We want H = arg min Y∈R r×n + X − X(:, K)Y 2 for some norm.

choosing the Frobenius norm results in a set of n nonnegative least squares (NNLS) problems: we can solve the NNLS problem with matrices of size n × n. After computing just the small R factor from the QR factorization, we can compute the entire nonnegative matrix factorization by working with matrices of size n × n. Analogous results hold for the SVD, where we replace Q by U, the left singular vectors.


Let us assume that we could run DISCARD with the unknown value = k. Then the solution we get satisfies f (A gd [m, k]) ≥ λQ.


Each random constraint is constructed in such a way that it rules out any given truth assignment exactly with probability 1 / 2 .

in expectation, after adding s such constraints, we will have a SAT instance with exactly one solution.


We aim to find the posterior P (W|D N ) in an online setting, where samples arrive sequentially.

after the n-th sample is received, the posterior is updated according to Bayes rule: for n = 1, . . . , N . Note that the MNN is deterministic, so the likelihood (per data point) has the following simple form 3 the Bayes update in Eq.


The solution of the outer optimization problem ( 4) is then straight-forward.

the complexity of the algorithm is dominated by a matrix multiplication that is solved naively in O(m 3 ), but faster algorithms working in O(m 2.376 ) are known [16].


we first clustered together all flights with the same origin and destination that were scheduled to depart within 15 minutes of each other, under the assumption they would have the same on-time statistics.


However, to solve the pose estimation problem, we are interested in Z, not in y.

we need an additional step that transforms the quadratic variables into the original variables.


Moreover, we have � −e n−1 I n−1 � p = 0, −p < 0.

by Lemma 14, we have Thus by Theorem 13, we get CCdim(� 0-1 ) ≥ n − 1.


To understand their behavior, it is often necessary to analyze functionally related components of the system, corresponding to subsystems.

the analysis of subnetworks may provide additional insight into the behavior of the system, not evident from individual components.


Note that throughout this discussion X and W played symmetric roles.

in cases with known prior density and unknown additive noise density, one can formulate the estimator entirely in terms of the prior.


Observing that the placements of all the operations in the network can be configured independently in practice, the device selection for each operation can also be independent.

the joint distribution f (d|u), i.e., the placement distribution for the neural network, can be expressed as a product of marginal distributions for each operation: To obtain the optimal distribution which results in the optimal placement d * = arg min T (d) with probability 1, we iteratively update the parameters until the final convergence, starting from randomly generated values.


We therefore need to show that the norm of w, and evaluation K w on any two examples can still be performed efficiently.

first note that given the vector v we can compute the norm of w as follows, Next, given two vectors x i and x j the value of their inner-product is, although we cannot compute the vector w explicitly we can still compute its norm and evaluate any of the kernels from the class C. Synthetic data: We generated binary-labelled data using as input space the vectors in 100 .


In this paper, we assume a strict subset of first-order logic called finite Herbrand logic.

we assume that we have no function constants and finitely many object constants.


The missing values in time series data make it hard to analyze and mine [14].

the processing of missing values in time series has become a very important problem.


For clarification, we now write an NPG that employs the natural policy gradient and a VPG that employs the "vanilla" policy gradient.

nPG(w) and VPG(w) denote the use of the parameterbased exploration strategy, and NPG(u) and VPG(u) denote the use of the control-based exploration strategy.


Bottou and Bousquet [BB07] show that if is a bound on the sub-optimality of w on S i , then the total error is bounded by H(n i ) + .

in designing an efficient algorithm for streaming data, we focus on reducing the sub-optimality to asymptotically balance with H(n i ) -it does not pay to reduce the empirical risk even further.


We see that all the V2 bases in Figure 5 have maximal response when its strongest V1-basis components are aligned with the stimulus.

some of these bases do indeed seem to encode edge junctions or crossings.


( 19) can be efficiently obtained by solving the linear equation: (30) Empirical results show that the convergence is fast and only a few iterations are needed to converge.

the proposed method can be applied to large scale problem in practice.


Note that n − 2α 2α+d is the optimal learning rate to estimate a function in B α 2,∞ .

the densitydifference estimator with a Gaussian kernel achieves the optimal learning rate by appropriately choosing the regularization parameter and the Gaussian width.


It is clearly visible that ( 7) is also a forward Chapman-Kolmogorov equation.

the only differences between prior and posterior process are the jump rates for the telegraph process µ and the drift of the diffusion process x.


When v is exposed, we see not only that v belongs to T , but we also see v's path all the way back to the root r of T , due to the list of names on v's copy of the petition.

if the set of people who post their copy of the petition is {v 1 , v 2 , .


Furthermore, in our experiments, we observe that the neighborhood function f has a curvature κ f = 1.

it suffices to use the simpler versions of algorithm EA (i.e., algorithm EASSCc and EASKc).


In this paper we propose a new type of regularization that encourages the network representations to form clusters.

the learned feature space is compactly representable, facilitating generalization.


However, the running time of the online algorithm is independent of the past and only depends on the number of documents introduced in each timestep (which in this case is always 1000).

the running time of the online  1. algorithm is almost the same across different timesteps.


This model captures variance dependency structure explicitly, but treating variance as an additional random variable introduces another level of complexity and requires several approximations.

it is difficult to obtain a simple analytic PDF of source signal u and to apply the model for image processing problems.


It is well-known that the network structure corresponds to the non-zero pattern of the inverse covariance matrix, ⌃ 1 [1].

obtaining a sparse estimate of ⌃ 1 by using `1 penalty has been a standard approach to inferring a network, a method called graphical lasso [2].


Our goal is to characterize the scaling of the sample complexity for various reinforcement learning approaches (i.e., how many trials they require in order to learn a near-optimal policy) for large numbers of agents n. Thus, in our bounds below, no serious attempt has been made to make our bounds tight in variables other than n. Below we show that if an RL algorithm uses only the global reward signal, then there exists a very simple MDP-one with horizon, T = 1, only one state/trivial dynamics, and two actions per agent-on which the learning algorithm will require Ω(n) trials to learn a good policy.

such algorithms do not scale well to large numbers of agents.


Instead of a binary decision function where E y is the expectation over P y and ℓ 0-1 (z) is the zero-one loss: Since the zero-one loss is hard to optimize in practice due to its discontinuous nature, it may be replaced with a ramp loss (as illustrated in Figure 1): giving an objective function of To avoid the non-convexity of the ramp loss, the hinge loss is often preferred in practice: giving an objective of One practical motivation to use the convex hinge loss instead of the non-convex ramp loss is that separability (i.e., min g J R (g) = 0) implies ℓ R (z) = 0 everywhere, and for all values of z for which ℓ R (z) = 0, we have ℓ H (z) = 0.

the convex hinge loss will give the same decision boundary as the non-convex ramp loss in the ordinary classification setup, under the assumption that the positive and negative samples are non-overlapping.


However, optimization is difficult because we do not know a reasonable t in advance, and the gradient updates are expensive due to backpropagation through the entire chain.

we propose a more efficient approximation, called Markov GAN (MGAN): where 2 (0, 1), b 2 N + , m 2 N + are hyperparameters, x denotes "fake" samples from the generator and T m ✓ (x|x d ) denotes the distribution of x when the transition kernel is applied m times, starting from some "real" sample x d . We use two types of samples from the generator for training, optimizing ✓ such that the samples will fool the discriminator: 1.


The RT difference bewteen easy and hard items does not fully disappear in mixed blocks.

rT depends on both the stimulus type and the composition of the block.


As a result, truncating higher-order mixture components has little impact on the approximation on shorter time scales.

we simply need to pick a value of N that allows for a representation of many orders of magnitude of time.


Our interest in the presented analysis was to study pairwise interactions in the data (cell vs. cell, and marker vs. marker).

we are interested in the matrix X = A ⊗ A, the Kronecker product of A with itself.


Observe that they intuitively measure discrepancy to the true label ±1.


Since we implement a bi-directional LSTM, this results in each embedding vector h t as size 64 and embedding vector m j as size 32.

we set the context vectors, W b and W s , to size 64 and 32 respectively.


A related issue is that we would also like to avoid the problem of separating a single outlier (or very small group of outliers) from the rest of the data.

to mitigate these effects we will impose a constraint that the difference in class sizes be bounded.


We additionally consider multi-scale image features to make some attribute detectors to focus on smallscale features and others to focus on large-scale features.

our conditional discriminator is rewritten as: where v j is the image vector of j-th layer, and β ij is a softmax weight that determines the importance of the layer j for each word w i .


For each variable y j in the model, each level of the cascade filters a current set of possible states Y j , and any surviving states are passed forward to the next level of the cascade by substituting each state with its set of descendents in the hierarchy.

in the pose estimation problem, surviving states are subdivided into multiple finer-resolution states; in the image segmentation problem, broader object classes are split into their constituent classes for the next level.


From [Berg et al., 1984] (Theorem 3.2.2, p.74), the Persistence Fisher kernel is positive definite.

our proof technique is not only independent and direct for the Fisher information metric on the probability simplex without relying on the geodesic distance on S + , but also valid for the case of infinite dimensions due to [Schoenberg, 1942] (Theorem 2, p. 102).


Under mild assumptions on the correlation function, which is unknown, [27] showed that, by using a bimodal type of kernel K such that K(0) = 0, the residual sum of squares (RSS) approximates the asymptotic squared error uniformly over a set of bandwidths.

choosing the bandwidth ĥb (of the bimodal kernel) minimizing the RSS results in an optimal bandwidth that minimizes the asymptotic squared error asymptotically.


Specifically, consider t1 = 1, t2 = 2, and t3 = 3, giving z 1 = ψ( t1 ) = (0, 1, 2) � , z 2 = ψ( t2 ) = (1, 0, 1) � , and T here is a convex set and ψ : � T →R 3 is a convex function.

moreover, for t1 = 1, we have we can use Lemma 8 to compute N S ψ (z 1 ).


DA-GAN generates the data in a completely different way from InfoGAN (5) and AC-GAN (22) which generate images from a random noise input or abstract semantic labels.

inferior to our model, those existing GAN-like models cannot exploit useful and rich prior information (e.g., the shape, pose of faces) for effective data generation and augmentation.


the fits minimize the block's squared loss.

we get the familiar characterization of the isotonic regression problem as one of finding a division into isotonic blocks.


