Basically, the t-entropy is a normalized version of the Tsallis entropy,  One can recover the SBG entropy by setting t = 1.0.

we can equivalently rewrite the Rényi Entropy as: The t-entropy of p(x) (when t � = 1) is equal to when α = t, When t and α → 1, both entropies go to the SBG entropy.


Theorem 8 (Expected Uncertainty Sampling Gradient).

if Assumptions 2, 5, 6, and 7 hold and θ ∈ Θ regular and b(θ) = 0, then if z (t) is chosen via uncertainty sampling with parameters θ, similar to how random sampling yields preconditioned SGD on the population surrogate loss L (9), uncertainty sampling yields preconditioned SGD on the population zero-one loss Z.


Lemma 1 states that the screening matrix Φ = A SIS X for SIS will eventually converge to the covariance matrix Σ in l ∞ when n tends to infinity and log p = o(n).

the screening performance of SIS strongly relies on the structure of Σ.


Although the six sets of tasks were chosen by Wolfe to represent key experiments in the literature, most are abstractions of the original experimental tasks because the retina of GS-and its descendent EGS-is greatly simplified and cannot accommodate the stimulus arrays used in human studies.

wolfe never intended to quantitatively model specific experimental studies.)


As noted by Sarpeshkar in describing the properties of analog and digital computing circuits, "In a cascade of analog stages, noise starts to accumulate.

complex systems with many stages are difficult to build.


In other words, as ˆ t converges to ⇤ , G t and H t are both consistent estimates of the Fisher information.

by using dropout instead of L 2 -regularization to solve linearized problems in online learning, we end up with an AdaGrad-like algorithm.


the second moment of w can be expressed as follows in terms of the Rényi divergence: the variance of w is given by σ For any hypothesis h ∈ H, we denote by R(h) its loss and by R w (h) its weighted empirical loss: We shall use the abbreviated notation L h (x) for L(h(x), f(x)), in the absence of any ambiguity about the target function f . Note that the unnormalized importance weighting of the loss is unbiased: The following lemma gives a bound on the second moment.


8 Finally, the constraint on dual variables ⌫ i1,...,i N are automatically enforced by the inherent log-barrier, ln( ⌫ i1,...,i N ), in (5) [6].

we do not need a separate method to enforce the constraint.


Let Q j be the set of the j-th rows of all matrices in Q, and take A j = span(Q j ) ∩ P d . Then any vector m ∈ A j is affine combination of vectors in Q j and, thus, m T l(η) = l(η).

if span(Q i ) has dimension n i , we can take a basis V i ∈ Q i of n i linearly independent vectors such that A i = span(Q i ) ∩ P d . By construction l(η) = (V T ) −1 U l(η).


Finally, we notice that V (0; w, λ, I, J) merely corresponds to the cost (3) at s−1 t=1 α t G t (•).

we have shown that taking a step α s decreases the cost (3).


The multi-location Bayesian observer may be computed by observing that the target-present event is the union of the target-present events in each one of the locations, while the target absent event implies that each location has no target.

the likelihood can be computed as the weighted sum of local likelihoods at each location in the display.


As one would expect, pairs of features directly linked by causal relationships are correlated in the common-cause schema F I is correlated with its effects and in the common-effect schema F4 is correlated with its causes.

cMT predicts that combinations of features serve as evidence for category membership to the extent that they preserve these expected correlations (i.e.


In contrast to USPS, where the classification error on the training data without regularization was zero, on IAM, the state-classification error on the training data ranges from forty to sixty percent.

the impact of regularization on the performance of the classifier is only minor.


The RGS (m,k) algorithm is useful in situations where the amount of time to produce an answer may be limited and/or the limit unknown beforehand: it is easy to show that the growing phase of GS (m) produces an an upper-bound of the m-boundary of X.

the RGS (m,k) algorithm, if interrupted, will return an approximation of this upper bound.


Evidently, the target epochs with a TPT< 0.5 sec display a classification accuracy that approximates chance performance.

the target epochs with TPT< 0.5 sec, constituting about 20% of all target epochs in a RC code, do not appear to be useful for transmission [10].


However, the number of variables and constraints does not increase with the number of data points to be labeled.

the TSVM and the STSVM can be applied to image segmentation problems, where a large number of pixel labels have to be assigned simultaneously.


A multi-step test can say more about the current state than a schema, but is not as useful for regression planning because there is no way to extract the probability that a particular one of its observations will be obtained.

pSRs are more useful as Markovian state for reinforcement learning, while schemas are useful for explicit planning.


We prove that there cannot exist any single pass streaming algorithm that can achieve a non-trivial approximation  of SSC, using sublinear memory, if the utility have to be met exactly.

we develop an efficient approximation algorithm, ESC-Streaming, which finds solution sets, slightly larger than the optimal solution, that partially cover the desired utility.


However, when there are d outputs, the rank of L is d and the method trains ridge regression d times, but the kernel PCA step first decorrelates the outputs.

in the special case of multiple outputs regression with a linear kernel , the method is also related to the work of [2] (see [4, page 73] for an overview of other multiple output regression methods.)


We note that invariance is a special case of the condition of independent changes, as a constant is independent from any variable.

our idea introduced in Subsection 2.1 can be applied to the case of existence of invariant parameter across domains.


By integrating against this measure, we obtain a new probability density which is denoted by where f b is a particular enumeration of the densities in the support of Λ and λ b is the probability of the bth density.


Then, for k uniformly sampled from {1, . . . , K}, the following holds: where θ * is a minimizer of U (Q) and G γ is the gradient mapping We note that computing the Lipschitz constant β is out of reach.

we choose γ small enough such that the algorithm converges.


Under conditions (5) of Theorem 1, T 2 = o p (1).

to show that 1 λ n V ∞ / 2 < 1 with high probability, it suffices to show that T 3 < γ with high probability.


From Lemma 2, we know that Γ is a metric graph.


To obtain a risk bound for G i ab , we need to find a closed-form expression for R S (G i ab ).

for this task, let U [a, b] denote the uniform distribution over [a, b] and let σ i a,b (x) be the probability that a ball with center x i assigns to x the class label y i when its radius ρ is drawn according to U [a, b]: Consequently, the risk R (x,y) (G i ab ) on a single example (x, y) is given by G i ab (x) if y = 0 and by 1 From this expression we see that R S (G i ab ) is small when G i ab (x j ) → y j ∀j ∈ i.


It is important to realize that although the mapping is one-to-one, we still need to satisfy the constraint in equation 4 because a volume element dx will change under the mapping.

wormholes are sampled from P EQ and proposed moves are accepted according to equation 6.


Also, X y(x) represents the set of samples that have labels equal to the label of x and X ¬y(x) the data that have labels opposite to the label of x.

the estimator for the entropy of the data that is classified with projection π is as follows: From 3 and using the fact that = π] for which we use the notation we estimate the objective as the contribution of each data point to the objective corresponds to a distance ratio on the projection π * where the class of the point is obtained with the highest confidence (data is separable in the neighborhood of the point).


At decision time, a policy is selected based on these predictions.

the self-similar arrival process is approximated by a quasi-stationary Poisson process.


According to Equation (4), we have which contradicts Assumption A1.

the reverse direction does not admit a low-cardinality hidden representation.


Without any assumption, it turns out that this information is actually useless [10].

to make use of unlabeled data, one needs to formulate assumptions.


Once the parameters are learned it performs inference over the unlabeled and partially labeled examples to query for the next random variable to label.

it requires a separate inference step for each active learning iteration.


Being parallelizable (d = 1, 2, 4 or 8), manifold of unit quaternions enjoys an injective homomorphism to the orthonormal matrix ring composed of the orthonormal basis [35].

we define V : S 3 → R 4×4 as follows: It is easy to verify that V(q) is orthonormal for every q ∈ S 3 .


MVAB estimates the dipole source time series by ŝn = W M V AB y n , where yy and R yy is the measured data covariance matrix.

mVAB also has computational complexity equivalent to a single-dipole scan, on the order O(K 2 + S).


An algorithm is called p,-margin successful if, for any input labeled sample, it outputs a hypothesis half-space that classifies correctly as many sample points as any half-space can classify correctly with margin p, (that is, discounting points that are too close to the separating hyper-plane).

a p,-margin successful algorithm is required to output a hypothesis with close-to-optimal performance on the input data (optimal in terms of the number of correctly classified sample points), whenever this input sample has an optimal separating hyper-plane that achieves larger-than-p, margins for most of the points it classifies correctly.


As in Section 3.1, filtering and resampling batch boosters perform poorly on Covertype.

while FilterBoost is competitive in classification, its best performance is in regression.


Screening aims to identify the zero coefficients in the sparse solutions by simple testing rules such that the corresponding features can be removed from the optimization.

the size of the data matrix can be significantly reduced, leading to substantial savings in computational cost and memory usage.


We establish that K(δ; p, d, r) is O p 2η+2 d 2η r 5 δ −1 poly log(p, d, r, δ −1 ) in [15].

we require the number of samples in (9) scaling as n = Ω p 4η+4 d 4η r 10 δ −2 −2 poly log(p, d, r, δ −1 ) .


We further use the Cholesky decomposition to parametrize the shared covariance structure across voxels: U = LL T , where L is a lower triangular matrix.

β i can be written as where α i ∼ N (0, I) (this change of parameter allows for estimating U of less than full rank by setting L as lower-triangular matrix with a few rightmost-columns truncated).


(Note that N ij  k has been implied by N ⌫ 0 and ) Since dropping the rank constraint eliminates the constraints B 2 Im(N ) and D 2 Im(N ) in ( 16) when N 0 [32], we obtain the following relaxed problem, which is jointly convex in N , B and D: Unfortunately, nonlinear semidefinite optimization problems in the form (20) are generally thought to be too expensive in practice despite their polynomial theoretical complexity [33,34].

we develop an effective training algorithm that exploits problem structure to bypass the main computational bottlenecks.


The previous result generalizes to a sigmoidal unit O = σ(S) = 1/(1 + ce −λS ) trained to minimize the relative entropy error The terms O and I i are not independent but using a Taylor expansion with the N W GM approximation gives with S EN S = j w j p j I j .

as in the linear case, the expectation of the dropout gradient is approximately the gradient of the ensemble network regularized by weight decay terms with the proper adaptive coefficients.


At a simple level, there is intuitively more behavioural control if, when repeating one action, the same outcome occurs again and again, than if this were not true.

at a very first level, control might be related to the outcome entropy of actions (see Maier and Seligman 1976 for an early formulation).


