The results show that our model outperforms both VFAE and DANN at nine out of the twelve tasks.

our model can also be used effectively for domain adaptation.


For implementation details, we allow the model to accept 3 state merges every third iteration, propose 5 cluster merges every third iteration, and split one state and one cluster every third iteration.

every iteration may affect the truncation level of either the number of states or clusters.


We make mild assumptions on the noise distribution.

in this situation, the data has intrinsic dimension equal to the ambient dimension.


In practice, however, the identity of the core component is unknown for each object.

we use the EM algorithm to automatically estimate the parameters of the above model.


In other words, we show that in the limit of T → ∞ the noisy network can correct any error pattern that can be corrected by the noiseless version and it can also get out of stopping sets that cause the noiseless network to fail.

the supposedly harmful internal noise will help Alg.


[7], as we build directly on top of their code 2 .

our encoders are 2048-bidirectional-LSTMs [15] with max-pooling, resulting in a sentence representation dimension of 4096.


During training, phone models are constructed that are pieced together for the test vocabularies.

method 2 above is also not possible for this data.


Third, we leverage on existing machinery to perform model selection in significantly higher dimensions than typically considered in the copula literature.

our work opens the door for numerous applications where the flexibility of copulas is needed but could not be previously utilized.


2 Lower Bound on Sample Requirements: The above result states that roughly samples are required for structural consistency under any estimation method.

when ρ = Θ(1) (constant fraction of observed nodes), polylogarithmic number of samples are necessary (n = Ω(poly log p)), while when ρ = Θ(m −γ ) for some γ > 0 (i.e., a vanishing fraction of observed nodes), polynomial number of samples are necessary for reconstruction (n = Ω(poly(p)).


Thus maximal mutual information is obtained as the rates become infinite.

to get sensible results, a constraint has to be placed on the rates.


In theory, particles can be arbitrarily placed within an object.

less complex parts can be described with fewer particles (e.g.


From the training data, we learn the relative spatial distribution map of the presence of the k th 2 object, given the k th 1 object in the center.

we then find out the top M highest response regions on the map, each of which has a relative location ∆  and co-occuring response w. the parameters of the k th 2 object that satisfy these relative locations, have semantic edges with θ k1,l1 .


In order to deal with non-uniform error corruption, our technical proof introduces a new weighted norm denoted by l w(1) , which involves the information of both localized µ 0 and µ 1 and is hence different from the weighted norms introduced in [4] for matrix completion.

our proof necessarily involves new technical developments associated with such a new norm.


It is well-known [4,7] that Γ is both elicitable and identifiable as the expectation of a vector-valued random variable X(ω) = (ω, ω 2 ), using for example L(r, ω) = r −X(ω) 2 and V (r, ω) = r −X(ω).

we can recover σ 2 as a link of the elicitable and identifiable Γ : P → R 2 , and as no such Γ : P → R exists, we have iden(σ 2 ) = elic(σ 2 ) = 2.


Defining potential function φ t (v, u) = exp w T f (v, u, x, t) , the recurrence of the normalized forward/backward score is written as, where It is easy to show that Z(x) = t s(t), and now the marginal has a simpler form of p(y t |x, w) = α(t, y t )β(t, y t ).

the gradient of the marginal is, where G α and G β are the gradients of the normalized forward and backward score, respectively.


Although S-CODE and LDC [15] achieve essentially the same level of performance on taggings that induce 17, 45, or 50 labels (Table 1), S-CODE proves superior for the induction of very fine-grained taggings.

we compared the performances of S-CODE and LDC on the task of inducing 300 labels.


We also get a very high degree of parallelism due to the large size of the data.

it is essential for the algorithm to be reducible.


However, it contains images of 100 categories compared to 10.

we have a smaller number of training samples per category.


For high-dimensional models like that considered in this paper, however, moves proposing large numbers of unique features have low acceptance rates.

mixing rates are greatly affected by the beta process hyperparameter α.


This works well for small turns, but for large turns, the thrust from the tail rotor also tends to cause the helicopter to drift sideways.

we enriched the policy class to allow it to correct for this drift by applying the appropriate cyclic pitch controls.


#grad / n   Given 2B IFO calls, SGD implements updates on two fresh batches while SCSG replaces the second batch by a sequence of variance reduced updates.

figure 1 shows that the gain due to variance reduction is significant when the batch size is fixed.


Comparable values were already obtained after 1 and 2 runs respectively for the CNNs with frequency spectrum input and the values shown in Figure 1 (left) were obtained after 45 and 105 runs respectively.

the frequency spectrum appeared to be the clearly preferable choice for the input representation.


In all three plots, we can clearly see the memorization effects of networks, i.e., test accuracy of Standard first reaches a very high level and then gradually decreases.

a good robust training method should stop or alleviate the decreasing processing.


The same equality holds true for group B.

p ⇤ gives the equal loss of z ⇤  ẑ for two groups.


The fact that the data collapses implies this probability depends on T and T in a special way, (1) Here Pinv is the function which describes the shape of the scaled curves.


While the regularization adds bias to the model to improve generalization performance, this same bias may conflict with the distribution of the data.

to maintain interpretability, it is important, along with prototypical examples, to deliver insights signifying the parts of the input space where prototypical examples do not provide good explanations.


Decision theory recommends, after observing x A , to choose the decision d * that minimizes the risk, i.e., the expected loss, namely A natural goal in Bayesian active learning is thus to adaptively pick observations, until we are guaranteed to make the same decision (and thus incur the same expected loss) that we would have made had we run all tests.

we can reduce the noisy Bayesian active learning problem to the ECD problem by defining the equivalence classes over all test outcomes that lead to the same minimum risk decision.


In our triangular model, the main property of interest is the distribution over ∆ 3 and ∆ 2 , analogous to how latent factor models and MMSB model distributions over 0-and 1-edges.

subsampling strategies that preserve ∆ 3 /∆ 2 distributions (e.g.


This result provides additional support to previous informal conjectures as to why SP is so effective near the satisfiability threshold: SP concentrates all its efforts on finding a satisfying assignment when they are scarce and "scattered" across the space of possible assignments.

sP assumes that the set of satisfying assignments has in fact special structure.


The noise measure from VAD-based or perfect speech pause detection cannot be updated while speech is active.

an incorrect spectrum is subtracted and leads to artifacts and degraded recognition performance.


In this task, the subjects have been explicitly advised to terminate the task as soon as they discover the direction.

there is an incentive for the subject to terminate the trial sooner.


As a direct consequence of Theorem 1, spectral clustering requires  min (for a large enough constant C).

the noise threshold for spectral clustering does not match the lower bound.


Since V is convex, consider a line joining x * and x * (N ).

for any point f is constant on the line joining x * and x * (N ).


The necessity of the more complex error term in Theorem 2 (b) (as opposed to that in part (a)) will become apparent below.

when d is large, σ 2 is small, and θ, η, τ, γ ∈ C, for some compact subset C ⊆ (0, ∞), Theorem 2 suggests that consistency of ŷpcr and ŷbc in the one-shot regime hinges on asymptotic properties of The following proposition is proved in the Supplementary Material.


2 correspond to the channel at each pixel with the maximum value in the selection map I(n), remember that these maps themselves are soft.

we also report the mean entropy of the underlying I(n) for each pattern in Fig.


By arguments similar to those in Theorem 4, it can be shown that any stable non-positive steady-state is unique, attains the minimum total opinion among steady-states, and is smooth and convex for decreasing h. The above result paints a significantly simplified picture of the MF-IIM problem when J is stronglyconnected and there exists a stable non-negative steady-state m(b).

given a budget H, for any feasible marketing strategy h ∈ F H , m(b + h) is the unique stable non-negative steady-state, attains the maximum total opinion among steady-states, and is smooth in h. the objective Algorithm 1: An -approximation to MF-IIM Input: System (N, J, b, β) for which there exists a stable non-negative steady-state, budget H, accuracy parameter > 0 Output: External field h that approximates a MF optimal external field h * t = 0; h(0) ∈ F H ; α ∈ (0, 1 L ) ; repeat allowing us to write down a gradient ascent algorithm that approximates a local maximum.


According to the above theoretical analysis, the local Rademacher complexity (the tail sum of the eigenvalues of the kernel) leads to tighter generalization bounds than the global Rademacher complexity (the trace).

we add the local Rademacher complexity to restrict H mkl : where λ j (K µ ) is the j-th eigenvalues of K µ and ζ is free parameter removing the ζ largest eigenvalues to control the tail sum.


It is given by 1 The variance of u due to the external input can be obtained by noting that ui(t+dt) = ui(t) exp( dt/⌧ )+ R t+dt t ds exp((s t dt)/⌧ )(bi + P j Wi,jXj(t))/⌧ .

in the weak coupling regime The weak coupling approximation amounts to replacing spikes of the latent neurons by intensities plus Gaussian noise.


is a sequence of actions from the root of the search tree, and we also use p to denote the state reached after executing the path p from the root.

d p is the observation distribution of the state at the end of the path p. We use p a to denote a path formed by executing all actions in p and then executing action a, and we use |p| to denote the length of the path.


However, this procedure was too expensive for finding the best parameters in SVM+.

we used the performance on a separate validation set to guide the search.


In each successive (discrete) timestep, the node transmits the message to the next neighbor in its ordering.

if a node has d neighbors, all d neighbors will receive the message within d timesteps.


Commonly, we need n convolution filters {f i } n i=1 of size d × d × c. The space complexity for storing these filters is O(d 2 cn), and the computational complexity for generating y is In contrast, the proposed spatial versatile convolution operation can extract s = d i /2 sets of feature maps simultaneously.

for generating N feature maps, the space complexity for storing the proposed spatial versatile convolution filters is The computational complexity for generating feature maps using the proposed spatial versatile filters in different scales is various, which affects by the size of convolution filters, i.e., the number of non-zero elements in each M i .


Moreover, when k-means++ initialization is used, the expected number of iterations required for Lloyd's algorithm to converge is only logarithmic [1].

ignoring the time required to embed the points, we can find a solution in O(n √ nk) time and space instead of the O(n 2 k) cost required by the exact method, with a strong O( √ n) improvement.


But the weights are dependent on the query point and hence the best grouping of the data will also be dependent on the query point.

the problem we now face is, given query point, how to quickly divide the dataset into groups such that data points in the same group have similar weights.


when each action is appropriate (rules that would be inaccurate and quickly become stale), we wanted Cobot to learn the individual and communal preferences of users.

we provided a mechanism for users to reward or punish Cobot, and programmed Cobot to use RL algorithms to alter his behavior on the basis of this feedback.