We may do so by applying a simple sigma-point quadrature ("unscented") approximation [16], in which the standard deviation of the convolved function m ts (x s ) is estimated by applying the same nonlinearity (13) to the standard deviation of the error on the incoming product M ts .

similarly to (18) and ( 19), we have The steady-state solution of (20) yields an estimate of the variances of the log-belief log p t by σ 2 t = u∈Γt σ 2 ut ; this estimate is typically much smaller than the bound (18) due to the strict sub-additive relationship between the standard deviations.


However, the first inequality in (3.2) requires t ≤ 1/β, because its proof in Lemma 3.2 needs (3.1).

we need to perform backtracking on t until (4.1) is satisfied, and use the same t to find x k by Algorithm 1 or Algorithm 2.


For computing the FID, we propagated all images from the training dataset through the pretrained Inception-v3 model following the computation of the Inception Score [44], however, we use the last pooling layer as coding layer.

for this coding layer, we calculated the mean m w and the covariance matrix C w . we approximate the first and second central moment of the function given by the Inception coding layer under the real world distribution.


Assuming P = N P , no polynomial-time algorithm can guarantee E[d(j, S)] ≤ cr j for c < 1 + 2/e, even if all r j are equal to each other.

the approximation constant in Theorem 3.2 cannot be improved.


As a result, instead of a single FV, describing the whole image, the image is represented by a large number of densely computed semi-local FVs, each of which describes a spatially adjacent set of local features, computed by the previous layer.

the new feature representation can capture more complex image statistics with larger spatial support.


The actual reason for the failure of the iterative TD(O) approach lies in an incompatible combination of the RL algorithm and the function approximator.

the idea is that either a change in the RL algorithm or a change in the approximator may yield a convergent iteration.


As mentioned in the introduction, the sample complexity of margin-error minimization can be upperbounded in terms of the average norm Alternatively, we can rely only on the dimensionality and conclude m(ǫ, γ, D) ≤ Õ(d/ǫ 2 ) [7].

although both of these bounds are tight in the worst-case sense, i.e.


Nonetheless, it is interesting to notice the major difference between the true and pseudoregret when compared to the standard bandit problem.

in fact, it is possible to show in the risk-averse case that the pseudo-regret is not an unbiased estimator of the true regret, i.e., to bound the expectation of R n we build on the high-probability result from Lemma 1.


We tried different set of parameters for RBSUMM ρ i 2 +1 i+t (0 ≤ i ≤ 5, ρ = 0.01, 0.1, 1) or sufficiently small step size, but could not see the convergence of the objective within 5000 iterations.


p MK) convergence rate, Algorithm 1 achieves ✏-approximation solution when mK / 1/(n✏ 2 ).

that is, the amount of computation required on each node is inversely proportional to n. linear speedup can be achieved by Algorithm 1 asymptotically w.r.t computational complexity.


Let k be as calculated by Algorithm 1 at iteration k. Then, for any k  k ⇤ the following holds: We are now ready to prove the remaining conditions.

for any k < n. Lemma 3.4 implies that k k+1 for any k < k ⇤ (minimizing the same objective with stricter constraints yields a higher optimal value).


If ρ k < ρ k however, the UCB algorithm will select k less often and the underestimation persists.

e[ρ k ] < ρ k . The following theorem is the main result of this section, which shows that the bias of the CTR estimators can critically affect the search engine's revenue.


Hence increasingly many particles are needed to accurately represent the posterior.

the estimation of "off-line" quantities such as p(Xk IY1:K) and x~:K becomes very costly since one has to store all past trajectories.


However, in the examples cited, the authors evaluated the performance of their methods on data simulated from a fixed finite number of components or populations, suggesting that they found this to be more realistic than a DPM for their applications.

it is important to understand the behavior of this posterior when the data comes from a finite mixture -in particular, does it concentrate at the true number of components?


(-), which approximates the ideal marginal Gaussianization w (.

), can be derived from univariate Gaussian mixtures The parameters are chosen by minimizing the negentropy of the transformed variable X(k+1): A,1r,f-t,u (3) after each iteration, the transformed variable becomes as close as possible to the standard Gaussian in the Kullback-Leibler distance.


In particular, the proof of Theorem 2.2 suggests that as the number of tests are increased the performance also improves.

in this section we empirically evaluate this observation.


For example, if "polar bear" is the only class in the dataset that contains snow, a classifier may be able to get a 100% accuracy on this class by simply detecting the presence of snow and ignoring the bear completely [6].

even with perfect accuracy, we cannot be sure whether our model actually detects polar bears or just snow.


Otherwise, the two models are said to be -far and ∆(M, N ) > .

two models M and N are close according to the above definition if there is no intervention which can make the resulting distributions differ significantly.


Splitting data does not generate anomalies.

we can evaluate methods while controlling the anomaly rate properly.


Consider the simplest loss, viz., the squared "loss" r(x, θ) = − x − θ 2 2 (note -ve sign to keep reward interpetation).

it can be easily seen that r(x, θ * p ) = E θ∼p [r(x, θ)] = − x 2 2 + 2x • θ * p − 1 • θ * p , if the relevance vectors θ are in {0, 1} n . the Lipschitz condition is satisfied.


By contrast, in the analogous graph frequency model of the present work, y is fixed, and we grow the network by repeatedly sampling the number of edges g ij between vertices i and j and summing the result.

in the Caron and Fox [12] model, a latent infinity of vertices (only finitely many of which are active) are added to the network each time y increases.


If it has an effective support of size 1, then the event ω for which q ω = 1 cannot be from a level set in L or H (otherwise, part (i) of Lemma 5.1 would be violated).

therefore it must belong to the single level set in M . if M = {S m } (for some index m), then there are feasible solutions Q such that q ω = 1 (for ω ∈ S m ), all of which have ρ(Q) = r m . If, on the other hand, Q has an effective support of size 2, then it is not hard to show that one of the two events must be from a level set S l ∈ L, and the other, from a level set S h ∈ H (since all other combinations result in a violation of either part (i) or part (iii) of Lemma 5.1).


Further, any graph sampled from a graphon is almost surely dense or empty [20].

vertex-exchangeable random graph models are misspecified models for sparse network datasets, as they generate dense graphs.


There is no significant difference in behavior between the squared error and the Huber loss in these experiments, despite the fact that the squared error is effectively unbounded.

the parallelization works in the sense that many machines obtain a better objective function value after each machine has seen k instances.


Correspondingly, the label accuracy A is estimated as In our Bayesian inference algorithm, workers' scores, PoBCs and the true label distribution are all estimated by comparing the true label samples with the collected labels.

t To prove the convergence of our algorithm, we need to bound the ratio of wrong samples.


Many practical combinatorial problems can be effectively translated into SAT.

one of the current most successful approaches to solving hard computational problems, arising in, e.g., hardware and software verification and planning and scheduling, is to first translate the problem into SAT, and then use a state-of-the-art SAT solver to find a solution (or show that it does not exist).


We nd that this extension (say, GBVS Multiresolution, or GBVSM) improves performance with little added computation.

we have presented a method of computing bottom-up saliency maps which shows a remarkable consistency with the attentional deployment of human subjects.


A Gaussian model for y can certainly describe correlations between the cells, but is ill-matched to discrete count observations.

as with the derivation of the RLM from the Kalman filter, we derive here a new generalisation of a low-dimensional, structured Gaussian model to spike count data.


Since the trajectories of the output of a linear dynamical system live in the so-called observability subspace, the intensity trajectories of pixels associated with a single dynamic texture lie in a subspace.

the set of all intensity trajectories lie in multiple subspaces, one per dynamic texture.


When Q satisfies Property A, then by Theorem 3.3 there exists a monotone optimal rejection function.

the following corollary of Lemma 4.2 establishes the optimality of any LDRS.


By the contraction map fixed point theorem [2] there exists a unique fixed point of O in the space of vectors q.

repeated applications of the on-policy operator converge to a vector q Π such that q Π = Oq Π ; that is, q Π satisfies the policy based Bellman equation.


Boolean relaxations afford us the ability to control the degree of sparsity explicitly in our predictors akin to exploiting an `0 regularization, unlike the `1 based methods that may require extensive tuning.

our approach harnesses the best of both worlds.


[2015] proved that the estimation error of the GTD algorithms can be upper bounded by their corresponding primal-dual gap error multiply a factor.

we are going to derive the finite sample primal-dual gap error bound for the convex-concave saddle point problem firstly, and then extend it to the finite sample estimation error bound for the GTD algorithms.


From the table, we confirm that EM for Model 2 (with unknown weights) has a higher success probability than EM for Model 1 (with known weights).

over-parameterization helps in all four cases.


, which means the enlarged kernel gives a larger margin and a smaller objective value.

on one hand, the large margin preference guides the scaling of the learned kernel to be as large as possible.


However, we need to sample m p = O( 1 /p) valuations to see a single nonzero value.

there is no bound on the sample size m p as a function of the expectation, which is simply 1.


The generative process of the model in Figure 1 can be described as: 1.

for each document d, choose the topic proportions according to a Dirichlet distribution:  the entire model can be described by the following joint probability where w d,1:N and z d,1:N denotes all the words and the associated topics, respectively, in the d-th document.


To assure, though, that this approximation has rich representational capacity we proposed to recover the variational parameters of the posterior via a non-linear recurrent recognition model.

the joint approximate posterior resembles the behaviour of the true system, which facilitates the effective learning of the GPSSM.


The failure is due to long-range spatial correlations between many thousands of pixels which overcome the noisy correlations between the few dozen pixels belonging to each cell.

the independent components extracted by the algorithm of [2] 1 have large spatial domains as can be seen in supplemental figure 1.


DIAL works as follows: during centralised learning, communication actions are replaced with direct connections between the output of one agent's network and the input of another's.

while the task restricts communication to discrete messages, during learning the agents are free to send real-valued messages to each other.


For the block stacking environment specifically, the robot should only need to pay attention to the position of the block it is trying to pick up (the source block), as well as the position of the block it is trying to place on top of (the target block).

a properly trained network can learn to match the current state with the corresponding stage in the demonstration, and infer the identities of the source and target blocks expressed as soft attention weights over different blocks, which are then used to extract the corresponding positions to be passed to the manipulation network.


