The random matrix W = W * in S n×n is a Gaussian Wigner matrix with variance σ 2 W if its entries on and above the diagonal are independent, zero-mean Gaussian variables (real or complex depending on context) with variance σ 2 W . Besides Assumption 1, another important assumption for our results is that the search space C (1) of (SDP) is compact.

in that scenario, there exists a finite constant R such that for all where G † is a continuous function of Y as in Lemma 2.2.


. . , x x x (n) } rather than a local window, due to the nature of structural dependencies among tags (e.g., graphical models like CRFs).

in structured prediction a local classification on k should be denoted as g(x x x (1) , .


While the optima of both bounds vanish with n → ∞, their behavior is qualitatively different.

in particular, the bound of [8] is U-shaped, and diverges for k = n, while ours is L-shaped (no tradeoff), and thus also convergent for k = n. when compared with [8], our results suggest that no regularization is required from a statistical point of view though, as clarified in the following remark, it may be needed for purposes of numerical stability.


However, it is required to incorporate new laser data and update w using SGD at each iteration.

gP predictions and probabilistic feature embedding can be skipped by setting Σ = 0, whenever it is not required to predict future or past maps as the uncertainty of knowing the current location for any laser reflection is zero.


In many practical applications, only a reasonable and limited number of data passes is performed due to time and resources constraints.

we also study how the structure of a network affects the largest possible batch size to converge within a fixed number of epochs/data passes to a pre-specified accuracy.


However, we note that our algorithms aggregate full rankings under PL, while the RC algorithm aggregates pairwise comparisons.

it is quite hard to directly compare our GMMs and RC fairly since they are designed for different types of data.


Recent research also reveals that deep features eventually transition from general to specific along the network, and feature transferability drops significantly in higher layers with increasing task dissimilarity [28], hence the sharing of all feature layers may be risky to negativetransfer.

it remains an open problem how to exploit the task relationship across different deep networks while improving the feature transferability in task-specific layers of the deep networks.


In essence, we build on the fact that vector quantizers project data into a finite set of N elements, the index space, and on the simple observation that kernels on finite sets are fully specified by the N×N Gram matrix of these elements (the kernel matrix), which we propose to learn directly.

qKs are piecewise constant but otherwise arbitrary, making them very flexible.


Olshausen & Field [11,12] try to find bases that result in a code s that preserves information about the original image x and that is sparse.

they minimise the following cost function with respect to A, where A, denotes a constant determining the importance of sparseness [11] : The algorithm proposed in this paper is similar, but it takes into consideration the temporal order of images.


Table 1: Summary of algorithms for Proj R n ↓ (y) (x) It can be easily verified that gradient of 1 2 P Ω (X) − z 2 2 is 2-Lipschitz continuous.

from standard results on convegence proximal gradient descent [30], Algorithm 1,converges to within an error in objective in O(1/ ) iterations.


Moreover, their gradients rf (w, z) are Lipschitz continuous with constant M krf (w, z) rf (w 0 , z)k  M kw w 0 k, for all z.

the conditions in Assumption 1 imply that the average loss L(w) and the empirical loss L n (w) are convex and their gradients are Lipschitz continuous with constant M . the empirical risk R n (w) is strongly convex with constant cV n and its gradients rR n (w) are Lipschitz continuous with parameter M + cV n . So far we have concluded that each subproblem should be solved up to its statistical accuracy.


This leads to our first relaxation of the generalized eigenvalue bound for a model.

from Lemma 3.2 and Proposition 4.2 we see that this result suggests that to approximate the graphical model (Θ, Φ) we can search for a parameter matrix B ⋆ , with corresponding simple graph G(B ⋆ ) ∈ S, such that While this relaxation may lead to effective bounds, we will now go further, to derive an additional relaxation that relates our generalized graphical model support number to the "classical" support number.


Hence, with probability of at least We note that the selection function g(x) equals one only for x ∈ X \DIS (G) .

for any x ∈ X , for which g(x) = 1, all the hypotheses in G agree, and in particular h * and ĥ agree.


Unfortunately, solving (1) is non-convex, discontinuous, and NP-hard in general.

the convex surrogate referred to as principal component pursuit (PCP) is often adopted, where • * denotes the nuclear norm and • 1 is the 1 matrix norm.


Furthermore, the search for good parameters was much harder for the waveform representation because the value range for the kernel width was much wider ( [1,1152] instead of [1,45]).


Meanwhile, the space Ω 1 is formulated under the RKHS framework as a set of columns of the kernel matrix (denoted K 1 ).

we can write f = 1 T d + K 1 e, and so our formulation becomes: The solution to this optimization problem is for our coefficients d and e to be linear estimates on y, our vector of observed values.


Adaptive language models change their parameters according to the recent history.

they implement a form of domain adaptation.


Nonetheless, as we will show in this section, NW and PL are deeply connected to the screening rule, and when given a large enough regularization parameter, the application of the screening rule to NW and PL can be lossless in practice (see Section 5).

when applied to NW and PL, the proposed screening rule can be considered as a strong rule (i.e., a rule that is not foolproof but barely makes mistakes) and an optimal solution can be safeguarded by adjusting the screened solution to optimality based on the KKT conditions of the inexact problem [Tibshirani et al., 2012].


This is because such models are typically obtained as a result of training based on many hours of speech with files occupying hundreds of gigabytes of disk space, while the models derived require orders of magnitude less space.

combining raw labeled data sets is not possible in such cases.


For the base case, suppose V is a leaf with child W; we want to describe the Nash breakpoint policy for V. If for all w E [0,1], the function ~v(w) is non-negative (non-positive, respectively), V can choose 1 (0, respectively) as a best response (which in this base case is an upstream Nash) to all values W = w. Otherwise, ~ v (w) crosses the w-axis, separating the values of w for which V should choose 1, 0, or be indifferent (at the crossing point).

this crossing point becomes the single breakpoint in Fv(w).


Finally, if the gradient in this context is not stochastic (i.e., the parameter Q = 0), then linear convergence to the optimal value is achieved, which matches known rates of convergence with (centralized) gradient descent under strong convexity and smoothness assumptions.

, the sequence of objective function values are themselves upper bounded as follows: using Theorem 1 we can establish analogous convergence rates in terms of the true objective function values { N n 1 T F(x k )} as well.


Sometimes, of course, words are strongly associated with extremes on the response variable continuum regardless of underlying topic structure.

in addition to hierarchical regression parameters, we include global lexical regression parameters to model the interaction between specific words and response variables.


Then we have that DB1 is linearly independent and [DB1 J ~ HI{.


This is achieved by calculating the average percentage of instances in S 3 which switches between clusterings.

conditioned on the event appearing in the second summand of Eq.


Thus we obtain where the last inequality comes from Lemma 4.6.

for a pre-specified accuracy ✏, we need at most T = Moreover, Corollary 4.8 implies where the last inequality comes from (4.6).


However, as mentioned earlier for large dimensions, this is infeasible.

we need to have an alternative search approach similar to the version for non-parametric version, except that in each round, for each variable X 2 ⇡ t , quantity T IB X, 1 (D), given in line 6 of Algorithm 1, is calculated.


Then, via the subadditivity of g(•) we can reach a result stronger than Theorem 3.1 that d 2 (x, y) is a distance metric.

d(x, y) is also a distance metric based on Lemma 3.2.


We show that the problem of even deciding whether a given data set is separable is NP-hard, and thus learning in a strict sense is no easier than prediction.

we should not expect for our algorithm, or any other polynomial time algorithm, to always succeed at learning from an arbitrary finite data set.


Since v σ i (I) ≥ L(I) and v σ i (I, a) ≤ U (I, a), so from (4) we get r t (I, a) ≤ U (I, a) − L(I).

for iteration T 0 ≤ T ≤ T 0 + m, R T (I, a) ≤ 0.


To avoid confusion on this point, we also introduce the notion of the observed denotation set of a query, denoted q train , which corresponds to the set of nodes that exactly satisfy q according to our observed, training edges.

our goal is to train using example queryanswer pairs that are known in the training data, i.e., (q, v * ), v * ∈ q train , so that we can generalize to parts of the graph that involve missing edges, i.e., so that we can make predictions for query-answer pairs that rely on edges which are unobserved in the training data (q, v * ), v * ∈ q \ q train .


However, the authors do not show how to solve the general KWIK online regression problem efficiently, and it is not even clear whether this is possible.

though the result of [15] is interesting, it does not provide a provably efficient algorithm.


DHA is parametric and uses rank-m SVD and stochastic gradient descent for optimization.

dHA generates lowruntime on large datasets, and DHA does not require the training data when the functional alignment is computed for a new subject.


However, the other papers which have up to now discussed global optimality for neural networks [11,8] have not included any results on real datasets.

up to our Nonlinear Spectral Method for 1 hidden layer Input: Model n 1 ∈ N, p w , p u ∈ (1, ∞), ρ w , ρ u > 0, α 1 , .


log scale) where t * > 0 represents future.

embedding both mean and variance, rather than merely mean, is crucial for a higher accuracy.


However, this comes at a cost as well: the nominal error increases throughout the cascade (this is to be expected, since the cascade essentially tries to force the robust and nominal errors to match).

there is substantial value to both improving the single-model networks and integrating cascades into the prediction.


A weight-update rule f is (strictly) incentive compatible for groups with transferable utility if and only if f is a (strictly) proper scoring rule.

for online prediction with selfish experts, we get TU-GIC "for free."


Given the result of the previous section, it is clear that if the oracle is replaced by the tests described, then CLIN is a.s. consistent.


By construction, the 3 assignment has value K − 1.

the optimal assignment must either be 3 with value K − 1, or some combination of states 1 and 2, which has value at most K * .


However, Theorem 1 guarantees the invariance of the optimal policy with auxiliary rewards.

even though the agent may receive non-positive auxiliary rewards, it can still learn to query those critical negative symptoms.


Combining ( 25) and (26) yields In the s update we fix q \s , i.e., g s (q) = q \s .

b s is an identity matrix with its s th column removed B s = I :,s , where I the identity matrix and s, : means without the s th column.


This is because the hovering controller is always trying to "catch up" to the moving $ S£ ¡ £ $£ ¡ §¦ p£ ¡ £ ( .

` e£ may remain large, and the helicopter will continuously incur a ` S£ cost, even if it is in fact flying a very accurate trajectory in the increasing -direction exactly as desired.


