More formally, the view classifier G(E(x i t ); θg) → p t maps an encoding at timestep t to a probability distribution over possible views V . Learning with GRL is adversarial in that θ g is optimized to increase G's ability to discriminate encodings from different views, while GRL reverses the sign of the gradient that flows back to E, which results in the encoder parameters θ e learning representations that reduces the view classification accuracy.

essentially, we minimize the cross-entropy loss for the view classification task with respect to θ g , while maximize it with respect to θ e . we define the view classification loss as the sum of the cross-entropy loss for the entire sequence: where i is the ground-truth view of the input.


The currently fastest strongly polynomial combinatorial algorithm has a running time of O(n 5 T +n 6 ) [24] (where T is function evaluation time), far from practical.

the minimum-norm algorithm [7] is often the method of choice.


When using ENT, a small W N value may cause distortion to the feature set used by the LF-ASD.

the optimal W N was not selected in this range (< 40 samples).


( 1) is a tight equality at the optimum.

the maximum variance is achieved at the preset upper limit k of cardinality.


phase congruency measures [11,17] provide no information about blur.


By setting the other models in relation to the best joint model (cICA:= 100%), we are able to tell apart the relative contributions of bandpass filtering (HAD= 91%), particular filter shapes (SYM= 93%, ICA= 94%), contrast gain control (cHAD= 98.6%) as well as combined models (cSYM= 99%, cICA := 100%) to redundancy reduction (see Table 1).

orientation selectivity (ICA) contributes less to the overall redundancy reduction than any model with contrast gain control (cHAD, cSYM, cICA).


we compare the gradient descent based update, a closed-form solution for matrix inversion 2 and the proposed inner-free update using a pre-trained neural network.


is a monoid with identity element £ ; ¡ distributes over ; and ¢ is an annihilator for ¡ : for all X `Y T ¤ %X 0¡ ¢ 38 ¢ a¡ bX 58 ¢ .

a semiring is a ring that may lack negation.


As DL is becoming ever more prevalent, it is imperative to improve the robustness, especially in safety-critical applications.

various defenses have been proposed attempting to correctly classify adversarial examples [14,25,31,32,33,36].


In the first iteration fC(αq) equals f C(αq) .

since O q = SV bq and ν = 1 − α q , statements (2)-(3) follow directly from Theorem 1 1 .


In words, Proposition 1 implies that under the stated global-optimality preserving conditions, if any element of E converges to zero during optimization with an arbitrary descent algorithm, it will remain anchored at zero until the end.

if the algorithm prematurely errs in setting the wrong element to zero, meaning the wrong support pattern has been inferred at any time during an optimization trajectory, it is impossible to ever recover, a problem naturally side-stepped by MC where the support is effectively known.


During inference, for which ground-truth sentences are not available, words ahead can only be generated by conditioning on previously generated words through the representation vector.

decoding error compounds proportional to the length of the sequence.


Consider an agent situated in a partially observable domain: It executes an action; the action may change the state of the world; this change is reflected, in turn, by the agent's sensors; the action may have some associated cost, and the new state may have some associated reward or penalty.

the agent's interaction with this environment is characterized by a sequence of action-observation-reward steps, known as instances [7].


Regardless of the setting considered, it is not hard to construct a loss sequence such that obtaining sublinear dynamic regret is impossible (in general).

the problem of minimizing it is usually weakened in one of the two following forms: (1) restricting the benchmark; and (2) characterizing the niceness of the environment.


1)  Since the Lovász extension f is convex and the maximum of a convex function over a compact convex set is attained at a vertex of the convex set (Thm.

2), the maximum of f over R is attained at a vertex of R. we have From Eq.


object in a scene) to sensor input in the whisker system is much less direct.

a biophysically-realistic embodied model of the whisker array is a critical first component of any model of the vibrissal system.


Our results show that the average AUC is monotonically increasing as a function of the classification accuracy, but that the standard deviation for uneven distributions and higher error rates is noticeable.

algorithms designed to minimize the error rate may not lead to the best possible AUC values.


Intuitively, the joint distribution γ on the right-hand side of the above equation can be viewed as a transportation plan which transports probability mass from P to Q.

the Wasserstein metric between two distributions equals the cheapest cost (measured in some norm • − • ) of transporting probability mass from one distribution to the other.


We aim to learn a jigsaw such that, given an image set, pieces of the jigsaw image satisfy the following criteria: • each piece is similar in appearance and shape to several regions of the training images; • any of the training images can be approximately reconstructed using only pieces from the jigsaw (a piece may be used more than once in a single image); • pieces are as large as possible for a particular accuracy of reconstruction.

while allowing the jigsaw pieces to have arbitrary shape, we ensure that such pieces are shared across the entire image set, exhaustively explain the input image set, and are also large enough to be discriminative.


This represents a prohibitively large computational cost for many applications.

a substantial research effort has sought to develop efficient approximation methods that side-step these significant computational demands [1][2][3][4][5][6][7][8][9].


This approach permits a tradeoff between simple representations with little or no communication between agents and complex, computationally intensive representations with extensive coordination between agents.

we provide a principled means of using approximation to combat the exponential blowup in the joint action space of the participants.


So, E is approximately independent of G\E.

given v, r, and denoting by λ 1 = (a + (k − 1)b)/k and λ 2 = (a − b)/k the two eigvenvalues of P Q in the symmetric case, the expected number of intra-community neighbors at depth r from v is approximately 1 k (λ r 1 + (k − 1)λ r 2 ), whereas the expected number of extra-community neighbors at depth r from v is approximately 1 k (λ r 1 − λ r 2 ) for each of the other (k − 1) communities.


To get the new update equations for q(A k ) we examine L β (q) in terms of q(A 1:K ): This corresponds to the Bethe free energy of the factor graph described in (21), with as the CAP parameter [26; 12].

we can compute E[A k ] using loopy belief propagation.


Due to its limited-lifespan, after some periods many sensors will wear out, whereas some new sensors can be spread.

features corresponding to the old sensors vanish while features corresponding to the new sensors appear, and the learning algorithm needs to work well under such evolving environment.


ec where 0 is the order of the b-spline, it is the number of iterations, and ec is the number of computations required to update a particle using condensation.

in the case offourth order splines such as the ones we use, the increase in computation for a single particle is only four for a single iteration, and eight for two iterations.


In these tasks, perturbations are not applied to the system during the training phase.

the ability to generalize and resist perturbations come entirely out of the states visited by the agent during training.


The only difference is that in our case, there is a "1" in (17), whereas in the SVR case, we would have a "0".

we have to change the initialization of the optimization algorithm to ensure that we start with a feasible solution.


The lemma essentially states that for a numerically sparse vector, most of the 2 -mass of the vector is concentrated in its top few coordinates.

if a vector is numerically sparse then we can remove a few big coordinates from it and reduce its 2 norm considerably.


Due to current technical limit, however, establishing convergence properties of Async-MSGD for these highly complicated nonoconvex problems is generally infeasible.

we propose to analyze the algorithm through a simpler but nontrivial nonconvex problems -streaming PCA.


We denote by d : Y ×Y ×H → [0, 1] the distortion function, which measures the dissimilarity among two elements of the output space Y and one element of the latent space H. (Note that the distortion function is general in the sense that the latent element may not be used in some applications.)

the goal is to find a w ∈ W that minimizes the decoder distortion, that is: In the above equation, the angle brackets indicate that we are inserting a pair (ŷ, ĥ) = f w (x) into the distortion function.


Moreover, the model has a translational invariance in the skills and a scale invariance in β/σ 0 and τ /σ 0 .

we fixed µ 0 = 1200, σ 0 = 400 and computed the log-evidence L := P (y|β, τ ) for varying values of β and τ (see Figure 2 (right)).


Under covariate shift, standard learning methods such as maximum likelihood estimation are no longer consistent-weighted variants according to the ratio of test and training input densities are consistent.

accurately estimating the density ratio, called the importance, is one of the key issues in covariate shift adaptation.


In our model, semantic and temporal factors exert their influence via the context vector, while variation across subjects is expressed in the parameters drawn from the group prior.

our goal in inference is to compute the posterior distribution over the context trajectory and subject-specific parameters, given a sequence of studied and recalled words.


However, since different policy models give different generative models for sample sequences, we need to obtain new sample sequences to evaluate the measure each time the model is changed.

employing a strategy of model selection based directly on the expected reward would be hopelessly inefficient.


( 4) is as follows: a feature y is to be associated with a cluster c in proportion to a weighted, though flattened, geometric mean of the "W-projected centroids" p(y|c,w), priored by p(c).

2 This scheme overweighs y's that contribute to c evenly across W. clusters satisfying Eq.


We now study how pooling changes the requirements for efficient decoding by the target neuron.

mathematically speaking, pooling maps the vector r onto a scalar ρ equal to the sum of the individual activities: ρ = r k . the mutual information that can be extracted by any decoder that only pools it inputs is given by the following expression: where P (ρ|s) and P (ρ) are the the stimulus-conditional and stimulus-unconditional probability of observing a pooled population response ρ on a single trial.


Using soft assignment, we define the soft quantization of z as where we write the centers as a matrix , where e(z) is the center in C nearest to z.

we can now write: Now, instead of computing Ẑ via hard nearest neighbor assignments, we can approximate it with a smooth relaxation Z := C[φ(z (1) ), • • • , φ(z (m) )] by using the soft assignments instead of the hard assignments.


Note that the incremental gradient step (step 6 in Algorithm 4) only modifies the entries of C where X •k is nonzero.

we can parallelize the algorithm with respect to updating either the rows or the columns of C. We store X in large contiguous blocks of memory to encourage hardware prefetching.


Briefly, the infinite sequence of weights that defines the null distribution is identical to the sequence of normalized eigenvalues obtained in kernel PCA [26,27,7].

we show that the null distribution defined using finite sample estimates of these eigenvalues converges to the population distribution, using only convergence results on certain statistics of the eigenvalues.


Moreover, the average number of inner CG iterations per epoch in the iterative dogleg algorithm was quite small, 5.53 for (b) and 4.61 for (c).

the iterative dogleg worked nearly (b) nine times and (c) four times faster than the direct dogleg in terms of the average execution time per epoch.