In such case, it takes Θ(t) space to store the empirical CDF Fi of the observed outcomes from arm i, and both calculating the stochastically dominant CDF F i and updating Fi take Θ(t) time.

the worst-case space usage of SDCB in T rounds is Θ(T ), and the worst-case running time is Θ(T 2 ) (ignoring the dependence on m and K); here we do not count the time and space used by the offline computation oracle.


Note that if (G, s) does not contain (B 2 , s + ) as a signed minor then any wreductions of (G, s) also do not contain (B 2 , s + ) as a signed minor; we can check this property for each type of w-reductions, (w,1,2,3).

it is sufficient to show that if a w-reduced signed graph (G, s) does not contain (B 2 , +, +) as a signed minor then it is one of the five types.


In the soft setting, given any rejection function, the learner can reduce the type II error by rejecting more (i.e., by increasing r(•)).

for an optimal r(•) we have ρ(P ) = δ (rather than ρ(P ) ≤ δ).


We address the case of the linear time statistic in [7,Section 6], for which both the test statistic and the parameters of the null distribution can be computed in O(n), for sample size n. This has a higher variance at a given n than the U-statistic estimate costing O(n 2 ) used in [8,7], since the latter is the minimum variance unbiased estimator.

we would use the quadratic time statistic in the "limited data, unlimited time" scenario, as it extracts the most possible information from the data available.


Among GMMs with top-k breakings, the larger the k is, the more information we use in a single ranking, which comes at a higher computational cost.

it is natural to conjecture that for the same data, GMM G k T with large k converges faster than GMM G k T with small k. In other words, we expect to see the following time-efficiency tradeoff among GMM G k T for different k's, which is verified by the experimental results in the next section.


When using the full learning rule for both features, the learning rate needs to be decreased until an agreeable signal-to-noise ratio between the drift introduced by the first term and the diffusion caused by the other terms is reached.

it is desirable for faster learning to reduce the effects of these noise terms.


we compare MCAC, CCAC and AC (fAfC, AverageFlow are unable to model this domain).


There is no hope to initially provide, for example, an autonomous robot with sufficiently rich prior knowledge to solve any problem that it may encounter during the course of its life.

an important goal of machine learning research is to replicate humans' ability to learn from experience and to reuse knowledge from previously encountered tasks for solving new ones more efficiently.


From a practical point, the main contribution of this paper is to suggest how to select a good Universum set: it should be such that it contains invariant directions and is positioned "in between" the two classes.

as can be partly seen from the BCI experiments, a good Universum dataset needs to be carefully chosen and cannot be an arbitrary backdrop as the name might suggest.


Therefore, the behavior of the algorithm in general, and its updates and mistakes in particular, are independent to the choice of a.


Note that at every invocation -thanks to submodularity-DISCARD increases the value of the solution by at least λ(Q − f (A)).

by running DISCARD at most log(Q)/λ times we get Q.


For global descriptors, we used the color correlogram for image texture representation, and the edge histogram descriptor.

a visual feature vector is represented in a 540-dimensional space (D = 540).


In other words, once the algorithm decides to discard an element from its memory, this element is gone forever, which is a standard assumption for streaming algorithms.

the theorem does not apply to algorithms that use multiple passes over N , or non-streaming algorithms that use o(N ) writable memory, and their analysis remains an interesting open problem.


Non-reliable sequences are always rejected.

this technique falls within selective prediction but the selection function has been manually predefined.


Natural Parameters Recall that for Gaussian distributions the natural parameters are conventionally defined as (Σ −1 µ, 1 2 Σ −1 ).

to find the natural parameters of ( 12), it suffices to show that (Σ −1 µ, 1 2 Σ −1 ) of ( 12) can be written as an affine transformation of some finite-dimensional parameters.


The state vector of a PSR at time i is the set of predictions p(Q|h i ).

at each time step, the state vector is updated by computing, for each q j ∈ Q: in order to update the PSR at each time step, the vector m t must be known for each test t ∈ X.


The empirical risk minimizer (ERM) is exactly the average h * = 1 m i z i , and for every h = h * we have L S (h) > L S (h * ).

an agnostic sample compression scheme in this setup should compress S to a subsequence and a binary string of side information, from which the average of S can be reconstructed.


If the regularization parameter λ (≥ 0) is small, this can be expressed as where o p denotes the probabilistic order.

up to O p (λ), the bias introduced by regularization (i.e., the second term in the right-hand side of Eq.


However, critical differences with our proposed method are as follows: i) the above methods cannot use observed data (X and Y in this study) about heterogeneous nodes for prediction, because the algorithms are based only on co-occurence information (A uv in this study), and ii) we need to define a new representation of not only the objects in the training set but also additional objects outside of the training set.

it is not possible to directly apply the above methods to the bipartite graph inference problem.


We consider a four node graph, with vertex set V = {1, 2, 3, 4} and edge-set E = {(1, s) | s ∈ {2, 3, 4}}.

the covariance matrix Σ * is parameterized the correlation parameter ρ ∈ [−1, 1]: the diagonal entries are set to Σ * ii = 1, for all i ∈ V ; the entries corresponding to edges are set to Σ * ij = ρ for (i, j) ∈ E; while the non-edge entries are set as Σ * ij = ρ 2 for (i, j) / ∈ E. for this particular example, Assumption [A1] reduces to the constraint |ρ|(|ρ|+2) < 1, which holds for all ρ ∈ (−0.414, 0.414).


However, the new learning method is more computationally intensive than existing approaches, solving a QP at each gradient step.

in future work, we plan to explore faster algorithms that can implement the balanced policy learner, perhaps using alternating descent, and use these to investigate comparative numerics in much larger datasets.


Note that SGD-BB requires the averaged gradients in two epochs to compute the BB step size.

we need to specify the step sizes η 0 and η 1 for the first two epochs.


assignments, this goal can be simply stated as  For supervised learning problems, the class label for every trammg instance is known.

the given conditional distribution of the class label for every training instance is a delta function or jJ(y Yi) where Yi is the given class label for the i-th instance.


Note that for all datasets, we compare all models solely on the RC task.

for fair comparison we do not compare with algorithms that use a second-pass answer re-ranker [Wang et al., 2017b].


It is easy to see that lim ε→0 ρ ε = ρ in distributional distance, and all ρ ε are stationary ergodic.

if H 0 is the set of all stationary ergodic distributions on pairs, we have found a distribution ρ ∈ cl H 0 such that W ρ (H 0 ) = 0.


To spell a symbol, we predict the desired position, then we look up the symbol assigned to it.

in a standard P300 setup the desired text can be modified by altering the look-up table.


Intuitively, using more proxies may allow for a more accurate reconstruction of the confounder and thus may facilitate more accurate causal inference.

collecting a large number of covariates is beneficial for causal inference not only to avoid confounding effects but also to alleviate bias caused by measurement noise.


Indeed, thinking of the shortest path metric on an unweighted bipartite graph in which P corresponds to the left set of vertices, and Q corresponds to the right set of vertices, for each pair of points p i ∈ P and q j ∈ Q, one can choose d(p i , q j ) = 1 or d(p i , q j ) > 1 independently of all other distances by deciding whether to include the edge {p i , q j }.

there are at least 2 Ω(mn) possible distance matrices A, and since our algorithm reads o(mn) entries of A, cannot learn whether d(p i , q j ) = 1 or d(p i , q j ) > 1 for each i and j.


It should be noted that the matrices Λ i and V t are positive semidefinite (PSD).

the problem is an integer quadratic program (IQP) which is hard to solve.


Patterns in which the values of nodes that are connected by an edge agree are more likely, the likelihood being determined by the weights W ij of the edges.

the graph structure dictates the statistical dependencies in network measurements.


(7) for all possible values of ✓ in total time of O(m).

shareBoost can be implemented efficiently with decision stumps.


Generically, optimal estimation of x with a sparse prior is NP-hard [12].

most attention has focused on greedy heuristics such as matching pursuit and convex relaxations such as basis pursuit [13] or lasso [14].


In general, the optimal bound is an increasing function in τ .

the larger the current reliability of the momentary evidence, the more sense it makes to accumulate evidence to a higher level of confidence before committing to a choice.


For this reason, they provide a simple model to study some of the nonlinear behaviors observed during training neural networks.

independent of the specific choice of `, the training cost function have infinitely many local optima, none of these local optima is isolated in the parameter space, and the cost function is not strongly convex at any point in the parameter space.


4, in particular of the cross-correlation term, to the case of j = i + 1 as the modulatory factor only effects the weight of the following state.

since weight changes are in general slow, we can assume a quasi-static process ( the derivatives of ρ on the right hand side of Eq.


samples drawn from some unknown probability distribution P. Catoni [2012] builds the estimator θ of the mean as the solution of the equation where α > 0 is a positive parameter, and ψ(•) : R → R is a truncation function that is non-decreasing and satisfies (2) By choosing α = 2/nν, where ν is the variance of the random variable, Catoni [2012] demonstrates that with high probability, the deviation of θ from the mean is upper bounded by O( ν/n).

(1) can be applied to heavy-tailed distributions, as long as the variance is finite.


Because of this dependence on D, the running time of the Lloyd algorithm becomes prohibitive in high dimensions, and even for D < 10 it is an  order of magnitude slower than k-means.

we were compelled to also implement a greedy algorithm reminiscent of Kégl's algorithm for estimating the packing number [4]: supposing that k − 1 codevectors have already been selected, the kth one is chosen to be the sample point with the largest distance from the nearest codevector.


While the main objective of Algorithm 1 is to identify all arms with means above a given threshold, we note that prior to adding an arm to S t in the TPR setting (i.e., when ⇠ t = 1) Algorithm 1 behaves identically to the nearly optimal best-arm identification algorithm lil'UCB of [21].

whether the goal is best-arm identification or to identify all arms with means above a certain threshold, Algorithm 1 is applicable.


We reiterate that with high regularisation, the unhinged solution is equivalent to an SVM (and in the limit any classification-calibrated loss) solution.

we do not aim to assert that the unhinged loss is "better" than other losses, but rather, to demonstrate that its SLN-robustness is not purely theoretical.


However, the so-called "shrinking technique" of [9,11] for establishing such rates, requires the free parameters to be chosen a-priori, and in addition, the optimal values of these parameters depend on features of the data-generating distribution which are typically unknown.

[9,11] do not provide a practical method for learning with fast rates.


These results, however, are first steps, and indicate similar phenomena (e.g., relation to isoperimetric inequalities).

as pursued in [14], these works motivate a deeper study of such inequalities for real data sets.