Even though our algorithm is given the rank m as a parameter, the internal dimension is not revealed, if ever known.

we could choose the best m using model selection.


Co-authorship graphs give another such example.

there have been efforts to develop algorithms that do not make assumptions about the PSD-ness of the similarity functions used.


During inference, 2 the neuron will fire strongly for its preferred stimulus (or stimulus interpretation), but then its firing probability will decrease as its excitability and synaptic efficacy drop, allowing the network to discover potential alternative interpretations of the stimulus.

in the case of sparse activity, equations 6-8 implement a form of neuronal fatigue and synaptic depression.


Using 4 binary searches, we can find the two p q closest to each of these roots (giving us 8 candidates for p q , plus two more which are the smallest and the largest p q ), and evaluate I(q) for each of those 10 candidates.


We add initial structure to q φ by noticing that the prior p θ d (d 1:T |u 1:T , d 0 ) in the generative model is a delta function over d 1:T , and so is the posterior p θ (d 1:T |x 1:T , u 1:T , d 0 ).

we let the inference network use exactly the same deterministic state setting d 1:T as that of the generative model, and we decompose it as .


Now we evaluate the learning of similarities over object categories.

we classify object bounding boxes of the PASCAL VOC 2007 dataset.


Hence, using induction, we can argue that after the first mistake, the learning can be done with k − 1 mistakes and k(|H| = |H| 1/k+1 number of ⊥'s before the first mistake.

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


Note that the larger the region R ⊆ Q is, the higher its evidence score LES(R|H ref ) (and therefore it will also provide higher statistical evidence to the hypothesis that Q was composed from ref ).

for example, assume for simplicity that R has a single identical copy in ref , and that T is restricted to shifts with uniform probability (i.e., P (T |H ref ) = const), then P (R|H ref ) is constant, regardless of the size of R. On the other hand, P (R|H 0 ) decreases exponentially with the size of R. the likelihood ratio of R increases, and so does its evidence score LES.


A hypothesis test of whether the sampler has converged based on the standard permutation-based bootstrap leads to too many rejections of the null hypothesis, due to dependence within the chain.

one would require heavily thinned chains, which is wasteful of samples and computationally burdensome.


While previous approaches based on random features typically require a prefix number of features, our approach allows the number of random features, and hence the flexibility of the function class to grow with the number of data points.

unlike previous random feature approach, our approach applies to the data streaming setting and achieves full potentials of nonparametric methods.


4 can only vanish in the limit 0: --+ 0 if N(~-~ = 0 while N(~-l) > 0; in other Xi , 7r i 7ri words, the parent-configuration 7rf must already have occurred previously along the sequence (7rf is "old"), while the child-state xf occurs simultaneously with this parent-state for the first time (xf, 7rf is "new").

in this case, the leading polynomial order of the ratio (for given k and i) is linear in 0:, assuming P(Xi' IIi) > 0; otherwise the ratio (for given k and i) converges to a finite positive value in the limit 0: --+ O. the dependence of the marginal likelihood in leading polynomial order on 0: is completely determined by the number of different configurations in the data.


While we cannot say in advance what each factor will represent, we observed that when Z k is large, components along this factor correspond to topics.

we set Z 1 > Z k>1 and assume the first factor is topic.


Bowling and Veloso showed that in selfplay, a restricted version of WoLF-PHC always reaches a stationary Nash equilibrium in two-player two-action games, and that the general WoLF-PHC seems to do the same in experimental trials.

in the long run, a WoLF-PHC player achieves its stationary Nash equilibrium payoff against any other PHC player.


It adopts an orthogonal basis in the mean function to model the residues that cannot be learned by the standard coupled approach.

our method extends, rather than replaces, the coupled approach to achieve strictly better performance.


This renders the network as a collection of different paths, where each path is a unique configuration of neurons from each layer connecting input to output.

all paths through ordinary neural networks are of the same length.


The cavity method is remarkably successful and a number of its predictions have been made rigorous [MM09,Tal10].

we view Claim 3 as a conjecture, with strong positive evidence.


From optimization perspective, enforcing the Lipschitz constraint is challenging.

many heuristics, such as weight clipping [4] and gradient norm penalty [22], have been developed to impose such constraint on the discriminator.


The goal is to optimize these operators-along with embeddings for all graph nodes z v ∈ R d , ∀v ∈ V-so that the embedding q for any query q can be generated and used to predict the likelihood that a node v satisfies the query q.

in particular, we want to generate query embeddings q and node embeddings z v , so that the likelihood or "score" that v ∈ q is given by the distance between their respective embeddings: our goal is to generate an embedding q of a query that implicitly represents its denotation q ; i.e., we want to generate query embeddings so that score(q, z v ) = 1, ∀v ∈ q and score(q, z v ) = 0, ∀v / ∈ q .


Since probabilities assigned to individual data columns are static we cannot use them as importance measures in a deterministic selection scheme.

in order to apply importance sampling in the heterogeneous setting, we non-uniformly sample m data-columns to reside inside the GPU memory in every round of Algorithm 2 and have the CPU determine the new set in parallel.


A simple choice would be to choose all data independently with the same probability 1/N , but other possibilities can also be implemented within our formalism.

some y i in D 0 may appear multiple times in D and others not at all.


Different from the previous likelihood framework, the likelihood given in equation ( 1) contains a hidden representation with an unknown cardinality.

the Bayesian Information Criterion (BIC) [Schwarz and others, 1978] is introduced to control the complexity of the model, which provides a trade-off between the goodness of fit and model complexity.


Fix π ∈ Π HR , µ ∈ C S and a history h t where t < T , denote r = E µ (r), p = E µ (p), then we have: From Lemma 1, by backward induction, one can show the following lemma holds, which essentially means that for any policy, the expected performance under an admissible distribution µ only depends on the expected value of the parameters under µ.

the distributionally robust MDP reduces to a robust MDP.


Spikes from the top channel induces an anti-clockwise turn, and the bottom channel induces a clockwise turn.

the head-aim is directed towards maximum intensity.


However, according to our results on En→De, layer norm affects the performance of rPAG only marginally.

we decided not to train rPAG with layer norm on other language pairs.


On the other hand, many times functions defined over a small domain can not be decomposed without introducing a large error.

when trying to limit the error introduced, a significant amount of time is needed for such optimization.


By the strict convexity of f , we then have f α(q y , w) + (1 − α)(q y , w ) < αf (q y , w)+(1−α)f (q y , w ) = f (q y , w), for 0 < α < 1, contradicting the global optimality of f (q y , w).

any convex relaxation of (1) that uses a distribution q y over missing values and does not make arbitrary distinctions can never do anything but produce a uniform distribution over the hidden variable values.


We generate data from a 3-dimensional distribution is the normal distribution with mean µ and variance σ 2 .

the first two dimensions x 1 and x 2 are generated from a Gaussian mixture with four components, and x 3 is independent of the other two variables.


This supports the idea that APM only needs a small number of topics to capture many of the semantically meaningful word dependencies.

when increasing the number of topics beyond 5, the performance does not decrease as in Evoc-1.


This turns out to be a complicated nonconvex optimization problem for TGL.

our second major technical contribution is to show that this highly nontrivial nonconvex optimization problem admits closed form solutions (Section 5).


Graphically, we can see that both the exponential family Harmonium (EFH) [26] and its extension of dual-wing Harmonium (DWH) [28] are special cases of multi-view latent space MNs.

it is not surprising to see that multi-view MNs inherit the widely advocated property of EFH that the model distribution can be constructively defined based on local conditionals on each view.


Let N (i) = {j|{i, j} ∈ E} denote the neighbors of i in the graph.

the mean µ ≡ E{x} and covariance P ≡ E{(x − µ)(x − µ) } are given by: The partial correlation coefficients are given by: J ij = 0 if and only if x i and x j are independent given the other variables x V \{i,j} .


Since all competing methods are rule-based models, it is important that students understand the notion of rules before working with any of the models.

we designed a screening question on rules and students can only proceed with the survey if they answered the question correctly.


The middle layer consists of a number of gate functions, each acting as a local neuron or feature extractor to capture the nonlinear relationship between input and output.

conceptually CNF is much more expressive than CRF.


Furthermore, since x k+1 − x k ≥ ε/n, we get using (9), f (S) − x(S) ≤ nε.

f (S) ≤ i∈S:xi<0 x i + 2nε which implies the theorem due to Theorem 2.


it attempts to arrive at a model that conserves the entropy of the data, subject to the a priori constraints.

reducing entropy of the mixture weights P n (z) to obtain a sparse code results in increased entropy (information) of basis distributions P (f |z).


Likewise, for our system, these error rates are 0.28 and 0.22, respectively.

just finding the average scale of the scene accounts for a large fraction of the total error.


As in Figure 1, in each mini-batch data, each network selects its small-loss instances as the useful knowledge, and teaches such useful instances to its peer network for the further training.

the proposed algorithm is named Co-teaching (Algorithm 1).


To compute P * effectively, instead of diffusing on TPG directly, we show in Section 3.4 that FD process on TPG is equivalent to an iterative process on N × N matrices only.

instead of an O(n 6 ) time complexity, we obtain an O(n 3 ) complexity.


By Lemma 4, after t = 4(q−H(0)) r + 1 steps of the MW updates, H(t) > q, a contradiction.

we can conclude that for any q > 0, there exists a time T q such that for all t ≥ T q , H(t) > q.


This works when neighbors in the given input space have similar target distributions, but unlike kernel learning approaches, the features are fixed.

cOREG cannot adapt the space to a misspecified distance measure.


In contextual bandits, these issues do not arise since there is only one state, while, in tabular MDPs, they can be trivially resolved as the states are observed.

with minor modifications, LSVEE can avoid these dependencies for both special cases.


The intuition is that, for ordinal labels, we may expect the contiguous classes will be close to each other after projection, while the noncontiguous classes should be pushed away.

we put larger weights for the noncontiguous classes.


The HSLDS model also performs better than recent comparable models in predicting the firing rate of an isolated neuron based on the firing rates of others, suggesting that it captures more of the "shared variance" of the data.

the method is able to trace the dynamical processes underlying the coordinated evolution of network activity in a way that appears to reflect its computational role.