Finally, the outputs of all the nodes that are not used by any other nodes are concatenated to form the final output of the cell.

for each of the B nodes we need to: 1) decide which two previous nodes are used as the inputs to its two branches; 2) decide the operation to apply to its two branches.


Therefore, in our scenario, the data center deploys exchangeable sensors with fixed quality reproductions, ŷ1 , • • • , ŷL .

lastly, the tth symbol of the estimate, T , is to be calculated by majority vote [3], overall performance of the system can be measured by the expected bit error frequency for decisions by the majority vote (3), In this paper, we consider two limit cases of decentralization levels; (1) The extreme situation of L → ∞, and (2) the case of L = R. The former case means that the data rate for an individual sensor information vanishes, while the latter case results in the transmission without coding techniques.


The top eigenvalues of the Gram matrix measure the variance captured by the leading dimensions of this embedding.

one can compare the eigenvalue spectra from this method and PCA to ascertain if the variance of the nonlinear embedding is concentrated in fewer dimensions.


Therefore, solving k tasks independently would require k • m ,δ samples in the worst case.

we are interested in algorithms that utilize samples from all players and solve all k tasks with sample complexity o k d ln 1 + ln 1 δ . Blum et al.


For adjacent channels responding to unresolved harmonics, although their responses may be quite different, they exhibit similar AM patterns and their response envelopes are highly correlated.

for T-F units labeled as target speech, segments are generated based on cross-channel envelope correlation in addition to temporal continuity.


Training all 2 N models separately can easily over-fit to the data, but weight sharing among the models can prevent over-fitting.

the standout network, much like a gating network, also produces a distributed representation to stochastically choses which expert to  turn on for a given input.


Let the feature correlation matrix be R = (R st ) ∈ p×p , clearly, R = R T , R st represents the correlations between features s and t, i.e., To let the selected features uncorrelated as much as possible, for any two features s, t, if their correlations R st > θ, then we put them in an exclusive group.


Since one major contribution of this paper is learning a classifier for structured objects that is defined over the complete part space P, we now examine the differences of semi-supervised and transductive learning in more detail.

the most common approach to realize the smoothness assumption is to construct a data dependent kernel k S derived from the graph Laplacian on a nearest neighbor graph on the labeled and unlabeled input patterns in the sample S. k S is not defined on observations that are out of the sample.


As in the previous example, it is easy to check that L t (h, g|Z t 1 1 ) = |c c ⇤ | 2 t 2 .

in this case, one can show that (7) reduces to 1 3 (T + 1)(2T + 1)|c c ⇤ | 2 and therefore, if Periodic signals.


The former will influence the choice of the default ν, the latter the form of constraints P. When the number of labeled and unlabeled examples is sufficiently large, it is reasonable to assume that the empirical distribution π(x) over all examples (labeled and unlabeled) is a faithful representation of π(x).

we consider defaults with ν(x) = π(x), shown to work well in species distribution modeling [13].


Since S n is bilinear in p and q, the plug-in estimator for S n is unbiased.

that is, our estimator for p, q H s is Here, we present our main theoretical results, bounding the bias, variance, and mean squared error of our estimator for finite n. By construction, our estimator satisfies via (6) and Cauchy-Schwarz, the bias of the estimator Ŝn satisfies p − p n H s is the error of approximating p by an order-Z n trigonometric polynomial, a classic problem in approximation theory, for which Theorem 2.2 of [15] shows: In combination with (8), this implies the following bound on the bias of our estimator: Theorem 1.


Unfortunately, some ideas in Bassily and Smith [4] such as Johnson-Lindenstrauss lemma do not scale to population sizes of millions of users.

in order to have a smooth trade-off between accuracy and communication cost (as well as the ability to protect privacy in repeated data collection, which will be introduced in Section 3) we introduce a modified version of Duchi et al.


We derive a structure learning algorithm such that a hierarchy of independencies in the input distribution is encoded in a deep generative graph, where lower-order independencies are encoded in deeper layers.

the number of layers is automatically determined, which is a desirable virtue in any architecture learning method.


It is not the case for unnatural supervised tasks, where the goal could be, for example, to determine whether the number of red pixels in an image is even or odd.

to be useful, an unsupervised learning algorithm must in effect correctly guess the likely set of downstream classification tasks without being directly exposed to them.


To derive the theoretical guarantee, we first re-formulate (4).

in this model, we can rewrite by assuming M = V T V , and using the fact that kM k ⇤ = min V :M =V T V kV k 2 F , the Factorization HOI (4) can be converted to the following nuclear norm regularization problem: where f w,M (e + , e ) := w T (e + e ) + (e + ) T M e + (e ) T M e . We then derive the guarantee for the following equivalent hard-constraint form: where ⌦ is the set of observed group comparisons (there can be repeated pairs in ⌦).


In this case, no off-chip communication occurs and Co immediately resets itself and its three rivals.

the location 0 circuit is identical to those of locations 1-3 except that the inverted spike is fed directly back to the reset transistor M20 instead of to a communication circuit.


From the final log-product term, it can be seen (from Schur's determinant identity [5] and the definition of σ t,b (x t,b )) that the product of the last B − 1 terms is exactly the B − 1 principal minor of K t,1 formed by the indices corresponding to S = {x t,b } B b=2 .

it is straightforward to see that the UCB-PE algorithm is really just (B − 1)-DPP maximization via the greedy algorithm.


On the other hand, we have shown that there exists an connection between the rank-differentiable property and the low-noise setting (see Figure 1).

one may get confused by the contradicting results and may wonder what will happen if a probability space satisfies both the rank-differentiable property and the low-noise setting.


For the subproblem that requires inverting a big matrix, a conventional gradient descent algorithm was used, leading to an alternating update, iterating between feed-forward propagation through a network and iterative gradient descent.

an inner loop for gradient descent is still necessary in this framework.


2(a) shows an example evolution of Q x4 for k = 1.

to further save computation time, we exploit two observations: First, if A is a subset of a connected component Q ∈ Q S , adding hyperedge S (A) to Q S will not change Q S . we do not test any hyper-edge A which is contained in a connected component.


3 Note that the GSNN can be derived from the CVAE by setting the recognition network and the prior network equal.

the learning tricks, such as reparameterization trick, of the CVAE can be used to train the GSNN.


As opposed to batch strategies that attempt to find the optimal solution conditional on all the data, the sequential strategy includes the data one at a time, so that the poste-rior exhibits a tempering effect as the amount of data increases.

the difficult global estimation problem is effectively decomposed into a series of easier estimation problems.


We choose g(z) as the point where this line intersects with the affine hyperplane 1 T ẑ = 1 containing the probability simplex.

g(z) is set equal to αz + (1 − α)q, where α = 1+Kq i zi+Kq (we denote it as α(z) as α is a function of z).


In this case we say that the loss of the policy is infinite.

there may be adversarially chosen sequences of graphs for which the regret of a decision-maker is not well-defined.


This assumption constrains the timevarying knowledge state: it can make at most one transition from 0 to 1 over the sequence of trials.

the {K t } can be replaced by a single latent variable, T , that denotes the trial following which a transition is made, leading to the BKT generative model: where λ L is the probability that a student has mastered the skill prior to performing the first exercise, λ M is the transition probability from the not-mastered to mastered state, λ G is the probability of correctly guessing the answer prior to skill mastery, and λ S is the probability of answering incorrectly due to a slip following skill mastery.


From another point of view, in the feature (kernel) space, we want to maximize the variance of instances in a positive bag along w, the normal vector of the separating hyperplane.

the principal component analysis (PCA) [15] is just the technique that we need.


These differences are highly significant as the total number of choices is 29600.

we conclude that subjects were indeed sensitive to the manipulation of outcome entropy, and that they did look ahead in a tree.


Learning techniques can be used to choose the similarity measure, which could be adaptive and learned from a large number of examples of classes not involved in the test.

given a large number of training images of a large number of objects labeled with their actual classes, and provided two pictures of unknown objects (objects which do not appear in the training pictures), we want to decide if these two objects are actually the same object.


The simplest special case is the univariate regression problem (K = 1), in which case the function ζ(β * ) outputs an s-dimensional sign vector with elements z * i = sign(β * i ), so that ψ(β * ) = z * T (Σ SS ) −1 z * = Θ(s).

the order parameter of block 1 / 2 -regression for univariate regresion is given by Θ(n/(2s log(p − s)), which matches the scaling established in previous work on the Lasso [11].


Motivated by the challenge of computing a Lyapunov function L * such that its induced set of policies contains π * , in this section, we approximate * with an auxiliary constraint cost , which is the largest auxiliary cost that satisfies the Lyapunov condition: L (x) ≥ T π B ,d [L ](x), ∀x ∈ X , and the safety condition L (x 0 ) ≤ d 0 .

the larger the , the larger the set of policies F L . by choosing the largest such auxiliary cost, we hope to have a better chance of including the optimal policy π * in the set of feasible policies.


And so there is always some Σ = Σ sufficiently large such that all hyperparameters converge to exactly zero.

we can be reasonable confident that the pruning mechanism of ARD is not merely an empirical phenomena.


On the other hand, when d TV (p, C) ", since p(S) 1 O("), p and q must still be far on the remaining support, and we can show that Z is still sufficiently large.

a small modification allows us to handle this case with the same sample complexity of O( p n/" 2 ).


Moreover, spikes are known to consume a significant fraction of the metabolic budget [30,32] placing a premium on their total number.

we can postulate that neuronal spike trains find a trade-off between the mean squared error in the decoded spike train relative to the input signal and the total number of spikes, as expressed by the following cost function over a time interval T: where x is the analog input signal, s is the binary spike sequence composed of zeros and ones, and is the linear filter.


For kernels that map into a space with dimension p > n, however, two problems arise: (i) the subspace spanned by the mapped samples varies with different sample sizes; (ii) not the whole feature space is accessible for vectors in the input space.

it is difficult to find a "proper" normalization of the Gaussian density in the induced feature space.


In structured prediction, the output space may be a set of sequences, images, graphs, parse trees, lists, or some other (typically discrete) objects admitting some possibly overlapping structure.

we assume that the output structure can be decomposed into l substructures.