Because we use SDDs, we can efficiently compute the exact log-likelihood and its gradient using only two passes over the SDD.

we learn maximum-likelihood estimates of the weights.


To see this, we denote the slack variable y = (ρ − µ ρ ) 2 , and we have: where equality is attained when var(y) = 0, i.e, when y is constant.

m 4 is minimal when |ρ − µ ρ | is constant, which means, since ρ is not constant (σ 2 = α > 0), that p has two values with equal probability.


We can infer that if any x ∈ X (i.e., any column of Φ not in Φ) lies in this convex cone, then the associated coefficients v must all be positive by definition (likewise, by a similar argument, any x in the convex cone of − Φ leads to the same result).

theorem 2 ensures that we are not at a local minimum.


Y is a directed edge in the observable graph D, because every variable except X becomes independent of all other variables in the post interventional graph.

one needs n interventions, each of size n 1 to find out the parent set of every node.


According to the domain adaptation theory [5], the generalization error on the target domain depends on the generalization error on the source domain as well as the difference between the two domains.

one possible solution for domain adaptation is to learn a feature extractor φ(X) shared by both domains, which defines feature distributions p φ S and p φ T for both domains, and minimize some distance between the feature distributions [47] as a regularization.


If several nodes of a GGM are perturbed, then this will correspond to the of the corresponding row-column groups in Θ 1 − Θ 2 .

in order to detect node perturbations in a GGM (Figure 1), we must construct a regularizer that can promote estimates whose support is the union of row-column groups.


With each learned model we forecast wind speeds at all sites and all test-year hours at six horizons (1-6 hours).

for each geography (WI and PNW) we have 20 learned models (4 folds and C = 1, 2, ..., 5) and 120 prediction sequences (horizon 1-6 hrs for each of the 20 learned models).


In other words, if we can find an initialization point with small objective value then a descent algorithm minimizing the objective function guarantees that q always stays in R2C in proceeding iterations.

any descent algorithm that escapes a strict saddle point can be applied to find some a l , or some shift truncation of a 0 .


It also makes learning hard to parallelize due to significant impact of hardware and environment on overall runtime.

instead of runtime, we use the amount of work measured as the number of basic operations performed by the solver required to solve a formula (e.g., implemented via the rlimit counter in Z3).


That additional constraint significantly complicates optimization and yields a second tuning parameter.

instead of doing a 2D-grid search, we use fixed values given in [3] for known σ.


Indeed, while conditions such as convexity, continuity, and differentiability of φ are easy to verify and have natural relationships to optimization procedures, it is not immediately obvious how to relate such conditions to their statistical consequences.

in Section 2 we consider the weakest possible condition on φ-that it is "classification-calibrated," which is essentially a pointwise form of Fisher consistency for classification.


To the best of our knowledge, there is no analytical solution to the influence estimation given Weibull transmission function.

we compare CON-TINEST with Naive Sampling (NS) approach (see Appendix C) by considering the highest degree node in a network as the source, and draw 1,000,000 samples for NS to obtain near ground truth.


the production of leaf-nodes associated with or-nodes, to obtain the complete structure.

we insert one step between two original ones to perform the structure reconfiguration.


Given bounds on the variance of these estimators, it seems that algorithms which reduce to computing weighted sums of kernel evaluations can exploit concentration of measure.

randomized kernels appear promising as a general tool for speeding up kernel methods, warranting further investigation.


For every possible edge e = {i, j}, we draw the multiplicity of the edge e in this step as m e ind ∼ f (•|w e ) and add m e copies of edge e to F . Finally, E n+1 = E n ∪ F . This technique potentially introduces multiple edges in each step, in which edges themselves may have multiplicity greater than one and may be duplicates of edges that already exist in the graph.

this technique generates multigraphs, multiple edges at a time.


The cost of (normal) random projections would be O(nDk), which can be reduced to O(nDk/3) using sparse random projections [1].

it is possible that CRS is considerably more efficient than random projections in the sampling stage.


The reason for projecting out the DC component is, that it can behave quite differently depending on the dataset.

it is usually removed and modelled separately.


However, modeling videos (building a generative model) is still a very challenging problem because it often involves high-dimensional natural-scene data with complex temporal dynamics.

recent studies have mostly focused on modeling simple video data, such as bouncing balls or small patches, where the next frame is highly-predictable given the previous frames [29,20,19].


Our primary motivation in this work is to extend the visual vocabulary of existing captioning models by making large object detection datasets available for training.

as a proof of concept   Baseline: A zebra is laying down in the grass.


However, the meaning of single words in both questions and answers should be the same.

we share the weight matrix between the word-embedding layers of the first component and the third component.


This is remarkable because the dependency between an object's appearance and its structure is tangled in a complex manner with extrinsic nuisance factors such as viewpoint, illumination, and articulation.

learning the intrinsic structure of an object from images requires removing these unwanted factors of variation from the data.


At level n, we perform the same outer sum, but for each trajectory we sum the probabilities of the histories for agents k = j that would lead to the beliefs we are interested in.

the sequence distributions at time t are sufficient for computing any desired element of the infinite belief sequence B j, * (h j 0:t ) for any agent j and history h j 0:t . Updating the distributions.


Since e ✓ is obtained via a model selection procedure, it is irregular and its asymptotic distribution cannot be estimated [14,22].

we proceed to create a regular estimator of ✓ ab in steps 2 and 3.


To do so, we define the regularized empirical risk R n (w) := L n (w) + (cV n /2)kwk 2 and the corresponding optimal argument and attempt to minimize R n with accuracy V n . Since the regularization in ( 4) is of order V n and (3) holds, the difference between R n (w ⇤ n ) and L(w ⇤ ) is also of order V n -this is not immediate as it seems; see [16].

the variable w n solves the ERM problem in (2) to within its statistical It follows that by solving the problem in (4) for n = N we find w ⇤ N that solves the expected risk minimization in (1) up to the statistical accuracy V N of the full training set T . In the following section we introduce a class of methods that solve problem (4) up to its statistical accuracy faster than traditional deterministic and stochastic descent methods.


Here, KL f t g i t is the KL divergence between two Gaussians and π i is the mode weight of g t . The final objective function is: We now show that by modeling the prior as MoG and the posterior as Gaussian, minimizing the approximation results in KL divergence minimization.

let define the log-likelihood we have (see Supplementary material for full derivation): minimizing D var results in KL divergence minimization.


Note, if you would know R ¦ ¦ , you had already reached your goal of designing the inverse controller and learning would be obsolete.

normally a set of resonators © must be predefined in a somewhat arbitrary way and their weights shall be learned.


[5]): In what follows, CV, ∆CV and JK will generically be denoted by μ: where, slightly abusing notation, i Note that μ is the average of identically distributed (dependent) variables.

it asymptotically converges to a normally distributed variable, which is completely characterized by its expectation E[μ] and its variance Var[μ].


The F step then fits F to (Y, Z) and hence gives us the direct fit (which generally will have a positive error 2 , otherwise we stop with an optimal PE).

the beginning of the path is the direct fit to the free embedding.


It is well-known that f * (z) = 1 2 z 2 .

we only need to find Ω * , where the concept infimal convolution is needed.


The reason mutual information is widely used in neuroscience despite the doubtfulness of the optimal decoder is that we are completely ignorant of how information is decoded in the brain.

we simply evaluate the maximal amount of information that can be extracted from neural activities by calculating the mutual information.


Our problem can then be formulated as where g 1 (P ) = The objective function in Equation 1 is linear and its constraints define a convex set.

the optimal solution to Problem 1 can be found by solving its Lagrangian dual.


The eigenvectors of L provide a set of basis functions over the nodes of the graph, ordered by smoothness.

smooth functions f can be well approximated by linear combinations of the bottom eigenvectors of L. Expanding the sensor locations x i in terms of these eigenvectors yields a compact factorization for the inner product matrix X.


in document retrieval systems, only a subset of documents is relevant to a query.

we define Y x ⊂ Y to be a set of items relevant to context x.


Notice that when σ C jk is a Rademacher variable, it has the same distribution as z l C jk σ C jk since z l C jk ∈ {−1, 1}.

using the first result of theorem 6 we have that with probability 1 − δ over the samples S, all h in H satisfy: Now putting in equation ( 8) the third point of theorem 6, with H = H K,B as defined in theorem 6 with Z = X , we obtain the following theorem: Where n S + , n S − are the number of positive and negative instances in S, and σ and ν also depend on S, and are such that x σ(i) is the i-th positive instance in S and ν(j) the j-th negative instance.


In addition, even if it was possible to solve them analytically, they are still subject to the limitations of the original graphical models, such as the need to use exponential family distributions and conjugate priors for tractability.

the next step in the proposed method is to map the probability distributions and the corresponding fixed-point equations into a Hilbert space, where some of these limitations can be relaxed.


The orientation used for linearization greatly affects the resulting linear model.

the linear model becomes significantly worse an approximation with increasing orientation error.


As mentioned before, an MRF is sparse for image analysis problems, on account of the neighborhood structure in the pixel domain.

the multivariate Gaussian distribution associated with a GMRF has a sparse precision matrix.


This theory is not in FO 2 , not in RU , and is not domain-liftable using R. However, the first two clauses are of the form described in Lemma 1, the third and fourth are in FO 2 (and also in RU ), and the third clause, which contains Assigned(v, j), has no other PRVs with more than one LV.

this theory is in S 2 FO 2 (and also in S 2 RU ) and domain-liftable based on Theorem 1.


Formally, we aim to compute a Stackelberg equilibrium of the game in which the learner moves first by choosing a linear classifier g(x) = w T x and all the attackers simultaneously and independently respond to g by choosing x according to a query-based algorithm optimizing the cost function c(x , x A ) subject to query and cost budget constraints.

we term this approach Stackelberg game multi-adversary model (SMA).


Limitations of our method arise from the restriction of capsule poses to be elements of a group for which we have proper M and δ.

in contrast to the original capsule networks, arbitrary pose vectors can no longer be extracted.


Toward this we note that Corollary 3 holds for an arbitrary loss, and Lemma 5, and Theorem 6 hold for a sufficiently smooth loss with bounded 2nd derivative w.r.t.

w. The conversion to Bayes risk in Corollary 8 holds for any loss convex in p. the result of Corollary 8 holds more generally for any sufficiently smooth loss that has bounded 2nd derivative in w and that is convex in p. We provide an application of this more general result in the next section.