Since in open-ended domains the set of categories to be learned is not predefined, it is not feasible to assume that one can pre-program all object categories required by robots.

autonomous robots must have the ability to continuously execute learning and recognition in a concurrent and interleaved fashion.


. . , M T . To ensure this balance, we use the regularization term stated below: The trade-off parameter γ t controls the regularization of M t for all t = 0, . . . , T . If γ 0 → ∞, the shared metric M 0 reduces to the plain Euclidean metric and if γ t>0 → ∞, the task-specific metrics M t>0 become irrelevant zero matrices.

if γ t>0 → ∞ and γ 0 is small, we learn a single metric M 0 across all tasks.


We started by taking a step back from the typical setup of a hierarchical classification machine learning problem.

our focus was on the fundamental aspects of the hierarchical problem definition, rather than on the equally important algorithmic issues.


As mentioned in the introduction, the p-generalized Normal is the only factorial L p -spherically symmetric distribution [22].

marginal independence is such a rigid assumption that the output scale is the only degree of freedom left.


There are well-developed algorithms to solve (8) and obtain non-trivial ⇥.

unlike most existing temporal matrix factorization approaches where the strength of dependencies is fixed, ⇥ in TRMF can be learned automatically from data.


This is another similarity with ICM, which also converges to a maximum.

combining ours with ICM cannot improve the performance of ICM, and vice-versa.


Moreover, a REVERSE-PUSH(v, k) operation involves updating the residuals for d in (v) + 1 states.

note that t∈S p k e v [t] = 1 and hence, via a straightforward counting argument, we have that for any v ∈ S, t∈S 1 {p k e v [t]≥δr} ≤ 1/δ r . we have: Finally, we choose δ r = 2 δ max log( max/pf ) to balance T f and T r and get the result.


We also find that different with text classification and sequential labelling, both ρ > 1 and ρ < 1 can obtain optimal scores.

with the feature configuration of MSTParser, the value of w ⊺ Φ(x, y * ) may not always be positive during the online learning process, and it reflect the fact that feature space of parsing problems is usually more complex.


By Lemma 7, Z(t) satisfies the conditions in Lemma 5 with max = G + .


Thus for any i ∈ I + , o i • K − L 2 i is semi-definite positive.

the third set of constraints are all in the form of difference of two convex functions.


Our point estimate of f is the average of the draws.

our procedure captures ensemble learning (in which many trees are combined) both in the fundamental sum-of-trees specification and in the model-averaging used to obtain the estimate.


We discern from the graph that definition 2 is satisfied because: ( 1 2) P (X|R y , Z) = P (X|R y ) and R y is the minimal set such that X⊥ ⊥({R y , Z} \ R y )|R y and (3) P (Z|R y ) = P (Z) and ∅ is the minimal set such that Z⊥ ⊥R y |∅.

the order Y < X < Z < R y induces a general ordered factorization P (X, Y, Z, R y ) = P (Y |X, Z)P (X|R y )P (Z)P (R y ).


allowing us to argue that both events hold w.p.

at least Ωp1{kq from now on we assume that both "good" events E 1 and E 2 occur.


This leads us to the following maximum walk-sum tree problem: Finding the optimal such tree is combinatorially complex.

we develop a relaxation that minimizes a looser upper bound than (8).


It guarantees that for each individual node, we can correctly recover its neighborhood with high probability.

the following corollary can be implied so that we can asymptotically recover the underlying graph structure under given conditions.


Full details of the experiments can be found in Appendix A. k was chosen to be a random Gaussian mixture.

the overall density is a "mixture of Gaussian mixtures".


We believe that the OB can offer an over complete representation of any natural image.

there is great room for possibly (near) lossless content-based compression of the image features into a much lower-dimensional, but equally discriminative subspace where key semantic information of the images are preserved, and the quality of inference on images such as scene classification are not compromised significantly.


( 4), the expression is twice differentiable, with a uniformly bounded second derivative.


For instance, taking Euclidean distance as the divergence, we obtain normal distribution as our model f . In this work, we take models of the above form whose parameters m lie in the same space as data vectors.

we can restrict the set of mixture components to the distributions centered at the data points, i.e., m j ∈ X .


Instead of using η m to localize the training error of data points, we localize a model's contribution ŷm = ξ(x) T w m towards the global fit of training point y, similar to how LWR operates during test time (Eq.4).

already during training, local models must collaborate to fit a data point ŷ = ∑ m=1 η m (x)ξ(x) T w m . Our loss function is combining the localizer η m (x n ) and the This form of localization couples all local models, as in classical radial basis function networks [19].


Therefore, there exists i 0 such that if i > i 0 then |l(y, λ * i , x) − l(y, λ * ∞ , x)| < for any y ∈ Y and x ∈ X .

input: Countable set of experts {H k,h }, y 0 ∈ Y, λ 0 ∈ Λ, initial probability {α k,h }, For n = 0 to ∞ Play y n , λ n . Nature reveals x n Suffer loss l(y n , λ n , x n ).


ICA, Product-of-Experts and Fields-of-Experts), can be interpreted as a form of feed-forward neural network.

the terms {∂E/∂x i } and {∂ 2 E/(∂x i ) 2 } can be efficiently computed using a forward and backward pass: the first pass performs forward inference (computation of E(x; w)) and the second pass applies the backpropagation algorithm [3] to obtain the derivatives of the energy w.r.t.


In defining ∆ Ŝi = ( Ŝi − S i ) we find that The fluctuations of the sufficient statistics ∆ Ŝi is a random variable with mean zero, and the randomness of (g L i − g i ) enters only via λ i .

one can assume a very small statistical correlation between those two terms, the cross-term can be expected to be negligible.


The meaning of second term however can be understood as follows: N t s measures how often we have visited state s in the recent past.

if N t s ≪ N t st+1 then state s has a value estimate based on relatively few samples, while state s t+1 has a value estimate based on relatively many samples.


When there is a large number of agents, agents cannot differentiate valuable information that helps cooperative decision making from globally shared information.

communication barely helps, and could even impair the learning of multi-agent cooperation.


This can be seen as a semiparametric statistical model [22,23], which includes not only parameters of interest, but also additional nuisance parameters with possibly infinite DoF, where only the policy is modeled parametrically and the other unspecified part corresponds to the nuisance parameters.

model-free WLPS can be framed as a semiparametric statistical inference problem.


We want to show that given a sequence of matrices §¦ X £ £ s s £ ' £ £ ¡ ' £ s £ £ ¡ £ £ ¤ $ £ s I) .

¥ ¥ AH ¤¡ ¦ X ¤ $ £ ¡ ) £ ¤ $ £ s H) .


To address the above-mentioned issue, we extend the Rademacher random variables to the complex domain and consider all roots of z m = 1, that is, Ω = {ω j } m−1 j=0 where ω j = e i 2πj m . Suppose σ(i) is a Rademacher random variable with Pr[σ(i) = ω i ] = 1/m.

by elementary algebra, E[σ(i) p ] = 0 whenever m is relative prime to p or m can be divided by p. by setting m = 4 we avoid collisions of repeated entries in a 3rd order tensor.


As a result, we know t k ≥ t min := min i=0,...,k t i ≥ η/β.

it is easy to see that in the k-th iteration of Algorithm 4, x * is contained in two balls: we have the following convergence result for Algorithm 4, whose proof is similar to that for Algorithm 3.


This is manifested by that the decay of the network activity exhibits a long plateau around R * before dropping to silence rapidly (Figure 2B).

persistent activity of finite lifetime is achieved.


Inspection of the formula for T 2 k shows that its distribution is the same under both H 0 and H 0,proj .

rejecting the original H 0 when T 2 k ≥ t α is also a level α test for the original problem (1).


The nodes y are root nodes in H and hence do not descend from x i .

p(x i |pa H (x i ), y) = p(x i |pa H (x i )) and the theorem holds.


Furthermore, this relaxation can also be solved as a quadratic pseudo-Boolean optimization problem using a graph-cut method [25], which is orders of magnitude faster than using a general purpose LP solver (i.e., 10 sec for labeling a typical scene in our experiments).

we refer to the solution of this relaxation as ŷcut.


Using the same derivation as in Remark 2, the probability of error ℓ(n) for this strategy may be bounded as In the Uniform+UCB-E allocation strategy, i.e., a two-level algorithm that first selects a bandit uniformly and then pulls arms within each bandit using UCB-E, the total number of pulls for each bandit m is k T mk (n) = n/M , while the number of pulls T mk (n) over the arms in bandit m is determined by UCB-E.

the probability of error of this strategy may be bounded as where the first inequality follows from Theorem 1 in [1] (recall that H m = k b 2 /∆ 2 mk ).


If the distribution P was known, the best weight parameter β could be found by solving the stochastic optimization problem In practice, however, P is only indirectly observable through N independent training samples.

the distribution P is itself uncertain, which motivates us to address problem (4) from a distributionally robust perspective.


The geometrical interpretation is that q i,j requests whether the reference r σ is closer to object θ i or object θ j in R d . Consider the line connecting θ i and θ j in R d . The hyperplane that bisects this line and is orthogonal to it defines two halfspaces: one containing points closer to θ i and the other the points closer to θ j .

q i,j is a membership query about which halfspace r σ is in, and there is an equivalence between each query, each pair of objects, and the corresponding bisecting hyperplane.


For other choices of A, such as HOLP defined in next section, the estimator satisfies RDD even when predictors are highly correlated.


Moreover, at high dimensions, the relative error made as considering E||X|| instead of the random variable ||X|| becomes asymptotically negligible [2].

the following holds in high dimensions: Finally, E(max(W )) ≤ √ 2σ ln(N ) when W follows a Gaussian distribution [13], establishing the following: Eq.


Depending on the value of b, we will have n + 1 possible sums, respectively − k i=1 c r(i) + n i=k+1 c r(i) , k = 0, . . . , n. It is obvious that those sums only depend on the order of the products u • x i , i = 1, . . . , n. When u varies smoothly on the unit circle, as the dot product is a continuous function of its arguments, the changes in the order of the dot products will occur only when there is a pair (i, j) such that u • x i = u • x j .

there are at most as many order changes as there are pairs of different points, i.e., n(n − 1)/2.


Notice that e) where e is the identity permutation, a quantity which is also called the number of inversions of σ.

in contrast to the previous mappings, the norm φ L (σ) is not constant for any σ ∈ S K . Hence it is not possible to write the loss ∆ L (σ, σ ) as − φ L (σ), φ L (σ ) 2 .Moreover, this mapping is not distance preserving and it can be proven that 1 Wang et al.