This shows that limiting the neighborhood helps concentrate the correlation calculation on relevant features in the spatial vicinity and enhance correlation estimation.

it is necessary to choose a proper neighborhood size (rather than the whole image) for image restoration.


In this setting, we show a reduction to bandit linear optimization.

in fact, that algorithm does not need to see the weights of all edges at the end of the round, but only needs to know the loss it suffered.


Moreover, the only step that depends on the size of the state and action spaces is single-task MaxEnt.

for high-dimensional tasks (e.g., robotics tasks), replacing the base learner would allow our algorithm to scale gracefully.


( 19), and the diagonal elements of D are determined by κ.

substituting this into the definitions of μj , γi , ηj , ¯ i , we have that after preconditioning using JR if the conditions of Lemma 1 hold for X, β * , they will continue to hold after preconditioning using JR.


We eliminate A and consider , where g(X) := ψ 0 (X) + δ(X|≥ 0), and where each f t (X) for 1 ≤ t ≤ T is defined as where g t (a) := ψ t (a) + δ(a|≥ 0).

for simplicity, assume that (35) attains its unique 1 minimum, say a * , then f t (X) is differentiable and we have ∇ X f t (X) = (Xa * − y t )(a * ) T . we can instantiate (28), and all we need is a subroutine for solving (35).


Maximizing reward in such a sequence equals maximizing the reward rate, RR, per unit time [9].

the objective function for either task is given by where P C is the probability of performing a correct decision, and DT is the expected decision time.


The Part-Detector (Section 3.1) performance on our validation set predicts heat-maps that contain many false positives and poses that are anatomically incorrect; for instance when a peak for face detection is unusually far from a peak in the corresponding shoulder detection.

in spite of the improved Part-Detector context, the feed forward network still has difficulty learning an implicit model of the constraints of the body parts for the full range of body poses.


This is because in the proof of Theorem 1, the set D φ has the property that exactly one action V t ∈ D φ is awake in any round t, and t (V t ) = 1[y t = y t ].

if we consider a ranking where the actions in D φ are ranked at the top positions (in arbitrary order), the loss of this ranking exactly equals the number of errors made by the disjunction φ on the input sequence.


. . ) = b π for each label sequence l. The identity of the GGN and the Fisher information matrix [16] has been shown for the network using the softmax with cross-entropy loss [17,18].


In addition, the teacher may provide regression direction that is contradictory to the ground truth direction.

instead of using the teacher's regression output directly as a target, we exploit it as an upper bound for the student to achieve.


Metric-based methods for model selection are complementary to our approach in that they are desgined to prefer models/algorithms that behave similarly on the labeled and unlabeled data [Sch97, SS02, BC03], while disagreement is a measure of self-consistency on the same dataset (in this paper, unlabeled data only).

our method is also applicable to scenarios in which the test and training distributions are different.


This Markov chain has as limit operator H (0) aa , as the granularity of the job market increases along with the number of observations.

in trying to recover the geometry, distribution and vector field, we are actually interested in estimating the full advective effect of the diffusion process generated by H We obtain an embedding of the job market that describes the relative position of jobs, their distribution, and the natural time progression from each job.


The results (Figure 4) showed that SPEEDUP was consistently higher for composite probes.

the association between probe constituents was strengthened by training in each of the four conditions.


Finally, notice that, in view of the algorithm's definition, every b i corresponds to a rejected price.

by Proposition 3, there exist nodes n i (not necessarily distinct) such that p ni = b i and It is immediate that δ r n ≤ 1/2 and δ l n ≤ 1/2 for any node n, thus, we can write T. The last inequality holds since at most T prices are offered by our algorithm.


Let θ denote all the parameters of the policy, namely, the parameters of f R , f P , and π re , and note that ψ(s) is in fact a function of φ(s).

the policy can be written in the form π θ (a|φ(s)), similarly to the standard policy form (cf.


Defining the density-ratio over the two sets of samples, we can see that the density-ratio values for regular samples are close to one, while those for outliers tend to be significantly deviated from one.

density-ratio values could be used as an index of the degree of outlyingness [1,2].


By Theorem 5, the Bayesian risk is largely determined by ϕ −s(c,P ) .

g is a good approximation of f 2 B with reasonably large n. Formally, this is stated in the following theorem.


As all the solid edges have zero weight and zero length, it is not difficult to see that the total weight of the perfect matching is the same as the total weight of the resulting cycles.

the NWA detection problem is reduced into a problem of finding a perfect matching with negative total weight.


( The 0 − 1 loss is non-convex and difficult to handle; it has been shown that it is NP-hard to even approximately minimize the regularized risk with the 0 − 1 loss [2].

other loss functions can be viewed as convex proxies of the 0 − 1 loss.


To prove our theorem, we construct a joint distribution over pairs (A, P) such that when M is given too small a sample X, and A asks k − 1 statistical queries, then either M does not answer all the queries accurately or A outputs a k-th query q * such that q * (P) − q * (X) > ε.

m cannot be both accurate and satisfy post hoc generalization.


For our parametric model of interest, 2 C corresponds to the cause module, while a and 2 E correspond to the effect generation module.

pIC implies that 2 C changes independently from the pair (a, 2 E ).


[21] provide tighter analysis that yields convergence rates that scale cubically in the network size, but are independent of the network topology.

their analysis does not capture the intuition that distributed algorithms should converge faster on "well-connected" networks-expander graphs being a prime example-than on poorly connected networks (e.g., chains or cycles).


Then the computation of U i+1 is achieved by a multiplication of U i with an l × l matrix produced by the decomposition operation on D, which permits a natural parallelism.

we can further accelerate Algorithm 2 by using a parallel computing scheme.


for more then 25 place cells and more then two snapshots per location.

for a behaving organism exploring an unknown environment, this implies that a relatively sparse exploration strategy suffices to create a complete representation of the new environment.


More formally, assume that the data elements lie in an unknown ultrametric space pA,distq and so exhibit a natural hierarchical clustering defined by this ultrametric.

the input is a random graph generated as follows: an edge is added between nodes u,v P A with probability p " f pdistpu,vqq, where f is an (unknown) non-increasing function with range p0,1q.vertices that are very close in the ultrametric (and so very similar) have a higher probability to have an edge between them than vertices that are further apart.


The dissimilarity coefficient we employ was first introduced by Rao [23] and is defined for any non-negative symmetric loss function.

any such loss can be incorporated in our setting, allowing the user to tailor DISCO Nets to his or her needs.


Furthermore, it is easy to construct a value oracle for Φ t simply by using the value oracles for the f τ .

the optimization in step 3 is poly-time solvable given oracle access to Φ t . While the algorithm itself is a simple extension of Hannan's [5] follow-the-perturbed-leader algorithm, previous analysis (such as Kalai and Vempala [9]), which rely on linearity of the cost functions, cannot be made to work here.


Importantly, the hippocampus can also encode these conjunctions rapidly, and therefore it shares some of the same functionality as the phonological loop mechanism (i.e., rapidly encoding arbitrary conjunctions required for tasks).

it is likely that the hippocampus and the prefrontal-mediated working memory system (including the phonological loop) are partially redundant with each other, and work together in many tasks .


In this case, rare level sets of [19] correspond to distributions' tails and anomalies to extreme values.

we focus on the kind of anomalies that are characterized by their outburst of events or extreme values, as in the setting of [22] and [17].


On this large dataset with ≈ 60.000 recordings from ≈ 8.000 neurons we were still able to fit the model on a single GPU and perform at 90% FEV (data not shown).

we conclude that our model scales well to large-scale problems with thousands of nonlinear and diverse neurons.


Therefore, we minimize the following normalized cut objective value [10]: In (11), d(V c ) is the volume of the vertex set V c for balancing cluster sizes, and the numerator is the sum of off-diagonal entries, which corresponds to E 2 F . As shown in [10,3], minimizing the normalized cut is equivalent to finding cluster indicators x 1 , .

. . , x c to maximize where D is a diagonal matrix with a common way for getting cluster indicators is to compute the leading k eigenvectors of D −1/2 S λ D −1/2 and then conduct kmeans on these eigenvectors.


( 2), as given by The sufficient condition for the GGN approximation to be exact is that the network makes a perfect prediction for every given sample, that is, J L = 0, or [N ] i stays in the linear region for all i, that is, G L•N has less rank than KT and is positive semidefinite provided that H L is.

l is chosen to be a convex function so that H L is positive semidefinite.


While best performance is achieved when testing on the same instrument used for training, these models can also achieve reasonable performance when applied to new instruments.

we also compare to a discriminative baseline, a simplified reimplementation of a state-of-the-art supervised approach [7] which achieves slightly better performance than the original on this task.


In such a situation, it is not possible to tell if one test has greater asymptotic power than another.

it is standard to derive asymptotic power results under the extra condition that H 0 and H 1 become harder to distinguish as n grows.


Then, all round-two card sequences that shared a round-one bucket are partitioned into ten equally sized buckets based on the metric now applied at round two.

a partition of card sequences in round two is a pair of numbers: its bucket in the previous round and its bucket in the current round given its bucket in the previous round.


Due to specularities and intensity changes of structured light imaging, the sensor's output depth is often noisy.

refining the depth to generate a cleaner, more accurate depth image is an important task.


Individual nodes are allowed to instantiate connectivity patterns of multiple blocks.

the goal of the analysis with a Mixed Membership Blockmodel (MMB) is to identify (i) the mixed membership mapping of nodes, i.e., the units of analysis, to a fixed number of blocks, K, and (ii) the pairwise relations among the blocks.


For any 0 < < 1, Algorithm 1 will obtain a fractional x s. Proof.

see Appendix C. we have a approximate solution to the concave maximization problem and using this, in concert with Lemma 2, we arrive at the following which offers a guarantee of our proposed method.


This implies that for any θ and for all non-zero ϑ, If AdEFGM has minimal representation at θ * , then there must exist sufficiently small constants δ > 0 and β > 0 such that for any θ ∈ {∥θ − θ * ∥ ≤ δ}, ϑ ⊤ ∇ 2 L(θ; X n )ϑ ≥ β∥ϑ∥ 2 .

assumption 2 holds true when AdEFGM has minimal representation at θ * .