To accommodate prior information on the class labels, we generalize the previous framework so that the estimated label distribution jJ(y I Xi) has low relative entropy with the prior on the class labels.

the objective function (1) and its EM -bound (4) can be modified to be where " i,y is the prior probability for the i-th training example to have class label y.


Here, we introduce a new approach using a near-maximum entropy model, that makes this type of analysis feasible for very high-dimensional data-the model parameters can be derived in closed form and sampling is easy.

our NearMaxEnt approach can serve as a tool for testing predictions from a pairwise maximum entropy model not only for low-dimensional marginals, but also for high dimensional measurements of more than thousand units.


Since by Assumption 4.1, the diagonal entries of A are 1, both of these distributions reduces to p i = 1 n , for all i 2 {1, . . . , n}.

in the rest of the paper, we consider the RCD algorithm with uniform and independent coordinate selection at each iteration.


By doing so we can learn the relative importances of variations in appearance and scale with regard to variations in shape per se.

even knowing that we are in a near-isometric setting, we will capture the eventual variations in appearance and scale into our matching criterion in order to produce a robust nearisometric matcher.


Provided accurate image correspondences, depth can be recovered deterministically in the stereo case [5].

stereo depth estimation can be reduced to developing robust image point correspondences -which can often be found using local appearance features.


The data available for solving each learning task is assumed to be sampled from modified versions of a common multivariate distribution.

we are interested in sharing the "common pieces" of this generative model between tasks, and use the data from each individual task to detect, learn and adapt the varying parts of the model.


In their seminal paper, Spielman and Teng proved the existence of a simplex algorithm that solves σ-smooth LP in polynomial time (polynomial also in σ −1 ).

vershynin [Ver09] presented a simpler algorithm and significantly improved the running time.


So we can directly derive the convergen ce order as D,, (8) = Op (nl / 2 ) and D(f 0) = Op( nl ).

by using t he rela tion (2.4) , it follows that R,, (8 ) -L* = Op( nl / 2 ) and R(f 0) -L * = Op (nl ).5 t he TOP kernel h as a much b etter convergen ce rate in R(f 0)' which is a strong motiva tion to use the TOP kernel instead of plug-in estimate.


By the central limit theorem, z i , z j / √ n d ⇒ N(0, 1).

σ 2 z i , z j /n ≈ N(0, σ 4 /n), and we need to choose the (rescaled) threshold proportional to √ σ 4 = σ 2 .


Randomly select M training samples indexed by ξ k,1 , ξ k,2 , ...ξ k,M ; 3: Randomly select i k ∈ {1, 2, ..., n} with uniform distribution; 4: For the shared memory platform, one can exactly follow ASYSG-CON on the computer network using software locks, which is expensive 4 .

in practice the lock free asynchronous parallel implementation of SG is preferred.


Observe that since the population size of the M/M/∞ queue is unbounded, we cannot upper bound the event rates in the system.

uniformization is not directly applicable to this system.


This is because in the transformed bandit problem, there exists an arm that has a higher expected reward than arm K, and Bob will mostly pull that arm.

the heuristic constant attack has to know an unknown quantity to guarantee a successful attack.


The concept of Reisz energy also exists in higher dimensions.

generalizing this result, we choose our optimal N -point set on ∂S which tries to minimize the Reisz energy.


For any signed graph, the complete w-reductions are obtained by first using reductions (w1,w2) and then reducing the bridges (w3) because (w3) always makes the degree bigger and does not make a loop.

the following two claims complete the proof.


The semi-smooth Newton method for the solution of ( 10) can be stated as the iteration α t+1 = α t − g(α t ) −1 p(α t ) , (13) where g is a generalized derivative of p. Proposition 12 (Generalized derivative of p).

if prox h•D −1/2 is Newton differentiable with generalized derivative G, then so is the mapping p with a generalized derivative Furthermore, g is nonsingular with a uniformly bounded inverse on R. as p is Newton differentiable with nonsingular generalized derivative whose inverse is also bounded, the general semi-smooth Newton convergence theorem implies that (13) converges superlinearly to the unique root of (10).


As we optimize (7) via gradient descent, the policy changes only locally.

we can achieve better data-efficiency by restricting the exploratory actions u with (x, u) ∈ S n to be close to π n , u ∈ [π n (x) − ū, π n (x) + ū] for some constant ū. Computing the region of attraction by verifying the stability condition on a discretized domain suffers from the curse of dimensionality.


Additionally, most formulations of apprenticeship learning are actually harder than reinforcement learning; apprenticeship learning algorithms typically invoke reinforcement learning algorithms as subroutines, and their performance guarantees depend strongly on the quality of these subroutines.

these apprenticeship learning algorithms suffer from the same challenges of large state spaces, exploration vs. exploitation trade-offs, etc., as reinforcement learning algorithms.


While a naive method will include all interactions of a sampled node as the observed pairs, we can leverage network sparsity for efficiency; in many real networks, only a small fraction of the node pairs are linked.

for each sampled node, we include as observations all of its links and a small uniform sample of m 0 non-links.


Note that this is different from training a prior for x only using training data x i .

the trained network for x is confined to the specific tasks constrained by the pairs (x, y).


In practice, features from different brain regions are often extracted, but not all the regions contribute to a certain disease.

it is desirable to determine which features (regions) are the most relevant and the most discriminative to use.


Furthermore, the Gibbs empirical risk R S (G i ab ) is small when the training points are located far away from the Bayes decision surface G i ab (x) = 1/2 (with G i ab (x j ) → y j ∀j ∈ i).

the Gibbs classifier with the smallest guarantee of risk should perform a non trivial margin-sparsity tradeoff.


The first formulation, termed non-stationary model, is to treat the system as having infinitely many states, each visited at most once.

we consider an equivalent MDP with an augmented state space, where each augmented state is defined by a pair (s, t) where s ∈ S and t, meaning state s in the t th horizon.


Another important issue is that inpainting can take multiple candidates to fill holes.

optimal results should be constrained in a spatially variant way -pixels close to area boundary are with few choices, while the central part can be less constrained.


Taking the logarithm and eliminating constant terms, we obtain the following objective: In most cases, the same prior is used for all features, so we have β k = β for all k. This objective is convex, and can be optimized efficiently using methods such as conjugate gradient or BFGS, although care needs to be taken with the discontinuity of the derivative at 0.

in principle, we can simply optimize this objective to obtain its globally optimal parameter assignment.


5 In a digital image, both domains D and Ω are discretized into a lattice, and dt is fixed.

spatial and temporal derivative operators are approximated, typically, by first-order differences.


. . , ξ n ) and m a observations are given for each ξ a .

the model can be written as We defined s a , r, and ψ as κ is estimated with equal weight for every observation.


As expected, using a distributionally robust approach renders the logistic regression problem more 'risk-averse', which results in uniformly lower CVaR values of the logloss, particularly for smaller confidence levels.

increasing the radius of the Wasserstein ball reduces the right tail of the logloss distribution.


This feature is expressive enough to differentiate whether an object just grazes over table's edge (steep change in vertical distance) versus, it first goes up and over the table and then moves down (relatively smoother change).

the features obtained from object-environment interaction are φ obj−env (•) ∈ R 20 (3*4+2+6=20).


However, we have the added complication that the streams are only roughly aligned in time.

we also need to consider a range of time shifts between the streams.


Considering the complexity of the inverse optimization problem (IOP), it will be extremely difficult and time consuming to extract user's PR from large, noisy data sets using conventional techniques.

incorporating traditional inverse optimization into RSs is impractical for real time elicitation of user's PR.


This ensures that the K-dimensional embedded representation is contained within a ball of unit norm centered at the origin.

we have proved that with only an M -dimensional projection of the data (with M ≪ N ) we can perform ID estimation and subsequently learn the structure of a K-dimensional manifold, up to accuracy levels obtained by conventional methods.


In contrast, P b is defined to Granger cause P a when: λ a (t|H a (t)) = λ a (t|H Q (t)), where Q = P a ∪ P b .

given two processes P a and P b whose intensity functions follow Eq (3), Granger causality arises when α ba = 0 2 4 Learning GRANGER-BUSCA Recall from Figure 1 that GRANGER-BUSCA exhibits a cluster like behavior.


Moreover, we find that removing instance normalization from the discriminator at that stage is helpful as we do not want its final prediction to be influenced by zero values coming from the background.

we update the adversarial energy L adv of Equation 2 to: Algorithm 1 summarizes the training procedure for learning F S→T ; training F T →S is similar.


This invariance must be taken into account in our analysis.

we assume the weights are rescaled such that i w * i = 1.


By this definition, it can be seen that the entry (t, Thus, the entries of matrix A are multivariate polynomials of entries of B.

furthermore, where A S and A Y are the rows corresponding to set S and Y in matrix A, respectively, and matrix ⇤ i is a diagonal matrix where the entries of vector      )), then we will reject the null hypothesis H ij 0,Y,S .


In contrast to Caltech101, O2P with SQ performance is similar to our implementation of Bag-of-Words (51.14%).


The amount of training patterns used to learn the prior seems to be a key aspect in the goodness of this prior and thus in the tightness of the bound.

ongoing research includes methods to systematically determine an amount of patterns that provides with suitable priors.


Second, the shrinkage step will in general not satisfy the all-1-row constraints in (3).

we add a third projection step at the end of each iteration to project Z k+1 back to the feasible region, by simply setting its last row to all 1's.


In other words, the optimal solution of the module k is dependent on the output of the upper modules.

minimizing the problem (1) with the backpropagation algorithm is equivalent to minimizing the problem (4) with the first K − 1 subproblems obtaining optimal solutions.


Indeed, in each synapse, the eligibility trace is generated by a forward model of the closed-loop system that is exclusively driven by the basis signal.

our main result is that by deriving a gradient descent algorithm for the CFPC cerebellar control architecture we have obtained an exact definition of the suitable eligibility trace.