If the columns of the true mixing matrix M are members of the ambiguity set, the resulting loss can be used to estimate posterior class probabilities from the observed partial labels.

a general procedure to design a loss function for learning from partial labels is: 1.


In the following, we localize the data depending on the position of the robot.

the partitioning of training data is performed in a 7-dim space (7 joint angles).


When this is not the case, they may vary since segmentation is then based on function approximation boundaries, and hence two trajectories segmented independently may be poorly aligned.

when segmenting two trajectories sequentially in anticipation of a merge, we may wish to include a bias on changepoint locations in the second trajectory.


This model of computation in recurrent networks, however, receives its inputs as sequences.

computing time is not an issue since the network halts when the input sequence terminates.


Consequently, a method for approximating ∂J(θ) would consist in estimating I h for a sufficiently small h. We know that J(θ) can be numerically estimated by 1 M M m=1 J N ωm (θ).

it seems natural to estimate I h by where we used independent random numbers to evaluate J(θ + h) and J(θ − h).


In this case, we are able to show that if we allow the low rank matrix B output to be of rank k + 4, then one can achieve the relative error guarantee of (1.2) with respect to the best rank-k approximation, namely, that F . Further, our algorithm runs in a sublinear n • k • poly(1/ ) amount of time.

our lower bound ruling out sublinear time algorithms achieving (1.2) for bicriteria algorithms cannot hold for this class of matrices.


Due to the lack of harmonicity and weak energy, frequency channels containing unvoiced speech often have significantly skewed distributions of target-dominant and interference-dominant units.

an accuracy-maximizing classifier tends to output all 0's to attain a high accuracy.


We prove that the commonly used non-convex objective function for positive semidefinite matrix completion has no spurious local minima -all local minima must also be global.

many popular optimization algorithms such as (stochastic) gradient descent can provably solve positive semidefinite matrix completion with arbitrary initialization in polynomial time.


Clearly, this is a non-convex optimization problem.

we cannot use existing convex optimization methods to obtain an efficient solution.


However, the results are representative for only a couple of hours due to drift in the system.

the desire for a method which has an acquisition function that can be efficiently optimized in high dimensions is paramount.


It is well known, in fact, that these models are prone to overfitting.

deep networks seem less useful when the goal is to learn a new concept on the fly, from a few or even a single example as in one shot learning.


To present the result, recall from Section 3.2 that, by a re-normalization of the kernels, we may equivalently express H 2 by H2 , as defined in (6).

the algorithms presented enjoy the following bound on the local Rademacher complexity, which was shown in [19] (Theorem 5).


The margin is the distance between the closest elements of different clusters, and hence we will be interested in finding k-clusters that maximize the margin.

we seek an element in O k (S) = arg max {S1,S2,...,S k }∈C k (S) seperation({S 1 , S 2 , .


Thus, the additive noise model may not be a proper representation of the causal mechanism for categorical variables.

a proper description of the causal mechanism for discrete data that helps in causal discovery remains under explored.


( 5), we also obtain the Gaussian likelihood given x, In contrast to GP-LVM, the precision matrix ) depends on the graph Laplacian matrix through W and Λ.

in LL-LVM, the graph structure directly determines the functional form of the conditional precision.


Using this notation, the spectral method can be described as follows.

given the desired number of states n, it consists of first computing the truncated SVD of H corresponding to the n largest singular values: U n D n V n . matrix U n D n V n is the best rank n approximation to H with respect to the Frobenius norm.


Our work is primarily motivated by the non generality of the approaches in [4] and [5], as they make Gausssianity assumptions about the underlying data distribution.

they tend to overfit for clusters that are uniformly distributed, or have a non-Gaussian unimodal distribution.


However, this distribu-tion reflects the full range of exemplars that lie within the category boundaries, and presumably in the production task, subjects attempt to produce a prototypical exemplar.

we exclude the intrinsic category variance, v c , from Equation 2 in generating exemplars, leaving variance only via uncertainty about the prototype.


The neural network will maximize its representational capacity by learning consistent bases that reflect this shared functional structure, leading in turn to consistent sparse function encodings.

in our setting, consistent functional bases emerge from the network without explicit supervision.


In Section 3, we show this conjecture to be true using a low-dimensional example.

our low-dimensional study provides insight into AdaBoost's large scale dynamical behavior.


A promising future direction to follow would be to enhance our formulation with domain-specific knowledge encoded, for example, in more sophisticated covariance functions or in the way that data are being preprocessed.

we can obtain application-oriented methods to be used for tasks in areas such as robotics, computer vision and finance.


4 Then there exists an adversarial contextual bandit algorithm with margin regret bound Õmax a K(R(F a , T ) ) 23 T 13 .

for margin-based contextual bandits, full information learnability is equivalent to bandit learnability.


Existing class-based approaches infer latent structure that is a special case of what can be inferred by this model.

our model is strictly more expressive than these approaches, and can use the solutions produced by these approaches for initialization.


On the other hand, the similarity graph built by our method has a locality preserving property by the definition of the weights.

we can use the adjacency matrix, W [i], of the i-th cluster as a similarity between points in the corresponding manifold and obtain a low-dimensional embedding of the data by taking the last few eigenvectors of the normalized Laplacian matrix associated to W [i] [3].


As mentioned previously, the performance of generative model learning depends on the gap between the variational and the posterior distributions.

the amortized inference has two sources of this gap: the approximation and amortization gaps [Krishnan et al., 2018, Cremer et al., 2018.


Because J is strongly-connected, the Perron-Frobenius theorem guarantees a simple eigenvalue equal to ρ(J) and a corresponding all-positive eigenvector.

when β crosses 1/ρ(J) from below, the Perron-Frobenius eigenvalue of f (m) crosses 1 from below, giving rise to a supercritical pitchfork bifurcation with two new stable steady-states corresponding to the Perron-Frobenius eigenvector.


A larger value of η improves the regret bound, but also increases the runtime of the algorithm.

η provides a trade-off between the computational complexity of the algorithm and its regret guarantee.


Unlike many existing parametric survival models, our model neither assumes a parametric form for the interactions between the covariates and the survival times, nor does it restrict the distribution of the survival times to a parametric model.

it can flexibly describe non-proportional hazard rates with complex interactions between covariates and survival times, which are common in many diseases with heterogeneous phenotypes (such as cardiovascular diseases [2]).


Unlike many other machine learning algorithms, tight bounds are known for the computational and statistical complexity of traditional perceptron training.

we are able to rigorously show different performance improvements that stem from either using quantum computers to improve traditional perceptron training or from devising a new form of perceptron training that aligns with the capabilities of quantum computers.


Similarly, if one were to assume that a simple operation (such as computing a probability vector over categories using Bayes rule) would take on the order of £ § sec per word, then a full pass through £ ¨ § © words will take 1000 seconds.

algorithms that make multiple passes over this sized corpus (such as occurs in many clustering and classification algorithms) will have run times in days.


We consider a positive vector-valued step-size ∈ ℝ + and "•" denotes component-wise (Hadamard) product of two vectors.

again, by exploiting (4), since where is an estimated eigenvalue of J as given in (2), when H is a symmetric matrix, its eigenvalue is given by eig we can update the step size component-wise by Since the mapping ℳ in SGD involves the gradient g(w ( ) ; B ( ) ) of a randomly selected training example B ( ) , ℳ is itself a random variable.


This result improves upon [9] where it is proven only for a global minimizer.

proof: If g is any function in H such that g(x j ) = 0, j ∈ IN m and t a real number such that |t| g ≤ , for > 0, then we have that f 0 2 ≤ f 0 + tg 2 from which it follows that (f 0 , g) = 0.


As mentioned above, there are still considerable filters in the ResNet whose sizes are larger than 1 × 1.

its memory usage and FLOPs were reduced obviously by exploiting the proposed versatile convolution filters.


We call a weighted data set C a (k, ε)-coreset for another (possibly weighted) set  Hereby φ(C | θ) is generalized to weighted data sets C in the natural way (weighing the contribution of each summand x j ∈ C by γ j ).

as ε → 0, for a sequence of (k, ε)-coresets C ε we have that Further, under the additional condition that all variances are sufficiently large (formally λ∈spec(Σi) λ ≥ 1 (2π) d for all components i), the log-normalizer ln Z(θ) is negative, and consequently the coreset in fact provides a multiplicative (1 + ε) approximation to the log-likelihood, i.e., ( More details can be found in the supplemental material.


We also introduce skip connections from the encoder to the decoder in the CL model, where each intermediate blocks' outputs are upsampled and concatenated with o t as well, so the decoder can take advantage of the localized memory provided.

with k blocks, we use a stride of 2 in each block, so the output of the k-th block must be upsampled by a factor of 2 k . only in the CL model do we also concatenate the the intermediate LSTM cell outputs to the input to the decoder D. These skip-connections from the intermediate blocks are a way to propagate spatio-temporal memory to the decoder at different spatial scales, taking advantage of the specific structure of this problem.


However, we operate deblurring in a space with the same spatial resolution as the original image rather than a downscaled space as conventionally done.

it avoids the additional estimation error caused by interpolation between spatial scales in the pyramid.


9 arms are picked and projected by a random Gaussian matrix to a two-dimension plane in Figure 3.


Let T}e f and T/ft denote the random variables that correspond to Tj , before and after the merger, respectively.

the values of £ before and after the merger are calculated based on Tr f and Ttt .


The MAP problem is then equivalent to the following linear program: where The extreme points of the marginal polytope are integral and correspond one-to-one with assignments x.

there always exists a maximizing µ that is integral and corresponds to x M . Although the number of variables in this LP is only O(|E| + |V |), the difficulty comes from an exponential number of linear inequalities typically required to describe the marginal polytope M(G).


When building statistical models of natural language, one reduces the difficulty by taking advantage of word order, and the fact that temporally closer words in the word sequence are statistically more dependent.

n-gram models construct tables of conditional probabilities for the next word, for each one of a large number of contexts, i.e.


On the other hand if α is too low, it becomes possible for each model to assign all of its weight to a single point, which may well be an outlier tending to infinity.

it is well expected that the robustness result below requires α to belong to a data dependent range.