Hence, the error between the distribution at the t th iteration p t and the true stationary distribution π is dominated by the second term in equation (4).

in order to finish the proof of Theorem 1 we require bounds on ∆ 2 and 1 − ρ.


The contribution of LTD reduces the storage capacity of our model to half.

in terms of the storage capacity, the covariance learning is better than TAH.


This expression can be simplified because e T i is a row of W * = C −1 , so that e T i C is the unit vector of the form (0,0,...,1,0,...) where the position of the "1" indicates the solution number 0.

we have e T i CΛ(τ )C T e k = Λ(τ ) ik .


. . , w K ) ∈ R K . Note minimizing the cross-entropy loss is equivalent to maximizing the likelihood of , which is biased towards fitting neither the data with y i = 1 nor these with y i = 0, since NLL(x i ) = ln(e −yiw ln(1+e Bx i ) + e (1−yi)w ln(1+e Bx i ) ).

while iSHM is structurally similar to an MLP, it is distinct in its unbounded layer width, its positive constraint on the weights r k connecting the hidden and output layers, its ability to rigorously define whether a hyperplane is active or inactive, and its inductive bias towards fitting the data labeled as "1."


The last term was also dropped since it is also independent of z.

optimizing z given (x, Q u , Q z , µ) is equivalent to the first step of the algorithm in section 2, which can be implemented with efficient gradient descent.


In the general case, it is not obvious what aspects of the quality of a population code are captured by the above measure.

both Fisher Information and the class-conditional second-order approximation used by Averbeck and Lee have shortcomings: The latter does not account for information originating from changes in the covariance matrix as is quantified by J cov (cf.


It is shown that, GTD algorithms have achieved the lower-bound of the storage and computation complexity, making them powerful to handle high dimensional big data.

now GTD algorithms are widely used in policy evaluation problems and the policy evaluation step in practical RL algorithms (Bhatnagar et al.


Then, for any learning example (x i , y i ), we define f v (x i , y i ) = e #yi where #y i = j if e j is mapped with the class y i .

we have exactly |Y| virtual points, each one corresponding to a unit vector and a class label.


Otherwise, setting the student's subsequent structure to be the same as teacher's, the student would achieve identical results as the teacher.

directly matching a student layer to a teacher layer [2,3] is unlikely to perform well.


This is the natural structure imposed on the cascaded conditionals by factor analysis.

we adopt the same constraint on S in the case of generalisedlinear observations.


The second term is an additive offset, which becomes relevant only for t large enough such that θ t − θ 2 = O(δ 2 ).

the result guarantees a globally geometric rate of convergence up to the tolerance parameter δ.


We formalize perceptual grouping and context by a graphical model aimed at capturing statistical dependencies among random variables (i.e., labels or attributes) associated with different pixel neighborhoods.

we derive a unified framework for combined object recognition and segmentation, as a graph-structured prediction of all random variables in a single, consistent model of the scene.


By applying the Taylor expansion to KL divergence between p(x; η) and p(x; η + dη), ds 2 can be represented as follows: , where G * is the FIM.

by comparing these equations, it is clear that the metric matrix consisting of the inner products of the basis vectors corresponds to the FIM: the FIM is a proper metric for the feature vectors (the expectation parameters) obtained from the GMRF.


However, as mentioned above, we expect different image regions to contain cues informative towards different aspects of surface depth.

instead of over-committing to a single value, our network outputs parameterized distributions for each derivative, allowing it to effectively characterize the ambiguity in its predictions.


Observing Eq.3, L i can be reformulated as: Xi + e Wy i Xi ), then performing the first order Taylor Expansion for the second term, a term of W yi X i shows up.

minimizing Eq.3 is to minimize W yi X i to some extend, and it can be viewed as a coupling decay term, i.e.


Some straightforward calculation shows that the penalties reduce to ∑ p ∥ * = ∥B∥ * for the second.

in the linear case the first penalty is encouraging B to be column-wise sparse, so that many of the B j s are zero, meaning that X j doesn't appear in the fit.


Our approach draws parameters for important dimensions from distributions with multiple modeswhile unimportant dimensions are drawn from a uni-modal distribution.

our model is more expressive than approaches in which all unimportant dimension values are drawn from the same distribution.


But, you may tell when you just look at an image of actions, under the hints of the backgrounds, objects and postures.

the aforementioned famous networks cannot produce the state-of-the-art performances on gesture recognition, without including multimodal fusion.


At each iteration during the inner loop, the only update is to swap the i-th and j-th slices in the D-th dimension of W . Thus, for S, we only need to swap the i-th and j-th rows without recomputing the tensor contraction.

we can move the computational intensive tensor contraction to the outer loop.


Note that the predictive error P E(s, a) = KL T (s, a) T (s, a) + H T (s, a) has the entropy of the true distribution as a lower bound, which is unknown.

we propose to drive exploration based on the learning progress instead of the current learner accuracy.


Given the asymptotic normality of G m , the same arguments used to derive the multivariate delta method can be used, with some modification, to show that ).

asymptotically, f i ( G m ) is normally distributed around its true value with a variance that depends on both the gradient of f and the covariance matrix of the approximated arguments in G m . This being the case, uniform sampling of the recursively estimated f i is equivalent to sampling from a distribution Pf that gives weight 1 n to a normal distribution centered on each f i .


Embedding approaches have many advantages including simplicity, ease of implementation, strong theoretical foundations, the ability to handle label correlations, as well as adapt to online and incremental scenarios.

embeddings have proved to be the most popular approach for tackling XML problems [6,7,10,4,11,3,12,9,5,13,8,14].


Note that if (ρ) is strictly decreasing for ρ < 0, and there are training examples for every rank, the change ( 6) is strictly negative.

the optimal θ * for any g * is always ordered.


The main benefit of treating invariances in this way is that models with different invariances can be compared using the marginal likelihood.

parameterised invariances can even be learned by backpropagating through the marginal likelihood.


From the definition of message → ( ), for any ⊆ , we also have where is the subtree of rooted at and containing .

we can recursively compute functions , → by applying the above updates for each node in , starting from from leaf nodes of and up to the root node .


Convexity of R(•) yields, R(w) ≤ t α t R(z t ) ≤ max t {R(z t )} .

if w attains the supremum sup w∈Q R(w) then so does at least one vertex.


However, this incurs the penalty of blurring the functional data within and across distinct cortical regions.

the functional alignment of multi-subject fMRI data remains an important problem.


The higher dimensionality and rich visual content captured by those features make the volume of their feature space much larger than that of traditional local features.

a much larger number of Gaussian mixtures will be needed in order to model the feature space accurately.


The size of each cropped image is 32 × 32 pixels, with 256 grey levels per pixel.

each image is represented by a 1024-dimensional vector.


However, there exists no constant C < ∞ independent of ε such that (d = 0, C) characterizes the Brownian motion.


Factor nodes of the same color are grouped together to form A j ⊂ I and denote Ī = {A 1 . . . A M }.

given a stable partition (coloring) we can define This simple symmetry will help us organize higher order symmetries throughout the LWMB algorithm.


By definition of GREEDY for any pair of points v, v ∈ ∪ m i=1 C i it must be that d(v, v ) ≥ d X (u) > 2OPT.

in the set {u} ∪ X there are k+1 points all of distance greater than 2OPT from each other.


The other kernel functions we used, Embedded Gaussion and Gaussian RBF, has a little improvements for performance.

we choose the dot production as our main experimental configuration for other tasks.


Large margins are not sufficient on their own and what is needed is a way to also control the spread of the data after projection.

rather than just maximizing the margin, a trade-off regularizer should also be used to minimize the spread of the projected data.


The total influence by seed set S is then |cc G (S i )| where cc G (S) is the set of nodes connected to S in G and we drop the subscript when it is clear from context.

the objective function is f (S) := E G [|cc(S)|].


Note that the segmentation algorithm might replace a number of recent pdf prototypes (and also recent segmentation bounds) during the on-line processing in order to optimize the segmentation each time new data is presented.

a relabeling of all segments that have changed is necessary in each update step of the labeler.


matrix at the first iteration is the similarity matrix induced by the matrix of true preferences, R true .

we see that R obs is an implicit function of R true and the set of similarity matrices S k , S k−1 , .


While strong faithfulness requires that for a node pair (i, j) that are adjacent in the DAG, the partial correlation corr(X i , X j |X S ) is bounded away from zero for all sets S 2 {S ✓ [p] \ {i, j}}, RSAF only requires non-zero partial correlations with respect to a subset of sets in {S ✓ [p] \ {i, j}}.

rSAF is strictly weaker than strong faithfulness.


In contrast to other works, we do not specify the base measure of the natural exponential family distribution, but instead treat it as a nuisance parameter.

our model is semiparametric and we are interested in estimating the underlying score matrix, which corresponds to the parametric component of our model.


In this case, the SVD operation adds significant overhead.

the fast approximation of Section 2.4 and the adaptive methods result in vast improvements.


( 5) is a simple fixed-point type update; in each iteration, only one component (i.e., p t s ) is modified while the others remain the same.

it is motivated from using a descent direction to strictly decrease l(p): where p t,n s − p t s is a descent direction in optimization since a sufficiently small step along this direction guarantees the strict decrease of the function value.


We observe that the third case is symmetric to the second case, and thus we can carry out very similar calculations in the third case to show that the right hand side is at most 6 hñ .

we have that for any T and T , provided the event E holds, |q(F (T, h, α, R), V ) − q(F (T , h, α, R), V )| ≤ 6 hñ The theorem now follows by combining Equation 21with Equation 19.


In summary, the memory requirement of this algorithm is 2m and the number of kernel calculations is not more than d•m.

the computational requirement of this algorithm is no more than the computational requirement for the evaluation ofthe margin ' Y% (WSVM)!


far from the threshold), where the probability of emitting a spike is so low (Figure 2 C, until -50 mv).

even if the exponential link-function overestimates the firing probability at these low voltages it rarely produces extra spikes.


For ML estimation, the lower bound on the number of measurements required by ML decreases to k − 1 as SNR → ∞.

1 In contrast, with thresholding, increasing the SNR has diminishing returns: as SNR → ∞, the bound on the number of measurements in (12) approaches even with SNR → ∞, the minimum number of measurements is not improved from m = Ω(k log(n − k)).


Without scaling the update steps by R , eventually random search would take update steps which are a thousand times larger than in the beginning of training.

r adapts the step sizes according to the local sensitivity of the rewards to perturbations of the policy parameters.