In the case of Problem (1.1), performing linear optimization over a trace-norm ball amounts to computing the top (left and right) singular vectors of a matrix, which can be done much faster than full SVD.

projection-free algorithms become attractive for convex minimization over trace-norm balls.


A general EM framework for continuous-time dynamic Bayesian networks, of which CT-HMM is a special case, was introduced in [6], but that work did not address the question of efficient learning.

there is a need for efficient CT-HMM learning methods that can scale to large state spaces (e.g.


Note that A's full eigenvalue decomposition can be written as A = V n ΣV ⊤ n , where Σ is diagonal.

it suffices to generate random orthogonal matrix V n and set with g i being standard normal samples, i.e., g i ∼ N (0, 1).


One thing that needs to consider is that due to the limitation in the image retouching area, it is hard to obtain the intermediate process and we only have the terminating results to obtain the rewards, also known as the sparse reward issue.

the AE function applies only in the final step.


However, the learning problem becomes convex once the latent variable h is fixed for positive examples.

we can train the LSVM by an iterative algorithm that alternates between inferring h on positive examples and optimizing the model parameter w. Dual form with fixed h on positive examples : Due to its nature of non-convexity, it is not straightforward to derive the dual form for the general LSVM.


Here, it should be noted that although axillary layers are fast compared to convolutions, they are not that efficient from an implementation viewpoint.

to make a fast network, we have to also consider these auxiliary layers.


As a direct consequence of the row normalization of W, each block of M has exactly one eigenvector composed of all ones, with eigenvalue O.

there is an m-dimensional eigenspace with eigenvalue 0, and there exist a basis of it, each vector of which has value 1 on a certain component, 0 otherwise.


Note that some of the trials required movement of the left hand while keeping the right hand steady and vise versa.

although we considered only movement periods of the trials, we had to predict both movement and non-movement for each hand.


Next, observe that J |S| is convex due to the convexity of the D KL on Bernoulli distributions and is minimized at x = |S|/|X| (KL Divergence of identical distributions is 0).

j |S| (x) monotonically decreases as x increases.


The random variables inherit the tree structure from the corresponding model segments.

we obtain a multiscale random field with random variables (RVs) the relational structure (RS) G α = (S α , E α ), and trees T 1 , .


That means g is a contraction mapping around the fixed point (0, 1) (the mapping is depicted in Figure 2).

(0, 1) is a stable fixed point of the mapping g. The norm of the Jacobian also determines the convergence rate as a consequence of the Banach fixed point theorem.


Rather than relying on learners to discover both sparse features and a visual hierarchy fully automatically, we use external "real-world" knowledge expressed in hierarchical structures to bias which sparsity patterns we want the learned discriminative feature representations to exhibit.

our end-goal is not any sparsity pattern returned by learners, but the patterns that are in concert with rich high-level semantics.


From the time change property [10] of point processes, if the sequences come from the point process λ(t) , then the integral Λ = tt+1 ti λ(s)ds between consecutive events should be exponential distribution with parameter 1.

the QQ plot of Λ against exponential distribution with rate 1 should fall approximately along a 45-degree reference line.


In our experiments, γ takes the concatenation of X t and H t−1 as the input and is fixed to be a one-hidden-layer convolutional neural network with 5 × 5 kernel size and 32 feature maps.

γ has only a small number of parameters and adds nearly no cost to the overall computation.


A more complex instance of this kind of approach is the Population of Linear Experts (POLE) model [7], in which hidden units each represent different linear functions, but the weights from input to hidden nodes indicate which linear function should be used to make predictions for particular input values.

the model can learn non-linear functions by identifying a series of local linear approximations, and can even model situations in which people seem to learn different functions in different parts of the input space.


Let v⇡ 2 R |S|⇥d be the matrix of gradients (with respect to the policy parameters ✓) of v ⇡ for each state s, and G 2 R |S|⇥d the matrix where each row corresponding to state s is the vector g(s).

then we obtain We can prove a similar result for the deterministic policy gradient objective, for a deterministic policy, ⇡ : S !


A useful property of CVaR, which we exploit in this paper, is its alternative dual representation [1]: where E ξ [Z] denotes the ξ-weighted expectation of Z, and the risk envelope U CVaR is given by U CVaR (α, P) = ξ : ξ(ω) ∈ 0, 1 α , ω∈Ω ξ(ω)P(ω)dω = 1 .

the CVaR of a random variable Z may be interpreted as the worst-case expectation of Z, under a perturbed distribution ξP.


While all previously mentioned approaches retain certain stability properties of the original representation with respect to common metrics in TDA (such as the Wasserstein or Bottleneck distances), they also share one common drawback: the mapping of topological signatures to a representation that is compatible with existing learning techniques is pre-defined.

it is fixed and therefore agnostic to any specific learning task.


In addition, the standard of error of E-C Bandit, hLinUCB, Logistic Bandit and PBMUCB's relative CTR performance among 100 trials are 0.032, 0.031, 0.030, 0.041, respectively.

the variance of our offline evaluation is small and the improvement from our solution to the baselines are statistically significant.


However, this method suffers from 1) greatly increased time complexity and computational burden, 2) inability to exhaust all possible dimensionalities and 3) lack of consensus between different functionality tests as their results can differ.

we need a universal criterion that can reflect the relationship between the dimensionality and quality of word embeddings in order to establish a dimensionality selection procedure for embedding methods.


. . , R |R| by noting that in moving between instances k and k + 1, only r k is added to the conditioning set.

by reassigning r k as the root for the BP instance associated with r k+1 (i.e., A ∪ R k as the conditioning set), only the second half of the message passing schedule (disseminating messages from the root to the leaves) is necessary.


The key difference is that in a general MDP, a state can be revisited several times even within the same episode, which is impossible in a tMDP.

our approach neglects correlations between values of future states.


For the case of on-policy sampling, the answer here is an affirmative one, as formalized in the following theorem.

[17], Lemma 6) Let w ⋆ D be the unique solution to where D is the stationary distribution of P . Then for on-policy sampling with linear function approximation, not only does TD converge to its fixed point, but we can also bound the error of its approximation relative to Π D V − V D , the lowest possible approximation error for the class of function approximators.


In the following part, we first prove that when m ∈ [1, t − 1], we have   According to the formulation of the diagonal matrix in Eq.

( 2) we have the assertion is satisfied.


Of course, the events keep happening despite their diminished capacity to reward because they are needed to achieve the more complex events.

the agent continues to turn the light on and off even after it has learned this skill because this is a step along the way toward turning on the music, as well as along the way toward turning on the monkey noise.


The input X t has a spatial size of W × H with C in channels, and ConvLSTM has a convolutional kernel size of K × K with C out channels.

the parameter size of ConvLSTM can be calculated as 3 : The parameter size of ConvLSTM is very large, partly due to the convolutional structures.


However, if the radius is too small, then the boundaries of the respective neighborhoods of the two chosen vertices would have a small intersection, so that estimating the similarities based on the small intersection of datapoints would result in high variance.

the choice of r is critical to the algorithm and analysis.


This also means that it may not be possible to design efficient online algorithm with sublinear regret without making any assumptions on either A or X because an efficient online algorithm with sublinear regret would imply an efficient algorithm for solving (1) in the offline case.

we focus on obtaining regret bounds for the dictionary update, assuming that the at each timestep the sparse codes given to the batch and online algorithms are "close".


Let {x(t)} be a set of measurements lying in the moving hyperplane Notice that the main difference between linear regression and hyperplane estimation is that in the latter case the parameter vector b(t) is constrained to lie in the unit sphere S D−1 .

instead of applying standard gradient descent as in (2), we must follow the negative gradient direction along the geodesic curve in S D−1 passing through b(t).


For instance, when π m is implemented according to x m starting from time τ m , the intensity x m+1 := f (x m , π m (x m )) at time τ m+1 depends on x m and the control π m (x m ), but is also random due to the stochasticity of the process during time [τ m , τ m+1 ).

the design of π needs to take future uncertainties into considerations.


While human-made controllers often exhibit a reasonable performance, they are not automatically optimized to reduce costs.

in many cases, it is possible to create a better controller, that would improve the performance of the system.


However, the drawback with this approach is that the identification of corresponding points with minimum distortion is also computationally expensive.

the aim of our research is to build an efficient image representation without using codewords or explicit correspondences between local features, while still achieving high classification accuracy.


Our results show that the upper bound of the generalization error depends on the sum of reciprocal of square of the margin over the labels.

we can answer the second question: the globally optimal CC exists only when the minimization of the upper bound is achieved over this CC.


2(a) and (b), each solid edge e can only be adjacent to a set of dashed edges, say {e 1 , e 2 , • • • , e K }, in a solid-dashed graph, and no two solid edges are adjacent to each other.

we can directly merge the solid-edge weight and length to its adjacent dashed edges by shares one vertex with e as in Fig.


Given that ψ is a constant function, we have the following, due to Lemma 3.3: = arg max This is the dual of the optimization problem with Lagrangian L, for which the costs c(s, a) serve as dual variables for equality constraints.


Over time, skills represented internally as options and their models also become available to the agent as action choices.

q B maps states s and actions a (both primitive and options) to the expected long-term utility of taking that action a in state s. Salient Events In our current implementation we assume that the agent has intrinsic or hardwired notions of interesting or "salient" events in its environment.


In our experiments, we observe that even with arguably the state-of-the-art implementation [Geng et   2017], PL without screening still has a significantly larger memory footprint compared with that of NW.

the experiments for PL without screening are not fully conducted in Figure 1b,1c, and 1d for networks with thousands of nodes.


That is, if a basis function is learned at one position and scale, then it is assumed to be repeated at all positions (spaced apart by two positions horizontally and vertically) and scales (in octave increments) .

the entire set of basis functions for tiling a large image may be learned by adapting only a handful of parametersi.e., the wavelet filters and the scaling function that is used to expand them across scale.


Similar to Lemma 1, for i > 1, q i = q i +τ •p i and p i = p i .