However, such a bound is applicable only to the bounded-support class P B , and helps to remove the dependence on √ d but at the cost of sacrificing the rate (i.e., from m −1/2 + n −1/2 to m −1/4 + n −1/4 ).

such an upper bound matches the lower bound in Corollary 2 for ReLU networks over the network parameters, but not in terms of the sample size, and is interesting only in the regime when d max{n, m}.


Y t where Y If the matrix that we would like to learn is asymmetric, we can transform it to an equivalent symmetric model by defining a new data matrix Y = 0 Y Y T 0 .

for the remainder of the paper, we will assume a n × n symmetric matrix which takes values in [0, 1] (real-valued or binary), but as argued above, our results apply more broadly to categorical-valued asymmetric matrices.


When changing the means of arms below, the triggering probability of arms above is not changed.

if we change µ to µ backwards, the triggering probability of each arm is unaffected before its expectation is changed, and when changing the mean of an arm i, the expected reward of the action is at most changed by p D,S i The proof of Lemma 2 is more complex, since the bottom-up modification does not work directly on graphs with cycles.


If the algorithm converges, i.e., the sequence (ŝ e,t ) converges to a point s * , and f i (•) is continuous, the variance of sEM-vr will converge to zero, while that of sEM will remain positive.

sEM-vr has asymptotically smaller variance than sEM, and we will see that this leads to better asymptotic convergence rates.


The usual approaches based on the probabilistic losses, such as the KL-divergence that leads to MLE, ignore this dependency.

they might be less efficient than an approach that considers how Planner is going to use the learned model.


Different with NAO that tries to reduce the huge computational cost brought by the search algorithm, weight sharing aims to ease the huge complexity brought by massive child models via the one-shot model setup [5].

the weight sharing method is complementary to NAO and it is possible to obtain better performance by combining NAO and weight sharing.


The output X of G reproj will be a 2D feature map with dimension d at each position (i, j).

x is compatible with a regular convolutional neural network.


If there are two sources, they are assumed to be independent.

we use the graphical model depicted in Fig.


From Lemma 1 we deduce that if the per-tree budget is at least we obtain 1−γ ≤ for any policy π. ASOP aggregates the trees and computes the optimal policy π of the resulting empirical MDP whose transition probabilities are defined by the frequencies (over the m S3-trees) of transitions from state-action to successor states.

π is actually a policy maximizing the function If the number m of S3-trees and the per-tree budget n are large, we therefore expect the optimal policy π of the empirical MDP to be close to the optimal policy π * of the true MDP.


Kingman shows that the partition on the leaves of a coalescent tree when its edges are cut by a Poisson point process is the same as that of a DP (Figure 4).

the partition structure along every row and column is marginally the same as a DP.


For each state s, we define a continuous observation probability distribution, for observing a pdf Pt(x) in state s. Next, the initial state distribution {1fsLES of the HMM is given by the uniform distribution, 1fs = liN, with N = lSI being the number of states.

each state is a-priori equally probable.


Parameter choices: As with previous GP-based algorithms that use confidence bounds, our theoretical choice of (i) in TRUVAR is typically overly conservative.

instead of using ( 14) directly, we use a more aggressive variant with similar dependence on the domain size and time: , where t (i) is the time at which the epoch starts, and a is a constant.


For space exploration, communication delays between the rover and the operator on Earth can be prohibitive.

it is important that the robot can act autonomously and explore the environment without risking unsafe behavior.


In the case when v is random, the matrix A has independent columns and dependent rows.

a 0 A is a spiked covariance matrix and it is possible to use the existing theoretical results on the first eigenvalue to characterize the left singular vector of A.


If V is true the constraint is satisfied, otherwise it is violated.

if V is true then Θ lies in the positive region of Fig.


The event pixel address was used to produce a 40-dimensional embedding via a learned embedding matrix [9], and combined with the polarity to produce the input.

the network architecture was 41-110-10 for the Phased LSTM and 42-110-10 for the BN-LSTM, with the time given as an extra input dimension to the BN-LSTM.


Hence, if one employs a model with a reasonable number of DoF (degrees of freedom), DPS could find a reasonable policy efficiently even when the target decision making task has a huge number of DoF.

the development of an efficient model selection methodology for the policy is crucial in RL research.


By the classic analysis of greedy algorithm for submodular maximization, we can conclude that every item we add has an added value that is at least (1 − )(L − f (S))/|OPT|.

after adding |OPT| ln(L)/(1 − ) items, f (S) becomes at least L. To upper bound rounds, we divide the rounds into two groups.


This invariance follows from the decomposition of equation ( 1): in particular, the distribution pi(n) (x; Tn) is left invariant by reparameterization; and TRP does not change terms in ri(n) (x; Tn).

the overall distribution remains invariant -i.e., p(x; Tn) == p(x; TO) for all n. By continuity of the map T f-7 p(x; T) , it follows that any fixed point T* of the algorithm also satisfies p(x; T*) == p(x; TO).


)we can show that that the uniform convergence of er S to er S in the transductive setting (for f = 1/2) is a necessary and sufficient condition for the uniform convergence of er S to the generalization error.

the results presented here can be extended to the supervised setting.


Also, the entries of Z can be represented with limited independence, and so Z can be stored with a short random seed.

we obtain the first algorithm with d poly(log(nd)kϵ −1 ) memory for this problem in the standard turnstile data stream model [19].


From Lemma 5.4 in the appendix, we know that The Eqn.

we cannot guarantee that the mechanism is approximately truthful.


It shows that Algorithm 1 achieves the optimal rate of convergence of first-order methods, but it does not take into account the complexity of solving the subproblems (5).

we need the following proposition: Proposition 3.4 (Inner-Loop Complexity, Non-Strongly Convex Case).


This assumption implies that the row player applies the same function to each row of a given game's utility matrices.

in a normal form game represented by the utility matrices U (r) and U (c) , the weights associated with each row of U (r) and U (c) must be the same.


Hence Algorithm 1 is a variant of simulated annealing.


Since e where e is a vector of 1's, it is straightforward to show that the following inequality In addition, the equality holds if and only if Therefore, |V 1 | β |V 2 | β achieves the maximum value when two clusters are of equal weighted size.

minimizing SRcut is equivalent to minimizing the similarity between two clusters and, at the same time, searching for a balanced partition.


These deep denoisers typically train their networks by minimizing the mean-squared error (MSE) between the output of a network and the ground truth (noiseless) image.

it is crucial to have high quality noiseless images for high performance deep learning denoisers.


The model allows straightforward variational optimisation of the posterior distribution on coordinates and locally linear maps from the latent space to the observation space given the data.

the LL-LVM encapsulates the local-geometry preserving intuitions that underlie non-probabilistic methods such as locally linear embedding (LLE).


Since the restricted iteration for θ new is expensive, only a few cluster merges may be proposed at a time.

merges are proposed for clusters with the smallest earth mover's distance [23] between their spectral densities.


If G : [0, ∞) → R denotes the cumulative distribution function of the squared deviation of F (p 1 , .

. . , pk ) from its mean, then We then have a mean squared error of It should be noted that the constants in both the bias bound and the variance bound depend exponentially on the dimension d. Lower bounds in terms of d are unknown for estimating most density functionals of interest, and an important open problem is whether this dependence can be made asymptotically better than exponential.


This observation can be explained by the fact that DRLR accounts for uncertainty in the label, whereas RLR does not.

there is a wider range of Wasserstein radii that result in   an attractive out-of-sample logloss and CCR.


Such problems are not wholly decoupled, however, as the actions available to each agent at any point depend upon the global system state.

agents interact by making choices that restrict or broaden the range of actions available to others.


Clearly, B = Ψ X Ψ T is feasible in 6 because of A3.

we can show that any solution B of (6) accurately estimates B using existing results on nuclear-norm minimization.


Furthermore, we found that the learned representation of the MNIST data allows for good classification in a semi-supervised setting, when only a small fraction of the data is labeled.

our model provides a starting point for constructing novel clustering and classification algorithms following the general approach in [9].


Because of memory limitations, [25] actually solved an approximate version of the PE objective function, where rather than using all N 2 pairwise point interactions, only BN interactions are used, corresponding to using minibatches of B = 5 000 points.

the solution obtained is not a minimizer of the PE objective, as can be seen from the higher objective value in fig.


If a coefficient is zero, it remains small after one online update, and the rounding operation pulls it back to zero.

rounding can be done only after every K steps (with a reasonably large K); in this case, nonzero coefficients have sufficient time to go above the threshold θ.


in this configuration, the network can already implement the first step of the Kalman filter through its recurrent connectivity.


Each QD-PageRank q is a vector of d q values.

the space required to store all of the PageRanks is S, a factor of S/N times the query independent PageRank alone (recall N is the number of web pages).


We further complicate matters by dropping the covariate indicating rural, inner-city, urban or suburban from all subsequent analysis.

we have significant unmeasured confounding in the CONF population, and also the unconfounded ground-truth in the original, ALL population.


This and other little variations don't help much.)

even in the most benign situations, we cannot expect that every target hypothesis will be identifiable using o(m) labels.


The hyperparameter 0 controls the impact of the GAN objective functions.

we use a VAE-like objective function to model the cycle-consistency constraint, which is given by where the negative log-likelihood objective term ensures a twice translated image resembles the input one and the KL terms penalize the latent codes deviating from the prior distribution in the cycle-reconstruction stream (there are two KL terms).


for the number of CI tests, the ratio between B-RAI and RAI is k i=0 s i .


Note that this exponentially large set of linear constraints can be replaced by a single, equivalent, non-linear constraint: In a factored MDP, the reward function is represented as the sum of local rewards 0 R A R . Furthermore, the basis B C and its backprojection R V are also functions that depend only on a small set of variables.

the right side of the constraint can be viewed as the sum of restricted scope functions parameterized by p . For a fixed p , we can compute the maximum over I % using a cost network, as in Section 2.


We adjust this regularization term so that it concentrates on increasing the entropy of the particular pairs of binary vectors that are not correlated in high-dimensional space.

we keep the balance between propagating the Hamming distances between the layers for correlated vectors and increasing the diversity of the binary vectors in the low-dimensional feature space.


Denote the probability Pr(e x ) by P x .

p x is the product of p(y) over all nodes y that are the ancestors of x (including x itself).


We also introduce the meta-scale-invariance (MSI) property, a natural alternative to scale invariance, and show how to design a single-linkage clustering algorithm that satisfies MSI, richness and consistency.

we avoid Kleinberg's impossibility result and achieve provably good meta-clustering.