Next, we show that π λ1 λ2 (•) can be directly derived from π 0 λ2 (•) by soft-thresholding.

we only need to focus on the case when λ 1 = 0.


provides an exact solution for the ground state of a 2D lattice Ising model without a ferromagnetic field [13,8,9,10] in O(N Computing a lower bound: At a given iteration, prior to adding a newly generated set of constraints we can compute the total residual constraint violation over all layers of hierarchy by ∆ = l (θ l + λ l + ω l−1 − ω l ) • z l . In the supplement we demonstrate that the value of the dual objective plus 3 2 ∆ is a lower-bound on the relaxed closest ultrametric problem in Eq 9.

as the costs of the minimum-weight matchings approach zero from below, the objective of the reduced problem over Ẑ approaches an accurate lower-bound on optimization over ΩL Expanding generated cut constraints: When a given cut z l produces more than two connected components, we found it useful to add a constraint corresponding to each component, following the approach of [20].


Assume that the second moment of the rewards of each arm are bounded uniformly as E X 2 i ≤ V and let c > 0 be arbitrary.

then, the regret of Boltzmann-Gumbel exploration satisfies Notably, this bound coincides with that of Theorem 3, except that σ 2 is replaced by V . by following the proof of Corollary 1, we can show a distribution-independent regret bound of order √ KT log K. Let us now present the proofs of our main results concerning Boltzmann-Gumbel exploration, Theorems 3 and 5.


Similarly, the distance between hyperspheres of dissimilar examples is γ−1 = min v,v ,v =v v − v 2 − γ1 .

even if we do not use cannot-link constraints our algorithm is able to push reasonably far away dissimilar examples.


Using the law of total probability, we have the following equation for the filtering distribution: We can think of the integral in (5) as a sum over a distribution of experts, where each expert corresponds to a single pose history, u 1:t−1 .

based on its hypothesis about pose history, each expert has an opinion about the current pose of the object, U t , and the texture maps of the object and background, V t and B t . Each expert also has a credibility, a scalar that measures how well the expert's opinion matches the observed image y t . (5) can be interpreted as follows: The filtering distribution at time t is obtained by integrating over the entire ensemble of experts the opinion of each expert weighted by that expert's credibility.


Each (z, t) 2 Q denotes the (j, t) entry of Z where j = iters[t].current() and z = z jt = h jt w t . Note that there are most k elements in the max-heap at any time.

we can implement Q by a binary heap such that 1) Q.top() returns the maximum pair (z, t) in O(1) time; 2) Q.pop() deletes the maximum pair of Q in O(log k) time; and 3) Q.push((z, t)) inserts a new pair in O(log k) time.


Similarly to [12], treating each block of columns I j in the BCD-IC algorithm is equivalent to updating the elements outside the active set Active Ij (A), followed by an update of the elements in Active Ij (A).

in (15), we set where the step index t corresponds to the block j at the iteration k of BCD-IC.


Given that the number of inducing variables determines the amount of the approximation, or compression of the DPP model, we examine different settings for this number and see whether the corresponding trained models provide similar estimates for the overdispersion measures.

we train the DPPs under different approximations having m ∈ {50, 100, 200, 400, 800, 1200} inducing variables and display the estimated overdispersion measures in Figures 3(a) and 3(b).


It contains two types of edges: 1) entity-entity edge connecting every two entity variables (y i and y j for 1 ≤ i = j ≤ n. 2) entity-relation edges connecting every relation variable y k (where k > n) to its two entity variables.

our graph is not a complete graph and our goal is to design an architecture that will be invariant to any automorphism of the graph, such as permutations of the entity variables.


by the null hypothesis H 0 , we can derive a global evidence score on the hypothesis that Q was generated from ref via the segmentation S (for simplicity of notation we use the symbol H ref also to denote the global hypothesis): Namely, the global evidence induced by S is the accumulated sum of the local evidences provided by the individual segments of S. The statistical significance of such an accumulated evidence is expressed by: Consequently, we can accumulate local evidence of non-overlapping regions within Q which have similar regions in ref for obtaining global evidence on the hypothesis that Q was generated from ref .

for example, if we found 5 regions within Q with similar copies in ref , each resulting with probability less than 10% of being generated by random, then the probability that Q was generated by random is less than (10%) 5 = 0.001% (and this is despite the unfavorable assumption we made that the rest of Q was generated by random).


It is known that the joint distribution induced on V by a structural causal model factorizes with respect to the causal graph.

the causal graph D is a valid Bayesian network for the observed joint distribution.


However, in this form we see that p(v j |u σ(t) , π 1:t−1 ) is directly related to the change in the partial energy from π 1:t−1 to H(v j , u σ(t) , π 1:t−1 ) -the larger the change the bigger the resulting probability will be.

the matching choices will be made solely based on the changes in the partial energy.


2 represents a mixture of 15 characteristic timescales.

the top 10 orthonormalized dimensions can make use of up to 15 timescales.


On the other hand, based on general techniques developed in [15] and greatly extended in [1], it can be shown that online learnability is information-theoretically possible for such W. However, these techniques do not provide a computationally efficient algorithm.

to the best of our knowledge, there is currently no efficient (polynomial time) online algorithm, which attain non-trivial regret.


(ii) According to (8), smaller requires smaller error ✏ for solving discriminator to obtain the same convergence guarantee.

solving the discriminator problem requires more effort.


The construction we give for SGs may replace undirected edges by directed edges in a way that may break the symmetry of the underlying interference relationship.

directed edges in a SG do not have a straightforward causal interpretation.


Note, however, that computing either ι tv or the mutual information is as difficult as inference itself.

we define a computationally efficient variant of influence that we term the influence trail score.


This will introduce zero elements on the diagonals of the metrics, which will in turn deselect the corresponding feature dimensions, because the corresponding rows and columns of those elements are necessarily zero due to the positive semidefinite constraints.

metrics that have orthogonal diagonal vectors will use non-overlapping subsets of features.


In signal detection theory, the sensitivity index, conventionally denoted d , is a measure of how well the annotator can discriminate the two values of z i [14].

it is defined as the Mahalanobis distance between µ 0 and µ 1 normalized by s, the lower σ j , the better the annotator can distinguish between classes of z i , and the more "competent" he is.


Intuitively, this occurs because the now finite spines of the prior p(w; H), which bound the variational approximation, do not allow us to place infinite prior density in any region of weight space (as occurred previously when any γ i → 0).

if any γ i goes to zero with a, b > 0, the associated approximate prior mass, and therefore the approximate evidence, must also fall to zero by ( 16).


Without any specific form of f ✓ , it is difficult to pick a worst case ✓ that will produce a tight bound.

in this section we consider single parameter exponential families (defined formally below) to provide us with a class of distributions in which we can reason about different possible values for ✓.


Second, it can be proved that the pairwise losses in Ranking SVM, RankBoost, and RankNet, and the listwise loss in ListMLE are all upper bounds of the essential loss.

we come to the conclusion that the loss functions used in these methods can bound (1−NDCG) and (1−MAP) from above.


We add the point x as another training point with weight 10 times larger than the weight of all training points combined.

the penalty for misclassification of x is very large and the optimizer finds a solution that doesn't violate the constraint.


For instance, object points used for SfM may be visible only in some cameras and only in particular object poses.

different nodes will be frequently exposed to missing data.


Each neuron that spikes at time step k will randomly evoke, with probability W ij , one recurrent excitatory post-synaptic potential (EPSP) at time step k + 1, after some network delay.

we define the number of recurrent EPSPs received by neuron l in the j-th sub-population as a j l . a j l is the sum of N k independent (but not identically distributed) Bernoulli trials: where The sum a j l follows the so-called "Poisson binomial" distribution [22] and in the limit approaches the Poisson distribution: The detailed analysis of the distribution of a i l and the proof of equation 6 are provided in the supplementary materials.


An egocentric map is altered dramatically after changes in viewpoint, whereas an allocentric map is not.

the PH and hippocampal representations could evolve via an unsupervised learning procedure that discovers a temporally stable, generative model of the parietal input.


Likewise, influence is proportional to variance, which increases quadratically with scalar coefficients.

we recommend against setting and δ to a value much higher than 0.05.


However, for many GPs -and ordinary kriging in particular -the correlation between two points decreases as a function of distance.

the full GP model can be approximated well by a local GP, where only the k nearest neighbors of the query point are used to compute the prediction value; this reduces the computation time to O(k 3 log(n)) per prediction, since O(log(n)) time is required to find the k-nearest neighbors using spatial indexing structures such as balanced kd-trees.


By mapping these short reads to databases of known bacterial species, we can deduce both the source species and gene from which each short read originated.

upon mapping a collection of different samples, we obtain a matrix of estimated relative species abundances for each person and a matrix of the estimated relative gene abundances for each person.


Therefore, both z +ε = (y + ε, x 1 , ..., x d ) and z −ε = (y − ε, x 1 , ..., x d ) are, with probability α, on the correct side of the regression surface, defined by z = (ŷ, x 1 , ..., x d ).

z +ε differs from z by at most +ε in the first dimension, and z −ε differs from z by at most −ε in the first dimension.


Lastly, note that we derived (14), an unbiased estimator for MSE, assuming a fixed θ.

there is no guarantee that the resulting estimator (denoiser) that is tuned by SURE will be unbiased [38].


By Theorem 1, our policy is learned directly by minimizing the discrepancy between π E and π θ which has a closed form expression.

we convert the original IRL problem to a minimization problem with only one set of parameters with respect to the policy.


This root finding procedure will take no more than O@nA steps in the worst case.

stepping beyond a single threshold potential, we now assume that the submodular function to be minimized can be written as a nonnegative linear combination of threshold potentials and a modular function, i.e., f @AA a c ¡ e A C j d j © wj;yj @AA: we have the smoothed Lovász extension, and its gradient: f @xA a c ¡ x C j d j © wj;yj @xA and r f @xA a c C j d j r © wj;yj @xA: We now wish to use the accelerated gradient descent algorithm of [18] to minimize this function.


When the value of V w decreases (weight increase) the current through P5-P6 increases, but P5 is maintained in the linear region by the high gain transistor.

a current proportional to the value of the weight is subtracted from I bpot .


However, in many real-world applications, collecting paired training data is laborious and expensive [37].

in many applications, there are only a few paired images available or no paired images at all.


Assume W has been first normalized to have unit-norm columns, w i , w j is essentially the (i, j)-the element of the Gram matrix W T W , and i = j requires us to consider off-diagonal elements only.

we propose the following mutual coherence (MC) regularization term inspired by (3: Although we do not explicitly normalize the column norm of W to be one, we find experimentally that minimizing (4) often tends to implicitly encourage close-to-unit-column-norm W too, making the objective of (4) a viable approximation of mutual coherence (3) 2 .