First, since the size of D t is small, we need to find a way to augment it.

second, we need to somehow use the label information of D t . we create pairs of samples.


The learning goal was to correctly classify email messages into user defined folders.

the instances in this dataset are email messages, while the set of classes are the user defined folders denoted by {1, . . . , k}.


That is, the unbiased assumption on gradient estimates used in SVRG no longer holds.

a careful analysis of ZO-SVRG is much needed to ensure its optimization performance.


There are two differences between DSTs and our model: (1) In DSTs, the parent chain has its own Markovian dynamics, while the team state of our model is not directly influenced by the previous team state.

our model captures the emergent phenomena in which the group action is "nothing more" than the aggregate behaviors of individuals, yet it provides a useful level of representation beyond individual actions.


In the case where x ⋆ not set variables X i and X j to the same state, the value c-map ⋆ gives at least an upper bound that is no worse than the bound given by the relaxation alone.

in particular: at least in the case where a single equivalence constraint is relaxed, the compensations implied by Equations 1 and 2 do indeed tighten a relaxation (see the auxiliary Appendix for further details).


Similarly, it can be applied to measure similarity between two different groups of signals.

similarity-by-Composition is suitable for detection, retrieval, classification, and clustering.


update the set S ui ← S ui ∪ {x oi,ui } 5: obtain the feedback let k i (x) be a vector-valued function that outputs an i-dimensional column vector with j-th entry k((x oj ,uj , S oj −1,uj , φ uj ), (x, S oi−1,ui , φ ui )) 7: let K i be an i×i matrix with (j, j )-entry k((x oj ,uj , S oj −1,uj , φ uj ), (x o j ,u j , S o j −1,u j , φ u j )) 8: update Suppose that by the end of the T -th round, an algorithm has selected T j items for the payoff function f φj ; therefore the cardinality of S j by the end of the T -th round is T j .

we have T j = t≤T 1{u t = j} and T = m j=1 T j .


On the other hand, most methods for reducing dimension will also bring H 0 and H 1 "closer together," making them harder to distinguish.

mindful of the fact that the Hotelling test measures the separation of H 0 and H 1 in terms of δ Σ −1 δ, we see that the statistical distance is driven by the Euclidean length of δ. we seek to transform the data in such a way that the dimension is reduced, while the length of the shift δ is mostly preserved upon passing to the transformed distributions.


DirBN is a self-contained module on φ, leaving θ untouched.

it can be used as an alternative to the simple Dirichlet prior on φ in many existing models.


Finally, for a partition P , let Z be the indicator variable of a partition, Z = s iff X ∈ P s . With this notation, one can write where H are the full and conditional Shannon entropies.

dER algorithm can be interpreted as seeking a partition that maximizes the information between current known state (Z), and the next step from it (Y ).


This is even more extreme in the medical domain, where obtaining extra labels is sometimes not even possible, e.g., when tests are not available.

reducing the amount of labeled examples required for learning the scoring function is key for the success of structured prediction in real-world applications.


This implies that lim n→∞ nE[f ( θn ) − f (θ * )] has a finite limit.

this interpretation explains why the averaging produces a sequence of estimators which converges to the solution θ * pointwise, and that the rate of convergence of Note that (a) our result is stronger since it is independent of the lowest eigenvalue of H, and (b) for other losses than quadratic, the same properties hold except that the mean under the stationary distribution does not coincide with θ * and its distance to θ * is typically of order γ 2 (see Section 3).


While the information-theoretic limits of the Stochastic Block Model have been heavily studied (in slightly different ways), none of the other models considered in this paper have been studied before.

we provide new information-theoretic results for the Exponential Random Graph Model (ERGM), the Latent Space Model (LSM), the Directed Preferential Attachment Model (DPAM), and the Directed Small-world Model (DSWM).


In equations ( 7)-( 11), we obtain an integer quadratic program.

we relax the boolean constraints, allowing y and z to take any value between 0 and 1.


Since we are thresholding by mean, the graphs have high connectivity.

from Corollary 4.3, the function class associated with LS labellingis rich and expressive, and hence it outperforms previously proposed regularizers.


• For q = 2: When we fix i, j, s, t which are all different, and take expectation over ⇡, then all products of off-diagonal elements of A appear the same number of times in E ⇡ A ⇡(i),⇡(j) A ⇡(s),⇡(t) .

this quantity is proportional to P p6 =r v p v r , where v is the vector of all non-diagonal entries of A.


Then by Proposition 1, there exists at least one i ∈ {1, • • • , n} such that lim inf k∈K,k→∞ We next consider the following two cases: (a) There exist a subsequence K of K and an integer k > 0 such that for all k ∈ K, k ≥ k, the GD-step is adopted.

then for all k ∈ K, k ≥ k, we have by the optimality condition of the above problem and properties of the Clarke subdifferential (Proposition 4 in Supplement A), we have Taking limits with k ∈ K for Eq.


For program synthesis [5,34,10], the search spaces are enumerable and deterministic.

we propose to conduct systematic exploration, which ensures that only novel trajectories are generated.


That is, the firing rate is then determined by a combination of both the external stimulus and the spiking-history of the neuron.

the model can account for typical effects such as refractory periods, bursting behavior or spike-frequency adaptation.


In contrast, our model is designed to infer the structure of its environment through experience, and gain modulation is a byproduct of this inference.

our model has no distinct control mechanism, leading to a novel perspective on executive control processes in the brain.


From the above derivation, we can see that the essential part of the CVI algorithm is to compute the expectations and variances of RMN features, both of which can be done by any inference procedures, including the mean field as we have used.

in principle, the CVI algorithm can be extended to use other inference methods like belief propagation or exact inference.


Further, the probability that one dimension falls outside of this interval increases exponentially with the number of dimensions.

the neighborhoods are even larger for high dimensional problems.


Slight rotations of the viewer around the exact viewing positions would dramatically change the observed image.

these views are unstable to perturbations.


In image registration, retrieval and 3D reconstruction, for instance, nearest neighbor search builds on such measures to establish point correspondences.

the choice of similarity or kernel impacts the performance of a system as much as the choice of visual features [2,16,18].


Furthermore, note that before making any mistake, we remove at least |H| of the functions from the pool each time we answer ⊥.

the total number of ⊥'s cannot exceed 2 |H|.


In addition, Laurent and von Brecht [24] show that an RNN model with state that asymptotically converges to zero achieves a level of performance comparable to that of LSTMs and GRUs.

our constraint that the GRU is locally stable at h * = 0 is effective for learning.


Eye-movements lead to temporal changes of the detectability.

the integral of the ROC curve, which we will call the "detectability index" (DI), then varies over time.


For local consistency, we enforce the predicted value for close locations to be similar via spatial Laplacian regularization.

we can formulate the forecasting task as the following optimization problem over the model coefficient tensor W: Though cokriging and forecasting are two different tasks, we can easily see that both formulations follow the global and local consistency principles and can capture the inter-correlations from spatialtemporal data.


However, for each FoE model, the Hessian of the energy function log(1 + 1 2 (J T i x r ) 2 ) is lower bounded by − J T i Ji 8 (proof in the supplementary material).

we simply add an extra term γx T r x r with γ > J T i Ji 8 to obtain the concave-convex decomposition log(1+ 1 2 (J T i x r ) 2 ) = {log(1+ 1 2 (J T i x r ) 2 )+ γx T r x r } − γx T r x r . We utilize a pre-trained 5 × 5 filterbank with 24 filters, and conduct experiments on the BM3D benchmark 1 with noise level σ = 15.


As there are exactly (2;;) distinct choices of dE {I, ... , ko} training examples from the 2m examples Z there are no more than (2emlko)kO different subsamples to be used in w. For each particular subsample z ~ Z the weight vector w is a member of the class of linear classifiers in a ko (or less) dimensional space.

from (7) it follows that for the given subsample z there are no more (2emlko)kO different dichotomies induced on the double sample Z E z2m.


It follows from Lemma 3 and Lemma 1 that, if u i = 0, then x * i = 0.

by applying the above iterative procedure, we can find the minimizer of ( 5) by solving a reduced problem that has p ′ ≤ p variables and g ′ ≤ g groups.


For example, Tσ 2 (p) = IE(X 2 ) − IE(X) 2 is the familiar plug-in statistic for the variance, while T unbiased Obviously, a plug-in statistic yields an unbiased estimate concerning the distribution that is plugged in.

when the empirical distribution is plugged in, a plug-in statistic typically does not give an unbiased estimate concerning the (unknown) true distribution.


Our main result, stated precisely in the next section, establishes hardness of approximating the backward mapping for the hard-core model.

despite the problem being statistically feasible, it is computationally intractable.


let ζ i (u, v|C) ∈ (−∞, 0] denote the warped MI between the enter and exit r.v.s of interstitial vectoral node i on Pu:v , with conditioning set C ⊆ V \ {u, v}.


Computing the exact worst-case adversarial error is not computationally tractable.

to provide a comparison, we also compute a lower bound on the adversarial error-the error obtained by the PGD attack.


The equivalence between the two problems can be verified using convex analysis [8] by noting that the Lagrangian for the constrained optimization ( 4) is exactly the objective in the optimization (5) (plus a constant that does not depend on θ), where κ is the Lagrange multiplier.

(4) can be solved by solving either (5) or ( 6) for an appropriate κ or γ.


The log likelihood is computed by where S = 1 k k i=1 (y i − µ)(y i − µ) T . To facilitate the following analysis we first assume V projects Y to a 1-D space, i.e., p = 1 and V = v T , and thus Note that v T Cv is the variance of the object samples in the projected space, and we need to impose a constraint, e.g., v t Cv = 1, to ensure that the minimum likelihood solution of v does not increase the variance in the projected space.

let v T Cv = 1, the optimization problem becomes we obtain an equation exactly like the Fisher discriminant analysis for a binary classification problem.


Such approaches measure performance relative to the single best set (e.g., of articles).

they are not suitable for many retrieval settings since the set of available articles can change frequently (e.g., news recommendation).


Common activation functions, such as the Sigmoid, Tanh, and ReLU all behave close to linear over large ranges of their domain.

for a randomly sampled input to break linearity, layers must be wide and the network deep to ensure some elements lie in non-linear regions of the activation function.


