) regression is a full Bayesian treatment of the linear regression problem.


In some situations, it may not be relevant to consider conditional swaps for every possible action, either because of the specific problem at hand or simply for the sake of computational efficiency.

for any S ⊆ N 2 , we define the following competitor class of functions: See Figure 2 for a transducer interpretation of this scenario.


On the other hand, once we have expanded the vectors, the task is merely computing inner products, for which we can use VW.

in the run-time, after we have generated the sparse binary vectors of length 2 b k, we hash them using VW with sample size m (to differentiate from k).


However, this approach increases the inference and training times and consumes more memory.

a large network might not be implementable in environments with limited resources, such as mobile applications.


Not only is the optimal recommendation set problem somewhat easier computationally, it is submodular, admitting a greedy algorithm with approximation guarantees.

it can be used to determine approximately optimal choice queries.


2 Let ( P , ẑ) be a solution to the SDP: ) of the LP: linear program in the variables i (with the u i 's fixed).

an extreme point solution ( ¯ , z ⇤ ) is defined by n + 1 equalities, at most three of which can be constraints in ( 6)-( 8) and the rest (at least n 2 of them) must be from the Given the upper bound of d on the sum of the ¯ i 's, this implies that at least d 1 of them are equal to 1, i.e., at most two are fractional and add up to 1.


Perturbations along any other directions do not yield equilibria because then, either ✓ D is no longer in an all-zero state or ✓ G does not match the true distribution.

we consider a setup where the rank deficiencies of K DD and K T DG K DG if any, correspond to equivalent equilibria (which typically exist for neural networks, though in practice they may not correspond to 'linear' perturbations as modeled here).


For instance, if one of the players has revealed her sample, other players may just notify everyone that they hold the same (or a correlated) sample, using O(1) extra bits.

our lower bound excludes the possibility of non-trivial protocols that do better than essentially having each machine transmit its entire sample.


However, if the agent only sees one task, all such states will be within some small fixed range of the other two lights; a termination set built from such data would not transfer to another task, since the relative positions of the lights would change.

generalization across tasks requires each option to terminate when it is close to the proper light, regardless of the observed positions of the other two lights.


The MAB formulation models the trade-off between exploration and exploitation, where exploration concerns finding the potential best arms, but can result in pulling sub-optimal arms, while exploitation aims at choosing arms with the current best performance and can lose reward if that arm is in fact sub-optimal.

optimizing this trade-off is very important for any controller seeking to minimize regret.


Often f 0 is a high-cost model that requires most, if not all, of features so c(f 0 , g) can be considered a large constant.

to meet the budget constraint, we would like to have (a) low-cost g and f 1 (small c(f 1 , g)); and (b) small fraction of examples being sent to the high-accuracy model (small E Sn [q(0|x)]).


As describe in Section 1, our model assumes that there is no complete data in training phase.

the loss function was computed based only on pixels from outside the mask.


It was also necessary to adjust the sampling of the nonlinearities (location of µ jk 's) because, as the fixed-norm filters rotated through input space, the variance of the projections can change drastically.

whenever data fell outside the range, the range was doubled, and when all data fell inside the central 25%, it was halved.


. . , 1/D), the matrix R is D j=1 (N/D) × D(D − 1)(e j − p)(e j − p) T > 0.

all the points on the critical lines are saddle points.


The procedure detailed in this paper can be easily generalized to multiple constraints.

for the rest of the paper, without loss of generality we consider the problem of the form, This is a special case of the general formulation in (1).


In general, an abstention stump is defined by the pair h ✓1,✓2 (X), r ✓1,✓2 (X) where, for Figure 4-left, h ✓1,✓2 (X) = 1 X✓1 + 1 X>✓2 and r ✓1,✓2 (X) = 1 ✓1<X✓2 .

our abstention stumps are pairs (h, r) with h taking values in { 1, 0, 1} and r in {0, 1}, and such that for any x either h(x) or r(x) is zero.


Because the bound in (1.1) is tight, the performance loss may grow with the discount factor as fast as γ/(1−γ) 2 while the total return for any policy only grows as 1/(1−γ).

for γ sufficiently close to 1, the policy π computed through state aggregation may be no better than a random policy even when the approximation error of the optimal policy is small.