Indeed, in (3) all summands corresponding to m > max i=1,2 n i equal 0; moreover, all summands corresponding to l > s min are equal, where s min corresponds to the partition in which each cell has at most one point in it s min := min i,j∈1..n, Xi =Xj |X i − X j |.

even with a most naive implementation the computational complexity of the algorithm is at most polynomial in all arguments.


Indeed, as pointed in [10] a convex minimization problem can be solved via boosting techniques which avoids the choice of features.

each intermediary convex problem of DCA could be solved via a boosting technique and hence make DCA non-parametric.


As another example, Alice may interfere with clinical trials [16] to funnel most patients toward certain treatment, or make researchers draw wrong conclusions on whether treatment is better than control.

adversarial attacks on MAB deserve our attention.


For ¡ ¢ , we can see intuitively that a boosting algorithm will not let any observation have exceptionally low probability over time since that would cause this observation to have overwhelming weight in the next boosting iteration and hence the next selected ¢ is certain to give it high probability.

after some iterations we can assume that we would actually have a threshold ¨independent of the iteration number and hence the loss would decrease at least as the sum of squares of the "weak learnability" quantities # ¢ .


If the derivatives of F exist and are continuous in a small neighborhood of h t , this also implies that the derivatives 'V F(ht) = O.


It is important to note that we do not make any assumptions on the "wrong" models (those that do not have Markovian dynamics).

we are not able to test which model is Markovian in the classical statistical sense, since in order to do that we would need a viable alternative hypothesis (such as, the model is not Markov but is K-order Markov).


We select two coordinates in row r of Z i and determine if the optimization problem can be improved by moving a words from topic to topic q.

we only need to solve a series of simple univariate problems.


Kernels have been widely used before in reinforcement learning (Ormoneit & Sen, 1999), however, they were used for approximating the values of different policies in a search for an optimal policy.

the kernels should span not only the optimal value function, but also the values of intermediate policies.


No single frequency band dominated classification accuracy -see Figure 4.

the regularization must have selected diverse features.


Note that the two graphs are not Markov equivalent.

to obtain the same joint distribution we must violate a condition called faithfulness.


(3), the joint distribution P (m, z | φ, ψ, λ) has a fully factorized form (see Lemma 1 in Appendix D).

marginalizing φ, ψ, and λ out according to the prior construction in Eq.


We used a recursive leastsquares update to learn M o , which took 9.1 seconds.

learning an LOEM model is faster than learning a UOM for a single fixed reward function, but the UOM can produce an accurate option return quickly for each new reward function.


f does not depend on i), and g i is a convex function.

after t online rounds, the amount of intrinsic strong convexity we have in the primal objective t i=1 t (w) is at least σ t. In particular, this explains the learning rate of 1 σ t proposed in the gradient descent algorithm of Hazan et al.


For example, in Pseudo Relevance Feedback [17,8], we manage to rank documents on the basis of not only relevance of documents to the query, but also similarity between documents.

the use of a model solely based on individual documents would not be sufficient.


When the difference between two stimuli is small, the conditional probability p(r|s + ∆s) can be expanded with respect to ∆s as p(r|s+∆s) = p(r|s)+p ′ (r|s)∆s+ 1 2 p ′′ (r|s)(∆s) 2 +..., where ′ represents differentiation with respect to s. Using the expansion, to leading order of ∆s, we can write mutual information I as where dr p ′ (r|s) 2 p(r|s) is the Fisher information.

we can see that the mutual information is proportional to the Fisher information when ∆s is small.


Consider also that there exists a reward function, g, which informs the agent of the scalar goodness of taking an action with respect to the goal of some multi-step decision task.


These distributions can give a good sense of what structures are and are not being learned.

to supplement the parsing scores above, we examine these distributions.


As a result, the number of parameters in a channel-wise convolution with a kernel size of d c is simply d c and the computational cost is By employing sparse connections, we avoid the m ⇥ n term.

channel-wise convolutions consume a negligible amount of computations and can be performed efficiently.


This approximation mitigates multiple issues that would arise with sampling approaches: By forcing π βp m (θ m |y 1:t ) into the conjugate closed form, steps (II) and (III) are solved analytically.

inference is orders of magnitude faster, while the resulting approximation error remains negligible (see Figs 2B, 3).


A model without this knowledge, MLGA-Local, fails to explain data.

the interrelatedness of categories appears to provide a source of constraint that individuals use in learning about the structure of the world.


Moreover, if s i is close to 0.5, a mixture of both control and treated units can be found around the unit i.

propensity score can kind of reflect the relative location of units in the covariate space, and we choose it as the indicator to select six data points.


However, in this model atomic read and write of x cannot be guaranteed.

for the purpose of analysis, each update on a single coordinate accounts for an iteration.


The algorithm stops when the value function improves less than the stopping criterion Γ .

at every iteration at least one of the beliefs must improve by at least Γ .


In many real-world classification problems, it is not equally easy or affordable to verify membership in different classes.

class proportions in labeled data may significantly differ from true class proportions.


Note that the stability measures the disagreement between labels on training data and test data, both assigned by C . This asymmetry arises naturally and directly measures the generalization performance of C . Furthermore, the stability can be interpreted as the expected empirical risk of C with respects to the labels computed by itself (compare (1) and ( 2)).

stability measures the self-consistency of C . This interpretation is also valid in the semi-supervised and unsupervised settings.


Furthermore, (13) implies aT(x -y) 2: O.

we can restrict a to be such that aT(x -y) = 1.


Unfortunately, [3] shows that arbitrary adaptive adversaries can easily force the regret to grow linearly.

we need to focus on (reasonably) weaker adversaries, which have constraints on the loss functions they can generate.


Suppose H k is a reproducing kernel Hilbert space (RKHS) with reproducing kernel k : X × X → R [4,6].

if k is translation invariant (i.e., there exists κ ∈ L 2 µ such that, for all x, y ∈ X , k(x, y) = κ(x − y)), then Bochner's theorem (see, e.g., Theorem 6.6 of [51]) implies that, up to constant factors, in the setting of Theorem 1, we have It is well-known known that MMD can always be estimated at the parametric rate n −1/2 [18]; however, to the best of our knowledge, only recently has it been shown that any probability distribution can be estimated at the rate n −1/2 under MMD loss [41], emphasizing the fact that MMD is a very weak metric.


to obtain a duty cycle of 112, Vb ,i HI,1 might have to differfrom Vb,j HI,r .


We can compute its mode, its marginal distributions, and draw samples from it, all using similar message passing algorithms.

by optimizing the variational bound, we obtain the desired approximate posterior over discrete and continuous variables q(z s , x s , h s ) = p θ (z s | x s , h s , i s , y s ) q φ (x s , h s ).


Both the PD and the TEP decoder sequentially reduces the LDPC graph by removing check nodes of degree one or two.

the decoding process yields a sequence of residual graphs and their associated DD.


• Spatial exclusion: The objects are assumed to be solid objects which cannot intersect.

the volumes occupied by different object are mutually exclusive.


Even if the small component that "owns" most of it predicts it is very unlikely, that term has very little effect on the overall density.

our goal is to eliminate the mixture weights from the equation.


In real scenes composed of many different objects, the spatial configuration of one object can facilitate recognition of related objects [1], and quite often ambiguities in recognition cannot be resolved without looking beyond the spatial extent of the object in question.

algorithms which jointly recognize many objects at once by taking account of contextual relationships have been quite popular.


The measurements z cannot be used in this respect, as the main goal is not to predict the EEG/MEG measurements, but the vector field v(x) as accurately as possible.

the only way to evaluate inverse methods is to assess their ability to reconstruct known functions.


17: Repeat 2-16 until no increment in average margin of bottom n ′ examples.

the maximum average margin can only happen at two ends of the interval.


Prediction reliability is orthogonal to prediction accuracy,  and [22] showed that state-of-the-art deep networks are not reliable as they are not well-calibrated to correlate model confidence with model strength.

to demonstrate the reliability of our uncertainty-aware attention, we evaluate it for the uncertainty calibration performance against baseline attention models in Table 3, using Expected Calibration Errors (ECE) [22] (Eq.


We will conduct our primary evaluations on multiclass classification problems, where output vectors y encode target classes by indicator vectors y 2 {0, 1} m⇥1 such that y 0 1 = 1.

although it is common to adopt a softmax transfer for f 2 in such a case, it is also useful to consider a perceptron model defined by f 2 (ẑ) = indmax(ẑ) such that indmax(ẑ) = 1 i (vector of all 0s except a 1 in the ith position) where ẑi ẑl for all l. for multi-class classification, we will simply adopt the standard large-margin multi-class loss [29]: (3) Intuitively, if y c = 1 is the correct label, this loss encourages the response ẑc = y 0 ẑ on the correct label to be a margin greater than the response ẑi on any other label i 6 = c. Second, consider the first layer model.


( 4) is unconstrained and strictly convex in x and γ and given [x, γ], it is unconstrained and strictly convex in C and convex constrained in A and Z.

under these conditions block coordinate descent (BCD) is guaranteed to produce a sequence of solutions that converge to a stationary point [22].


The sequential variance-reduced stochastic gradient updates cannot be parallelized in this way, and must be performed using queries to the gradient oracle in just one of the M available parallel chains, requiring I/K rounds of synchronization.

each outer iteration of SVRG requires , and n = min .


This procedure is guaranteed to converge to a local optimum or saddle point [16].

one potential approach to solve the optimization problem in ( 5) is to use successive linear lower bounds on H(y, y ) and to solve the resulting decoupled problems in p(y|x) and p(y |x ) separately.


This scheme works as follows: Let k be a parameter with k being a divisor of m, and pick a random hash function h : To estimate the similarity of two sets A and B we simply take the fraction of indices, i, where This is, however, not an unbiased estimator, as there may be empty bins.


Intuitively, if the desired property of preserved similarity can be achieved for the hard cases, it will hold for other cases as well.

we focus on preserving such a property for the hard cases (e.g., triplet pairs) in the latent space, and employ PDDM to achieve this goal.


Similarly, as the shapes of observed objects change over time (because the only measurement in laser is depth), morphology based object tracking algorithms and optical flow [11,12] which are commonly used in computer vision are unsuitable.

we devise as a method that is robust to occlusions and noise without relying on the shape of the objects present in the scene.