On the other hand, the extent to which redundancy grows with cue length is really a product of the specific words in memory and the cues chosen.

the exact shape of the performance curve will also depend on these factors.


negative) labeled points tend to co-vary.

there is a reason to prefer the GP with a graph kernel, and we do expect humans to learn on manifold in 4 l moons U h. We also explore the convex combination of the seven GPs as a richer model for human behavior: k(λ) = 7 i=1 λ i k i , where λ i ≥ 0, i λ i = 1.


In addition, the number of elements in M with large values increases as the size of the blur kernel increases.

the matrix M contains the most essential information of the input blur kernel.


Interaction history is very important to generate the logical form of the following utterance.

we incorporate a dialog memory to maintain information from interaction history.


Each action cell is associated with angle φ i , all φ i being distributed uniformly in [0, 2π].

an activity profile on the level of place cells (i.e.


Submodular functions are a natural model for several summarization problems [7,8].

given a submodular summarization model g, and a set of existing diverse summaries A 1 , A 2 , .


We find that all three measures of sparseness decrease significantly with age 1 .

during a period when the cortex actively adapts to the visual environment, the representation in primary visual cortex becomes less sparse, suggesting that the optimization of sparseness is not a primary objective for learning in the visual system.


We stop whenever the algorithm corrects all external errors or declare a recall error if all errors were not corrected in 40 iterations.

the corresponding areas in the figure where the number of iterations reaches 40 indicates decoding failure.


Updating the model with observations in the current interval results in a higher probability for the observed edges, and a lower probability for the edges that have not been observed recently.

we do not need to consider an aging factor to take into account the older cascades.


Equipped with the results in Theorem 2 and Lemma 3, we can now show that the fixed point solution of T[V ](x, y) = V (x, y) is unique, and equals to the solution of the CVaR MDP problem (3) with x 0 = x and α = y. Theorem 4 (Optimality Condition) For any x ∈ X and y ∈ (0, 1], the solution to T[V ](x, y) = V (x, y) is unique, and equals to Next, we show that the optimal value of the CVaR MDP problem (3) can be attained by a stationary Markov policy, defined as a greedy policy with respect to the value function V * (x, y).

while the original problem is defined over the intractable space of history-dependent policies, a stationary Markov policy (over the augmented state space) is optimal, and can be readily derived from V * (x, y).


Specifically, in the high SNR regime, thresholding requires at most 4/MAR more measurements than lasso.

the benefit of lasso over simple thresholding is its ability to detect the sparsity pattern even in the presence of relatively small nonzero coefficients (i.e.


[5] have considered the error bound for such problems and their results imply that there exists c > 0 such that dist(x, where Ω * denotes the optimal solution set.

the HOPS also has a linear convergence for cone programming.


In (ii) & (iii), negating any ρ ij , it still holds that J = I − R 0 : (ii) negating ρ ij doesn't affect the eigenvalues of J (remove edge {i, j} and, in each eigenvector, negate all entries in one subtree); (iii) negating ρ ij preserves J {i,j} 0 in (4) so J 0.

making all ρ ij > 0, we find I − R 0 and R ≺ I.


Interestingly, the algorithmic transition ∆ alg is finite if at most one prior is has zero mean.

the general tensor decomposition case is qualitatively different than the symmetric case in that an easy regime can exist even when a tensor mode has zero mean.


In this case, the loss term in ( 6) is simply kY W H T k 2 F . Thus, setting rf (H) = 0 is equivalent to solving the following Sylvester equation for an n ⇥ k matrix H: (7) admits a closed form solution.

however the standard Bartels-Stewart algorithm for the Sylvester equation requires transforming both W T W and L h into Schur form (diagonal in our case where W T W and L h are symmetric) by the QR algorithm, which is time consuming for a large L h . we consider applying conjugate gradient (CG) to minimize f (H) directly.


In contrast, the recognition network has been designed to approximate the posterior distribution of hidden variables with more fidelity [18,29,25].

it can be a better candidate for the variational distribution in our task.


Although this formalization is very general, it is difficult to apply directly because of the embedded invocation of the reader's inference procedure and the use of the MENTION modality.

we sidestep this problem by manually "compiling" the maxims into ordinary first-order Markov Logic as follows.


The probability of an unseen fragment f is given by p The model deviates from reality in that it may generate fragments that may not be aggregateable into a consistent sequence of code positions.

non-zero probability mass is given to impossible events, which is a potential source of inaccuracy.


The location of the sound source was limited to be in front of the head (¥ © © ¢¡ "¦£ # U# V# ¡ "¦£ ) at a fixed constant elevation and in jumps of I §¤£ .

the number of classes, © , for training is I &¡ .


We say that such a property is elicitable, if for some loss function it can be represented as the unique minimizer of the expected loss under the distribution.

the study of which properties are elicitable can be viewed as the study of which statistics are computable via ERM [1,2,3].


Proof: From (1) and ( 4), the equation − Fs,a (θ, ι)ω * holds.


This term appears in F, either directly or in expectations, in terms t through T of the summation: steps on which q(zt|x ≤t , z<t) depends terms in which q(zt|x ≤t , z<t) appears However, the filtering setting dictates that the optimization of the approximate posterior at each step can only condition on past and present variables, i.e.

steps 1 through t. of the T terms in F, the only term through which we can optimize q(z t |x ≤t , z <t ) is the t th term: q * (z t |x ≤t , z <t ) = arg min q(zt|x ≤t ,z<t) Ft . Optimizing Ft requires evaluating expectations over previous approximate posteriors.


It usually takes a certain amount of time for fingers to finish extension or flexion.

the duration of certain movement patterns will deviate from the distribution described by Equation 1.


between expanding a leaf following a path that is optimistic according to ρ+ t (step 7,8,9), or according to ρ− t (step 11.)

at a high level, the algorithm performs on each cell (h, i) ∈ P of the given partition two hierarchical searches, one for the maximum value of ρ in region R(h, i) and one for its minimal value.


It is easy to understand why this occurs by considering the prior distribution over amplitude and phase implied by our choice of prior distribution over x t (or equivalently over z t ), Phase and amplitude are dependent in the implied distribution, which is conditionally a uniform distribution over phase when the amplitude is zero and a strongly peaked von Mises distribution [15] when the amplitude is large.

the model favors more highly variable IFs at low amplitudes.


However, all these invariance inducing criteria suffer from a similar drawback, which is they are defined to measure the divergence between a pair of distributions.

they can only express the invariance belief w.r.t.


While enumeration of weak learners is possible in the case of decision stumps, it poses serious difficulties in the case of weak learners such as decision trees, ridge regression, etc.

vadaBoost is the more versatile boosting algorithm for sample variance penalization.


If our proposal is deterministic, then f ✓ (f ✓ (x, v)) = (x, v) should hold for all (x, v), a condition which is difficult to achieve 2 .

we introduce a proposal which satisfies Equation ( 9) for any ✓, while preventing random walk in practice by resampling v after every MH step.


As AoA and frequency become less correlated, the significance of frequency increases, and vice-versa.

as frequency and AoA become less correlated, frequency is able to begin making an independent contribution to the SSE of the network.


Furthermore, using the techniques in [10] it is straightforward to show that (11) is equivalent to a * (λ) ≤ c 2 λ 2β 1−β .

if H approximates P with exponent β then the rate in Theorem 3.1 becomes n − 4β (2+p)(1+β) which is the rate we established in Theorem 2.3 for the original SVM.


Throughout this section, we assume that F is a non-negative submodular set-function that is equal to zero at ∅ and V . This immediately implies that f is invariant by addition of any constant vector (that is, f (w + α1 V ) = f (w) for all w ∈ R p and α ∈ R), and that f (1 V ) = F (V ) = 0.

contrary to the non-decreasing case [3], our regularizers are not norms.


Note that the wave function ψ describes a real physical system that must collapse to exactly one basis state.

the probabilities governed by the amplitudes c i must sum to unity.


This procedure works because the space F n of nth-order monomials has a very special property: it has the structure of a reproducing kernel Hilbert space (RKHS).

the dot product in F n can be computed by evaluating a positive definite kernel function k n (x 1 , x 2 ).


Our first result shows that some common methods based on regularization using graph Laplacians do not lead to faster minimax rates of convergence.

the estimators that use the unlabeled data do not have smaller risk than the estimators that use only labeled data.


(c) Numerical experiments show a strong tendency to avoid entrapment in a learning plateau 3 , especially with large numbers of states.

the development of a tractable algorithm for NSG is important, and this is one of the goals of our work.


Note that λ z is part of the prior knowledge and belief about the problem domain.

the objective of learning the generative model can be written as min W (P r , P g ) or equivalently: In GAN terminology f w is called the discriminator and g θ is known as the generator model.


Because the ultimate goal is to compute q(X|Y new ), not p(X|Y new ), we only need to compute the posterior transitions p(x t |x t−1 , Y new ).

we circumvent having to first find p then project it onto q.


In the kernel calculation with the statistical feature selection, if χ 2 (u) < τ holds, that is, u is not statistically significant, then u is eliminated from the features, and the value of u is presumed to be 0 for the kernel value.

the sequence and tree kernel with feature selection (SK,TK+FS) can be defined as follows: (2) The difference with their original kernels is simply the condition of the first summation, which is τ ≤ χ 2 (u).


Accordingly, learning a single domain-invariant latent space with aforementioned assumption [37,20,24] prevents the model from capturing domain-specific information.

a sketch can only be mapped into one of its corresponding photo-realistic images.


Because there is no subtraction in the numerator of the right-hand expression, the complexstep derivative does not suffer from the numerical problems of the finite-differencing method that result from cancellation.

much smaller step sizes can be used, allowing for greater accuracy.


The noise would have little effect on the next neuron's input because the input is a summation over many weighted neurons.

the real-valued version could be updated by the back propagated error by simply ignoring the binarization noise in the update.


This means, that after the training phase the student has achieved some overlap to the teacher, their weight vectors are correlated.

the student can classify an input pattern which does not belong to the training set.


This may not be optimal, as a noun-phrase is often very informative to infer the POS tags belonging to the phrase.

it is more desirable to jointly model and infer both the NPs and the POS tags at the same time.


Accumulated till d max basis functions are added, this leads to a T selection that has O(n 2 d 2 max ) complexity, which is disproportionately higher than T basic .

smola and Bartlett (2001) resorted to a randomized scheme by considering only κ basis elements randomly chosen from outside I during one basis selection.