Comparing 9 ¡ @ and @ ¡ b9 ` should tell us which of 9 and @ is more likely to have its samples mis-classified by the other model.

the difference 9 ¡ @ ¨ @ ¡ 9 W , when positive, indicates that samples of class @ are more likely to be mis-classified as class 9 than samples of class 9 are to be mis-classified as class @ (and vice-versa when the difference is negative).


However, the rate of this method is dependent on the problem dimension m, n, and converges quite slowly when m, n are large.

they leave an open question on whether other efficient splitting methods exist, resulting in convergence analysis in the space with lower dimension m or n. In this paper, we propose a new splitting method for LP, which splits the equality and inequality constraints into two blocks.


Any Vn that lies outside the hull is modeled with error.

the objective of the model is to identify P (f |z) such that they form a convex hull surrounding the data distributions.


We would like to encode these symmetries directly into the model rather force an algorithm to learn them from scratch.

model identification is typically done not in the spatial (world) coordinates , but instead in the helicopter body coordinates, in which the , £ , and ¦ axes are forwards, sideways, and down relative to the current position of the helicopter.


vectors with small ℓ 1 -norm show k −1 decay).

we expect the reconstruction algorithms will have to contend with the sparsity error of the target.


Even though the evolving graph model requires the algorithm to issue a low number of queries, our analysis shows that (under reasonable assumptions on p and q) this small number of queries is sufficient to maintain a high-quality clustering.

our theoretical results hold for large enough n. we also perform simulations, which demonstrate that our final theoretically optimal algorithm is able to beat the other algorithms even for small values of n. Clustering and community-detection techniques have been studied by hundreds of researchers.


It crucially depends on the previously selected items and contextual information S oi−1,ui , φ ui .

the only avenue through which we can learn about the payoff functions f φ (that we try to optimize over) is via noisy feedbacks y i .


This new activity vector represents exactly the same point on the plane: Da = [0 1].

the percept, the point on the plane, remained constant despite changing neuronal activity.


In other words, if two labels u i and u j are close i.e., |u i − u j | ≈ 0, then the difference between the row slices |w(u i , •) − w(u j , •)| and the column slices |w(•, u i ) − w(•, u j )| should also be small.

to measure the similarity between two labels using the graphon slices, we define the following distance d ij is small only if both row and column slices of the graphon are similar.


Note that interventions in game-theoretic settings are not new (see Appendix B for a survey).

we use causal strategic inference simply as a convenient name here.


At the start of the darkness period there is a positive upward and a negative downward disturbance inferred by the system (Figure 1C, reversal).

by the end of the dark period, the estimate has become gain-up, the gain learned in the initial session.


Note that Equation 3 is a special case of Equation 4, using the instrumental distribution q = p. The estimate Ĝn,q given by Equation 4 depends on the selected instances (x i , y i ), which are drawn according to the distribution q(x)p(y|x).

ĝn,q is a random variable whose distribution depends on q.


This data is often given or represented as samples embedded in a high dimensional Euclidean space, point cloud data, though it is assumed to belong to lower dimensional manifolds.

in recent years, there have been significant efforts in the development of methods to analyze these point clouds and their underlying manifolds.


For sufficiently large n and m and small p, there exist parameters for our algorithm such that it outputs a clustering with accuracy (1 − ε) and the expected number of comparisons is Note that n 1+O(p) n 2 /m Ω(1/p) in the expression above whenever the reads are long enough, that is, when m n cp (where c is some small constant).

for a large range of n, m, p, and ε, our algorithm converges in time proportional to n 1+O(p) , which is sub-quadratic in n, the number of input reads.


Then, where the Lévy measure ν is Furthermore, where {T (t) : t ≥ 0} is a Gamma subordination and each T (t) has density As we see, T (t) follows Gamma distribution Ga(T (t)|t/γ, α).

the {T (t) : t ≥ 0} is called the Gamma subordinator.


There are no gold standard messages for images.

a message can be treated as a variable-length sequence of discrete latent variables.