In this case we may assume that the non-linearity has adapted to give an approximately exponential distribution of the firing rate Y before w can change much.

from (6), ∆w can be seen as a weighted sum of the inputs u, with the activities Y acting as weights that follow an approximately exponential distribution.


considered as members of another class which represents all classes of D u , making the total number of classes for the purpose of this evaluation C + 1.

this measure can be highly sensitive to the dataset bias since the number of samples belonging to D u may be many times higher than those belonging to D c . One may argue that a simplified weighted accuracy might solve this issue but it still would not be able to provide a measure of the number of samples from D u that were classified as belonging to a known class c ∈ C. Algorithms are Incomparable: Since confidences across algorithms cannot be meaningfully related, comparing them based on their individual confidences becomes tricky.


For each object category, high confidence predictions (th == .1) were made in at least 50% of the tested scene pictures and the presence/absence of each object class was correctly predicted by the model on 95% of those images.

for those images, we do not need to use local image analysis to decide about the presence/absence of the object.


Sample {ξ n1 } n1 , {η n2 } n2 We propose new ξ n1 , η n2 values from the uniform prior distribution.

the acceptance probability is purely based on the likelihoods of the proposed and current configurations.


In the monochromatic case, the trees are identical and the traversal alternates between them.


After an initial series of trials with this set of targets, the targets were switched to be a green X and a red Y.

the task clearly requires binding of color and letter information, and updating of these bindings after the switch condition.


Training a neural network with the gradient descent algorithm gives rise to a discrete-time nonlinear dynamical system.

behaviors that are typically observed in these systems emerge during training, such as convergence to an orbit but not to a fixed point or dependence of convergence on the initialization.


For example, if the distributions of P and Q were perfectly correlated, meaning that knowledge of Q allowed perfect prediction of P , then the conditional distribution would be degenerate, having entropy zero.

the mutual information of P and Q would be H(P ), meaning that all of the entropy of P was eliminated through knowledge of Q.


the space is the unit hypercube and the mean-payoff function is locally Hölder with known exponent in the neighborhood of any maxima (which are in finite number) and bounded away from the maxima outside of these neighborhoods.

we get the desirable property that the rate of growth of the regret is independent of the dimensionality of the input space.


In many different large-scale optimization problems, applying complex optimization methods is not practical.

popular optimization methods for those problems are usually based on a stochastic estimation of the gradient.


The primary goal of this experiment is to evaluate the importance of modeling the dependencies between components.

following [3], we evaluate the predicted velocities of the balls.


Z, it is a convex function of M := ZZ T (since it is a maximum over linear functions of M ).

the intractability of (4) only lies in the combinatorial constraint M = ZZ T for some Z ∈ {0, 1} N ×K .


H and pre @ £ jj £ A a H. Therefore @ £ jjA !

h and @ £ jj £ A a H. If P OPT ; £ , then @ £ jjA a H. pre @ £ jjA a H for any pre P ¢ pre .


By contrast, the number of linearly independent output vectors of a nonlinear function can be much greater than the number of linearly independent input vectors.

if the other nonlinear functions are replaced with exponential functions, the logarithm of such functions can be nonlinear and the softmax bottleneck can be broken without additional parameters.


Hence, we expect localization on low order global eigenvectors, meaning that class separation will not be evident in the leading global eigenvector, but instead will be "buried" further down the spectrum.

this will illustrate how semi-supervised eigenvectors can represent relevant heterogeneities in a local subspace of low dimensionality.


In typical real-world data, even if the data lies in a high-dimensional space, there may be far fewer degrees of freedom.

our theoretical results suggest that when this is true, then our methods will enjoy far better convergence rates -even when the data overall has full intrinsic dimension due to factors such as noise.


This contradicts the assertion that the comparisons for these element pairs are all correct.

g(C, π) is a hitting set and the claim is established.


While content-provider fairness is essential, users are the driving force of the RS.

one may advocate for other mediators that perform better with respect to user utility, albeit not necessarily satisfying S. If S is discarded and a mediator satisfying Complete adopted, would this result in better user utility?


Notice that φ(rkev) (s) also contains components in φ(x (si) ) from eigenvectors beyond the M selected kernel eigenvoices for adaptation.

robust KEV adaptation may have the additional benefit of preserving the speaker-independent projections on the remaining less important but robust eigenvoices in the final speaker-adapted model.


In this paper, we propose a new importance estimation method that has a closed-form solution; the leave-one-out cross-validation score can also be computed analytically.

the proposed method is computationally very efficient and numerically stable.


When a ij = 1, no matter q jz is 1 or 0, we have Suppose the groups found by the base models have balanced size, i.e., n i=1 a ij = γ where γ is a constant for ∀j.

then the objective function can be approximated as: when the classification and clustering algorithms generate balanced groups, with the same set of constraints in Eq.


However, the automated transformation will reorganize it into CPS format, realizing delimited continuations.

the cumbersome example of Figure 2 can be simplified by using shift and reset in construction.


we have a contradiction, which gives x * ∈ ∂U.


This set is an important characteristic of a regularizer, since it contains all mappings which are not penalized.

the null space is the set of mappings which we are free to fit the data with -only deviations from the null space are penalized.


In several experiments it has been shown that a single spike is not sufficient to induce synaptic modification ( [10,8]).

we introduce a spike-induced postsynaptic activation factor z which enhances the contribution of a postsynaptic spike to the postsynaptic trace, e.g.


In addition, the method can easily cope with chains with arbitrary numbers of bonded cysteines.

it overcomes one of the major limitations of previous approaches restricting predictions to chains containing no more than 10 oxidized cysteines.


This is similar to how A* traces the shortest path back from !U ¥W H QX to ¨E GF IH QP RF using the search tree it has constructed.

we only need to determine the start   (1) A vertex is called locally consistent iff its g-value equals its rhs-value.


Hence, if d Q is close to 1/2, Var(W Q ) must be small as well.

any slack in the bound of Var(W Q ) has a multiplicative effect on each of the three proposed PAC-bounds of R(B Q ).


The leading contribution to µ is then given by µ(x; J) ∼ J • tanh(βx) as β goes to zero; The expression is valid in the asymptotic region L 1 for a fixed R. Then, the formula for the delta function yields [8]  where we have used ρ(x; x) = x − tanh(βx).

we have for given β 2 C. Inserting (12), ( 13) into (11), we get where we rewrite x = βx.


One can verify that the following equality holds: In view of that, it follows that ( 7) is equal to for any coordinate j 2 [1, N].

for any coordinate j 2 [1, N], by Hölder's inequality, we have ) 2 is the empirical variance of the j-th coordinate and where r = sup y (y) 2 is the empirical `1-norm radius of the data.


For example, for F (X) = log det X and A(θ) ∈ R d×d , the exact computation of ∇ θ F (A(θ)) can take as much as O(d 3 k), where k is the number of parameters in θ.

it is desirable to avoid computing, or at the very least reduce the number of times we compute, the gradient of F (A(θ)) exactly.


A motivating example of this problem scenario is in the medical setting where the time of human experts is precious, and the machines must aim to make the best use of the time.

having a data efficient predictive model that can generalize well with few labels is of critical importance in addition to having a good acquisition function.


Again by multiplying the state evolution equation by matrix C the resulting equations are where F := CB.

the loss can be generally expressed as The optimization would now be over the variables C, E, , D, F, where the optimization could additionally include regularizers on D and F to control overfitting.


Consider sets S and S 0 , and assume g(•), b(•), and p(•) are such that Since s P  t P for S ✓ T , we obtain f S (e) f T (e) 0.

it is also easy to see that we get f T (e) f S (e) f T (e) (1 c)f S (e) 0 for all S ✓ T and e 6 2 T . Finally, for poor elements,  We consider several benchmarks.


And in this setting, asynchronous variants of SGD remain indispensable [2,8,13,21,28,30].

a key question is how to extend the synchronous finite-sum VR algorithms to asynchronous parallel and distributed settings.


Let v k (s, a) be the number of times action a has been chosen in episode k when being in state s, and denote the total number of episodes by m. Then setting ∆ k := s,a v k (s, a)(ρ * − r(s, a)), with probability at least 1 − δ 12T 5/4 the regret of UCCRL after T steps is upper bounded by (cf.

section 4.1 of [11]), Next, we consider the regret incurred when the true MDP M is not contained in the set of plausible MDPs M k . fix a state-action pair (s, a), and recall that r(s, a) and pagg (•|s, a) are the estimates for rewards and transition probabilities calculated from all samples of state-action pairs contained in the same interval I(s).


For α = ∞, we keep drawing from the same flat outcome distribution.

α very simply determines the entropy of the actions' outcome distribution.


The algorithm tends to escape from the saddle point through reducing 2 p,⌘ (t), since this yields the largest eigengap, q p . When we have q = r and p = r + 1, the eigengap is minimal.

it is the worst situation for the algorithm to escape from a saddle point.


Maybe not so known is the fact that considering a simplified, "statistical average" of noise correlations may lead to dramatically different results in the estimation of certain quantities such as discriminability.

inferring a noise correlation structure must be done with as much care as possible in sticking to the available structure in the data.


They found anomalies based on the inconsistency of clustering results across multiple views.

they cannot find inconsistency within a cluster.


It is important here to remember why we care about single trial or small number of trial estimates, since we believe that in general the neural processing on repeated trials is not identical.

we expect this signal to be difficult to find with or without trial averaging.


Following the KM, we use a Gaussian random matrix M for the memory, and approximate samples of M using its mean R. We use subscripts t for M t , R t and U t to distinguish the memory or parameters after the online update at the t'th step when necessary.

we use a neural network encoder e(x) → z to deterministically map an external input x to embedding z.


In a typical setting, we are given as input a protein interaction data set, specifying a set of proteins P and a set of observed interacting pairs T.O.

we are also given a set of potentially relevant motifs, and the occurrences of these motifs in the different proteins in P. all the variables except for the O variables are hidden.


Notably, it was found that when a rodent was moving passively, i.e., a situation where no internal motor command is available, the head-direction encoded by ADN cells was still leading the actual position of the rodent's head by around ∼ 50ms, even larger than that in a free-moving condition [5].

extra anticipation strategies may exist in neural systems.