From Example 3 and the proofs of Theorems 2 and 3 we can see that the non-submodularity only occurs in the case when two or more cascades arrive at one node at the same time.

if such a scenario does not happen frequently, the Max-M and Min-M problems will be close to submodular optimization problems, and consequently, the greedy algorithm is effective.


From this definition, one can see that the potential at x is negative (positive) if x is in a neighborhood of many negative (positive) charges.

the potential indicates the sign and amount of charge in the local neighborhood.


Intuitively, Algorithm 1 relies on the fact that the corresponding functions { ft } T t=1 are memoryless and convex.

standard regret minimization techniques are applicable, yielding a regret bound of O(T 1/2 ) for { ft } T t=1 .


Let the stationary distribution for this Markov chain be denoted by, v = [p 0 , p 1 ] that satisfies v = vP .

which describes the long term expectation of visiting a nominal or adversarial state.


This consists in a bi-level problem [3] for which it is challenging to derive an optimization approach [4].

a regularized version of the Sinkhorn approximation is usually considered [5,6,7,8,9], for which it is possible to efficiently compute a gradient and thus employ it in first-order optimization methods [8].


Using the additive property of the ΓP, posterior sample for the sum of λ k over unused atoms is obtained as .

we obtain a closed-form posterior sampler for the rate parameter c 0 of the ΓP as c 0 | − ∼ Gamma(e 0 + γ 0 , f 0 + λ γ0 + K+ k=1 λ k ).


As convolutional layers are invariant to location, we can convolve each layer depending on the length of the input.

in our architecture, we only use convolutional and pooling layers.


This can be done via standard maximum likelihood estimation (MLE), but the latent grouping pattern among the parameters is totally ignored, and the model can be overparametrized.


1 to zero we find that E q * [(V * 0 − V ) K ] = 0, where q * ≡ q( • ; λ * ) is the variational distribution at the optimum.

the lower bound at the optimum is , where the sum runs only to K − 1 because the term with k = K vanishes at V 0 = V * 0 .


We observe that the online shortest path problem is a special case of online MDP learning.

the above reduction also shows that, short of a major breakthrough, it is unlikely that there exists a computationally efficient algorithm for the fully adversarial online MDP problem.


In this framework, we generate deep GNNs by "stacking" GNN layers in a hierarchical fashion (Figure 1): the input nodes at the layer l GNN module correspond to the clusters learned at the layer l − 1 GNN module.

each layer of DIFFPOOL coarsens the input graph more and more, and DIFFPOOL is able to generate a hierarchical representation of any input graph after training.


We also show that the performance of NBS-H is sensitive to the hyper-parameter k in the k-nearest neighbor search.

the soft assignment is preferable if computational resources allow.


The 2N distances can determine the location of the vertices along the dimension up to a shift, through use of the law of cosines: For each subsequent dimension, two new vertices are chosen and new dissimilarities (δ ai , δ bi ) are computed by Dijkstra's algorithm.

the subsequent dimensions are assumed to be orthogonal to previous ones, so the distances for dimension N are computed from the dissimilarities via: each dimension accounts for a fraction of the dissimilarity matrix, analogous to PCA.


Although the theorem gives an upper bound for the local minimax complexity, this does not guarantee the existence of an algorithm that achieves the local complexity for any function.

it is important to design an algorithm that adapts to this benchmark for each individual function.


Fortunately, by taking D(Q||P ) ≥ Λ, we can effectively constrain the adversary.

we note, as usual, that the learner can (and should) reject with probability 1 any null events under P . an adversary would be foolish to choose a distribution Q that has any probability for these events.


This is because b > 0 for all integers k ≥ 1 with 4k ≤ M . So choosing M = 4k, k ≥ 1, leads to a pathological approximation of φ logit where the log-likelihood can be made arbitrarily large by taking θ 2 → ∞.

a reasonable polynomial approximation for logistic regression requires M = 2 + 4k, k ≥ 0.


What makes it slightly unusual and perhaps of more general interest is the fact that our sensor data are recorded as averages over each minute (our analysis is off-line, for the purpose of making recorded data useable for biomedical research), whereas the events of interest-in this case, re-zeroings, line flushes, and blood draws-can occur at any time and have durations ranging from under 5 seconds to over 100 seconds.

the natural time step for modeling the sensor state transitions might be one second, whereas the measurement interval is much larger.


The contributing response will change only due to a component of motion along a circle centered on the heading, such as is the case when heading direction or rotation is varied.

the contributing response will always be from the motion sensor oriented along the circle around the template's preferred heading.


The cluster tree of W consists of a single cluster at all levels, whereas the ideal bipartite clustering has two clusters.

consider applying a transformation S to W which maps it to a "similarity" graphon.


A mode of B with small potential may cause big penalty when it is extended to a labeling of the whole graph.

when defining a local mode, we select a boundary region of the subgraph and consider all possible label configurations of this boundary region.


The basin of attraction becomes larger than that of the fixed threshold condition, θ = 0.52 (Figure 2).

the network becomes robust against noise.


Also note that Theorem 3 establishes consistency of misoKG for the special case that D is finite, since then d = 0.

interestingly, we can compute K p given Σ and p. we can control the additive error K p • d by increasing the density of A, leveraging the scalability of our parallel algorithm.


Note that our algorithm treats the beginning and end of words symmetrically.

the fact that it performs better when the first letters of the word are given than when the last are given is due to regularities in English spelling, and is not built into the algorithm.


However, these systems are often modeled with MDPs with continuous state and action spaces, with limited access to expensive data.

there is a need for control of systems with unknown dynamics and large or continuous state, action, and parameter spaces in data-poor environments.


As the model learns new features, the distance metric changes to accommodate these features.

consistency targets guide learning in two ways.


Synchronized batch-normalization slows down training significantly and training on 128 GPUs is also impractical for most people.

group normalization [35] has been recently proposed so that instance level recognition tasks can benefit from another form of normalization in a low batch setting during training.


Remark Because g(x, y) = ∇ x log p(x)k(x, y) + ∇ x k(x, y), and ∇ x log p(x) is often unbounded if the domain X is not unbounded.

the condition that g(x, y) must be bounded in Lemma 3.1 suggests that it can only be used when X is compact.


The second term grows at most as Õ( √ n), but also becomes a constant for realizable problems.

we attain a logarithmic label complexity in the realizable setting.


Error correction is still the most costly step, but it would be 10× more costly without restricting to the locations found by the error detection network.

the cost of error detection is more than justified by the subsequent savings during the error correction phase.


The kernel for Ĝr = I and F = r F r (whose entries are f ij ) should be positive semidefinite [15].

choosing F = 11 T (all entries 1) and multiplying the kernel by 1 and using K RBF as the kernel on vectors, we get back the kernel defined in (1).


In order to measure similarity between the input images, the applied transformations must be similar.

along the lines of Ahmed et al., we perform two stages of convolutions and pooling on both the input images by passing them through two input pipelines of a "Siamese" Network, that share weights [2].


After establishing that δ is an unbiased estimator, we show that the probability of a large deviation between δ and δ decreases exponentially in |V − L|, the number of internal nodes of T δ .

to show a high probability bound for our size estimate, we need to establish a lower bound on the number of internal nodes of T δ , which will be the final step in the analysis.


A detailed proof of this theorem will be given elsewhere.

the merger cost is a multiplication of the weight of the merger components (P(tj)) with their "distance" given by d(t; , tj).


A multi-class binary decision tree in dimension d can be defined by a pair (t, h), where t is a binary tree with a variablethreshold question at each internal node, e.g., X j ≤ θ, j ∈ [1, d], and h = (h l ) l∈Leaves(t) a vector of distributions over the leaves Leaves(t) of t. At any leaf l ∈ Leaves(t), h l (y) ∈ [0, 1] for all y ∈ Y and y∈Y h l (y) = 1.

for convenience, we will denote by t(x) the leaf l ∈ Leaves(t) associated to x by t. the score associated by (t, h) to a pair (x, y) ∈ X × Y is h l (y) where l = t(x).


Humans can easily reason about the entities in a complex scene by building multifaceted mental representations of the scene contents.

any plausible attempt for Artificial General Intelligence must also be able to reason about entities and their relationships.


Note that V (x, y) in Lemma 2.1 represents the amount of violation from the hard constraint in the original dual formulation (3).

by adding the regularizer in the primal, we are relaxing the hard constraint in the dual representation to a soft one in the objective function.


The other stream aims at computing the rest pixels Î(t+1) bg .

the output end of OODP, Î(t+1) , is calculated by, bg .


For simplicity, let us assume that kl/δ = O(1) and ε = o(1).

by Lemma 5.2 we know that unless v ≤ ε, there are Ω(1) fraction of r such that ∂L(w)/∂w r 2 = Ω(ε).


In this case, we exactly solve the asymptotic dynamics and prove that w(t) = log(t) ŵ + O(1) where ŵ is to the L 2 maximum margin separator.

the margin (affecting generalization) improves slowly (as O(1/ log(t)), even while the training error is very low.


We show that, for all odd K, L (K) (λ, V 0 ) is a nontrivial lower bound on the marginal likelihood p(x).

we propose the perturbative black box variational inference algorithm (PBBVI), which maximizes L (K) (λ, V 0 ) over λ and V 0 with stochastic gradient descent (SGD).


Since LS based algorithms need O(d 2 ) storage and computational complexity while GTD algorithms are both of O(d) complexity, gradient based algorithms are more commonly used when the feature dimension is large.

in this paper, we focus on GTD algorithms.


In other words, if the summary S contains private data of a user v = u, the user u cannot have access or evaluate v's private part of S, i.e., S ∩ V v . In public-private data summarization, we would like to find the smallest subset S ⊆ V such that all users reach a desired utility A naive way to solve the above problem is to find a separate summary for each user and then return the union of all summaries as S. A more clever way is to realize that problem (1) is in fact equivalent to the following problem [13,14] OPT = arg min Note that the surrogate function f (•) is also monotone submodular as a thresholded submodular function remains submodular.

finding a set S that provides each user with utility Q is equivalent of finding a set S with f (S) ≥ L . = Q × |U|.


As a result, we can approximate (M u − M u D(λ)M u ) −1 as follows: .

we can write ω(f l , ρ) as follows: where φ(f l , ρ) is the output of the following optimization problem max We can simplify the above problem by approximating As the above problem is a linear programming problem, the solution for λ can be computed as: From the above formulation, we find that ρ plays the role of a threshold of selecting the unlabeled examples.