To model this, we follow the Bayes-Adaptive MDP framework, and include the φ and ψ vectors in the state of the BAPOMDP.

the state space S ′ of the BAPOMDP is defined as S ′ = S × T × O, where T = {φ ∈ N |S| 2 |A| |∀(s, a), s ′ ∈S φ a ss ′ > 0} represents the space in which φ lies and O = {ψ ∈ N |S||A||Z| |∀(s, a), z∈Z ψ a sz > 0} represents the space in which ψ lies.


More problematically, the entropy is defined in a high-dimensional feature space, which is changing dynamically throughout the training process.

it is difficult to estimate and optimize the generator entropy in the feature space in a stable and reliable way.


Despite leading to a looser upper bound of the Gibbs decoder distortion, if one could control the statistical accuracy under this approach then one could obtain a fully polynomial-time evaluation of the objective function, even if |H| is exponential.

whether this method is feasible, and under what technical conditions, are potential future work.


At each training step, all the EMA predictions of the examples in that minibatch are updated based on the new predictions.

the EMA prediction of each example is formed by an ensemble of the model's current version and those earlier versions that evaluated the same example.


Theorem 2 guarantees that to get an ε guarantee for the accuracy of the optimization it suffices to perform n(ε) = O(1/ε) evaluations of f , where we are now treating log(δ 2 ) − log(δ 1 ) as a constant.

the number of functions evaluations is O(1/ε), and, since the time complexity of every evaluation is O(h|S| 2 |A|), the total time complexity of the algorithm is O(h|S| 2 |A|/ε).


This takes the same values as X i,t , but its distribution is given by a different exponential parameter ( θi instead of θt but for all θ ∈ [ θt , θt+1 ], var(X i,t, θ ) ≤ e ηB var(X i,t ).

hence, for η < 0.567/(B + λ), Q(ᾱ t ) is non-increasing in t. The proof is in [1].


Nevertheless our aim is not to correct the dynamical system forever, but to teach it to generate the desired output itself without the error feedback.

the matrix M needs to be modified over time.


After all, the effect of expanding one node depends not only on local information such as the local bounds it obtains, but also on how many pruned nodes it may lead to and many other interacting strategies such as branching variable selection.

given our single goal of finding a good solution quickly, we design an oracle that finds the optimal solution without a proof of optimality.


In the parallel setting, we cannot investigate L * directly and we may only analyze it through L N . However, calculating the total loss function directly and exactly requires excessive inter-processor communication, which carries a huge computational burden as the number of processors increase.

we must approximate the true parameter θ * = arg min θ∈Θ L * (θ) by an empirical risk minimizer θ = arg min θ∈Θ L N (θ).


One may imagine, however, that training data for the handling of a load can be obtained along a fixed reference trajectory T r for calibration purposes, and also along a trajectory typical for that load, say T m for the m th load.

for each load, 2000 random training samples are acquired at a common reference trajectory T r = (p 2 , s 3 ), and an additional 2000 random training samples are acquired at a trajectory unique to each load; Table 1 gives the combinations.


The proof follows from the fact that λ max (D) < 1 implies D N → 0 as N → ∞.

given any fixed, stable controller, if N is sufficiently large, it will satisfy (10).


( 1), that on any iteration t, kX t . if we consider replacing the choice of X t+1 in Algorithm 1 with the following update rule: then, following basically the same steps as in Eq.


their relevance to the network output) and introduce a regularization term that gradually lowers the absolute value of parameters with low sensitivity.

a very large fraction of the parameters approach zero and are eventually set to zero by simple thresholding.


Let Z denote the random variable w(x) L h (x) − R(h).

then, |Z| ≤ M . By lemma 2, the variance of the random variable Z can be bounded in terms of the Rényi divergence d α+1 (P Q): by Bernstein's inequality [4], it follows that: Setting δ to match this upper bound shows that with probability at least 1−δ, the following bound holds for the importance weighting method: Using the sub-additivity of √ • leads to the simpler expression These results can be straightforwardly extended to general hypothesis sets.


These neurons fire after a particular key is pressed.

the feedback signal y m with k = 1, 2, .., K is partitioned by K 0 : The first part (k ≤ K 0 ) represents the motor state, and the second part (k > K 0 ) represents the context.


Furthermore, if a sparse regression algorithm falsely detects one variable, then SWAP can potentially recover the correct support.

using SWAP with other algorithms does not harm the sparse recovery performance of other algorithms.


One can verify that H 2 is equivalent to H up to a logarithmic factor: H 2  H  log(2n)H 2 (see [1]).

by setting the probability of error (the RHS of Eq.


We first note that, as desired, our model selects features that have large variation in their probabilities across the clusters (rows in Figure 3).

it is straight-forward to read what makes each column different from the others: the mammals in the third column do not lay eggs; the insects in the fifth column are toothless and invertebrates (and therefore have no tails).


We also found diminishing returns for sampling large neighborhoods (Figure 2.B).

despite the higher variance induced by sub-sampling neighborhoods, GraphSAGE is still able to maintain strong predictive accuracy, while significantly improving the runtime.


Region segments, on the other hand, have large spatial support, but lack the structure that contours provide.

additional grouping of contours can give us both qualities.


This notion is readily extended to non-linear classifiers obtained by kernel transformations, by noting that in the feature space, the optimal discriminant resides in the span of the transformed data.

for both linear and nonlinear classification, searching the space of dyadic hypercuts forms an efficient strategy for exploring the space of all hypotheses.


Let X = X be the compressed design matrix, and let Y = Y be the compressed response.

the transformed noise is no longer i.i.d..


The non-robustness of (online) PCA is independent of used optimization method.

we only compare with the basic online PCA method [1], which is enough for comparing robustness.


The module also has 6 parametric bias nodes, 4 context nodes, 50 hidden nodes, and 10 prediction output nodes.

no a priori knowledge about the meanings of words is pre-programmed.


In practice, it is more convenient to compute a fixed amount k of nearest neighbors.


Let us now consider the family of distributions P with marginal distribution PX and regular conditional distributions Q x := P (• − m(x))/σ(x) x , x ∈ X, where m : X → R and σ : X → (β, 1/β) are as in the previous example.

then Q x has a τ -quantile of type αβ, because we obtain for s ∈ [0, αβ] the inequality P has a τ -quantile of p-average type αβ if and only if P does have a τ -quantile of p-average type α.


Soft-thresholding the eigenvalues of the input covariance matrix, Figure 1A: eigenvalues below the threshold are set to zero and the rest are shrunk by the threshold magnitude.

the number of output dimensions is chosen adaptively.


Each iteration requires one matrix-matrix multiply with K XX , and the subsequent work to derive these inference terms takes negligible additional time (Appendix B).

p iterations of mBCG requires O(nt) space (see Appendix B) and O(p Ξ( K XX )) time, where Ξ( K XX ) is the time to multiply K XX by a n × t matrix.


5) as it does not promote sparsity as an L 1 term would.

the observed sparsity in the trained network must be a genuine consequence of having optimized the connectivity for robust stability.


Rearranging the product from equation 2, we achieve a likelihood in terms of recipient states x : x |p e −q x |p Tp (3) where we define p as {x}×{u}×(X \(X ×U X )) in each partition P x , and likewise: q x |p = q xx |u , M x |p = M xx |u , and T p = T x|u .

cTBNs are one instance of partition-based CTBNs, with partitions corresponding to a specified donor state x and parents setting u.


1 and 2) but not between cells, and A is an arbitrary constant as absolute firing rates are not shown.

in this model, a place cell's response is simply determined by the parameters d i and ifJi chosen for the set of BVes connected to it.


The task of VI is to find the best approximation, in terms of KL divergence, to this unknown function within a family of tractable distributions Q.

vI can be written as the following optimization problem: It should be obvious that the quality of the approximation directly depends on the expressivity of the family Q.


The benefit of RSGD is that it calculates only Riemannian stochastic gradient gradf i (x) corresponding to a particular i-th sample every iteration.

the complexity per iteration of RSGD is independent of the sample size n, which leads to higher scalability for large-scale data.


We remove the samples which fail the MRI quality controls and have missing entries.

we have 6 tasks with each task corresponding to a time point and the sample sizes corresponding to 6 tasks are 648, 642, 293, 569, 389 and 87, respectively.


Experimentally, it was found that queries consisting of more than about three terms returned very few images.

the terms are generated by appending the polysemous word to its synonyms and first-level hypernyms.


Further, by calculating the posterior distribution of stimuli given an observed neural response, the model can be used to decode stimuli and yields a natural spike-train metric.

extending the framework of maximum-entropy models to continuous variables allows us to gain novel insights into the relationship between the firing patterns of neural ensembles and the stimuli they are processing.


The twist from the standard setup is that the losses of all the experts are not uniformly bounded in the same range.

indeed, for the setup described for the oracle inequality (3), class W k will produce predictions with norm as large as 2 k . here, we assume that expert i incurs losses in the range [−c i , c i ], for some known parameter c i ≥ 0.


It is clear that these various stopping conditions qualitatively trade off certain of the properties in Theorem 2.1.

for example, the k-cluster stopping condition does not attempt to produce all possible partitions, while the distance-r stopping condition builds in a fundamental length scale, and hence is not scale-invariant.


The overall cost function c i of player v i is defined as where ∆ G(s) (u, v) is the shortest distance between u and v in G(s).

in this game player i wishes to minimize c i (s), which requires balancing edge costs and shortest paths.


(I8) and ( 27), we see that the second term plays the same role as the prior knowledge in MAP.


Sample-MIPS, as mentioned earlier, is essentially a sampling algorithm with replacement scheme to draw entries of Z such that (j, t) is sampled with the probability proportional to z jt .

sample-MIPS can be thought as a traversal of (j, t) entries using in a stratified random sequence determined by a distribution of the values of {z jt }, while the core idea of Greedy-MIPS is to iterate (j, t) entries of Z in a greedy sequence induced by the ordering of {z jt }.


By letting , we can fit (6) to X = F Y A in Definition 2.

learning {O i } i∈ [3] is reduced to selecting k rank-one matrices from {X t,i ⊗ X t,j } t∈[n] indexed by A whose conical hull covers the q matrices {Y (t) } t∈ [q] .


i = 1, . . . , N . i is a length-L vector.

the feature group hi is represented by a feature map of L channels, where h (x,y) i contains L channels of features at location (x, y).


In practice, this process is performed in milliseconds, resulting in millions of transactions recorded daily by the AdExchange.

one might expect that the AdExchange could benefit from this information by learning how much an advertiser values the right to display his ad and setting an optimal reserve price.