A finite-albeit large-set of expectations does not itself fully specify the probability distribution p θ (z|x).

the recognition outputs {r (i) l } i=1...K l are interpreted as representing an approximate posterior q(z l |x) defined by the distribution of maximum entropy that agrees with all of the encoded expectations.


Therefore, we selected the final parameters based on the tuning split and computed the final evocation scores on the test split.

even when selecting the parameter settings, the modeling process never sees the test data.


Hence, the weak classifiers selected most of the time, i.e., effective for most of mini-batches, will have higher weights.

the overall loss of IB-CNN is calculated as where Compared to the B-CNN, the IB-CNN exploits the information from all mini-batches.


Fitting exactly the parameters of a Boltzmann machine to the observed data has exponential cost in the number of dimensions of the learning problem.

when compared to the proposed model, the model considered in [16] is particularly difficult to train.


When µ B is small (note that µ B 1), this model for the low-rank matrix L ensures that its singular vectors are not sparse.

this can be seen in the case of the a singular vector u by noting that 1 = Rearranging terms enables us to write kuk 0 µ B controls the sparsity of the matrix L by bounding the sparsity of its singular vectors.


This is due to the fact that given functions f 1 , f 2 , .

. . , f k for the personalized setting, where f i is the function assigned to player i, we can combine these functions into a single function f ∈ F k for the centralized setting by defining f (⊥) = 0 and f without loss of generality we focus below on the centralized setting.


However, we are reporting overall runtime which is also a function of the number of iterations.

total runtime (a function of both) may not always decrease/increase with k. Additional latent-likelihood results For completeness, Figure 5 depicts two additional data-sets to complement Figure 2.


If this could be done reliability on data, then the analysis of probabilistic neural computation would no longer be restricted to regions for which we have good ideas about the mathematical form of the underlying distribution, but could be applied to any cortical circuit of interest.

11 our coding scheme opens exciting avenues for multiunit data analysis.


For non-uniform absolutely-continuous P , P(r 1 (x) > ε) → 0 as n → ∞ for any x in the support of p(x) and any ε > 0.

as n tends to infinity p(arg 2 )) → p(x) and the limiting distribution of p(x)/ p 1 (x) is a unit-mean exponential distribution.


However, by Carathéodory's theorem [32, §17], any matrix K ∈ S can be written as the conic combination of finitely many rank one matrices of the form {cc : c ∈ C(Γ)}.

conceptually, the recovery problem has been reduced to finding a sparse set of non-negative weights, µ, over the set of feasible basis vectors, c ∈ C(Γ).


The saccadic gain increased faster during the second gain-up session than during the first(Figure 3B).

the reversal learning did not washout the system.


Large deviation principle [16] states that, the probability that an opponent with strategy q behaves like p * is roughly exp (− .

we need to continue exploration of the actions until ∑ i N i (t)D(p * i ∥S i q) ∼ log t holds for any q ∈ C c 1 to reduce the risk to exp (− log t) = 1/t.


(1) We assume that f (x i ) < f (x j ), and define f ′ as the function such that f According to the conditions of RDPS, the requirements of the weight function D in Section 2 and the assumption that ϕ is a non-increasing function such that ϕ(z) < ϕ(−z), ∀z > 0, we can obtain This is a contradiction with R Φ (f )=inf h∈F R Φ (h|x).


Synchronization occurs for normalized weights, unnormalized ones do not synchronize [6].

for discrete weights, we introduce a restriction in the space of possible vectors and limit the components w In order to obtain synchronization to a parallel -instead of an antiparallel -state w A = w B , we modify the learning rule (3) to: Now the components of the random input vector x are binary x i ∈ {+1, −1}.


In the context of source code repair, or more generally sequence correction, we need to constrain our generated samples y to be a corrected versions of x.

we have the following two requirements: (1) correct sequences should remain unchanged when passed through the generator; and (2) repaired sequences should be close to the original corresponding incorrect input sequences.


Thus the structural complexity of the graph can be conveyed by the collective dynamics of these basic cycles.

we may characterize a cluster with the global combination of structural cycles in the associated graph.


Word vectors are dense and generally have positive and negative entries whereas rectifier activation causes the resulting intermediate vectors to be sparse and nonnegative.

when leaves and internals are represented in the same space, a discrepancy arises, and the same weight matrix is applied to both leaves and internal nodes and is expected to handle both sparse and dense cases, which might be difficult.


We will say that a matrix loss function (W , X) is proper if for any distribution P on density matrix outcomes, the expected loss with respect to P is minimized by predicting with the mean outcome of P , i.e.

minimization of the matrix entropic loss leads to well-calibrated forecasting, as in the classical case.


Every time we say ⊥, it must be the case that |S ∩ P | = 1.

the label of the example is positive iff S ∩ P is contained in the target function, and so the algorithm correctly moves S ∩ P to P + or P − .


( 2) satisfied for perturbations with 2 -norm larger than c, we require the following condition.

we add √ 2cL F to all elements in logits except for the index corresponding to t X .


Given the low inter-annotator score and the fact that generated explanations almost match the inter-annotator BLEU-score, we conclude that this measure is not reliable for our task, and we further rely on human evaluation.

we manually annotated the first 100 datapoints in the test set (we used the same partial scoring as in Section 3).


( 1)) is a regularly-defined smooth manifold works in pair with the ambition that the result should hold for (almost) all cost matrices C. The starting point is that, for a given non-convex smooth optimization problem-even a quadratically constrained quadratic program-computing local optima is hard in general [Vavasis, 1991].

we wish to restrict our attention to efficiently computable points, such as points which satisfy first-and second-order KKT conditions for (P)-see [Burer and Monteiro, 2003, §2.2] and [Ruszczyński, 2006, §3].


However, in the situation of domain adaptation, we only know the information of source domain, while the expectation EN (F, ξ, 1 (Z N 1 )) is dependent on both distributions of source and target domains because z = (x, y).

the covering number is no longer applicable to our framework to obtain the generalization bounds for domain adaptation.


The vector form of the matrix W is formed by arranging its rows one after the other, and then taking the transpose (i.e., the vector ).

for two matrices V and W, V • W denotes their inner product in their vector form.


In such domains, using all available features may be prohibitively expensive, unnecessarily wasteful, and may lead to poor generalization performance, especially in the presence of irrelevant or redundant features.

selecting a subset of features of the domain for use in subsequent application of machine learning algorithms has become a standard preprocessing step.


We propose a generalization of a well known extremality result for orthogonal univariate polynomials on the real line (or the complex plane) [16,Theorem 3.1.2].

the distinguished SOS polynomial of interest in this paper is understood as the unique optimal solution of a convex optimization problem: minimizing an average value over a structured set of positive polynomials.


As a result, an arm with the smallest single-period risk might not be the optimal choice over an horizon of n rounds.

the performance of an algorithm should be compared to the smallest risk that can be achieved by any sequence of arms over n rounds, thus requiring a new definition of regret.


Applying Chernoff bounds, Ç ÐÓ ´½ AEµ ¯ examples are sufficient that, with probability at least ½ AE ¾, this classifier is incorrect on at most a fraction ¯ ¾ of the training data.

the training error of the hypothesis output by is at most ¯ ¾ with probability at least ½ AE ¾.


All the information regarding data point correlations is captured in the KL term and the connection with the observations comes through the variational distribution.

the first term in (14) has the same analytical solution as the one derived in [8].


Recently there have been several contributions towards solving the question of equivalence of different rules [3,4,5,6], which presented specific solutions to be discussed later (see section 4).

there is more and more evidence emerging that Hebbian learning and reinforcement learning can be brought together under a more unifying framework.


Moreover, as shown in [13], the Euclidean distance between points mapped to these eigenvectors is equal to a so called 'diffusion distance' between points on the graph, where p(z, t | x) is the probability distribution of a random walk at time t given that it started at x, π s is the equilibrium distribution, and • L2(w) is the weighted L 2 norm with weight w(z).

the eigenvalues and eigenvectors {λ j , v j } for j > 1, capture the characteristic relaxation times and processes of the random walk on the graph towards equilibrium.


If G is amiable, though Rλ G is expected to be small, the complexity penalty of G is larger than that for G * .

only G * can simultaneously achieve a small Rλ G * and a relatively small complexity.


It is straightforward to compute the gradient of object position from rendered RGB images and ground truth.

this simple graphics engine is also differentiable, making our system end-to-end trainable.


3 Because the Hessian of L(W ) is Lipschitz continuous, if the Hessian at a saddle point has a negative eigenvalue, it starts appearing as we approach the saddle point.

theorem 2.3 and Corollary 2.4 suggest that for 1-hidden layer networks, training can be done in polynomial time with a second order method or even with a modified stochastic gradient decent method, as discussed in (Ge et al., 2015).


The objective of variational methods is equivalent to minimizing a description length.

on our image classification tasks, variational methods do not have good results even for their own objective, compared to prequential codes.


However, the structures of the parameter space and the control space are not always identical.

the sampling strategy generates controls that are not likely to be generated from the current controller, even if the exploration variances decrease.


Unlike the random policy, which includes both an unstable spiral fixed point and limit cycle structure and requires a 3-dimensional embedding to preserve locality, the learned policy exhibits a 2-dimensional unstable spiral fixed point.

the fixed-point structure (embedding structure) of the combined policy-environment system changes during learning.


for variable E and any set S, if , then S is not the parent set of E. Similarly, the authors of [WSBU18] consider the case that a (i) 6 = a (j) and either E . under this assumption, in this case for variable X 2 {C, E} and set , then S is the parent set of X.


Indeed, to solve this problem it suffices to have consistent estimates of some distance between time series distributions.


This is our second important system of spaces.

to understand the interplay between modules and window spaces, observe that each coset µ i1...i k S n−k has an internal notion of left-translation which fixes S i1...i k . S i1...i k must be decomposable into a sum of irreducible Furthermore, the modules of different window spaces can be defined in such a way that is an S n−k -module in the sense of being invariant to the internal translation action (4), and this action depends on i 1 . . . i k .) Now, for any fixed t, the space U = i1,...,i k M i1...i k t , is fully S n -invariant, and therefore we must also have U = α∈A M α , where the M α are now irreducible S n -modules.