Transfer learning The beauty of kernel methods is, however, that a kernel function can be used across a wide range of tasks and consistently result in good performance.

in the next experiment, we investigate whether the resulting kernel retains this generalization capability when it is task-adapted.


However, it is unlikely that an autonomous robot is able to keep all this data.

we instead focus on a streaming setting for lifelong learning, where the learner can only retain learned models from previously encountered tasks.


Again, analogous to support vectors, rules with non-zero u i j components are called support constraints [6].

in the final classifier the advice sets are incorporated with advice error (advice set 1) or are completely ignored (advice set 2).


As we remarked earlier, we cannot make a statement that holds uniformly for all events E; we can only make a guarantee in expectation.

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


We also make use of a result due to Koo and Lee (2007), who proved a rate of convergence result that holds under more stringent conditions.

the main thrust of the present work is show-ing that these conditions hold for p-power weighted nearest-neighbor graphs.


Proposition 15 Consider the set of points that are solutions of Eq.

3 for some seed For the reverse direction, suppose that w * ∈ W G , in which case there is some v ∈ V such that w * G v and L(w * ) ≤ L(w) for any w G v. Since G is a pre-order, it must hold in particular that L(w * ) ≤ L(w) for any w G w * G v. we also have that w * ∈ arg min w G w * L(w), i.e., w * ∈ W G . We describe the results of numerical experiments when regularizing with the permutahedron (symmetric group) and the signed permutahedron (hyperoctahedral group).


We can treat that sequence is one of the special form of trees if we say sequences are rooted by their last symbol and each node has one child each of a previous symbol.

in this paper, the word 'tree' is always including sequence.


This is an unrealistic case (π * is unknown), but since we work with known MDPs we can compute this quantity (the model being known), for the sake of a complete empirical analysis.

pS(d µ,π * ) and RPS(d µ,π * ) are compared in Fig.


Note that the bounds generally depend on distribution-specific properties of the Markov chain (e.g., P , t mix , γ ⋆ ), which are often unknown a priori in practice.

much effort has been put towards estimating these unknown quantities, especially in the context of MCMC diagnostics, in order to provide datadependent assessments of estimation accuracy [e.g., 11,12,[17][18][19].


For each cluster k, we independently drew M (k) * samples from N H (k) * (0, 10I H (k) * ), where N d (µ, Σ) denotes the d-dimensional Gaussian, and projected them into the observed L-dimensional space by , each entry of which follows N 1 (0, 1).

we obtained a noiseless matrix for the k-th cluster.


On the other hand, by standard concentration argument our test statistic Note that the latter term is determined by the angle of θ(M :t , U k ).

our test statistic in Step 2 is indeed an effective measure of θ(M :t , U k ), or θ(L :t , U k ) since L :t ≈ M :t , as proven by the following novel result.


We now sketch the proof of Theorem 1.1; almost all the details are relegated to the appendix.

recall that for n ≥ n o , we take F n to be the sigma-field of all outcomes up to and including time n, that is, An additional piece of notation: we will use u to denote u/ u , the unit vector in the direction of u ∈ R d . for instance, the Rayleigh quotient can be written G(v) = v T A v. We first bound the expected improvement in Ψ n in each step of the Krasulina or Oja algorithms.


Note that switching to a motion path without any motion constraint (equation ( 2) with WI = 0) will never cause any additional discontinuities, because the smoothness constraint is the only one left.

we solve our problem by precomputing a cost function that does not constrain the motion for a couple of transitions, and then starts to constrain the motion with the new motion direction.


It is worth mentioning that this paper aims at developing a turbo learning approach to training LSTM-based CaptionBot; therefore, it is directly comparable to an LSTM baseline.

in the experiments, we focus on the comparison to a strong CNN-LSTM baseline.


The difference is that we assume a hierarchical Bayesian model where the disease rates q in , q out , and q all are themselves drawn from Gamma distributions.

under the null hypothesis H 0 , we have q = q all for all s i ∈ G, where q all ∼ Ga(α all , β all ).


In this case, the boundary is a horizontal line.

the velocity boundary depends upon the spatial frequency.


Stochastic SVM algorithms have been recently developed that have convergence time with no dependence on input size [19].

we develop a large-scale algorithm based on projected stochastic subgradient descent.


In the case of the Gaussian, the amount of overlap between the two likelihoods decreases (Figures 3Ai,3Aiii).

the samples are from a somewhat smaller region in space and hence the variability also decreases.


(high likelihood, or far from the margin).

importantly, however, the samples are tied together in the objective through the parameter w. no sample is considered independently easy; rather, a set of samples is easy if a w can be fit to it such that the corresponding values of f (.)


Furthermore, assume grandpaOf, ABE and BART represent the indices of the respective symbols in a global symbol vocabulary.

then, the new proof state constructed by unify is: the output score of the neural network S 0 ⇢ will be high if the subsymbolic representation of the input s is close to grandpaOf and the input i is close to BART.


Improper handling of missing data can bias outcomes and potentially distort the conclusions drawn from a study.

accurate diagnosis of the causes of missingness is crucial for the success of any research.


In an input network, links (i, j) ∈ E are observed as positive training examples a ij = 1.

their latent inner product z i W z j should be increased to reflect such close linking relationship.


As shown in [9], PCA reveals that the Origin-Destination (OD) flow matrices of ISP backbones have low intrinsic dimensionality: For the Abilene network with 41 links, most data variance can be captured by the first k = 4 principal components.

the underlying normal OD flows effectively reside in a (low) k-dimensional subspace of R n . This subspace is referred to as the normal traffic subspace S no .


Note that in the "realizable" setting where these constraints can be simultaneously satisfied, they do not uniquely determine the parameters {Φ c }, which can be scaled to yield arbitrarily large margins.

as in SVMs, we propose a convex optimization that selects the "smallest" parameters that satisfy the large margin constraints in eq.


Thus, for a GMM, each of Gaussian essentially models a small partition of the feature space and many of them are needed to depict the whole feature space.

the number of mixtures needed will be determined by the volume of the feature space.


As the prune ratio increases, information flow flows through an increasingly small subset of features.

no feature maps are dropped out when the prune ratio is equal to zero, and the whole residual branch is dropped out when the ratio is equal to one (i.e., the information of the identity branch passes through directly).


To account for fuel usage, the stage-wise cost of each move until reaching the destination is 1, while the reward achieved for reaching the destination is 1000.

we would like the agent to reach the destination in the shortest possible number of moves.


By construction, a proportion of measurable motion directions will be into one specific direction, but, through the random relocation of other dots, the RDM display will also contain motion in random directions.

the brain observes a distribution of motion directions at each time point.


However, since spherically symmetric models are invariant under orthogonal transformations, they are agnostic to the particular choice of basis in the whitened space.

there is no role for the shape of the filters in this model.


Unfolding the definition of cost now using cash invariance, we have ρ .

we may view a potential-based market maker as a constant-risk agent.


Note that L 1 (ω) is continuous and twice differentiable.

standard gradient-based optimization techniques can be applied to minimize it.


sample, and approximation error ( a ) resulting from the approximate feature map.

to get a better handle on the excess error, we decompose it as follows.


The disturbance, however, enters the system only after having been filtered by the environmental transfer function ¨¦ .

compensation of requires to reverse this filtering by a term R ¦ ¦ which is the inverse environmental transfer function (hence "inverse controller").


To compute Q w (z, s) for a rational loss, recall that Q w (z, s) ∝ y : y s−q+1:s =z e L U (y,yi)+w•Ψ (xi,y) .

we will design two WFAs, A and B, such that A(y) = e w•Ψ(xi,y) , B(y) = e L U (y,yi) , and their composition C(y) = (A • B)(y) = e L U (y,yi)+w•Ψ(xi,y) .


For such cases, our algorithm only needs to take a few samples in each round of the elimination step and drops half of vertices with high confidence.

the experimental result shows that the sample cost of ME-AS is much less than AMCV.


This guarantees that we only need to set the step size ⌘ t = 1 and the objective value is monotonely decreasing without the sophisticated backtracking line search procedure.

the proximal Newton algorithm enjoys the same fast convergence as in low dimensional optimization problems [12].


Let f w (x) be the function computed by a d layer feed-forward network with parameters w and Rectified Linear Unit (ReLU) activations, where φ(z) = max{0, z}.

let h i be the number of nodes in layer i, with h 0 = n. for any layer i, we have W i ∈ R hi×hi−1 .


Similarly, the tree-to-sequence model has two variants, and our tree-to-tree only has one form.

we have 9 different models in our evaluation.


f (x i ) = g(y i ) ∀(x i , y i ) ∈ (X, Y ).

the finite-sample version of Equation ( 1) is constant and equal to "1" for continuous random variables.


Moreover, the relations between nodes are usually directed.

we should employ a (1) directed, (2) multi-labeled, and (3) hierarchically structured graph to model structured natural language data.


[12] showed that such networks can express any function computed by a Turing machine, similar to networks with ReLU or sigmoid activations.

following [12], we consider the class of 2-layer polynomial networks with inputs from R d and k hidden neurons: If we write A = k i=1 a j w j w j , we have the following equivalent formulation: if replace the hard rank constraint with trace norm A * ≤ θ, the task of empirical risk minimization (ERM) given training data {(x 1 , y 1 ), .


We introduce the learning the learning rate (LLR) algorithm, which satisfies (2), ( 3) and ( 4) and in addition guarantees a regret satisfying for any ε > 0.

lLR performs almost as well as the learning rate ηT ∈ [η ah t * , 1] that is optimal with hindsight.


t ) in an online fashion, only presenting the true label y ?

t after the online learner A has chosen a prediction y t . setting `t(y t ) = `(y ?


Here, we show that L p -nested functions allow for an analytical expression of the partition function.

the corresponding distributions constitute a flexible yet tractable subclass of ν-spherical distributions.


Similarly, we assume that the distribution of features of different phones (i) is a Gaussian distribution with covariances ~i' Therefore, h(X/Y1) = ~ l: p (y~)Iog (2net I~il (6) yiCYi Finally, we assume that the distribution of features of different phones spoken by different speakers is also a Gaussian distribution with covariances ~ij.

h(X/Y1,Y2 ) = ~ l: p(yLYOlog(2netl~ijl (7) y;CY1,y~CY2 Substituting equations 6 and 7 in equations 4 and 5, we get J(X ' Y;)=~lo I~I .


In fact, the activations at each intermediate layer could be interpreted as some intermediate representation of the data for the following part of the network.

we can define the margin based on any intermediate representation and the ultimate decision boundary.