In other words, marginalizing over an observation and its associated marginal's parameter (which can be taken to be x N and θ N without loss of generality) still produces a similar isd posterior on the remaining observations X /N and parameters Θ /N .

we need: We then would recover the posterior formed using the formula in Equation 1 with only N − 1 observations and N − 1 models.


That is, we can reproduce S l just from the original random seed.

with a proper algorithm, S l will not incur extra model storage and transfer cost.


Next, we showed the mechanism that TAH qualitatively had the same effect as the covariance learning by analyzing the stability of the stored pattern sequence and the retrieval process by means of the statistical neurodynamics.

the variance of cross-talk noise didn't diverge, and this is another factor for the network learned by TAH to store and retrieve a pattern sequence.


For queries such as 'michael jackson' traditional ranking functions that value features such as high page rank will not work since documents new to the web will not have accrued sufficient inlinks.

a search engine's ranking function should not be fixed; different results should surface depending on the temporal context.


First, it is easy to see that when z = 0, these polynomials attain a value of unity (i.e., Q m (0) = 1).

secondly, it can be shown that for any s and z ∈ choosing the approximate solution as 7 d (m) s = −H(λ s ) −1 (I − Q m (H(λ s ))) g s guarantees an -close solution.


The larger the value of δχ Ci∪Cj is, the more likely the two data groups C i and C j are perceived to be one cluster.

δχ Ci∪Cj may be exploited to measure the structural affinity between two groups of samples from a whole set of samples.


a linear search can be implemented for the value of µ that leads to the tightest bound.


It should be noted that SURE does not require any prior knowledge on images.

potentially it can be applied to the measurement domain for different applications, such as medical imaging.


Note, however, that if we still use all the tensor entries and intensionally impose the Kronecker-product structure in the full covariance, our model is reduced to InfTucker.

from the modeling perspective, the proposed model is more general.


Given a bandwidth σ > 0, for a set Θ, one can smooth and normalize µ Dg as follows, where Z = Θ u∈Dg N(x; u, σI)dx, N is a Gaussian function and I is an identity matrix.

each PD can be regarded as a point in a probability simplex In case, one chooses Θ as an entire Euclidean space, each PD turns into a probability distribution as in [Anirudh et al., 2016, Adams et al., 2017.


Surrogate Upper Bound of Composite System: We can get better insight for the first two terms of the objective in (OPT) if we view z ∈ {0, 1} as a latent variable and consider the composite system Pr(y|x) = z Pr(z|x; g) Pr(y|x, f z ).

a standard application of Jensen's inequality reveals that, − log(Pr(y|x)) ≤ E q(z|x) (f z (x), y) + D KL (q(z|x) Pr(z|x; the conditionalentropy of the composite system is bounded by the expected value of our loss function (we overload notation and represent random-variables in lower-case format): This implies that the first two terms of our objective attempt to bound the loss of the composite system; the third term in the objective together with the constraint serve to enforce budget limits on the composite system.


We prove that the difference in the tightness of the bound on the likelihood of a document decreases as O(k − 1) + log m/m, where k is the number of topics in the model and m is the number of words in a document.

the advantage of CVB over VB is lost for long documents but increases with the number of topics.


For the purpose of the here presented classification system, the integration of visual information is set to take place over the complete image space.

the resulting estimate represents the global visual motion perceived.


It is then natural to approximate the distribution of n t=0 f (s t ) as a Normal distribution if n is sufficiently large. .

x s0,π is approximately normally distributed for each s 0 ∈ S if H is sufficiently large.


Let the estimate of the model be m. Let us assume that for every model in M the solution to (1) is unique.

we can define a mapping τ * : L × Z × T → T |Z| , where τ * (l, z, t) is the optimal recommended screening time at l, at time t following z.


To see the importance of this proxy, let us first describe a different approach to the problem at hand that in particular extends directly the approach of [8] who show that if an upper-bound on V T is known in advance, then the optimal regret is attainable.

one might guess that having an unbiased estimator for V T will eliminate the need in this prior knowledge.


We leave the the AP and time results in Appendix F. SEAL is flexible with what GNN or node embeddings to use.

we choose a recent architecture DGCNN [17] as the default GNN, and node2vec [20] as the default embeddings.


This value is too large for an exhaustive search.

we considered only possible combinations of up to 20 neurons, which required several weeks of computation on a 30-node cluster computer.


If we have more binarized weight bases/activations, the network has the capacity to approximate the full-precision one more precisely.

it can be deduced that when M or N is large enough, the network learns to tune itself so that the combination of M weight bases or N binarized activations can act like the full-precision one.


As we mentioned previously, the GDELT data set was extracted from news corpora.

patterns in the data primarily reflect patterns in media coverage of international affairs.


Superposition: Sample F ∼ DP(ν), and sample (c D , c F ) ∼ Dir(T qµ t−1 (Ω), ν(Ω)).

then set D t to be the union of (θ, c D π) for all (θ, π) ∈ D t and (θ, c F π) for all (θ, π) ∈ F . D t is a random convex combination of D t and F , where If the DDP is used as a prior over a mixture model, these three operations allow new mixture components to arise over time, and old mixture components to exhibit dynamics and perhaps disappear over time.


5 and 6 we know that the difference in diffusion value between the target location and the distractor location grows linearly in time.

the longer the process lasts, the better the approximation.


Then, it needs at most enP max expected iterations to increase J max .

after k • enP max expected iterations, J max must have reached k. By the procedure of POSS, we know that the solutions maintained in P must be incomparable.


x), ... , k (xm, x)) and aopt is the value of a that maximizes (3).

it suffices to compute aopt before any predictions are required.


This distribution is a mixture over all possible permutations that could have generated the original document v. Now, we can use the fact that sampling uniformly from V(v) can be done solely on the basis of the word counts of v, by randomly sampling words without replacement from those word counts.

we can train DocNADE on those generated word sequences, as if they were the original documents from which the word counts were extracted.


As we wish the foreground of mapped images to be in the target domain T and the background to remain in the source domain S, a good mapping should have a low KID value when computed using both the target and the source domains.

we report the mean KID value computed between generated samples using both source and target domains in Table 1.


. . , S k } have been fixed, then each pair (C i , T i ) can be chosen independently for different i.

the above maximization can be written as By applying Definition 2 to the inner maximization the claim follows.


Since K H φ j = λ j φ j , we have ⟨ϕ(f (x)), κ j ⟩ = λ j ⟨ϕ(x), κ j ⟩.

from the assumption, ⟨ϕ(f (x)), Ua j ⟩ = λ j ⟨ϕ(x), Ua j ⟩ .


We assume that w is piecewise Lipschitz, i.e., there exists a sequence of non-overlaping intervals , and a constant L > 0 such that, for any (x 1 , y 1 ) and (x 2 , y 2 ) For generality we assume w to be asymmetric i.e., w(u, v) = w(v, u), so that symmetric graphons can be considered as a special case.

a random graph G(n, w) generated by w is directed, i.e., G[i, j] = G[j, i].


Note that a larger value of results in a sparser solution with greater regularized expected loss and a smaller number of nonzeros.

for each subfigure of Figure 2, the order of the curves in the lower plot is reversed from that of the upper plot.


Unfortunately, this update is generally intractable for large networks, mainly because we need to store and update an exponential number of values for P (W|D n ).


The likelihood model is: , where π ∼ Dir(α).

(4) It is well-known that a Dirichlet sample can be generated by sampling from C independent Gammadistributed random variables with shape parameters α i and rate λ = 1; realizations of the class probabilities can be generated as follows: , where the noisy Dirichlet likelihood assumed for each observation translates to C independent Gamma likelihoods with shape parameters either α i = 1 + α , if y i = 1, or α i = α otherwise.


However, in practice we compute κ empirically by running several steps of the chain as discussed in Remark 2.7, making κ depend on x and T 0 .

we typically assume T 0 to be known in advance in some other way.


Finding optimal split functions at different levels of a decision tree according to some global objective, such as a regularized empirical risk, is NP-complete [11] due to the discrete and sequential nature of the decisions in a tree.

finding an efficient alternative to the greedy approach has remained a difficult objective despite many prior attempts.


The trick is that in learning, the weight can be applied to the gradient.

it is sufficient only to remember the gradient, or indeed the corresponding change in synaptic weights.


This can be achieved using Huffman Codes for example, and we refer the reader to Cover and Thomas ([5], Chapter 5) for more details.

given a distribution q, the firm pays L(q; i) = − log q(i) for each character i.


Since the drift is 0, the expected return of either stock in any trading period is 1; and since the returns in each period are independent, the expected final change in wealth, which is the product of the returns, is also 1.

in expectation, any CRP (indeed, any portfolio selection strategy) has overall return 1.


We show that for regimes of practical interest, randomly initialized neural networks well-approximate a rich family of hypotheses.

despite worst-case intractability of training neural networks, commonly used initialization procedures constitute a favorable starting point for training.


Note that in case of SMaLL, some features might be selected in more than one prototype.

to be fair to the other methods, we included the multiplicity while computing the total feature count.


As explained above, it is useful to retain spatial structure so that we can perform spatial pooling and convolutional feature extraction.

rather than applying our algorithm to the entire input, we apply the receptive field learning to 2-by-2 spatial regions within the 9-by-9-by-K 1 pooled representation.


The selectivity of this ORN is then defined by the orientation of the affinity vector i Κ r . A close look at the OB also shows that neighboring GLs respond to similar odors [10].

we model the ORN-GL projection with a Kohonen self-organizing map (SOM) [11].


uX a ¥ $# for any stationary distribution .

the flow is caused by the eigenvectors x ) with ) &% ¥ § , and hence we analyze the rate of decay of these eigenflows !


By adding a pairwise region in between the original high-order factor node and the variable nodes we are able to reduce computational complexity since the high-order factors are now singly connected.

we can guarantee that the complexity of the local message-passing steps run in each machine reduces from the state-space size of the largest factor to the size of the largest newly introduced region in each computer.


However, because this method requires to list all the sets in the SLT-region, the complexity of generating an LFP increases exponentially with the number of variables in the MRF.

although this method provides more flexibility to deal with any arbitrary structure, its computational cost does not scale with the size of MRFs in contrast with the structure specific SLT-bounds of section 4.2, that are assessed in constant time.