Recall that uPU overfitted seriously on all the benchmark datasets, and note that the larger π p is, the more different nnPU is from uPU.

the replacement of π p with some π p > π p introduces additional bias of R pu (g) in estimating R(g), but it also pushes R pu (g) away from R pu (g) and then pushes nnPU away from uPU.


Conventional sorting algorithms are guaranteed to return the elements in correct order.

the critical difference between them is in their runtimes, and runtime depends primarily on the number of elements to be sorted and their presortedness.


Our memory cell vector c(t) is a deterministic function of the past history (H i , t − t i ).

5 the event intensities at any time are also deterministic via equation (4).


Every maximal region provides evidence to the fact that all points within the region should be clustered/segmented together.

the value LES(R [q] |H ref )) is added to all entries (i, j) in an affinity matrix, ∀q i ∀q j ∈ R [q] .


We only require the loss function is a proper norm in a normed vector space.

we have the following Theorem for a general form of Equation (1).


As mentioned before, let o l i (•) denote the output function of this locally learned KM, then the good performance of local learning methods mentioned above implies that o l i (x i ) is probably a good guess of f l i , or the proper f l i should be similar as o l i (x i ).

a good SPM F should have the following property: For any x i (1 ≤ i ≤ n) and any cluster C l (1 ≤ l ≤ c), the value of f l i can be well estimated based on the neighbors of x i .


end for end while data that survived upto stage k. On the other hand, the optimization problem for g k is exactly in the form where Lemma 2.1 applies.

we can also reduce this problem to a supervised learning problem: where The composite classifier F (x) based on the reject and region classifiers can be written compactly as follows: Observe that if the kth region classifier correctly classifies the example x i , i.e., f k (x i ) = y i then this would encourage g k (x i ) = 0.


Rearranging the terms yields which is exactly the linearization of ψ jt (x jt ) at x t jt .

if solving (5) exactly becomes difficult due to some problematic terms, we can use the Bregman divergence to linearize these problematic terms so that (5) can be solved efficiently.


Then, by the iterative applications of Eqs.

( 2) and (3), we obtain where f l is the l-time compositions of f . λ j characterizes the temporal behavior of the corresponding Koopman mode u j , i.e., the phase of λ j determines its frequency, and the magnitude determines the growth rate of the dynamics.


There is a trade-off between the two goals: we may be able to return the optimal solution faster if we do not invest the time to prove that all other solutions are worse.

we will aim only to search for a "good" (possibly optimal) solution without a rigorous proof of optimality.


If µ is a compactly supported distribution satisfying the two-subspaces criterion for γ > 0, then there exists a constant α 0 ≡ α 0 (µ, D, ) > 0 such that for any > 0 and N > 2(D −1) the following estimate holds: where .

(1.9) Intuitively, α 0 represents a lower bound on the directional second derivatives of F . α 0 should affect sample complexity because the number of random samples taken to approximate a minimum of F should be affected by how sharply F increases about its minimum.


The main premise of ResNets is that they allow the training of each layer to focus on fitting just the residual of the previous layer's output and the target output.

we should expect that the trained network is no worse than what we can obtain if we remove the residual layers and train a shallower network instead.


Finally, we randomly select κ fresh instances (from outside of I and the vectors that define C) to replace x j and the κ − 1 cached instances with the lowest score.

in each basis selection step, we compute the criterion scores for c instances, but evaluate full kernel rows only for κ fresh instances.


It is of course possible to select more than one K, but only if the data indeed supports both clusterings.

indirectly, the BTR can provide a theoretically sound method of selecting K. The author gratefully acknowledges Maryam Fazel for her early interest in this work, for reading a previous version of this paper and for many discussions; the Simons Institute for the Theory of Computing where part of this research was performed, and partial support from the NSF DMS PD 08-1269 and NSF IIS-0313339 awards.


Deep learning for precipitation nowcasting and video prediction For the precipitation nowcasting problem, the reflectivity factors in radar echo maps are first transformed to grayscale images before being fed into the prediction algorithm [23].

precipitation nowcasting can be viewed as a type of video prediction problem with a fixed "camera", which is the weather radar.


Nevertheless, algorithms that aim to illuminate learning in cortex should be able to perform well on difficult domains without relying on any form of weight sharing.

our results offer a new benchmark for future work looking to evaluate the effectiveness of biologically plausible algorithms in more powerful architectures and on more difficult datasets.


the learning rate, dfl/dt, is equivalent to -dL1 Vfgldt.


At the level of generality at which (VC) has been stated, it is unlikely that any convergence rate can be obtained, because unlike in the convex case, one has no handle on measuring the progress of mirror descent updates (recall that in (VC), only non-negativity is guaranteed for the inner product).

we focus here on the class of strongly coherent problems (a generalization of strongly convex problems) and derive a O(1/ √ T ) convergence rate in terms of the squared distance to a solution of (Opt).


Let us re-parametrize the error term in Propostion 3. becomes a candidate that can serve to build up undesirable shortcuts.


The second KL term vanishes as the two distributions are equivalent.

our final maximization objective is: where we first sample random weights with dropout masks ω ∼ q M (ω|X, Y) and sample z such that z = g(x, ε, ω), ε ∼ N (0, I), with a pathwise derivative function g for reparameterization trick.


Applying the feature-based model analysis from Experiment 1 to the current results revealed that the weighted feature model required an average of 6.0 features to match the performance of the absolute utility model for participants in Cluster 1, and an average of 3.9 features to match the performance of the relative utility model for participants in Cluster 2.

although a single model did not fit all participants well in the current experiment, many participants were fit well by one of the two inverse decision-making models, suggesting that this general approach is useful for explaining how people reason about negative effects as well as positive effects.


For example, using a standard covering argument and a union bound [23] shows that if (9) holds for a sufficiently large constant c 1 > 0 then we have (4) for a sufficiently small constant δ 2k with probability exceeding 1 − 2e −cm for some constant c > 0 that depends only on δ 2k .

theorem 1 yields the desired result which holds with probability exceeding 1 − 2e −cm − e −C3n ≥ 1 − 3e −c2m for some constant c 2 > 0 depending only on δ 2k .


Intuitively, the difference between w (t−τ ) and w (t) will be small when reaching a stationary point.

also the change in gradients will vanish.


However, hardware has no such requirements.

we ask whether bitwidths can be fractional.


Let -AI > -A2 > -A3 > ... be the poles of J (z) and mk be the order of -Ak.

then, by using the inverse Mellin tansform, it follows that v(t) has an asymptotic expansion with coefficients {Ckm}, k=lm=1 also Fo (n) has an asymptotic expansion, by putting A = Al and m = ml, which ensures the asymptotic expansion of F(n) by eq.


For the task of object detection, we can only acquire bounding-box annotations instead of pixel-level labeling.

it is natural to apply our latent e-SVM in this task to provide complementary information for the current object detection system.


Moreover, raw features are typically not realizable, i,.e., the assumption that e t (s t , a) = θ * a T x(s t ) + η t does not hold.

we propose learning a set of features φ(s t ) that are realizable, i.e., e(s t , a) = θ * a T φ(s t ), using neural networks (using the activations of the last layer as features).


While this approach is mathematically elegant from a Bayesian point of view, we have so far only been able to obtain results for simple, artificial domains.

to cope with large-scale data sets, we propose in Section 3 a maximum a posteriori (MAP) approach using the Expectation-Maximization (EM) algorithm to maximize the posterior.


As argued earlier, the proximal resembles projection onto the manifold of visually plausible images.

one can interpret P ψ as a denoiser that gradually removes the aliasing artifacts from the input image.


Similarly we can show that our model can be reduced to vanilla encoder-decoders (without attention) by constructing a case where r = c and f t = 0.

our model is more expressive than (attentive) encoder-decoders.


If Φ i,j = 0, the regularization doesn't induce interaction between s i and s j in the objective function.

this regularization term makes SLFA produce a collaborative reconstruction based on the conditional dependencies between latent factors.


However, the recursion makes the depth of the neural network too deep for an efficient joint training.

we first learn the CNN parameters (θ CN N ) using the raw image and the ground truth per pixel labels.


These special words are called anchor words.

separability occurs when each topic is essentially identified uniquely by its anchor word.


However, in a theoretical part we show that using more classes can be worth the effort if a suitable accuracy of all pairwise classifications is available.

extensions to multi-class settings are worthwhile for a BCI system, if and only if a suitable number of effectivly separable human brain states can be assigned.


When α = 0, the above method is reduced to a direct density-ratio estimator called unconstrained least-squares importance fitting (uLSIF) [14].

the above method can be regarded as an extension of uLSIF to the α-relative density-ratio.


Therefore, it would be interesting and highly desirable to train a unified model for all of these sub-systems.

the molecules in the dataset contain at most 4 different types of atoms, namely C, H, O, and N. we need 4 sub-networks corresponding to the four types of atoms with different environments.


Our GroupSAM only needs to construct one component function for each variable group, but the model in [26] needs to find the component functions for each variable in a group.


As a first step, we translate the Boolean alphabets Z = {0, 1} to the "Ising" ones, S = {+1, −1}.

we need to translate the additive operations, such as, Similarly, we translate the Boolean y i (t)s into the Ising J i (t)s. For simplicity, we omit the subscript i, which labels the L agents, in the rest of this section.


First consider the case when n i ≤ ρi/2.

we have: for this case, I is c-risk-competitive to Algorithm ERM.


. . , q ri ) is an orthonormal basis to the first r i columns in S. Thus, v i (r i ) is the (squared) error of approximating x i with the first r i columns in S. In each pass the algorithm identifies the l candidate columns x i corresponding to the l largest values of v i (|S|).

that is, the v i values are computed as the error of predicting each candidate by all columns currently in S. The identified l columns with the largest v i (|S|) are stored in the list L. In addition, the value of the l+1'th largest v i (|S|) is kept as the constant B F . after a pass is terminated the following condition holds: The list L and the value B F can be calculated in one pass using a binary heap data structure, with the cost of at most n log(l + 1) comparisons.


Similar results hold for postsynaptic spikes.

the weight change depends in a complex way on the whole temporal pattern of pre-and postsynaptic spikes.


In other words, ConQ[sa] is a partition function defined on some partition of [s → a] (rather than all of Θ), and represents Q-values of cells that are consistent with [s → a].

if is the partition function over Θ obtained by collecting all the "restricted" action value functions.