Since B = XAY for positive diagonal matrices X and Y , Lemma 1 implies B is the optimal solution to min By Lemma 7, there exists a matrix P ∈ U r ,c such that P − P * 1 ≤ 2 ( r − r 1 + c − c 1 ).

moreover, since B is an optimal solution of (3), we have by Hölder's inequality where we have used the fact that 0 ≤ H(B), H(P ) ≤ 2 log n. Lemma 7 implies that the output P of ROUND(B, U r,c ) satisfies the inequality B − P 1 ≤ 2 ( r − r 1 + c − c 1 ).


The expected profit function Π(r, c) is quasi-concave in r for each c. The previous result shows that the reserve price of the single-period constrained model is larger or equal than the one of the multi-period constrained model.

in the multi-period constrained model items are allocated more frequently and the social welfare is larger.


Our model assumptions based on the PL model imply some regularity properties for the pairwise marginals, such as strong stochastic transitivity and stochastic triangle inequality (see Appendix A of [28] for the proof).

the INTERLEAVED FILTER [28] and BEAT THE MEAN [29] algorithms can be directly applied in our online framework.


Further, the analysis in these papers seems to have gaps (see Neu et al., 2010).

currently, the only result for the case considered in this paper is an asymptotic "no-regret" result.


Note that for distinct agents j and k, Opt j and Opt k are disjoint.

by summing over all agents, we can upper-bound the value of Opt by n i=1 α i + m j=1 β j .


The Radon point in each of the h iterations can then be calculated in parallel in time r 3 (see Appendix C.1).

the runtime of the Radon machine with N = r h n samples is T R,h (N ) = T L (n) + hr 3 .


We can then recover the familiar upper bound on the Rademacher complexity: R n (F)  p Tr(k)/n.

but, when , as in the case of Gaussian kernels [14], then by Theorem 8 (shown in the supplemental material), we have , which yields a much stronger learning guarantee.


Different from segmentation, the filter kernels in our network only eliminate the corruptions, which is not sensitive to the orientation of image contents.

we can rotate and mirror flip the kernels and perform forward multiple times, and then average the output to obtain a smoother image.


However, most existing MAB algorithms only focus on the identification of the best arm; if naively applied to search advertising, there is no guarantee to get an accurate estimation for the click probabilities of the top two ads.

search advertising, with its special model and goals, merits specialized algorithmic design and analysis while using MAB algorithms.


Taking calibration measurements is a very costly process, in particular if larger areas need to be covered.

one is very much interested in keeping the number of calibration points as low as possible.


In each point v ∈ V, we have a probability distribution p atlas (•|v) ∈ R T ×A which informs about the tissue type and the atlas region in v. Without any loss of generality, one can assume that the tissue information is encoded in the atlas.

we consider the probability p atlas (•|v) ∈ R A .


Note that private data is usually small and private functions are fast to compute.

the function evaluation is mainly affected by the amount of public data.


Moreover, decades of painstaking work have revealed a plethora of RGC types [16].

it is natural to characterize these cells in terms of their type and the location of their receptive field center.


The § ¡ 43 ¢ -mismatch kernel is based on a feature map to a vector space indexed by all possible subsequences of amino acids of a fixed length ¡ ; each instance of a fixed ¡ -length subsequence in an input sequence contributes to all feature coordinates differing from it by at most ¢ mismatches.

the mismatch kernel adds the biologically important idea of mismatching to the computationally simpler spectrum kernel presented in [14].


The neurons encoding specific features in one dataset are not affected much during training on another dataset, whereas neurons encoding common features can be reused.

lWTA may be a step forward towards models that do not forget easily.


Compared to the binary case, the classConstraints only grows linearly by the number of classes, while the majority of the constraints are the ones encoding the activations.

the time should not grow significantly, as we shall see in subsection 4.2.


For a solution to be optimal, the Karush-Kuhn-Tucker (KKT) conditions must be satisfied [3].

which can be rearranged as For being a probability measure, the variables π ijk have to satisfy Eliminating λ ij by jointly considering Equations ( 5) and ( 6), we obtain a labeling model in the exponential family: Plugging Equation ( 7) into (4) and performing some algebraic manipulations, we prove Theorem 2.1 Let π ij be the maximum entropy distributions in (3).


Our sum assumption corresponds to a situation where the feature map Φ(x) and feature space F for k is the concatenation of the feature maps Φ v (x) for each kernel k v , i.e, F = v∈V F v and Φ(x) = (Φ v (x)) v∈V .

looking for a certain β ∈ F and a predictor function f (x) = β, Φ(x) is equivalent to looking jointly for β v ∈ F v , for all v ∈ V , and As mentioned earlier, we make the assumption that the set V can be embedded into a directed acyclic graph.


We first notice that, by using (4) in (1), we obtain q(h 1:S 1:M ) = i q(h i 1:M ).

we then observe that we can rewrite the update for q(h i 1:M ) as proportional to the joint distribution of the following linear gaussian state-space model (LGSSM) where inference on q(h 1:S 1:M ) can be accomplished with LGSSM smoothing routines [7].


By comparison, for the greedy heuristic, a training example (X (i) , Y (i) = α k ) contributes log(λ k f k (X (i) )) to w(k, k ).

the greedy estimator can be seen as a "soft" version of MV that also greedily optimizes the MLE objective.


Our methods usually takes about 10 iterations to get convergence.

our methods are computationally more efficient than the AdLasso and MCP.


We can store X t as a decomposition into at most rank(X t ) ≤ kt rank-1 components.

6 for If we always use the first running time in Theorem 4.3, then Fact 4.4 implies: which roughly equals k-times the 1-SVD running time Õ nnz(∇) ∇ 2 /ε) of FW and Garber [6].


Running the observational learning algorithms like PC/IC after an intervention on a set S of variables, we can learn the new skeleton after the intervention, which allows us to identify the immediate children and immediate parents of the intervened variables.

if we perform a randomized experiment on a set S of vertices in the causal graph, we can learn the direction of all the edges cut between S and V \ S. This approach has been heavily used in the literature [13,10,31].


In practice, the maximum subregion density will be lower than our computed value of D max more often, since our estimates for d max and q are conservative.

though our algorithm is approximate, it is very likely to converge to the globally optimal mdr.


S(L) → 1 as m → ∞, then Rtrue − Remp converges to zero uniformly across all parametric loss functions.

algorithmic stability is sufficient for uniform generalization.


Because H is closed under union and finite, H 0 must be in H. If ∃z ∈ H 0 ∩ Z, then z ∈ h ∩ Z for some h ∈ M(Z).

this is a contradiction, because all h ∈ M(Z) are consistent with Z. H 0 is consistent with Z, and is in M(Z).


Designating by an overbar the average over randomly chosen states x and x , the mean C and variance δC 2 of the inner product are This proves that in the limit of large n and M , different columns of the random perceptron measurement matrix have inner products that concentrate around 0.

the matrix of inner products is thus orthonormal almost surely, Â Â → I. with enough measurements the random perceptrons asymptotically provide an isometry.


Program (P1) is equivalent to sequential CS.

we expect the behavior of conventional CS to persist for larger K.     We consider the problem of recovering the first 10 frames of a real video using our dynamic CS algorithm, and conventional CS applied to each frame separately.


However, CDL is a feedforward model that uses bag-of-words as input and it does not model the order-aware generation of sequences.

the model would have inferior recommendation performance and is not capable of generating sequences at all, which will be shown in our experiments.


Unlike in the convex case, in the non-convex case one can not expect a gradient-based algorithm to converge to the global minimum if only smoothness is assumed.

instead of measuring functionvalue suboptimality Ef (x) − inf x f (x) as in the convex case, convergence is generally measured in terms of the squared norm of the gradient; i.e., E ∇f (x) 2 .


We consider several types of tv-embedding learning task trained on unlabeled data; e.g., one task is to predict the presence of the concepts relevant to the intended task (e.g., 'desire to recommend the product') in the context, and we indirectly use labeled data to set up this task.

we seek to learn tv-embeddings useful specifically for the task of interest.


Both problems are known to be NP-hard [13,14].

by the chain rule and nonnegativity of MI, for any A ⊆ S. maximizing unfocused MI does not imply maximizing focused MI.


In general detection theory terms, the control neuronal response sets some locus in a high dimensional space, then the distorted neuronal response will project near that locus if it is perceptually equivalent, or very far away if it is not.

the distance between the control neuronal response and the distorted neuronal response is a function of intelligibility.


For BIC we use the G-Wishart posterior mode ΩG as the plug-in estimate, since the MLE is undefined for n < d. But we use a vague and proper prior (δ 0 = 3).

all three approximations will require finding the mode of a G-Wishart (for the posterior and/or the prior).


[30], such that where p T (x k ) and p T (x k , x l ) are the marginals of p T (x).

after running TRW-BP, we can calculate p T (x) via Because TRW provides a convex upper bound, it is often well-suited to the inner loop of learning algorithms [e.g., 28].


For α to be closed, it must satisfy the closure condition 2π 0 e jθ(s) ds = 0.

we represent curves by direction functions satisfying the average-π and closure conditions; we call this space of direction functions D. Summarizing, D is the subspace of L 2 consisting of all (direction) functions satisfying the constraints It is still possible to have multiple continuous functions in D representing the same shape.


As to the definition of B −1 z , the feasible path of each segment must start with several blanks, followed by the element corresponding to that segment more than once.

we can find the maximum of entropy for every segment According to Equation 2 and the definition of segmentation, the entropy of feasible paths satisfying a specific segmentation sequence can be disassembled into the summation of the entropy of segments.


However extracting such shaped rewards from the simulator is very challenging and not feasible for large number of agents.

we apply this reasoning to the critic (or action-value function approximator) Q w (n sa , d).


The two endpoints of any edge in the graph form a vertex cover of size k = 2, but C could conceivably be any edge.

under these constraints, the only set guaranteed to contain C is the entire node set V . In this negative example, once we put v 1 into a 2-node vertex cover, the other node can be arbitrary, since it is superfluous.


Since Theorem 3.4 ensures that x n visits B(x * , ε) infinitively often (almost surely), the hypothesis is guaranteed to be true.

this leads to the following claim: ∀ε > 0, ∃τ 0 (a positive integer), such that: Figure 3: SMD run on the objective function of Fig.