Notice that the left-hand side is the average cost occurred with initial state x(t) [5,4].


Define the matrix A such that A(:, I j ) = A j .

then since Φ i A T i = 0, the well-defined matrix A T is a basis of N (Φ), we can use A T to recover the global embedding T . A simpler alternative way is use a one-sided affine transformation, i.e., fix one of T i and affinely transform the other; the affine matrix is obtained by fixing one of T0i and transforming the other.


Note that given w (k) , b and a are independent.

this step can be partitioned into two parts.


Each value variable V ∈ V is associated to a bounded real-valued utility function u V over Ω pa V , which quantifies the (additive) contribution of the states of its parents to the overall utility.

the overall utility of a joint state x ∈ Ω C∪D is given by the sum of utility functions V ∈V u V (x ↓pa V ).


Moreover, every projection operator II : ~N -+ [<I>] reduces to the identity.

all the projection operators for the different algorithms are equivalent to the identity.


Given the linear Kalman filtering computation we have proposed as underlying tracking, equal magnitude noise in either will have the same effects.

to avoid the complexities inherent in allocating attention to space, we will consider memory limitations, but this resource limitation can be thought of as "attention gain" as well (though some of our work suggests that memory may be a more appropriate interpretation).


it is more realistic to study models which allow the parameter of interest to vary.


For simplicity, if we assume that every subset of N columns of Φ are linearly independent, the unique representation property (URP), then w ′ must necessarily have nonzero elements in locations that differ from w * .

any solution in the neighborhood of w * will satisfy w * 0 < w * + αw ′ 0 .


By Lemma 3.5 the starting point (x 0 , t 0 ) computed by GOODSTART and used by ISOTONICIPM is feasible and has symmetry sym(x 0 , D K ) ≥  Each call to HESSIANSOLVE fails with probability < n −3 .

by a union bound, the probability that some call to HESSIANSOLVE fails is upper bounded by O( √ m log(npw p max /δ))/n 3 = O(1/n).


An image in the source domain might have multiple representations in the target domain.

ambiguity in modeling of the conditional distribution arises, specially when the images in the source and target domains come from different modalities.


Notice, however, that these combinatorial structures are auxiliary, as required for proofs, and they do not appear explicitly in the algorithm descriptions.

they are much easier to implement than Blossom-V that maintains complex data structures, e.g., priority queues.


Many neural circuits are composed of numerous distinct cell types that perform different operations on their inputs, and send their outputs to distinct targets.

a key step in understanding neural systems is to reliably distinguish cell types.


In our experiments, this causes LB-2 to perform worse than LB-1 (for any δ) in terms of the state sequence reconstruction error, even while obtaining solutions outside the feasible region of LB-1.

we focus on LB-1 in our conceptual and qualitative comparisons as it is the strongest baseline available.


It also resolves ambiguity in object poses: e.g., cars share similar silhouettes when facing forward or backward.

our renderer can synthesize an object under different 3D poses.


Observe that R is non-empty, every set has a partition, and C is selected to be non-empty.

all three maximizations are over non-empty ranges and it remains to show that the product over i = 1, . . . , k is well defined.


Feature selection in the GLM neuron model can be done by the assumption of a Laplace prior over the linear weights, which naturally leads to sparse posterior solutions.

all weights are equally strongly pushed to zero.


We classified neurons as active if they spiked in the two second period over which we recorded.

we characterized PEP's pattern-recall performance as a function of the probability that the pattern in question's neurons are activated (Figure 6C).


We use the regret criterion to measure our performance against the best distribution of weights in hindsight, and wish to generate a series of weights {x t } T t=1 such that the regret is sublinear.

define the memoryless loss function ft (x) = f t (x, .


Nevertheless, we will see in the next section that we need to calculate the probability density function for likelihood-based estimation of differential entropy and mutual information.

we need an efficient way to calculate the probability density function that is tractable for many discrete variables.


Moreover, predefined communication architectures restrict communication and hence restrain potential cooperation among agents.

they cannot adapt to the change of scenarios.


[8]: where β takes the value that maximizes I * (β).

β is the value that satisfies ∂I * /∂β = 0.


If the neurons are uncorrelated, then the probability of two or more action potentials happening at the same time is zero.

at any time t k there is only one action potential from one neuron.


Since info-GAIL has to train a posterior distribution over the latent code to separate demonstrations, it requires more iterations for reaching all modes as well as prone to the mode collapsing problems.

we can conclude that the MCTEIL efficiently utilizes each mixture for wide-spread exploration.


The constructive proof of the Dirichlet processes [26] shows that a DP can be sampled by iteratively scaling an infinite sequence of Beta random variables.

posterior MCMC inference in a DPMM could, in theory, alternate between the following samplers where ∝ ∼ samples from a distribution proportional to the right side, x {k} denotes the (possibly empty) set of data labeled k, and f • (•) denotes a particular form of the probability density function of •.


As explained in 3.4, the hard-assigned matches correspond to the modes of the distributions of (7).

maximizing the score forces the distribution towards a Kronecker delta distribution, having the desirable effect of producing well-identified matches in positive image pairs.


The deep learning based image denoiser with the cost function of ( 14) can be implemented using a deep learning development framework, such as TensorFlow [34], by properly defining the cost function.

the gradient of ( 14) can be automatically calculated when the training is performed.


In practice, the notion of conditional independence is too strong.

as in previous work [21,8], we instead use conditional mutual information, denoted by I, to partition the set of variables.


In a practical optimization application, it serves the purpose of guiding the exploration of the domain, as locations x * with high prior value y 0 (x * ) are more likely to contain the maximizing argument.

given a set of data points D t , the prior ( 9) and the likelihood (8) lead to a posterior given by the particular choice of the likelihood function guarantees an analytically compact posterior expression.


The case of logistic regression is particularly interesting.

here, we can write the quadratic dropout penalty from (10) as just like additive noising, dropout generally gives an advantage to confident predictions and small .


A rigorous derivation of an objective function based on well-known estimation principles is very difficult, because the statistics involved are non-Gaussian, and the processes have difficult interdependencies.

instead of deriving an objective function from first principles, we derived an objective function heuristically, and verified through simulations that the objective function is capable of estimating the two-layer model.


Unfortunately, the UNION-SPLIT algorithm from the previous section requires an efficient algorithm for submodular function minimization, and no such algorithm exists in the constrained setting; submodular minimization is NP-hard even under simple cardinality constraints [19].

similarly, the BEST-B algorithm breaks down in the constrained setting; its guarantees carry over only if all the B i are within the constraint set C. for the constrained SH-min problem we instead propose a majorization-minimization algorithm.


The on-hull points tightly enclose the dataset but do not need to form a convex set.

nonconvex hulls can be more adaptive to capture practically complex data structures.


For our devices, these work out to effective modification thresholds of -2V and +2.3V.

we need waveforms where coincident activity causes a voltage rise above the positive threshold resp.


In this case, ∇ a r(a * ) = 0 and hence ∇ a r(A n ) = 0.

per the ROGD update rule, whether any agent updates or not does not matter: either the gradient is not received, in which case no gradient update happens; or a gradient is received, but at this Nash equilibrium, it is 0 and therefore nobody will want to make any update.


Although the function ( 4) is not complex-differentiable, if one identifies C n with R 2n and treats f (z) as a function in the real domain, f is still differentiable in the real sense.

we adopt the Wirtinger calculus [22], which can be thought of as a clean way of organizing the real partial derivatives [29,34].


Local reward independence, which was not present in the original problem, is ensured by using event dependencies to affect future rewards of the other agent.

local immediate rewards remain dependent only upon the actions of the individual agent, but the state in which that agent finds itself (and so the options available to its reward function) can depend upon events involving the other agent.


The unique path from x to the root contains a and the path from y to the root contains b, both paths contain c. Since weights decrease along paths, the assumption implies that a, b, c are pairwise distinct and c is an ancestor of a and b.

there must be a path from z via a to c and another path from z via b to c. Hence, T is not a tree, contradicting the assumption.


More in general, the analysis reported in the paper suggests that the performance of a upper confidence bound based algorithm such as UGapE is characterized by the same notion of complexity in both settings.

whenever the complexity is known, it is possible to exploit the theoretical analysis (bounds on the performance) to easily switch from one setting to the other.


However, a general parametrization of the transformation may violate the invertible requirement and result expensive or even infeasible calculation for the Jacobian and its determinant.

several carefully designed simple parametric forms of T have been proposed to compromise the invertible requirement and tractability of Jacobian [Rezende and Mohamed, 2015, Kingma et al., 2016, Tomczak and Welling, 2016, Dinh et al., 2016, at the expense of the flexibility of the corresponding variational distribution families.


This learning model is motivated by applications in which the cost of labeling data is high relative to that of collecting the unlabeled data itself.

the hope is that the active learner only needs to query the labels of a small number of the unlabeled data, and otherwise perform as well as a fully supervised learner.


This number has been attained after 53 cycles.

the execution time was approximately 50 times larger than for our procedure.


To construct a final feature representation for classification, the outputs of the first and second layers of trained features are average-pooled over quadrants as is done by [6].

our first layer of features result in 1600 × 4 = 6400 values in the final feature vector, and our second layer of features results in 3200 × 4 = 12800 values.


On the other hand, if the start timing is shifted from V G , arbitrary amount of fluctuation can be included in the bit comparison result.

we utilize quantum mechanical tunneling processes assisted by thermal noise in this structure, which is similar to a phenomenon known as stochastic resonance.


Note that the 'input' to the network now combines the actual input signal, x t , and the delayed recurrent input, r t−1 .

this extended input is neither white nor centered, and we will need to work with the generalized dynamics and generalized learning rules derived in the previous section.


3 we have F[q] ≤ log p(D), with equality holding if q(x) ≡ p(x|D).

maximization of the variational objective, Eq.


However, the performance of the learner is judged based on full ranked list and unrevealed, full relevance vector.

we have a PM game, where neither adversary generated relevance vector nor reward is revealed to learner.


In particular, sparse filtering resembles the 1 2 sparsity penalty that has been used in non-negative matrix factorization [35].

instead of the usual 1 penalty that is used in conjunction with sparse coding (i.e., s 1 ), one can instead use a normalized penalty (i.e., s 1 / s 2 ).