The membrane potential of each neuron tracks the component of the reconstruction error along the direction of its decoding weights.

the network is balanced (because the dynamics aim to bring the reconstruction error to zero) and membrane potentials are correlated, particularly in pairs of neurons with similar decoding weights [7] (see Fig.


Here, our approach is based on a simple theoretical connection between GDA and the softmax classifier: the posterior distribution defined by the generative classifier under GDA with tied covariance assumption is equivalent to the softmax classifier (see the supplementary material for more details).

the pre-trained features of the softmax neural classifier f (x) might also follow the class-conditional Gaussian distribution.


c j = c j−∆ if k t = 0 for t j−∆ ≤ t ≤ t j .

during a single oscillation period of length τ , the units only update during a duration of r on • τ , which will result in substantially fewer than n update steps.


Finally, this approach learns a restricted form of networks that requires many hidden variables to represent dependencies among variables.

it has limited utility in distinguishing "true" hidden variables from artifacts of the representation.


The fixed point equation for the Using the definition of U and S, we have is the ridge regression solution of where Tr is the trace.

after multiplying out all i T i in 1/n n i=1 i T i , we obtain: For the fixed point of Ψ, the update rule gives: diag (Ψ) = diag 1 n n i=1 i T i + W ΣW T . Thus, W minimizes Tr (Ψ) given µ i and Σ. Multiplying the Woodbury identity for W W T + Ψ −1 from left and right by Ψ gives Inserting this into the expression for diag (Ψ) and taking the trace gives for Ψ W W T the error is quadratic in Ψ. W U T = W SW T = U W T follows from fixed point equation U = W S. Using this and Eq.


This can happen only if distributions are represented by a sufficiently large dataset.

uDA approaches are in a position of weakness when we assume D t to be small.


In the function, α > 0 is a balance parameter.

in general, we expect d d. the output Y is called sparse lifting of the input X, and the matrix W is called sparse lifting operator.


Experiment S 4 The training entirely in an unsupervised fashion for 3D bounding box estimation would fail since each 2D pixel could correspond to an infinite number of 3D points.

we integrate some common sense into the unsupervised training by restricting the size of the object close to the average size.


While these methods provide fairly good performance improvements, directly transferring the teacher's outputs overlooks the inherent differences between the teacher network and the student network, such as the network structure, the number of channels, and initial conditions.

we need to re-interpret the output of the teacher network to resolve these differences.


However, we will see shortly that the PAC-Bayes theorem offers a better guarantee for another type of deterministic classifier.

with these choices for Q i (ρ ρ ρ), P I (i), and P i (ρ ρ ρ), the KL divergence between Q i and P is given by Notice that the KL divergence is small for small values of |i| (whenever p(|i|) is not too small) and for large margin values (b i − a i ).


While we prove that the learning bounded vertex cover Bayesian network structures can be done in polynomial time, the unavoidable dependence on k in the degree the polynomial makes the algorithm of our main theorem infeasible for practical usage when the vertex cover number k increases.

we investigate using an integer linear programming (ILP) formulation as an alternative way to find optimal bounded vertex cover Bayesian networks in practice (Section 4).


Recall that our generalization bounds use the posterior's divergence from a fixed prior to penalize the posterior for overfitting the training data.

to connect Algorithm 1 to our bounds, we analyze the adaptive posterior's divergence from a uniform prior on {1, . . . , n} T . This quantity reflects the potential cost, in generalization performance, of adaptive sampling.


Moreover, since the outgoing edges of v are divided by c, we get γ p ( w(t) , e) = γp(w (t) ,e) c 2 and ∂L ∂we ( w(t) ) = ∂L c∂we (w (t) ).

a similar argument proves the invariance of Path-SGDupdate rule for outgoing edges of v. Path-SGDis rescaling invariant.


Each observation n belongs to exactly one latent cluster k, indicated by z n . The binary variable i ng indicates whether group g is present in observation n. Finally, the probability of some observation w nd = 1 depends on whether its associated group g (indicated by l d ) is present in the data (indicated by i ng ) and the associated formula f g . The complete generative process first involves assigning dimensions d to groups, choosing the formula f g associated with each group, and deciding whether each group g is important: where DP is the Dirichlet process.

there are an infinite number of potential groups; however, given a finite number of dimensions, only a finite number of groups can be present in the data.


The magnitude of w determines the importance of each feature, and this importance is task-dependent.

w is estimated by supervised training described in Section 3.


In other words, Neural LP learns a distribution over logical rules, instead of an approximation to a particular rule.

we do not use hardmax to replace softmax during inference time.


Structures with the same edge sets but slightly different transition times will probably have similar likelihoods.

we can add a new move that proposes a local shift to one of the transition times: let d be some small positive integer and let the new time t i be drawn from a discrete uniform distribution t i ∼ DU (t i − d, t i + d) with the constraint that t i−1 < t i < t i+1 .


The only property of the prediction strategy that we used was that it plays in the eigensystem of the past sufficient statistic.

using the same line of argument, we can show that if for some classical prediction strategy we can obtain a meaningful regret bound by bounding each term in the regret δ t independently, we can obtain the same bound for the corresponding matrix strategy, i.e.


Finally, we define the probability of a particular dictionary sense given the image to be equal to P (s|d).

our model is able to assign sense probabilities to images returned from the search engine, which in turn allows us to group the images according to sense.


Note that the first column of T contains the impulse response curve of the closed-loop system, while the rest of the columns are obtained shifting that impulse response down.

we can build the transfer matrix T either in a model-based manner, deriving the state-space characterization of the closed-loop system, or in measurement-based manner, measuring the impulse response curve.


(In the experiments, subjects are well practiced and make few errors.

we assume the R's are correct or corrected responses.)


The key idea is motivated by the following: observations that preserve complete randomness when applied to Poisson processes result in a new process that remains Poisson.

one can obtain a Dirichlet process which is dependent on other DPs by applying such operations to their underlying compound Poisson processes.


To see why, note that the amount of work for a given thread is a linear function of the number of nonzeros over all of the features in its block.

the block with the greatest number of nonzeros serves as a bottleneck.


Applying this bound and (1.8) we obtain that if η > 2 is fixed, 1/η < < 1 2 and N ≥ D η , then (2.1) Since > 1/η the first term in the RHS of (2.1) decays exponentially as a function of D (or, equivalently, as a function of N ≥ D η ).

similarly, since 0 < γ < 1 and η > 1 the second term in the RHS of (2.1) decays exponentially as a function of D. Furthermore, since < 1 2 it follows that the error term for the minimizer, i.e., N − 1 2 + ≤ D η( − 1 2 ) , decays polynomially in D. in order to achieve low error estimation with high probability it is sufficient to take N = Ω(D η ) samples for any η > 2.


The GGN matrices of L = L c • N c and M = L p • N c are the same, i.e., G Lc•Nc = G Lp•Nc .

their Hessian matrices are approximations of each other.


(Interestingly, the model also produces below chance performance on presentation 2 if the object is not named correctly-23.5%-which is also found in the human data-20.0%.)

by the stringent criterion of 4AFC, the model shows no access consciousness, and therefore illustrates a dissociation between priming and access consciousness.


Result 3 in Property 1 shows that the inner product of two tangent vectors at Y k is equal to the Euclidean inner-product of two vectors U, V ∈ R d×d .

we can reformulate (4) as follows: where β = 4/ √ µL−1/L.


As the input x can have highly complicated structure, we employ a dedicated model or algorithm to extract an expressive representation h from x.

when we extract the representation h from x, we want the representation h to preserve variations that are necessary to predict y while eliminating information of s. To achieve the aforementioned goal, we employ a deterministic encoder E to obtain the representation by encoding x and s into h, namely, h = E(x, s).


Yet in practice it differs from standard generalization error (even with continuous feature spaces).


In addition to the O(nd) cost for calculating , our algorithm requires an O(n log n) cost for sorting the entries of , as well as an additional running time of O(k ⇤ ), where k ⇤ is the number of non-zero elements at the optimum.

the running time of our method is independent of any accuracy ✏, and may be significantly better compared to any off-the-shelf optimization method.


By setting β = 0, R 3 will reduce to the regularization term proposed in ASO if we also replace the tangent vectors by the task parameters.

this formulation is a generalization of ASO.


Even if available, such learning is often overkill.

instead of trying to learn h everywhere, we only seek to learn the parameters w of a function f w that lives within some family, parameterized by w, so that if B ∈ argmax A⊆V :|A|≤k f w (A), then h(B) ≥ αh(A * ) for some α ∈ [0, 1] where A * ∈ argmax A⊆V :|A|≤k h(A).


Recall that r G (p, q) = v p − v q 2 G (see equation ( 7)) and use n q=1 v q = 0 to obtain that n n q=1 v q Gv q which implies the left inequality in (8).

next, by Corollary 3.1, if T is the Laplacian of a tree T ⊆ G then r G (p, q) ≤ r T (p, q) for p, q ∈ V . from Lemma 3.2 we conclude that r G (p, q) ≤ d T (p, q).


Without loss of generality, let the work area (that fully contains the movement range) be an axis-aligned square The endpoint of the limb (wrist) is assumed to be connected to one end of four imaginary springs, the other end of which is sliding with no friction along rails forming the boundaries of the "work area".

at every time instance each spring is parallel to one of the axes.


In tractable learning, the regularization term reflects the cost of inference in the model.

we use the following score function: where T is the training data, M is the model extended with feature f , M is the old model, |SDD .


if we define the -margin loss function to be + (note that this is simply the well known hinge loss function scaled by a factor of ), we can define our goodness criterion as follows: Definition 12.


An important problem in the object recognition is the fact that an object can be seen from different viewpoints, resulting in differently deformed images.

the invariance to viewpoints is a desirable property for many vision tasks.


Likewise, if the diameter of the current version space is more than ǫ then any hypothesis chosen from it will have error more than ǫ/2 with respect to the worst-case target.

in a non-Bayesian setting, active learning is about reducing the diameter of the version space.


The long-term causal effect is the causal effect defined after the system has stabilized, and is more representative of the value of policy changes in dynamical systems.

in Figure 1 the long-term causal effect is the difference between the objective values at the top and bottom endpoints, marked as the "targets of inference".


One way of avoiding this situation is to make the S 3 I memory to work on the original unscaled version of the input current.

the adjustable-time-constant CNN core will be a current conveyor, followed by the S 3 I current memory and then the binary weighted current mirror.