Similarly to Theorem 2 in (Abbasi-Yadkori, Pal, and Szepesvari, 2011) 2 , for any state history and with probability of at least 1 − δ, it holds for all t > 0 that θT t,a x(s t ) − θ * a T x(s t ) ≤ we define δ = δ/k and bound this probability for all the actions, i.e., ∀a, t > 0 Recall that any valid action a at state s satisfies E[e t (s, a)] = θ * a T x(s t ) ≤ .

we can eliminate action a at state s t if This ensures that with probability 1 − δ we never eliminate any valid action.


Maximizing log P (Y, Z|X, θ) is straightforward if both Y and Z are observed, but here we are interested in the case where Y is observed and Z is "hidden".


For the methods we consider, the arithmetic costs of linear updates and psd approximation are roughly comparable.


This simple covariance calculation was much cheaper in our experiments than the moment conversion with cost O(pq 2 ) or the SVD with cost O(p 3 q 3 ), both of which are independent of the data set size N T . In contrast, each iteration of EM requires at least O(N T (p 3 + pq)) time.

the computational advantage of PLDSID is expected to be especially great for large data sets.


With this notation, we define feedback as feedback i.e., the difference between the predictions our model actually made and the predictions it would have made had it not had the chance to affect its environment by broadcasting predictions in the past.

statistical feedback is a difference in potential outcomes.


The effectiveness of spectral clustering hinges crucially on the construction of the graph Laplacian and the resulting eigenvectors that reflect the cluster structure in the data.

we construct an objective function that consists of the graph Laplacians from all the views of the data and regularize on the eigenvectors of the Laplacians such that the cluster structures resulting from each Laplacian look consistent across all the views.


Formally, suppose f in (4) is convex; for such f , FBS performs the iteration where {η k } is a suitable sequence of stepsizes.

the usual convergence analysis of FBS is intimately tied to convexity of f . to tackle nonconvex f we must take a different approach.


It consists of 2501 training images and 2510 validation images, and the task is to classify an image into one or more of the 20 categories.

this task can be casted as training 20 binary classifiers.


Although gains are expected from incorporating higher-order derivatives into the score-space, the size of the score-space dramatically increases.

practical systems may truncate the likelihood-ratio score-space after the 1st-order derivatives, and hence use linear approximations to the Taylor series expansions 3 .


However, the authors assume that the distribution satisfies a Lipchitz condition which technically allows them to bound the number of lies in the same way as in Proposition 2.

the regret bound achieved by their algorithm remains the same in our scenario.


Note that every row of this matrix is a simple hashed counter, and all rows are independent.

without the heaps, CMS are reducible summaries, i.e., different summaries with the same hash functions can be merged by simply adding the sketches.


However, instead of choosing an attribute that minimizes the size, we would like to choose one that minimizes costs.

given a tree, we need to come up with a procedure that estimates the expected costs when classifying a future case.


If perturbed beliefs are calculated from constant initial messages with a small step, one obtains the same result.

bBP and perturbation give the same gradient for the Bethe approximation.


From our analysis of multiplicative approxinlate greedy (see Lemma 3), given a multiplicative approximation factor (1 + f=) in the approximate nearest neighbor method, the approximate greedy coordinate descent has the convergence rate: K . 1\:1,,,3 for some constant K > O.

the num-'ber of iterations required to obtain a solution with accuracy fopt is given by, T greedy = ~:~.


This is achieved by avoiding zero weights, which would instead be allowed by linear or quadratic weightings.

the exponential weighting scheme can be used as weights associated with features for weighted distance computation D(x, y) = )2::7=1 Wi(Xi -Yi)2.


Instead, e-INFERSENT significantly outperforms InferSent on 4 tasks, while it is significantly outperformed only on 1 task.

we conclude that training with explanations helps the model to learn overall better sentence representations.


However, the expressiveness of GPs comes at a price: solving (approximate) inference for a GP with N data instances has time and space complexities in Θ(N 3 ) and Θ(N 2 ), respectively.

gPs have traditionally been viewed as a tool for problems with small-or medium-sized datasets Table 1: Comparison between SVDGP and variational GPR algorithms: SVI [10], iVSGPR [2], VSGPR [24], and GPR [19], where N is the number of observations/the size of a mini-batch, M , M α , M β are the number of basis functions, and D is the input dimension.


The advantage stems from the fact that with Green's function, the force between two nearby points goes to infinity as the points are pushed together, whereas with the Gaussian, the force goes to zero.

without Green's function, one might expect local optima in which clusters of points collapse onto a single location.


For dataset 'ecoli', 'wine' and 'iris', the averaged error rates of the 'EM Model' are very close to the cases when there are no distractor class labels.

we can conclude that the 'EM Model' is able to reduce the noise caused not only by random label ambiguity but also by some systematic label ambiguity.


Hence, the amount of the induced bias generally varies among the different edges in the graph.

the BIC has to be amended when applied to a bootstrap sample D (b) (instead of the given data D).


From (10), we observe that Σ i+1 Σ i for all i, so Σ n+1 Σ i+1 Σ 1 = I for all i.

the conditions on Σ * in ( 18) are satisfied by where the last equality holds since we assumed that an update was made for the ith example.


In other words, PCCs construct a perfect binary tree of height L in which every node except the root node corresponds to a binary classifier.

obtaining the exact solution of such a probabilistic tree requires to find an optimal path from the root to a leaf node.


Let Note that the choice of policies has no influence over future losses in C T (A, π).

c T (A, π) can be bounded by a reduction to full information online learning algorithms.


However, instead of integrating out Θ, Φ and Ψ, we use them as the parameters to approximate T's posterior distribution.

the estimation of the parameters is to minimize  Due to the convexity of negative logarithm, the second term in the RHS of Eq.


Here s = 4, and This gives N S ψ (z 1 ) = � p ∈ Δ 3 : p = (q 1 + q 2 , q 3 , q 4 ) for some q ∈ Δ 4 , q 1 − q 2 − q 3 − q 4 = 0 � = � p ∈ Δ 3 : p = (q 1 + q 2 , q 3 , q 4 ) for some q ∈ Δ 4 , q A similar procedure yields N S ψ (z 2 ) = Q ord 2 and N S ψ (z 3 ) = Q ord 3 .

by Theorem 7, we get that ψ is classification calibrated with respect to � ord over Δ 3 .


The first term in the general form of L (Equation ( 8)) has a non-zero diagonal, which can be removed by adding L ⊥ , a matrix with all its entries equal to one (times a scalar).

l is: This seems a rather unlikely candidate matrix to support persistent percepts.


Arguably one of the biggest differences between classification and detection is the ability to accurately localize objects and reject background.

we show that our method successfully adapts the classification parameters to be more suitable for detection.


Following the above analysis, we first approximate gradient field of the prediction function by a  vector field, then we require the vector field to be as parallel as possible.

we try to learn the function f and its gradient field ∇f simultaneously.


Furthermore, for every γ in this range, this tradeoff between m 1 and m 2 is necessary, up to constant factors.

our results smoothly interpolate between the known bounds of Θ(n 2/3 ) on the sample size necessary in the setting where one is given two equal-sized samples [6,9], and the bound of Θ( √ n) on the sample size in the setting in which the sample is drawn from one distribution and the other distribution is known to the algorithm [22,29].


The Bethe approximation can reflect dependence of random variables (dependence between U and V in p(U, V |A) in our problem) to some extent.

one can intuitively expect that performance of the AMP algorithm is better than that of the variational Bayes matrix factorization, which treats U and V as if they were independent in p(U, V |A).


The Bethe free energy is not necessarily convex, which causes unfavorable behaviors of LBP such as oscillation and multiple fixed points.

clarifying the region where the Hessian is positive definite is an importance problem.


This also means that our framework estimates labels using the adjacency matrix W optimized in the feature space instead of the output (label) space.

if input features and labels share the same adjacency matrix (i.e., sharing the same local structure), the minimization of the objective function (2) should estimate the adjacency matrix which accurately propagates the labels of graph nodes.


Compared with original convolution filters, the proposed spatial versatile filters can provide more feature maps without increasing the number of filters.

we further analyze the memory usage and computation cost of neural networks using the proposed spatial versatile filters.


However, if the sequence is very imbalanced and has many more 0's than 1's then this never predict strategy has a high regret with respect to the strategy that predicts the majority bit.

one is interested in a strategy that has a small regret with respect to predicting the majority bit and incurs no loss at the same time.


Therefore, with a union bound argument, the probability of any of these estimations being wrong is at most 2 • n • δ = δ.

from now we assume that all the estimations were correct, that is, the high probability events of Theorem 1 hold.


By (23), if Ẑ = CH, then On the other hand, if Γ C :,i 2 = 1 and H i,: = H i,: 2 C :,i Γ, then we have ||| Ẑ||| * = i H i,: 2 , implying the optimality of (C, H).

given Γ, the recovery problem (22) has been reduced to finding a vector µ and matrix C such that µ ≥ 0, C :,i ∈ C(Γ) for all i, and C diag(µ)C Γ = Ẑ.


Although both V b2 and V b1 converge to zero, the rate and the type of convergence are not the same: nV b2 converges in law to some random variable while the behaviour of nV b1 is unspecified.

tests that utilize V b2 usually give lower Type II error then the ones that use V b1 .


Of interest to us is the discrete entropy of a discrete-valued random vector x ∈ {−1, +1} n , whereas the Gaussian bound of Lemma 2 applies to the differential entropy of a continuousvalued random vector.

we need to convert our discrete vector to the continuous space.


Also, let λ = β P 2 for β > 0 and x 0 − x * small.

we can thus approximate the right parenthesis of the bound by the bound on the precision of the extrapolation is approximately equal to Also, if we use equation ( 8), it is easy to see that where ζ is defined in (9).


The term (s) = s T ⇤s is a slack variable-when ⇢ > s T P s or s = 0 (so the second term is nonpositive), it can be made sufficiently large so that the first constraint holds regardless of s T P f ⇡ (s), but when ⇢  s T P s and s 6 = 0 (so the second term is positive), we must have s T P f ⇡ (s) < 0 since s T ⇤s 0 by the second constraint.

properites (i) and (ii) hold from (1), and ( 2) verifies (iii) for all if a solution ⇢ > 0 is found, then V is a Lyapunov function with region of attraction U .


At a high level, a Markov logic network (MLN) can be seen as a template for generating ground Markov networks.

a natural way to answer inference queries over MLNs is to construct a ground Markov network and then use standard inference techniques (e.g., Loopy Belief Propagation) for Markov networks.


it has been suggested that position shifts can be used to "bias" the preferred disparity of a population around a rough estimate of the stimulus disparity, and then use a population of neurons tuned by phase shifts to obtain a more accurate estimation of the actual disparity.