Each project is represented as a 112-dimensional vector specifying the distribution of skills required for its delivery.

each feature is associated with a particular job role/skill set (JR/SS) combination, e.g., "data-specialist (oracle database)".


Then these concatenated features will be up-sampled and refined with details  Note that there is no Isolated convolution (I-conv) in the fish body and head.

the gradient from the loss can be directly propagated to shallow layers in tail, body and head.


Since g is also concave, x 0 is a global maximum, and thus g(x) ≤ g(x 0 ) = 0 for all x, implying that f (K) V0 (x) ≤ x.

for odd K, the function f V0 satisfies all requirements for Eq.


Recall that N ¬p wk represents processor p's belief of the counts of all the other processors with which it has already communicated (not including processor p's local counts), while N p wk is the processor's local word-topic counts.

the sampling of the z p 's is based on the processor's "noisy view" of the global set of topics.


In [Ng, Jordan, Weiss], we obtain δ ≥ 175.35.

the assumptions in these papers are all violated as well.


Furthermore, let di = j∈V A ij and in general, throughout this paper, we will denote computable quantities derived from the observed A with the same letter as their model counterparts, decorated with the "hat" symbol.


First, if a given optimization algorithm obtains an iteration complexity of an order of c(n, κ) ln(1/ ), up to logarithmic factors (including the norm of the minimizer which, in our construction, is of an order of 1/n and coupled with the accuracy parameter), for solving smooth and strongly convex finite sum problems with a stochastic oracle, then c(n, κ) = Ω n 2 ln(n 2 /(κ + 1)) .

the following holds for optimization of finite sums with smooth and strongly convex individuals.


As a consequence, if µ h corresponds to the l-th column of C c , then its l-th coordinate is expected to be greater than other coordinates.

we set the l-th column of C c to some vector µ h whose l-th coordinate has the greatest component (if there are multiple such vectors, then randomly select one of them; if there is no such vector, then randomly select a µ h ).


He proposes to use the unlabeled data to learn a mixture of models and he introduces the Mutual Information kernel which is defined in such way that two points belonging to different components of the mixture model will have a low dot product.

in the case of a mixture of Gaussians, this kernel is an implementation of the cluster assumption.


Using recent results on the properties of random projections, we proved that in sparse spaces, random projections of sizes logarithmic in the original dimension are sufficient to preserve linearity.

bEBFs can be generated on compressed spaces induced by small random projections.


The exponential weighting is more sensitive to changes in local feature relevance (3) and gives rise to better performance improvement.

(4) can be used as weights associated with features for weighted distance computation These weights enable the neighborhood to elongate less important feature dimensions, and, at the same time, to constrict the most influential ones.


In practice, assessing peptide identification accuracy is made difficult by the lack of ground-truth encountered in real-world data.

it is most common to estimate the false discovery rate (FDR) [1] by searching a decoy database of peptides which are unlikely to occur in nature, typically generated by shuffling entries in the target database [18].


Notice that for a good top-k diversified ranking list, it often requires the balance between the diversity and the relevance in order to fulfill the user's information need.

we also present the relevance score (measured by PageRank) captured by the entire top-k ranking list.


The population receptive fields and the characteristic patterns are found by a joint optimization in stimulus and response space.

one does not need to know-or assume-a priori what features the population is sensitive to, or what spike patterns convey the information.


The covariance matrix Σ could be estimated using the Hessian matrix of the logL [33].

for a given pairwise observation M, we could obtain the approximated prior information on s ∼ N (ŝ, Σ).


This paper observes that in the structured learning setting, the optimization with respect to energy can be formulated as a logistic regression problem for each factor, "biased" by the current messages.

it is possible to use any function class where an "oracle" exists to optimize a logistic loss.


For computational feasibility, in practice d Fnn (μ n , νm ) is typically adopted as an approximation (i.e., an estimator) of the true neural net distance d Fnn (µ, ν) for the practical GAN training, where μn and νm are the empirical distributions corresponding to µ and ν, respectively, based on n samples drawn from µ and m samples drawn from ν.

one important question one can ask here is how well d Fnn (μ n , νm ) approximates d Fnn (µ, ν).