Real causal learning tasks often involve uncertainty about both structure and parameters.

even when a task demands ratings of causal strength, the structural uncertainty should still be taken into account; we do this by considering a hierarchy of causal models.


Features replay denotes that immediate feature is input into the module k for the first time in the forward pass at iteration t + k − K, and it is input into the module k for the second time in the backward pass at iteration t. If t + k − K < 0, we set h t+k−K L k−1 = 0 .

the new problem can be written as: where denotes the gradient of the loss f (w t ) regarding ht L k with input ht L k into the module k + 1.


The random strings generated at the start of the algorithm are almost entirely perpendicular to the global optimum, projecting only an amount of order 1/..JL towards the optimum.

the situation is like that of the needle-in-a-haystack problem.


We then perform this procedure for time i, repeating until we reach time 1, to obtain the changepoints and models for the entire sequence.

at each time step t we compute P t (j, q) for each model q and changepoint time j < t (using P M AP j ) and then compute P M AP t and store it.


At such a point there is a Lagrange multiplier β such that ∇ q,λ L = 0 0 0 (compare with (11) and ( 12)) and this β solves problem (9).

for each I ∈ (0, I max ), there is a corresponding β which solves problem (9).


The individual GPs represent connections between different endpoints.

they do not represent observations of the exact same trajectory, but rather of distinct, nearby trajectories.


For this distribution, the equivalent covariance matrix can be given by: Now, the eigenvalues λ 1 , ..., λ n of the overall covariance matrix Σ * characterize the variance of the distribution across n dimensions.

since Σ * is positive-definite, all eigenvalues are positive (this can be shown using the fact that each covariance matrix is itself positive-definite, and diag(µ to describe the variance of the feature distribution we define Diversity.


But often it is necessary to make good decisions without much experience.

learning rules should be preferred that quickly lead to orders with good performance.


A policy, At each stage k, a stage-reward function r k : Z k × U k × W k → R, quantifies the benefits of applying a control u k to a state z k , subject to a disturbance w k . A final reward function r N +1 : Z N +1 → R, similarly quantifies the benefits of ending at a state z N +1 .

the expected reward starting from state z 1 and using policy π is: where the expectation is taken with respect to the disturbances.


. . , w q = 0 and ξ 1 = • • • = ξ ℓ = 1.

we have u∈Y w u 2 ≤ C in the optimum, and we deduce w u ≤ √ C for each u ∈ Y .


Our experiments suggest that a value of D in the order of thousands of units is sufficient for good accuracy.

o(nmDd) will dominate the computational cost for training, and O(mDd) the one for testing, since m is usually in the thousands, and d in the hundreds of units.


Recall that the truncation function ψ(•) satisfies Then, we have where the last step is due to the basic inequalities 1 + x ≤ e x and (a + b) 2 ≤ 2a 2 + 2b 2 .

by Chernoff's method [Lugosi, 2009], we have with probability at least 1 − δ, we have We complete the proof by taking the union bound over all w ∈ N (W, ε).


Indeed, we only require that x 1 is sparse, as well as the increments.

there may exist values of k such that x k is not a sparse vector.


However, if the lookup tables are sparse each row of the table could be stored in a sparse data structure.

when indexing the table with a certain index, we just need to update the scores of a small fraction of templates.


. . achieves the same risk as any other model selection criterion defined with respect to the same estimators, up to lower order terms; in other words, model averaging based on the switch-distribution achieves at least the same rate of convergence as model selection based on any model selection criterion whatsoever (the issue of averaging vs selection will be discussed at length in the full paper).

the theorem requires that the prior π in ( 4) is of the form (7), and satisfies ) π M , the prior on the total number of switch points, is allowed to decrease either polynomially or exponentially (as required for Theorem 1); π T and π K must decrease polynomially.


This makes generalization of capsule network to 3D very challenging.

it is crucial to optimize the routing procedure when scaling capsule networks to high dimensional inputs like videos.


However, we next transform this problem into a projection onto the region cone K G (v) (the proof is in the supplemental material).

then, the solution of the problem in line 5 is all is necessary is computing the arg-max associated with the matching function, and a black box that projects onto the region cone K G (v).


They follow from simple algebra and are proved in Appendix A in the supplementary material.

moreover, for all i ∈ {1, . . . , n} we have that we can choose a = 2b.


The Laplace mechanism does not apply when the sufficient statistics are unbounded, because ∆ s = max x,y t(x) − t(y) 1 = ∞.

we need a new release mechanism A and inference algorithm P. We present a solution for the case when x is univariate.


As a result, classic measures, such as log-likelihood on the test set, cannot be evaluated.

many researchers focused on qualitative comparison, such as comparing the visual quality of samples.


. . , u(x n ) and u(x * ) are jointly Gaussian, it is easy to see that the predictive (posterior) distribution P (u * |D), u * = u(x * ) is given by where .

if p = M −1 y, the optimal prediction under the model is û * = k T * p, and the predictive variance (of P (u * |D)) can be used to quantify our uncertainty in the prediction.


1b, we start oscillatory activity and we observe the following sequence of events: (1) the activity in the bottom WTA stage ramps up due to recurrent excitation, and when it is high enough it begins activating the middle WTA stage; (2) activity in the middle WTA stage ramps up and as activity in the inhibitory population of this stage rises, it shuts down the bottom stage activity; activity in the middle WTA stage keeps on increasing until it activates the top stage; (3) activity in the top WTA stage increases, shuts down the middle stage, and provides input back into the bottom stage via the plastic connections.

a bump of activity continuously jumps from one WTA stage to the next.


To achieve quick convergence, we want to ensure that every move in the move set efficiently jumps between posterior modes.

the majority of the next section is devoted to describing effective move sets under different levels of uncertainty.


If φ is a smooth map φ : R M → R M and this system is observed via function, y, such that where y : R M → R, then if φ is invertible, φ −1 exists, and φ, φ −1 , and y are continuously differentiable we may apply Takens' Theorem [24] to reconstruct the complete state-space of the observed system.

for each s(t), we can construct a vector s E (t), such that s E lies on a subset of R E which is an embedding of s. Because embeddings preserve the connectivity of the original vector-space, in the context of RL the mapping ψ, may be substituted for f (Eqn.


Second, the model enables derivation of a variational inference method that can efficiently compute joint posterior distribution over all the unobserved random variables.

we can infer labels not only for the test point but also for all the missing labels in the training set.


Finally, V (B1) and Î computed in Line 6 require O( k) and O(km) memory space, respectively.

when = 1 δ log m , this first part of the algorithm requires O(k(m + n)) memory.


for any fixed x , the following holds: Therefore we have Since this holds for arbitrary x , we establish the theorem.


Initially, when the policy is very far from the correct answer, even if the movement direction is not along the gradient (biased), it is beneficial to make consistent progress and not bounce around due to high variance.

high bias estimates of the policy gradient, corresponding to smaller λ values make fast initial progress.


Finally, we construct our estimator by averaging over the block matrices: We note that within k 2 entries of the diagonal, each entry is effectively the sum of k 2 estimates, and as we move from k 2 to k from the diagonal, each entry is progressively the sum of one fewer entry.

within k 2 of the diagonal, the entries are not tapered; and from k 2 to k of the diagonal, the entries are linearly tapered to zero.


. . , hk −1 that leads to error less than /2 on the problem i k. In other words: Now, by way of contradiction, suppose that .

by the definition of discrepancy and the assumption on the marginal distributions it follows that for all j: Therefore by Lemma 2: by using the triangle inequality: which is in contradiction with (15).


Here we represent the transition density between m states in the following form (Fig.

1): Here, χ(x) ⊤ = [χ 1 (x), ..., χ m (x)] represent the probability of configuration x to be in a metastable (long-lived) state i these functions are nonnegative (χ i (x) ≥ 0∀x) and sum up to one ( i χ i (x) = 1∀x).


Additional static information τ includes (a) terrain features, a H × W × C T tensor that includes elements such as walkability, buildability and ground height, and (b) the faction, f me and f op , that each player picks.

each input x t = (o t , τ, f me , f op ) Additionally, we pick a temporal resolution of at least s = 5 seconds between consecutive states, again aiming to model high-level dynamics rather than small but often irrelevant changes.


Whereas the latter requires problem-specific prior knowledge which is hardly available in practice, the former depends only on the variance of a linear model fit.

ε can be specified in a problem independent way.


We can adjust the width of the transition region independent of the threshold such that a step function would be achieved in the limit.

we have a sigmoidal function with an adjustable threshold, without reliance on cooperative binding of transcription factors as is common in biological systems [9].


This is intended to address, in part, the difficulty with interpreting RNNs where the recurrent weights feed past information to the hidden layer.

to consider both the visit-level and the variablelevel (individual coordinates of x i ) influence, we use a linear embedding of the input vector x i .


In the second and third equalities, we collect each category and perform a MLE to estimate each parameter in the distribution.

the optimum solution θ is given in the closed form, which will be used in the second step of the optimization procedure.


Not only did we want to push the λ σ values close to zero, we also wanted to account for the fact that most spans are not constituents.

4 we expect the distribution of the λ σ to be skewed towards low values.


Therefore, maximizing SNR ẑ j is equivalent to Due to the use of batch normalization, Eq.

( 12) is a constant with respect to each sample of z s j , and thus we have ∂ E ẑs j 2 /∂z s j (m) = 0, ∀m ∈ B, where B denotes a set of mini-batch samples, and z s j (m) denotes the value of z s j corresponding to sample m. we can now remove the feature correlation effect of multiplicative noise without causing optimization difficulties.


Note also that F expresses a conceptual space of objects, not necessarily a feature space.

to emphasize that objects, like z or p i , might not be feature vectors, they will not be printed in bold.


Conversely, the pure semantic model does a poor job of predicting recall of studied list items, because it does not scope recall to the study list.

each of these models is hobbled by crucial (but complementary) shortcomings.


We note that unlike the ReLU transformer, both Sigmoid and Tanh transformers always create a new noise symbol whenever l x = u x .

the runtime does not increase significantly with and is not plotted.


They are accumulated and passed through a non-linearity to obtain µ θ , which is the embedding of p(θ), the distribution over the topics for the document.

the embedding µ θ is determined (up to the non-linearity) by the average of the embeddings µ zi , as in the original LDA model.