This is because (i) the reference predictors do not need it; and (ii) if we did include the cost of observing the loss function for the reference predictors, then the loss of each predictor would just be increased by c d+1 T , and so the regret R T would just be reduced by c d+1 T , making it substantially easier for the learner to achieve sublinear regret.

we prefer the current regret definition as it promotes the study of regret when there is a price attached to observing the loss functions.


That is, if a state s is visited for multiple times, then each time the distribution (of uncertain parameters) µ s is the same.

mathematically, we can adapt the augmented state space as in the non-stationary model, and requires that µ s,t does not depend on t. the set of admissible distributions is The next theorem is the main result of this section; it shows that a sequentially robust policy is distributionally robust to both stationary and non-stationary models.


Matters are only slightly more complicated here.

proposition 10 If trial t is a mistake, and q maximizes entropy subject to membership in S t ∩ T t , then it is on the separating hyperplane for T t . Proof: Because q and p both satisfy S t , any convex combination of the two satisfies S t . if q was on the interior of T t , we could find a probability distribution with higher entropy that still satisfies both S t and T t by taking a tiny step from q toward p. This would contradict the assumption that q is the maximum entropy member of S t ∩ T t . This implies that the next hypothesis of a ROME algorithm is either on J t (the separating hyperplane T t ) only, or on both J t and H t (the separating hyperplane of S t ).


In particular, if we wish to minimize Hamming loss, which measures the number of incorrect labels, gradient-based optimization methods cannot be applied directly.

1 surrogate optimization problems, such as maximum likelihood or maximum margin training, are solved instead.


Such a model does not require any fine-tuning during voice cloning.

the same model can be used for all unseen speakers.


By contrast, rational metareasoning predicted that adaptive strategy selection would be evident from the choices of more than 70% of our participants.

the chosen experimental design was well suited to discriminate rational metareasoning from previous theories.


Not only so, but every once in a generation the training model is allowed to switch its gating vector that results the best fitness-score, which comes as a crucial feature of our optimization for highly multi-modal solution spaces.

among a population, an elite neural network model f (x; θ, µ elite ) is selected, where µ elite denotes the chromosome that causes an individual to yield the best fitness-score.


Consider what happens if we run a 1 -smooth boosting algorithm, where the initial distribution D satisfies D rel / U k = τ.

at each stage we will have D rel t (x) ≤ 1 • D rel (x) for all x ∈ {−1, 1} k , and consequently we will have by Lemma 2 each distribution D t will satisfy d T V (D rel t , U k ) ≤ 1− 2 /(4τ 2 ).


A principled way to achieve such a goal is to impose a boundedness constraint on 1 -like norms of these maps to achieve sparsity and simultaneously impose smoothness by penalizing their Laplacian.

we propose the following penalized dictionary-learning model The ingredients in the model can be broken down as follows: 2 measures how well the current dictionary V explains data X i from subject i.


However, we can construct a feasible solution Ŵ = W − G • E, where G ij = sign(W ij ) and • indicates the entrywise product of two matrices.

the assumption of this theorem implies that From Lemma 1 we have log det Ŵ ≥ log det W − p γσmin( W ) E F . Since W * is the optimal solution of P λ and Ŵ is a feasible solution of P λ , log det W * ≥ log det Ŵ ≥ log det W − p γσmin( W ) E F . Also, since W is the optimal solution of P Λ and W * is a feasible solution of P Λ, we have log det W * < log det W . By the mean value theorem and some calculations, we have |f , which implies (7).


The evaluation result is represented by where ¥ is a random variable with probability distribution function § ¦ ¨¥ T# .

the evaluation result ¢ H is also considered as a random variable.


However, we emphasize an important point: the reward, transitions, and attention can be defined by parametric functions, and trained with the whole policy 2 .

a rough design can be specified, and then fine-tuned by end-to-end training.


Moreover, in positive chip sampling, we do cover every ground truth sample.

for generating proposals, it is sufficient to train on just positive samples.


The problem setting for graph partitioning is relatively simple compared with other GNN applications.

it is suitable as a baseline for more complicated problems.


The functions that we consider are (x) = x and (x) = x 2 .

notice that any such function can be encoded by three coefficients a, b, c ∈ R. summing two functions amounts to summing their respective coefficients and takes time O(1).


While the synchronization time increases with L 2 [15], the probability of finding a successful flipping-attacker decreases exponentially with L, P ∝ e −yL as seen in Figure 4.

for large L values the system is secure [15].


Now, for such a utility measure, if it is true that for any two possibilities {x i , x j } and any two contexts {c k , c l }, then the choice behavior of an agent maximizing u(x, c) would be equivalent to one maximizing u(x).

for the inclusion of context to have any effect, there must exist at least some Note however, that the context in this operationalization is simply a collection of other possibilities, i.e.


Each network had the same number of nodes, and the drift of each network was given by (3) with K.i = 1, ~ = 0 for all units, and ~ being part of the adaptive vector A.

the number of hidden units was varied from one to 5.


They capture the semantic similarities of the words, and they also serve as the input into the amortization networks.

the global embeddings of words with similar pattern of across-group variation need to be in regions of the embedding space that lead to similar modifications by the amortization network.


Note that when α h = 1, p θ h (E|h) = e θ h , we have the well-known log-sum-exp (LSE) convex function.

this result generalizes the LSE function to a broader class of convex functions.


The Bregman divergence representation for the scaled distribution is given by where the last equality follows from Lemma 3.1 and the fact that, for a Bregman divergence, D βφ (•, •) = βD φ (•, •).

as β increases under the above scaling, the mean is fixed while the distribution becomes increasingly concentrated around the mean.


Here, our goal is to design a fully automatic approach to predict image memorability, while still exploiting the semantic information.

we use the automatic Object Bank [17] feature to model the presence/absence of various objects in the images.


The quantity R h (X m+u ) is called the full sample risk of the hypothesis h, R h (X u ) is referred to as the transduction risk (of h), and R h (X m ) is the training error (of h).

r h (X m ) is the standard training error denoted by Rh (S m ).


Resampling an MDP independently at each time period breaks the agent's intent to pursue a sequence of actions revealing the rare reward states.

the Thompson resampling agents do not commit.


Then, define the proxy-function d 2 (M ) = M 2 .

the proxy-function d 2 (•) is strongly convex on Q 2 with convexity parameter σ 2 = 2.


Lemma 2 Let us approximate (or assume) If the policy is non-degenerate for the task (so the dimension of θ, d, is equal to |S| i=1 (|A i | − 1), where |S| and |A i | are the numbers of states and the available actions at state s i , respectively) and ω * denotes the gNG(ι), then the 'near' optimal auxiliary function ρ * in the 'near' optimal estimating function g * ι,θ (s, a; ω) Proof Sketch: The covariance matrix for the criterion of the auxiliary function ρ is approximated as The |A i | degrees of freedom.

there exist ρ * and ∇ θ ln{d θ (s) ι π(a|s;θ)} ω that satisfy (14).


While performance on a specific dataset might be better, we would lose global convergence guarantees.

rather than obsess over the performance of a specific algorithm on a specific dataset, via this paper we hope to draw the attention of the community to the need for developing principled parallel algorithms for this important problem.


The importance measures Z j and Zj are determined by two factors: (1) the relative importance between X j and its knockoff counterpart Xj , encoded by filter weights z = (z 1 , • • • , z p ) T and z = (z 1 , • • • , zp ) T , respectively; and (2) the relative importance of the jth feature among all p features, encoded by the weight matrices, w = W (0) (W (1) W (2) W (3) ), where denotes the entry-wise matrix multiplication.

we define Z j and Zj as With the above introduced importance measures Z j and Zj , the knockoff statistic can be defined as W j = Z 2 j − Z2 j , and the filtering step can be applied to the W j 's to select features.


( 16) and local diffusions, and demonstrate that the proposed nonlocal blocks can be viewed as an analogue of local diffusive terms.

since we assume that the kernel matrix K has a finite Frobenius norm, we have the corresponding integral operator L is a Hilbert-Schmidt operator so that Eq.


In order to construct robust loss functions, consider the following two transformation functions: which, in turn, can be used to define the following loss functions: One can see that σ 1 (t) → ∞ as t → −∞, but at a much slower rate than σ 0 (t) does; its derivative σ 1 (t) → 0 as t → −∞.

σ 1 (•) does not grow as rapidly as σ 0 (t) on hard-to-classify data points.


Let t-rankpYq " r and rankpY k q " r max .

it can be seen from the definition of tensor rank that Y k " ř r i"1 c k r pa r b J r q. it follows from the subadditivity of matrix rank, i.e.


For example, in a chain graph, all the variables are correlated with each other but they have a much simpler dependency structure that can be encoded with O(n) parameters.

pMRFs more directly and succinctly capture the dependencies between words as opposed to other simple statistics such as covariance or pointwise mutual information.


The connectivity weight W ij will be zero whenever neuron j has no connection to neuron i.

determining W will determine which neurons are connected to one another and the strengths of those connections.


Under smoothness and strong convexity assumptions, the authors obtain convergence guarantees of essentially O(1/k 2 ) after k epochs, vs. O(1/k) using with-replacement sampling (with the O(•) notation including certain dependencies on the problem parameters and data size).

withoutreplacement sampling is shown to be strictly better than with-replacement sampling, after sufficiently many passes over the data.


Also, after passing x i and x j through T decision trees of the learned RF, suppose they reached leaves l t i and l t j in each tree t = 1, . . . , T . Then, we compute To estimate the ratio of the proposal distributions, q(B→A) q(A→B) , it is necessary to compute each individual probability p ij , since the nominator and denominator of q(B→A) q(A→B) do not contain the same set of "cut" edges, Cut A = Cut B , as specified in (3).

we compute In the following, we first present our empirical evaluation of (RF) 2 , and then derive the theoretical performance bounds of a simple, two-class (RF) 2 .


lSQ can use every single sample available to it no matter what policy is under evaluation.


We have outcome records for both twins and view them as two potential outcomes for the treatment variable.

the −2.5% difference between the average mortality rate of heavier twins and that of ligher twins can be viewed as the "true" ATE.


We have observed that, in most cases, the optimal multilabel y * is still on the K-best lists but the conditions of Lemma 7 are no longer satisfied, hence forbidding us to prove exactness of the inference.

working to establish alternative proofs of exactness is a worthy future research direction.