We observe in passing, that for any K ∈ K 1 , and for any independent set I in the graph, taking since for each term in the second sum, either (i, j) is an edge, in which case either α i or α j is zero, or (i, j) is a non-edge in which case K ij ≤ 0.

like ϑ(G), the Delsarte version ϑ 1 (G) is also an upper bound on the stability or independence number α(G).


Theorem 4 and Lemma 12 give an O( √ T log T )-upper bound of the regret of function optimization in Algorithm 1 under the same conditions as Theorem 4.

given the convexity of f , the average of the chosen actions of any dueling bandit algorithm āT : if an optimization algorithm has a sub-linear regret bound, the above online-to-batch conversion guarantees convergence to the optimal point.


Nonetheless, these operations are usually expensive to compute.

what is surprising about BayesOpt for approximate inference is that it disregards most of the computed posteriors and keeps only the posterior p(θ | X, η) that optimizes the marginal likelihood.


In this example, the gradients are backpropagated through the opposite direction of solid lines.

the delayed error 8 is passed to steps 5, 6 and 7.


F P R = P (C = 1|T ) and 1 − F N R = P (C = 2|T ).

we have: Similarly, when log R(T ) reaches the target absent threshold (Θ 1 ), we have: given desired FPR and FNR, we can analytically compute the upper and lower thresholds for the Full Bayesian model using Eqn.


With T = 2, simultaneous one has another 1% boost.

simultaneous optimization substantially outperforms alternative one in terms of accuracy and speed.


The slope of the regression line through the origin is 0.97.

the model shows a key signature of the behavioral data-symmetric blocking effects (Phenomenon P2).


In turn, μtgt is determined at the point in the diffusion process when a response would be initiated.

the model claims that selecting a response on trial k is key to influencing performance on trial k + 1.


In PSRL, each agent k samples an MDP M k,0 from the common prior at time t k,0 and computes the optimal policy π k,0 (•) with respect to M k,0 , which does not change throughout the agent's interaction with the environment.

the PSRL agents do not adapt to the new information in real-time.


The amplitude b (k) c 2 R + and phase shift c 2 [0, 2⇡] are channel-specific.

the convolutional filter in each channel will be a discretized version of a scaled and rotated Morlet wavelet.


We are more interested in investigating the underlying causal structure among the variances of the observed data.

in this paper, we consider the situation where the correlation in the variances of the observed data is interesting.


the basic block K tq can be recursively calculated in three different ways: ) by adding Eq.


As with the speech signals discussed in section 2.2, the value becomes smaller as ω increases, and thus, the gain of W (ω) tends to decrease.

the same frequency compensation scenario as found in section 2.2 may again be useful for the MMSE based dereverberation scheme.


Equivalence with the ECOC decision rule thus follows from the fact that 3 is the Hamming distance between the appropriate row of the coding matrix and the concatenation of the bits returned from the binary classifiers.

with the appropriate definitions of W 2 and o , the conditional model on the ranking poset is a probabilistic formulation of ECOC that yields the same classification decisions.


As discussed in Section 1.1, these methods can be viewed as different ways to optimize (5).

in addition to the normalized mean square error (nMSE) [19] in prediction, we report KUKA1 -Variational Lower Bound (10 the performance in the variational lower bound (VLB) (5), which also captures the quality of the predictive variance and hyper-parameter learning.


Here, we mean by sparsity that VB exhibits the automatic relevance determination (ARD) effect [19], which automatically prunes irrelevant degrees of freedom under non-informative or weakly sparse prior.

it is naturally expected that VB-LDA also produces a sparse solution (in terms of topics).


This implies that P r and P s have disjoint supports and are perfectly separable in theory.

there exists a 'trivial' discriminator which looks at each token independently and discriminates based on whether a sequence consists of one-hot vectors or not.


For Gaussian designs, which we also consider, their results guarantee that for n = O (s log d), where s is the sparsity index of the regressor w * .

note that for k = Θ(n), the right hand side behaves as Ω σ √ log n . the result is unable to ensure We have excluded some classical approaches to the RLSR problem from the table such as [18,1,2] which use the Least Median of Squares (LMS) and Least Trimmed Squares (LTS) methods that guaranteed consistency but may require an exponential running time.


[1] We consider a setting in which an online learning algorithm (also called an adaptive control algorithm) is used to design a controller.

on each time step t, an online algorithm may (based on the observed states and action sequence so far) propose some new gain matrix K t . If we follow the learning algorithm's recommendation, then we will start choosing controls according to u = K t x.


The monotonicity of ≤ allows us to detect suboptimal strategies during variable elimination.

to illustrate this, consider the variable elimination scheme in Figure 2 for two different strategies s and s , and let ψ s 1 , ψ s 2 , ψ s 3 be the valuations produced in the propagation step for strategy s and we can abort variable elimination for s after the second iteration.


Learning In AQM, the answer-generator network in the questioner and the answerer does not share the representation.


In addition, this method causes less noise during training, and the noise that is caused is more gentle as the law of large numbers ensures that µ[t] and µ[∇ t] are approximately normally distributed.

the added noise has much lighter tails than the highly kurtotic noise caused by the minibatch estimate of the variance used in full batch normalization.


We have proved that this can be accomplished in hardware in the address domain by presenting the network with stimulus patterns containing a set of correlated inputs and a set of uncorrelated inputs: neurons x 1 . . . x 17 are all stimulated independently with a probability of 0.05 per unit of time, while neurons x 18 . . . x 20 have the same likelihood of stimulation but are always activated together.

over a sufficiently long period of time each neuron in the input layer will receive the same amount of activation, but the correlated group will fire synchronous spikes more frequently than any other combination of neurons.


The safe set after n such expansions can be found by repeatedly applying the operator in (4): Ultimately, the size of the safe set is bounded by surrounding unsafe states or the number of states in S. As a result, the biggest set that any algorithm may classify as safe without visiting unsafe states is given by taking the limit, R ✏ (S) = lim n!1 R n ✏ (S).

given a tolerance level ✏ and an initial safe seed set S 0 , R ✏ (S 0 ) is the set of states that any algorithm may hope to classify as safe.


Our main goal is to bring tools from the machine learning community to solve this problem more efficiently.

we propose two algorithms to quickly and accurately estimate consumer preferences.


This implies that X 2 is the parent of X 3 .

the structure in Figure 1(b) is distinguishable using the proposed identification approach.


Further, observe that it was fairly painless to incorporate the previously known recharge while avoiding lava constraints.

in contrast to quantitative Markovian rewards, learning Boolean specifications enabled encoding compositional temporal specifications that are robust to changes in the environment.


Observe that W bag for a negative bag is always −1.

the weight for a negative instance, p ij , is the same that would result in a non-MIL AdaBoost framework (i.e.


For k ∈ {1, . . . , n}, define the U-statistics where the sum is over all S ⊂ {1, . . . , n} such that |S| = k. By Hoeffding's strong law of large numbers for U-statistics [19], for any k ∈ {1, 2, . . . }.

using Equations 3.1 and 3.2 we have that for any K ∈ {1, 2, . . . } and any n > K, R 2 (X 1:n ) = n 3/2 where H K is the K th harmonic number, and the last inequality follows from the standard bounds [20] on harmonic numbers: log K < H K ≤ log K + 1.


In this example perfect integration over p(x|N ) gives a distribution of width ≈ 3 over log( Ẑ).

improvements over (c) for approximating equation ( 5) are unwarranted.


of ( 4) is agent i's risk-neutral probability (Pennock [1999]) of {X = 1}, given her utility function, her action, and the current market state.

lemma 3 is a natural extension of the elicitation properties of a MSR.


Usually, we have m d and d > n. Similar to SMV, we can use W p,0 to measure the number of rows in W that contain nonzero entries.

the problem of finding the sparsest representation of the signal W in MMV is equivalent to solving the following problem, a.k.a.


On the other hand, since P * k X i , i = k, will be small and D * k is not trained to reconstruct X i , the residual F . In the testing phase, if the query sample y is from class k, its projective coding vector by P * k (i.e., P * k y ) will be more likely to be significant, while its projective coding vectors by P * i , i = k, tend to be small.

the reconstruction residual y − D * k P * k y 2 2 tends to be much smaller than the residuals y − D * i P * i y 2 2 , i = k. Let us use the Extended YaleB face dataset [22] to illustrate this.


A state-ofthe-art approach is the low rank model [16,12] where one can first conduct matrix completion on adjacency matrix and then use the sign of completed matrix for relationship prediction.

if features of users are available, we can also consider low rank model by using our model for matrix completion step.


The key challenge in generating graphs is that sampling directly from a joint distribution over all configurations of labeled nodes and edges is intractable for reasonably sized graphs.

a generative model must decompose this joint in some way.


Let {B h | h = 1, ..., H} be a set of background patches surrounding the target I 1 in frame I t−1 .

the weight of S α is defined as the smaller the values of S α , the larger is the weight ω α .


By measure concentration, it can be shown that holds with high probability whenever m ≥ Ω (n poly log n).

to show iterative contraction of both methods, based on ( 12) and ( 13), it is sufficient to show that for some constant η ∈ (0, 1), where θ = arg min θ∈[0,2π) z − xe iθ such that e iθ = x * z/ |x * z|.


As mentioned before, GDML does not scale well with the number of atoms and training examples.

it cannot be trained on 50,000 examples.


The scalar h 1 and h 2 are uniformly distributed in the interval [10,20].

most of the energy of the signal can be described by the sine function and most of the discrimination power is in the cosine function.


Thus, the designs that we found and the data we have collected in these experiments are not necessarily optimal for discriminating between, say, a power model and a logarithmic model.

aDO is best used as a tool for confirmatory rather than exploratory analyses.


DR-GAN is evaluated in a different manner: the feature vectors are directly extracted by their model.

no extra feature extractor is needed for DR-GAN.


If the dissimilarity measure is a true distance, the neighborhood averaged vector Φ nbd (x) stays inside the convex hull of the vectors in its neighborhood, all the neighborhood vectors stay within region R. In general, the cluster contracts inside R under the averaging operation.

under the new representation, different clusters can become better separated from each other.


While luminance dominates the structure of the visual world, color vision provides important additional information about our environment.


Also, notice that w i , which is the minimizer of the sample obtained by discarding the i'th example, is independent of z i .

we have Furthermore, we can write Plugging these expressions into Eq.