Moreover, the "outer" case introduces additional technical difficulties.

the proofs for Theorems 2.2 and 2.3, while still following the overall fundamental structure and ideas introduced by Nemirovskii, are significantly different in that context.


For example, when X is finite, then the denominator and enumerator of (4) are step functions, whence so is F (⌧ ).

gradient methods cannot be applied for finding ⌧ ⇤ .


The reason is that (14) allows the difference between r ij p j and r ji p i to get canceled first.

conceptually, (13) and ( 16) are more extreme -the former tends to underestimate the differences between p i 's, while the latter overestimate them.


When (K, ∆) is an infinite metric space, without additional constraints on the loss functions, the problem becomes ill-posed with a linear regret rate, even without movement costs.

one has to make additional assumptions on the loss functions in order to achieve sublinear regret.


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

thanks to Andrew Ledvina, David Blei, Yael Niv, Nathaniel Daw, and Francisco Pereira for useful comments.


However, existing subset selection methods ignore these relationships and treat items independent from each other.

there is a need for sequential subset selection methods that, instead of treating items independently, use the underlying dynamic models of data to select high-quality, diverse and compatible representatives.


When moving from a discrete to a continuous prediction space, it becomes very possible that the predictions, on a particular atom, of two randomly chosen structures will be close but not perfectly aligned.

instead of checking strict equality of these predictions, we need to modify our querying strategy to take into account the distance between them.


By Lemma 3.5, for every t ∈ [T ], we have .

then, for any measure-preserving bijection π : [0, 1] → [0, 1] and t ∈ [T ], we have we are left with the problem of sampling from P. Let S = {x 1 , .


Besides dopamine (DA), whose levels are known to be related to the TD reward prediction error [9], serotonin (5-HT), noradrenaline (NA), and acetylcholine (ACh) were discussed in relation to TDRL meta-parameters [10].

the knowledge of the characteristic meta-parameter dynamics can give an insight into the putative neuromodulatory activities in the brain.


Since M is assumed to be symmetric positive definite, it follows that {−1, 1} n ⊂ H(M).

the hinge loss L u,t := max(0, 1 − y t u, v it M ) of any classifier u ∈ {−1, 1} n with any example (v it , y t ) is either 0 or 2, since | u, v it M | = 1 by equation ( 3).


Finally, notice that using an indicator function over each s, a, s triple is equivalent to matching the (empirical) joint distribution p(S t , A t , S t+1 ) induced by the reference policy and the true dynamics.

even when we consider a different constraint for each triple, our solution implicitly couples the transition probabilities of different state-action pairs and differs from a rectangular formulation which focuses on matching the conditional distribution p(S t+1 |S t , A t ).


the subspace spanned by {w j } d j=1 if it deviates from the subspace, i.e., In the above definition, we assume that the basis w j and outliers o are both normalized (see Algorithm 1 step 1)-a) where all the samples are 2 -normalized).

we directly use inner product to define Γ o . Namely a sample is called outlier if it is distant from the underlying subspace of the signal.


In (episodic) partially observable multiagent environments, when the other players are fixed the environment becomes Markovian and computing a best response reduces to solving a form of MDP [30].

any reinforcement learning algorithm can be used.


Let s t be the state at time t and s c t be the first convention state the agents reach after time t. The transition probability is p ε h,h = P r{s c t = h |s t = h}, and it depends only on h, not t (for a fixed ε).


Our analysis is based on combinatorics (such as the complexity of sub-levelsets of GMMs) and probabilistic methods (non-uniform random sampling).

generalizations to other non-Euclidean distance functions, or error functions such as (non-squared) distances (mixture of Laplace distributions) is straightforward.


Unfortunately, some partial derivatives of ρ may diverge to infinity on the boundary of the simplex in the non-Lipschitz case, making the error hard to analyze.

to ensure that this error can be bounded, instead of b * , we will take a safe b ∈ B (far enough from the boundary) by using an intermediate point b in an inner simplex ∆ ε , where Thus, for a given b ∈ ∆ and ε ∈ (0, If we pick ε > δ B , then we are sure that b is not on the boundary of the simplex ∆, with a minimum distance from the boundary of η = ε − δ B .


As explained in [22], this was done using a special case of the widely applicable Expectation Propagation (EP) algorithm [19] -with an additional approximation that the fan-in of all neurons is large, so their inputs are approximately Gaussian.

given an error function, one can analytically obtain the Bayes estimate of the weights or the outputs, using the factorized approximation of the posterior.


Additionally, SIR only utilizes labeled (given response) data (predictors).

it is valuable to extend SIR to deal with transductive and semi-supervised learning problems by considering unlabelled samples.


The specific domain might care if each independent anomaly range is detected as a single unit or not.

we may also want to capture cardinality when measuring overlap.


Third, we show that one cannot use a DNF-based architecture with saturating dendrites.

we show that an important family of lnBFs implemented with a CNF-architecture can require an exponential number of saturating dendritic units, whereas the same family implemented with either a DNF-architecture or a CNF-architecture always require a linear number of spiking dendritic units.


One might raise the question as to whether the Laplacian regularizer helps.

we report the results for our formulation without Laplacian (ORTHONL) for comparison.


It is easy to verify that ∂C θ ∂u (v) = e −θu (e −θv − 1) e −θ − 1 + (e −θu − 1)(e −θv − 1) .

the quotient is constant and lim tցω C θ1,θ2,ω (t, v) = C θ1 (ω, v) − 0 + 0.


This is due to the fact that n IFO calls correspond to one PO call in PROXGD, while one IFO call in PROXSVRG corresponds to one PO call.

we do not gain any theoretical advantage by using constant minibatch PROXSVRG over PROXGD.  ))


In this framework, SVM and TSVM can be regarded as a learning machine without regularization and one with full regularization from the unlabeled data, respectively.

to supplement this framework of the regularization strength, it is necessary to introduce data-dependant partial regularization.


Direct democracy is just a representative democracy where each group has one voter.

a mathematical characterization of optimal representative democracies would also highlight the subcases where direct democracy is best.


The solutions of all the examined algorithms, TD(O), LSTD and the residual gradient algorithm, can be characterised as minimising different corresponding quadratic objective function.

each of the value functions, that one of the above algorithms converges to , can be interpreted as image of the optimal exact value function under a corresponding orthogonal projection.


However, EM produces poor results with topic models, which have many parameters and many local maxima.

recent work has focused on approximate inference algorithms [6,8].


)]0,8k2{1,2,...,m}} P T t=1 f t (x) be the best fixed decision in hindsight (knowing all loss functions f t (x) and the distribution of stochastic constraint functions g k (x; !)).

x ⇤ minimizes the T -round cumulative loss and satisfies all stochastic constraints in expectation, which also implies lim sup T !1 x ⇤ )  0 almost surely by the strong law of large numbers.


A reward of zero produced the best results in the reduced consistency case.


All valid fixing sequences for S yield the same CADMG G(V \ S, S), and if p(V) obeys the nested factorization for G, all valid fixing sequences for S yield the same kernel.

as a result, for any valid sequence σ for S, we will redefine the operator φ σ , for both graphs and kernels, to be φ S . In addition, it can be shown [16] that the above kernel set is characterized as: we can re-express the above nested factorization as stating that for any fixable set S, we have φ S (p(V); G) = D∈D(φ S (G)) φ V\D (p(V); G).


The absence of a linear correlation hints at factors extraneous to the user, i.e., the recommender.

we plot Rtrue (the deconvolved ratings) against the Robs , and search for characteristic signals that exemplify recommender effects (see Figure 2a and inset).


Then our assumption on L guarantees h 1 ≥ 0, and since we also have In addition, we obviously have E P (h 1 −E P h 1 ) 2 ≤ E P h 2 1 ≤ BE P h 1 .


Alternatively, the asymmetric score function S W (p i , p j ) in l W can be replaced with a symmetric score and used to derive an OASIS-like algorithm (which we name Dissim-Oasis).

the optimal update for this loss has a symmetric gradient if W 0 is initialized with a symmetric matrix (e.g., the identity) all W i are guaranteed to remain symmetric.


However, this has no influence as numerical reward values are arbitrary.

if learning follows this third factor differential Hebbian rule, weights will converge to the optimal estimated TD-values.


The expected number of repeat pulls to an arm before it expires is (1 − p)/p.

using Wald's equation [8], the expected number of different arms the algorithm must use for the repeat pulls is (t − τ (t)) • p/(1 − p).


However, an obstacle remains: spectral decomposition over different triplets {u, v, w} results in different permutations of the labels of the hidden variable H. To overcome this, note that any two triplets (u, v, w) and (u, v , w ) share the same set of eigenvectors in (5) when the "left" node u is the same.

if we consider a fixed node u * ∈ V as the "left" node and use a fixed matrix to diagonalize (5) for all triplets, we obtain a consistent ordering of the hidden labels over all triplet decompositions.


Equipped with an inner-product, we define the family of base kernels to be the possible outer-products K w = ww T between a vector w ∈ n and itself.

input: A distribution D t . Labelled and unlabelled sets:  Using this definition we get, the similarity between two instances x i and x j is high iff both x i and x j are similar (w.r.t the standard inner-product) to a third vector w. Analogously, if both x i and x j seem to be dissimilar to the vector w then they are similar to each other.


The need to cope with gradient sparsity becomes further pronounced in mixed-norm problems, as a single component of the gradient may correspond to an entire row of W . Updating the entire matrix because a few entries of g f t are non-zero is clearly undesirable.

we would like to extend our methods to cope efficiently with gradient sparsity.


The square-root transform is known to both stabilize the variance of Poisson counts and allow Poisson counts to be more closely modeled by a Gaussian distribution, especially at low Poisson means [16].

the two issues above can be largely resolved by applying PCA/PPCA to square-rooted spike counts, rather than raw spike counts.


It is possible that training robust neural networks on top of these features, or using such features for the middle layers of neural nets may give significantly more robust models.

our experiments should be considered mainly as a proof of concept, indicating that spectral features may be a useful tool in one's toolkit for adversarial robustness.