In other words, if communication is an expensive resource, then the solution proposed in [13] only works well when λ is at least order of 1/ √ m. In machine learning applications, the strong convexity parameter λ often comes from explicit regularization designed to prevent over-fitting, and needs to scale with the data size, usually between 1/ √ m and 1/m.

the solution above is communication-efficient only when λ is relatively large.


Canonical correlations ρ 2 are the solutions to the eigenproblem: where C xy = cov(X, Y ) and the matrices C xx and C yy are assumed to be invertible.

the largest canonical correlation ρ 1 between X and Y is the supremum of the correlation coefficients over their linear projections, that is: When p, q n, the cost of CCA is dominated by the estimation of the matrices C xx , C yy and C xy , hence being O((p + q) 2 n) for two random variables of dimensions p and q, respectively.


The MESA distance is robust to the additive local perturbations of its arguments such as independent noise or high-frequency signal as long as the integrals (counts) of these perturbations over larger region are close to zero.

it does not matter much how exactly we define the ground truth density locally, as long as the integrals of the ground truth density over the larger regions reflect the counts correctly.


Specifically, if a base class admits compression, then the class of mixtures of that base class, as well as the class of products of the base class, are compressible (Lemmas 3.6 and 3.7).

it suffices to provide a compression scheme for the class of single Gaussian distributions in order to obtain a compression scheme for the class of mixtures of Gaussians (and therefore, to be able to bound their sample complexity).


Worse, it may take exponentially many steps to reach an -equilibrium again.

stability of -equilibrium is an important consideration.


By spectral graph theory [2], dP can be estimated by the diagonal matrix D on the sample points.

the weighted data variance can be estimated as follows: To remove the mean from the samples, we define: It would be important to note that, if we do not remove the mean, the vector f r can be a nonzero constant vector such as 1.


In practice one would like to save this additional work.

the least-squares TD algorithm can fail due to matrix singularity and the iterative TD(O) algorithm can fail if the sampling distribution is different from the steady-state distribution.


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

when the sample size is large enough, we expect Ĝ = G * with high probability.


If both the sender and receiver have cached these keys, the sender then only needs to send the values with a signature of the keys.

we effectively double the network bandwidth.


Variational Bayes (VB) casts posterior inference as a tractable optimization problem by minimizing the Kullback-Leibler divergence between the target posterior and a family of simpler variational distributions.

vB provides a natural framework to incorporate ideas from stochastic optimization to perform scalable Bayesian inference.


, where S(µ, ϵ) is as in Lemma 2.6 for ϵ = δ/9mnk.

let m ≤ n. Then, by Lemma 2.5 and union bound, for any Y ∈ S ′ (µ, ϵ), where C 1 ≥ 0 is a constant independent of m, n, k. if p > Cµ 2 k 2 log n/δ 2 m, where C = 16(C 1 + 1), with probability at least 1 − exp(−n log n), the following holds ) As the statement of the theorem is invariant under scaling, it is enough to show the statement for all µ-incoherent matrices X of rank at most k and ∥X∥ 2 = 1.


It is known that using the total variation on graphs as a regularization functional in semi-supervised learning (SSL) leads to very spiky solutions for small numbers of labeled points.

one would like to have regularization functionals enforcing more smoothness of the solutions.


On the contrary, the motion representation, which we use trajectory as a medium, has long been considered to be invariant to appearance intuitively and empirically [17,28].

we cannot naïvely adopt the way of learning offsets in [2].


• The stopping time τ |Θ = 1 is accessible 2 , with a Markovian dependency on history, i.e.

unlike the stochastic deadline models in [10] and [15], the decision deadline in our model (i.e.


The first two terms in this sum are modular functions: Corollary 2 in Friedland and Gaubert [2013] states that the third term, m 3 , is a supermodular function.

the following function, consisting of the first few terms from the geometric series representation of f , is submodular: Monotonicity.


In this case, applying the GPI theorem to the newly-computed set of value functions will give rise to a policy that performs at least as well as a policy based on any subset of these, including the empty set.

this strategy satisfies our definition of successful transfer.


Results (displayed in Figure 3b) showed that participants again chose compositional extrapolations more frequently (68% vs. 32%, χ 2 = 172.8, p < 0.01), even if the ground truth happened to be generated by a spectral mixture kernel.

people seem to prefer compositional over non-compositional extrapolations in forced choice extrapolation tasks.


Its extreme points, however, are easy to find and characterize using the greedy algorithm [7].

we define a subclass of d with H f chosen so that it picks an extreme points of ∂ f (Y ), which we will call the permutation based lower bound submodular Bregman, henceforth referred to with d Σ f . The extreme points of ∂ f (Y ) can be obtained via a greedy algorithm ( [7,9]) as follows: Let σ be a permutation of V and define S i = {σ(1), σ(2), .


For large F the dynamics of @ corresponds to an Ornstein-Uhlenbeck process with a Gaussian stationary distribution of fixed unit variance.

if one chooses too large F initially the dynamics will become localised close to f 0 x (recall, f 0 @ !H A £ ).


There are also some "bridges" between different groups of data points.

this dataset is very challenging for any single clustering algorithm.


In our previous work we showed that concurrent actions formed an SMDP over primitive actions [5], which turns out to hold for all the termination schemes described above.

we can use SMDP Q-learning to compare concurrent policies over different termination schemes with the use of this method for purely sequential policy learning [7].


Note that the sum of each row in the underly score matrix needs to be zero for the sake of identifiability.

for each row of V , we subtract the sample mean of that row from each entry, that is, we compute Finally, we set Θ * = U V . Then, we generate n pairwise measurements following the mechanism specified in §2.


[7] typed their responses, this accounts for it taking longer for participants to type longer than shorter words.

according to the random walk models, the inter-item retrieval time (IRT) between animal k and k − 1 is where τ (k) is the first hitting time of animal X τ (k) and L(X) is the length of word X.


This will produce a graph G ∈ G s . In such a situation, the value of Ĝ is recovered by a general unipartite b-matching algorithm rather than a bipartite b-matching program.

the set of possible output solutions is strictly smaller (the bipartite formulation relaxes the symmetric one).


Suppose X S ⊆ X is a subset of X with size m, sampled uniformly at random without replacement.

let Ĉ = { Ŝ1 , • • • , Ŝ2 } be an (1 + )-approximation of optimal k-means subspace clustering computed on X S . If m = Ω( ) with γ < 1−802φ 2 800φ 2 − 2(1 + ), then we have: where applying Theorem 3.4 together with the sample-aggregate framework we obtain a weak polynomial-time ε-differentially private algorithm for agnostic k-means subspace clustering, with additional amount of per-coordinate Gaussian noise upper bounded by O( Our bound is comparable to the one obtained in [22] for private k-means clustering, except for the (ψ − η) term which characterizes the well-separatedness under the subspace clustering scenario.


If a signal resolution of 8b is pretended, then 0.5LSB = 2nA.

our current memory must be able to distinguish 2nA out of 46µA.


An important observation is that the sketch matrix M is enough to estimate the counts of any given item accurately but cannot identify the frequent items on its own.

without identifiability, we need another pass over every item, estimate its count, and then report top-K.


even if a priori the gradient has comparable algorithmic complexity as computing the original Wasserstein, it is reasonable to expect it to be more efficient in practice.


3, both k m (z, z ) = m(z) m(z ) and k δ (z, z ) = δ(z) δ(z ) are the inner product of two vectors and thus are positive definite kernels.

k h (P, Q) is a match kernel over sets (here the sets are image patches) as in [8,1,11,17,7].


However, does the sequence of delta distributions I=n converges to H ?

most people would answer yes based on the intuition that I=n 3 H and so does the sequence of corresponding delta distributions, even though the support of I=n never has any intersection with the support of H . convergence can be defined for distributions not only in a point-wise way, but in a way that takes consideration of the underlying structure of the sample space.


( 1), it is called an ε-coreset of D, and the approximate posterior has a marginal likelihood ẼN which approximates the true marginal likelihood E N , shown by Proposition 2.1.

from a Bayesian perspective, the ε-coreset is a useful notion of approximation.


However, all these methods need time series data where the temporal resolution of the measurements is high relative to the characteristic time scale of the feedback loops in order to rule out instantaneous cyclic relationships.

a significant practical drawback of this strategy is that obtaining time series data with sufficiently high temporal resolution is often costly-or even impossible-using current technology.


Since we need to make some assumptions about q s , the equality in (6) usually does not hold.

the tightest lower bound is an approximation of the intractable discriminant function Then, the margin is ∆F d (s; Θ) = θ ⊤ ∆f d (s) + max qs d L −θ (q sd ) − max qs L −θ (q s ), of which the linear term is the same as that in a linear SVM [7] and the difference between two variational bounds causes the topic discovery to bias the learning of the scene classification model, as we shall see.


Each term in the chain rule of probability can be conditioned on side-information y, decomposing any conditional density as p(x | y) = i p(x i | x 1:i−1 , y).

we can turn any unconditional autoregressive model into a conditional one by augmenting its set of input variables with y and only modelling the conditionals that correspond to x.


In a neural network, the primary requirement would be bidirectional connections between each feature (bigram) and each item in memory.

computational simplicity is gained by using a small feature set, at the cost of some approximation in the computation.


More interestingly, the two performance functions were similar, the brain slightly outperforming the external computation of XOR in this nonlinear task.

the brain can perform also nonlinear computations.


In other words, the variance of a correlation profile which has only high Fourier modes can still preserve its slowly varying components.

by projecting out the low Fourier modes of the spatial responses the spatial correlations are suppressed but the information in the response variance is retained.


An intuitive way of representing an observability model is through a directed graph over actions: an arc 1 from action i to action j implies that when playing action i we get information also about the loss of action j.

the expert setting is obtained by choosing a complete graph over actions (playing any action reveals all losses), and the bandit setting is obtained by choosing an empty edge set (playing an action only reveals the loss of that action).


We can now state our main result, which is a bound on the average loss R(f  The proof of this theorem is based on an algorithmic stability analysis.

we will consider two samples of size m, Z ∼ D m consisting of m i.


Note that, due to ( 5) and ( 6), it holds E(x (s−1) |M (s−1) ) = E(x (s) |M (s) ).

this initialization ensures that energy will continually decrease if the same is true for the optimization applied per scale.


Specifically the first four conditions in (9) identify an orthant in R n 2 where variables are allowed to move, while the last condition in (9) determines the variables to be held at zero.

the sets of free and active variables are defined as in (10).