However, for the tasks considered in Section 4, we observed that this led to overfitting the training data even when applying regularization techniques and reduced the performance at test time.

we will not further investigate this extension of the model in this paper, even though it might be beneficial for other tasks or larger data sets.


Compared to structural and functional connectivity patterns, anatomical profiles are more easily affected by variability across individuals.

the anatomical constraint energy is designed to provide constraint only to ROIs that are obviously far away from reasonableness.


Note that f l i is a sub-vector of f l , so equation ( 8) can be written in a compact form as: where o l and f l are the same as in (3), while the matrix A = [a ij ] ∈ R n×n is constructed as follows: ∀x i and x j , 1 ≤ i, j ≤ n, if x j ∈ N i , then a ij equals the corresponding element of α i in ( 9), otherwise a ij equals 0.

similar as α i , the matrix A is also independent of f l and the cluster index l. Substituting (10) into (3) results in a more specific optimization problem: subject to F is a scaled partition matrix (12 where based on the KRR algorithm, we have transformed the objective function (3) into the quadratic function (11).


For λ = 1, corresponding to standard segmental k-means, we obtain results similar to the standard HMM, which obtains an NMI of .57 and error of 1.16.

the parametric method offers some added flexibility via the new λ parameter.


All training data are needed for each machine to calculate Q(J k , k).

each machine must be able to store a local copy of the training data.


We can prove known facts trivially by a unification with themselves, resulting in no parameter updates during training and hence no generalization.

during training we are masking the calculation of the unification success of a known ground atom that we want to prove.


If the blue spike arrives just 5ms later, as shown in red, the EPSCs sum to a value already 20% less.

the extra precision of the digital signal may be used to communicate the amplitude of the analog signal.


In fact, this similarity underlies the usage of gradient information in the classical computer vision literature.

while the distances are expected to differ in the two domains, one can expect a high correlation.


The condition above implies that the policy distribution P has a value function that can be approximated by using F . Enforcing this condition results in a bias favoring policies with a low Bellman error.

we are interested in learning a distribution P (π 0:H ) that satisfies this condition, while maximizing its expected value.


whether to collapse out certain variables), or the choice of MCMC transition operators.

the ability to evaluate AIS-based inference should also yield insights which inform the design of MCMC inference algorithms more broadly.


Note that a uniform grid provides the loss n −α/D , which is polynomially decreasing only (and subject to the curse of dimensionality).

in this example SOO is always better than both Uniform and DOO except if one knows perfectly α and would use DOO with β = α (in which case we obtain an exponential loss).


-Type II: Consider a strict KL contraction operator with β = 1, denoted as Φ t , is parameterized by an auxiliary parameter t that is different from θ, and for any distribution p ∈ Ω d , we have Φ 0 {p} = p and Φ t {p} is continuously differentiable with regards to t. Then, the KL divergence Φ t {p} and Φ t {q θ } can be regarded as a function of t and θ, as: L(t, θ) = KL(Φ t {p} Φ t {q θ ).


Because the Readers and Updaters may reside on different machines, perhaps located in different parts of the world, communication between them is not instantaneous.

when making a prediction, a Reader will generally be using a coefficient vector that is somewhat stale relative to the most recent version being served by the Updaters.


The strength of this approach, however, comes at a price: the set X still must be specified beforehand.

when one faces a recovery problem without any prior knowledge of X , this approach cannot be implemented.


The condition b/(Cδ) > Ω/π in Theorem 1 is a Nyquist-type rate condition.

perfect identification of the projection of h onto Ξ can be achieved for a finite average spike rate.


[4] has shown that the PŁ condition is the weakest condition among the following conditions: strong convexity (SC), essential strong convexity (ESC), weak strong convexity (WSC), restricted secant inequality (RSI) and error bound (EB).

the convergence analysis for the PŁ condition naturally extends to all the above conditions.


In the new formulation (2.3), the coefficients of each individual sample (i.e., a row of the matrix R) is uniformly constrained.


Such plasticity may disrupt the recall of stored episodic patterns by changing the correspondence between the input areas and EP.

if the brain is still to be able to recall hippocampally stored episodes, it either needs to maintain the correspondence between the low-level and EP representations of the episodes by restricting neocortical learning (achieved in the previous model by having the hippocampus replay its old episodic patterns along with the new semantic patterns governing continued neocortical plasticity), or it needs to update the connections between the hippocampus and EP such that the hippocampally stored pattern continues to match the EP representation of the input pattern corresponding to the episode.


However, for this case, it is easily checked that any subgraph of G obtained by deleting exactly one edge results in a (k − 1) tree, and every sub (k−1)-tree results from this operation.

the optimal (k−1)-tree can be found using Algorithm 1, and in this case, the complexity of Algorithm 1 is O(n(k + 1) 2 ).


It is therefore obvious that the Shapley mediator satisfies EF.

4 one would expect that the Shapley mediator will be the only mediator that satisfies F and EF.


In our setting, where the adversary chooses a different entry at each round, [21,Theorem 6] implies that for the class W of all matrices with trace-norm at most r = O(n), it holds that R T (W )/T ≤ O(n 3/2 /T ).


The blocking effect demonstrates that the response time depends not only on information accruing from the current stimulus, but also on recent stimuli in the trial history.

any explanation of the blocking effect must specify the manner by which response initiation processes are sensitive to the composition of a block.


On the other hand, though theoretical bound on the pre-constant ζ(ε, ρ) as function of 1/ε and ρ is not very exciting, our simulations suggest (see Section 5) that even for hard problem setup, the performance is much more optimistic than predicted by these theoretical bounds.

as a recommendation for a system designer, we suggest use of smaller 'radius' distribution in algorithm described in Section 3 for obtaining good algorithm.


1 Specifically, consider the self-expressiveness term in (3), Z Θe − Z Θe C 2 F . Since each data point z i (in the latent space) is approximated by a weighted linear combination of other points {z j } j=1,••• ,N (optionally, j = i) with weights c ij , this linear operation corresponds exactly to a set of linear neurons without non-linear activations.

if we take each z i as a node in the network, we can then represent the self-expressiveness term with a fully-connected linear layer, which we call the self-expressive layer.


To parallel standard linear regression, {y i , x i } n i=1 are samples of (Y, X) linked through Y = X, β * + W . However, we assume each entry of x i is missing independently with probability ∈ (0, 1).

the observed covariate vector x i takes the form x i,j = x i,j with probability 1 − * otherwise .


Let us first look at samples originating from the same initial condition.

according to Theorem 1, the DMP equations are exact on tree graphs, and hence it is easy to see that since the expression inside the brackets sums exactly to one.


Selecting from large minipools would require too much data and sampling without replacement (as is usually done in practice) would converge trivially since the entire dataset would eventually be labeled.

instead of selecting points from the minipool subset and without replacement, we select points from the entire pool with replacement.


For pylons as well as for the pool-based approaches [11,18], the segment pool is much richer.

the constraints (1) and (2) that are trivial to enforce in the case of the flat random field, become non-trivial.


For simplicity, suppose this is the only strategy allowed by the set P. Then x t (ǫ) − E t−1 x t (ǫ) are independent increments when conditioned on the history.

further, the increments do not depend on ǫ t . V T (P 1:T ) ≤ 2E T t=1 Y t where {Y t } is the corresponding random walk.


Natural Images: Natural image patches have sparse underlying structure in the sense that they can be well approximated by sparse linear combinations of fixed bases, but they cannot be exactly reconstructed at a level of sparsity required by the theorems of CS and ACS.

cS and ACS cannot be expected to produce exact reconstructions of natural image patches.


For notational simplicity, we define another operator F =: I + ∂R(•; y).

the last condition in (6) indicates that x k+1 = F −1 (v).


Such a simple similarity measure could be computed more efficiently outside the random walk framework.

halting may affect both the expressivity and efficiency of these similarity scores.


The improvements of L2H&TF-IDF and L-LDA&TF-IDF over TF-IDF are clearer for Macro-F 1 compared with Micro-F 1.

features from both topic models help improve prediction, regardless of the frequencies of their tagged labels.


More precisely, let x p be the mean of X ∼ µ p and let us look at the limit of V µp (f (X)) when v p goes to 0.

assuming that f is differentiable, we get if we introduce Σ p to be the covariance matrix of the random variable X ∼ µ p , then we simply have lim Example with hyper-cubic regions An important example is when R p is a hypercube with side length v 1/d p and µ p is the uniform distribution over the region R p . In that case (see Lemma 1), we ), then it is not too difficult to show that for such hyper-cubic regions, we have On the other hand, by direct computation, the second term is such that � stays bounded away from 0 (unless ν is deterministic).


In order to improve the convergence of the algorithm, we draw the inputs from a zero-mean random Gaussian distribution.

the performance of the dimensionality reduction network is ultimately limited by the resolution of the memory cells, the reference noise, the learning rate of the LMS training stage and linearity of the multipliers.


By the Paley-Zygmund inequality, every time the algorithm takes Poi(m) samples, there is at least a c r > 0 probability of seeing an r-wise collision.

if we consider our algorithm to take samples in blocks of size Poi(m), the probability that we have not found an r-wise collision after t blocks is at most (1 − c r ) t . Thus, the expected number of blocks until we have an r-wise collision is O r (1).


Of course, the theoretical maximum of λ depends on unseen test data in reality.

we often have to set λ conservatively so that we can apply the trained model to any unseen graph data.


In fact, FLD is arguably a more principled whitened variant of the Prot classifier: Its weight vector can be written as w = S −1 W ( µ + − µ − ), where S −1 W is the within class covariance matrix of the two classes, and µ ± are the class means.

if we disregard the constant offset b, we can write the decision function as x , which is a prototype classifier using the prototypes µ ± after whitening the space with S −1/2 W . We combine the linear preprocessor (PCA) X = EB and the linear classifier (SVM, RVM, Prot, FLD) y( x) = w| x + b to yield a linear classification system: y = w T E T + b where b = b 1.


Although, our convergence guarantee applies to this objective, our smoothness analysis predicts that small weight regularization leads to an unstable algorithm.

this fact has also been observed in our experiments.we use Sinkhorn loss [19,41,5] for which our convergence guarantee holds.


The task given to experimental subjects specifies that there are four categories, and they have an ordering; the circle diameters associated with category A are smaller than the diameters associated with B, etc.

one might make a further assumption that the category prototypes are equally spaced.


This initial period is known as the "burn-in."

human perceptual inference may similarly require an initial burn-in period to reach the stationary distribution.