Our theory allows for possible nonconvexity in both the loss function L n and the regularizer ρ λ .

due to this potential nonconvexity, our M -estimator also includes a side constraint g : R p → R + , which we require to be a convex function satisfying the lower bound g(β) ≥ β 1 , for all β ∈ R p . any feasible point for the optimization problem (1) satisfies the constraint β 1 ≤ R, and as long as the empirical loss and regularizer are continuous, the Weierstrass extreme value theorem guarantees that a global minimum β exists.


Our method successfully captures domain-specific properties of the target domain.

we are able to generate diverse images from a single input sample.


We have shown that, for finite-length LDPC sparse graphs, the accuracy of the marginal estimation with the method proposed significantly outperforms the BP estimate for the same graph.

the decoding error rates are clearly improved.


Preprocessing We found the original data to exhibit significant random relative motion between the two faces throughout the entire sequences which could not reasonably be modeled.

we transformed the data with an affine transform on all markers in each frame such that a select few nose and skull markers per frame (stationary facial locations) were approximately fixed relative to the first frame of the source sequences.


Recall that the Laplace exponent ψ(s) is nonnegative, nondecreasing and concave on (0, ∞).

if we let s = |b|, then ψ(|b|) defines a nonconvex penalty function of b on (−∞, ∞).


However, there are serious scaling issues for these models when the number of timesteps is large, or even unbounded, such as in online reinforcement learning (e.g., in the real world).

some decisions need to be made regarding the size of the past-embedding buffer that should be stored, whether it should be a rolling window, how computations should be cached and propagated across time, etc.


However, θ is presumed to be slow-mixing, making µ(θ) difficult to compute.

this divergence is only practical on low-treewidth "toy" graphs.


The Bregman Lagrangian assumes that the underlying manifold is a subset of R n , which is not true of many manifolds including diffeomorphisms 1 .

we use the mechanics formulation, which provides a formalism for general metrics though the Riemannian distance.


This separation of randomness allows gradients to be deterministically backpropagated through deterministic nodes rather than stochastic nodes.

the variance of gradient estimation is largely reduced.


Let Y = Diag([y i ] m i=1 ) be a diagonal matrix.

using the Cauchy-Binet identity we have when Y = I S , the (diagonal) indicator matrix for S, we obtain AY A > = A S A > S . Consequently, in the summation above only terms with T ✓ S survive, yielding We are now ready to sketch the proof of Theorem 6.


Not only is the scale of such problems huge, but they might also involve missing values, due to sensor malfunctions, occlusions or simple human errors.

modern time series applications present two challenges to practitioners: scalability to handle large n and T and the flexibility to handle missing values.


These models are necessarily complex: the brain contains multiple, interacting memory systems whose decay and interference characteristics depend on the specific content being stored and its relationship to other content.

these computational theories are fairly flexible and can provide reasonable post-hoc fits to spacing effect data, but we question their predictive value.


One might hope to minimize (5a) and (5b) directly, perhaps using sampled states and next states, but this is not a straightforward optimization problem to solve in general, because the search space for Φ is the space of functions and because Φ appears inconveniently on both sides of 5(b) making it difficult rearrange terms to solve for Φ as an optimization problem with a fixed target.

without additional assumptions about how the states are initially encoded and what space of features will be searched, it is challenging to apply Theorem 3 directly.


8 equal to zero we obtain where we have absorbed all terms independent of i into a constant.

we find the solution (10) in which the constants Zi follow from normalisation.


This shows that uLSIF is computationally more efficient than KLIEP.

proposed uLSIF is overall shown to work well in covariate shift adaptation with low computational cost.


Note that, our reordering algorithm is orthogonal to existing structural pruning algorithm.

we use one typical structural pruning algorithm, block sparsity, as a case study to show how our algorithm can improve its sparsity and granularity.


This factorization, of course, assigns mass to pairs which are incompatible, either because they do not generate the same terminal string or do not embody compatible bracketings.

the total mass assigned to valid structures will be less than one.


Although the spike generation mechanism operates in continuous time, the high temporal precision of the spikegeneration mechanism may be viewed as a manifestation of oversampling, which is needed for the digital encoding of the analog signal.

the extra order of magnitude in temporal precision available to the spike-generation mechanism relative to somatic current, Figure 1B, is necessary to faithfully encode the amplitude of the analog signal, thus potentially reconciling the firing rate and the spike timing points of view [13][14][15].


For the initialization, it is important for the computation to be efficient since we need to evaluate all possible detection windows in the image.

for simplicity and speed, we use a spanning tree T to approximate the full graph G, as shown in Figure 2(b).


Because ICET assigns costs globally, they will have similar costs as well.

iCET will not be able to recognize which attribute is relevant in what context.


In RC systems however, the input and recurrent weights are not trained and drawn from predefined statistical distributions.

by rescaling the features, the impact of the inputs on the activations of the reservoir neurons is changed as well, which makes it compulsory to employ an appropriate input scaling [8].


This implies that when given the loss vector (1) for all the steps till the end of this block, A never plays arm 2 in the block with probability more than 1/2.

if we make the switch to the loss vector (2) = (1/2, 0) at the beginning of the block, then A with probability more than 1/2 still never plays arm 2 and never notices the switch in this block.


It is important to note, however, that the classifiers were always able to identify the segment containing "once again," and these segments were identified before all other segments.

if necessary, we could have had our system cut the file in half after first labeling this segment.


One way to achieve this is to use the fact that y|X ∼ N (0, Σ).

gradient-based methods can be readily applied to maximize the marginal likelihood.


To establish the equality between ( 14) and ( 15), since ˜ embeds the submatrix kI, for any ⇤ 2 R h⇥ t + there must exist an ⌦ 2 R t⇥ t + satisfying ⇤ = ˜ ⌦/k.

furthermore, these matrices satisfy the result (8) holds for the first layer loss (4), using L1 defined in Lemma 3.


Inserting this into 19 we find after some algebra δC = g log 1 + e ag − 1 Because all c i < 0, the first term must be smaller (more negative) than the corresponding term in the linear case implying that δC < 0 (the second term is unchanged w.r.t the linear case).

if the entire spectrum is log-convex the gap will be located on the right, resulting in PCs.


First, if d is large, the parameters of the R j are decoupled and the RESCAL model is retrieved (up to a change of loss function).

second, our model is also related to classical tensor factorization model such as PARAFAC which approximate the tensor [R k (S i , O j )] i,j,k in the least-square sense by a low rank tensor H of the form The parameterization of all R j as linear combinations of d rank one matrices is in fact equivalent to constraining the tensor R = {R j } j∈ 1;nr to be the low rank tensor R = d r=1 α r ⊗ u r ⊗ v r . the tensor of all trigram terms 4 can be written also as d r=1 α r ⊗ β r ⊗ γ r with β r = S u r and γ r = O v r . This shows that our model is a particular form of tensor factorization which reduces to PARAFAC (up to a change of loss function) when p is sufficiently large.


The optimal value of z equals the slope at the current γ of log |Σ y |.

we have ) This formulation naturally admits the following optimization scheme: Step 1: Initialize each z i , e.g., z i = 1, ∀i.


This trade off is expected, because block composition affects only the stopping point of the model and not the model dynamics.

any speedup should yield a higher error rate, and vice versa.


However, by choosing an encoder distribution q(z|x) with sufficient capacity, we can assume that the non-negative KL-divergence D KL [q(z|x), p(z|x)] is small.

we know that the right-hand-side is a lower bound on the log-likelihood log p(x), which can be maximized w.r.t.


Hope comes from two directions: first, there are many cue orders besides the exact validity ordering that can yield good performance; and second, research in computer science has demonstrated the efficacy of a range of simple ordering rules for a closely related search problem.

we find that simple mechanisms at the cue-order-learning stage can enable simple mechanisms at the decision stage, such as lexicographic one-reason decision heuristics, to perform well.


The KL-divergence between those Gaussians scales with kA(S n ) E[A(S n )]k 2 .

more precisely: bounding kA(S n ) E[A(S n )]k will give (via the PAC-Bayes bound of Theorem 4 above) a corresponding bound on the binary divergence between the average empirical risk R(Q, P n ) and the average theoretical risk R(Q, P ) of the randomized classifier Q. Hypothesis stability (in the form of our Definition 1) implies a concentration inequality for kA(S n ) E[A(S n )]k. This is done in our Corollary 8 (see Section 4.3 below) and completes the circle of ideas to prove our main theorem.


Following symmetrisation, it is shown in [17,8] that the empirical HSIC can be written as a degree four V -statistic with core given by where we denote by S n the group of permutations over n elements.

we can directly apply the theory developed for higher-order V -statistics in Section 3.


Let Assumption 1 hold, and choose a sufficiently small η > 0 such that m : Then Breg-SVRG enjoys linear convergence in expectation: For example, we may set 45 , and Z = O(L 2 ).

between epochs, the gap (z s ) decays (in expectation) by a factor of 45 64 , and each epoch needs to conduct the proximal update (21) for m = Θ(L 2 ) number of times.


When dealing with high-order region graphs, one can choose a region graph, e.g., the Hasse diagram, that has significantly less edges than a region graph that connects variables i to cliques α.

when considering many high-order regions, the formulation in the above corollary is more efficient than the one in Corollary 2.


Note that even if the exogenous state process has finite memory, it can be reduced as an equivalent Markov process on a higher-dimensional space.

the Markov assumption is unrestrictive for most practical purposes.


However, the friction changes only at one point of the track.

restraining the search to smooth policies yield faster convergence.


We then have: Since w is very large, the AA T block is relatively very small, and so MM T is nearly rank-2kit has a 'heavy' strip of elements in its last k rows and columns.

computing a relative-error rank-2k approximation to MM T recovers all entries except those in the AA T block very accurately, and importantly, recovers the wAC block and so the product AC.


the correlation between X j and X j ′ is t 2 /(1 + t 2 ) for j ̸ = j ′ .


There is no need to synchronize child nodes.

all child nodes stay busy and consequently their efficiency gets maximized.


To sample from R t , we sample U, V from the boosted posterior (U, V) t and then sample from N (U > V, I).

r t ⇠ N (U t > V t , I) where (U, V) t ⇠ P t i ↵ i s i (U, V) and s i (U, V) is the t-th iterate returned by the LMO.


"Content-Based Image Suggestion" (CBIS) [4] motivates the modeling of user preferences with respect to visual information under the influence of the context.

cBIS aims at the suggestion of products whose relevance is inferred from the history of users in different contexts on images of the previously consumed products.


As to be confirmed by simulation, neuronal firings in the state of persistent activity are irregular and largely independent to each other.

we can assume that the responses of individual neurons are statistically equivalent in the state of persistent activity.


In fact, Kingman's paintbox theorem [12,13] implies that any exchangeable partition of N, such as a partition obtained using equation 1, is either equal to the trivial partition in which each part contains one element or satisfies lim inf N →∞ M N / N > 0 with positive probability.

by Kolmogorov's extension theorem, a sequence of random partitions (C N : N = 1, 2, . . .) corresponds to an exchangeable random partition of N whenever (a) each C N is finitely exchangeable (i.e., its probability is invariant under permutations of {1, . . . , N }) and (b) the sequence is projective (also known as consistent in distribution)-i.e., if N < N , the distribution over C N coincides with the marginal distribution over partitions of [N ] induced by the distribution over C N . to obtain a nontrivial model that exhibits the microclustering property, we must sacrifice either (a) or (b).


Then, for sufficiently large K (specifically K ≥ c 1 n 1/(2s+1) for some constant c 1 ), with probability at least 1 − 2 exp(−t 2 /2), we have where the constant c 1 depends on R and the distribution of ε i , and the constant C depends on R. The above theorem includes an approximation error term f 0 − R f 0 2 2 which depends on the type of interpolation matrix R. For example, for linear interpolation of a twice continuously differentiable function, the approximation error scales as O(K −2 ).

for a sufficiently large K (particularly K = n), the approximation error will disappear.


10 we find, Next we note that the projections of this equation on the space spanned by W and its orthogonal complement should hold independently.

multiplying equation 11 on the left by either P W or P ⊥ W , and multiplying it on the right by RΛ −1 , we obtain the following two equations, Inserting eqn.


For example, if Ω = R p and the density p(x) consists of k highly separated Gaussian clusters of roughly equal size (k clusters), then there are exactly k eigenvalues very close or equal to zero, and their corresponding eigenfunctions are approximately piecewise constant in each of these clusters.

in this setting spectral clustering with k eigenvectors works very well.


Hence, total computational complexity of DPDS is O(K max(t, α 2 t )) at each period t. Under the assumption of Lipschitz continuity, Theorem 1 shows that the value of DPDS converges to the value of the optimal policy under known model with a rate faster than or equal to log t/t if the DPDS algorithm parameter α t = t γ with γ ≥ 1/2.

the regret growth rate of DPDS is upper bounded by O( √ T log T ).


However, we note that softmax ascribes unexplained variance in choice behavior entirely to "noise", when subject may indeed employ a much more strategic policy whose learning and decision components are poorly captured by the model(s) under consideration.

smaller fitted b does not imply subjects are necessarily more noisy or care about rewards less; it may simply mean that the model is less good at capturing subjects' internal processes.


Note that in the non-Bayesian model, if there are no observations for some arm i, players will have zero knowledge about its mean reward and they cannot make decisions.

we assume without loss of generality that in the first N time slots of the game, with some constant compensation, the long-term controller can control the short-term players to choose all the arms once.


distinguishing thousands of identities at a time).

the two supervisory signals emphasize different aspects in feature learning and should be employed together.


In contrast, BOHAMIANN needed 1335.7 ± 236.5 seconds per BO iteration, which was about 80 times more expensive.

we ran ABLR for 200 BO iterations, and limited BOHAMIANN to only 50 iterations due the high computational cost (see Figure 2a).


A Gaussian process f (x) (GP) is an infinite collection of random variables, of which any finite subset is jointly Gaussian distributed.

a GP is completely specified by its mean function m(x) and covariance kernel function k θ (x, x ), where x, x ∈ X denote the possible inputs that index the GP and θ is a set of hyper-parameters parameterizing the kernel function.


For reaching J max = k, it is sufficient that J max increases at least once in each phase.

we have We then only need to investigate our assumption that in the run of 2eBnk 2 log 2k iterations, when implementing lines 8-15, one solution with the best f value in Q is always kept.


This is surprising: because of the way the uniform train/test splits were created, all test pixels are close to multiple train pixels.

we reach a first striking conclusion: learning a smooth function from (x, y) to one-hot pixel is difficult for convolutional networks, even when trained with supervision, and even when supervision is provided on all sides.


Cough, cold and fever could imply both flu or pneumonia, however, the absence of sputum and chills leads to the diagnosis of flu.

sputum and chills are pertinent negatives, which along with the pertinent positives are critical and in some sense sufficient for an accurate diagnosis.


Our matrix Bk then consists of a single Y i for each pair in B 2 .

observe that Bk has rank at most k and 1 For an online SVD calculator, see http://www.bluebit.gr/matrix-calculator/ if Bob succeeds in solving f on input B, then, Comparing ( 3) and ( 4), we arrive at, conditioned on E: where c 1 > 0 is a constant that can be made arbitrarily small by making c > 0 an arbitrarily small.


Essentially, the estimates of A and B is based on the factorization of D and are independent of Y.

one can apply a matrix factorization method to D, and relate this factorization to DAB to find AB.


Dropout can regularize DNNs by adding randomness, while TernGrad also introduces randomness.

dropping fewer neurons helps avoid over-randomness.


We emphasize the property that Algorithm 2 combines two strategies S 1 and S 2 , improving on the performance of S 1 using S 2 whenever possible, essentially without introducing any loss with respect to S 1 .

this amounts to lossless boosting of one strategy's performance using another.


While these objectives are convex and in some cases admit closed-form (i.e., non-iterative) solutions, they require inverting a matrix of size m × m. Thus in the case of very large data sets (e.g., like the one we consider in section 5), it might not be feasible to use this approach.

an iterative update is employed in practice.


Of course, the time to produce such a partition cannot dominate the time to run the algorithm itself.

we propose a simple fast node ordering procedure (Algorithm 1) that can be run once before the parallelization begins.


Lemma 3 Under Assumption A1, let m be the number of schedules up to time T , we can show: where M j is the number of steps in the jth episode.

lemma 4 Given Assumption A2 we can show: Combining the above results, we have This concludes the proof.


To do this, we relate the affine transformation of the latent processes for the nth neuron, h n (•), to the non-negative intensity function of a pointprocess, λ n (•), via a static non-linearity g : R → R + .

the spike times of neuron n on trial r, t t t (r) Where λ n (t) = g(h n (t)), T r is the duration of the rth trial, and Φ(n, r) is the total spike-count of neuron n on trial r. The expected log-likelihood term in ( 4) can be evaluated as The resulting expected log-likelihood in (7) still contains an integral over the expected rate function of the neuron, which cannot be computed analytically.


EG2 prefers attributes with high information gain (and low test cost).

when the concept to learn hides interdependency between attributes, the greedy measure may underestimate the usefulness of highly relevant attributes, resulting in more expensive trees.


The latter is preferable, because it induces a chain-structured dependency graph amongst the z and x variables.

the distribution q(x, z) has two important properties.


The key challenge of this problem is the large search space.

we proposed a generalization of Minoux's 'lazy greedy' on B&B tree to speed up classical greedy.


For example, in the Biological taxonomy of Figure 1, for the category Dalmatian, this path includes the nodes (superclasses) canine and animal.

for class c, the linearly combined kernel is given by where n ∼ c indexes the nodes that are ancestors of c, which is a leaf node (recall that the first C nodes in every taxonomy are reserved for leaf class nodes).


Further analysis of each of the single disks by our measure shows that each is a coherent cluster.

combination of our coherence measure with normalized cut not only clusters correctly, but also automatically finds the correct number of clusters, regardless of cluster scale.


Therefore, it is also proper for any affine combination of matrices in R inside P d . But span(R) ∩ P d = Q.


. . , m M/2,k ) uniformly without replacement from the integers 1 to M , and we included the linear constraints α m j,k ,k = 0.

we constrained each sub-network f k to be a weighted sum of no more than M/2 units from the final hidden layer.


3 Due to the limited availability of the codes during this work, we are not capable to access all the source codes.

for different methods, we utilize different strategies: For methods Exposure, FI, and DPED, we use the already-trained model to run the results; for CycleGAN, we retrain it with MIT-Adobe 5K training dataset; for Deep Guided Filter and Deep Photo Enhancer, we use demos from their own websites (http://wuhuikai.me/DeepGuidedFilterProject/ and http://www.cmlab.csie.ntu.edu.tw/ project/Deep-Photo-Enhancer/, respectively) to derive imagery results.


Therefore, after mapping we obtain ⌘ 0 i (x) ⌘ 0 j (x).

for independent labels, the pick-one-label heuristic is consistent for precision@k. The section above has revealed that HSM cannot be properly adapted to multi-label problems by the pick-one-label heuristic.


For example, in our toy world the entropy trajectories for the word orders SOV, OSV and OVS (two of which are pictured in Figure 1) are perfectly horizontal at various points (equivalently, some words carry zero information) because knowledge of the object in this world uniquely determines the verb (since foods are strictly eaten and drinks are strictly drunk).

any word order that places O before V renders the verb entirely uninformative, in significant conflict with the UID hypothesis.


Note that in the classical linear regression model where y i = w> x i + " i for i 2 [n], the maximum likelihood estimator ŵmle satisfies Ek ŵmle wk 2  C p d/n, where C > 0 is an absolute constant.

the SNR requirement to approximately recover w up to (say) Euclidean distancek wk 2 /24 is SNR 24 2 Cd/n.


Studies have shown that having devicelevel energy information can cause users to conserve significant amounts of energy, but current electricity meters only report whole-home data.

developing algorithmic methods for disaggregation presents a key technical challenge in the effort to maximize energy conservation.


Although this decomposition restricts the covariance matrix representation, it dramatically reduces the number of variational parameters in the covariance matrix from M 2 X M 2 H to M 2 X + M 2 H . Thanks to the above decomposition, the lower bound can be rearranged to speed up the computation, Similarly, the KL-divergence between q(U) and p(U) can also take advantage of the above decomposition: As shown in the above equations, the direct computation of Kronecker products is completely avoided.

the computational complexity of the lower bound is reduced to , which is comparable to the complexity of sparse GPs with independent observations O(N M max(D, M )).


The probability model can be written as where We defined s, r, and ψ as Then, the estimating function is given by where we used To calculate the conditional expectation of log T 1 , let us use Bayes's Theorem: By transforming random variables, (T 1 , T 2 , T 3 , ..., T m ), into (s, T 2 , T 3 , ..., T m ), we have where the beta function is defined as Similarly, we have where the digamma function is defined as Note that E[log(T 1 )|s] does not depend on the unknown function, k(ξ).

we have The form of u I can be understood as follows.