By the right invariance of D, different choices of π 0 lead to the same set of isomorphism classes Q M . Hence we may assume that π 0 is the identity.

by Theorem 6.3 in [20] the (M × M )matrix with entries P (i, j) is a Toeplitz matrix, i.e., P (i, j) = P (i + 1, j + 1) for all i, j ∈ [M − 1], with entries strictly increasing along rows, i.e., P (i, j) < P (i, j + 1) for 1 ≤ i < j < M . by Theorem 2, thresholding leads to M different partial orders.


The left two panels show the metaphase phenomenon clearly, where y 0 ≈ 0.

w * can not be well approximated by w(γ) if γ is not small enough.


In Tarone's terminology, S is said to be untestable.

defining the set of testable feature subsets at level δ as I T (δ) = {S| Ψ(S) ≤ δ}, Tarone's testability criterion obtains the corrected significance threshold as δ tar = max {δ : FWER tar (δ) ≤ α}, where FWER tar (δ) = δ|I T (δ)|.


Direct maximization of the marginal likelihood is impossible because the marginalization involves integration: log p(x) = log R p(x, z) d z.

training is usually conducted by maximizing the expected complete data log likelihood (ECLL) over an auxiliary distribution q: max When using the conditional q(z|x) we emphasize the use of a recognition network in amortized VI (Kingma and Welling, 2014;Rezende et al., 2014).


The idea of the surrogate function starts from the Taylor series expansion of The global high-order derivatives ∇ s L N ( θ) (s ≥ 2) are replaced by local high-order derivatives So the approximation error is We can omit the additive constant L 1 ( θ) − L N ( θ) + 〈∇L 1 ( θ) − ∇L N ( θ), θ〉.

thus the surrogate loss function L (θ) is defined as the surrogate minimizer θ = arg min Θ L approximates the empirical risk minimizer θ.


However, some other types of divergences, such as MMD and Stein Variational Gradient, do not suffer from this problem.

in this paper, for L 1 int , to maximize I(x, v x1 ; θ) we can replace the first term in (2) with Maximum-Mean Discrepancy (MMD) [35], which always prefers to maximize mutual information between x and v x1 .


Since a i ≥ 0 and i a i = 1 for each a ∈ A, every action whose positive components are in I t yields the highest possible expected reward of 1/|I t |.

binary search minimizes expected regret in period t for this problem.


It is conjectured that this radiation permeated through the universe unchanged since its formation 15 billion years ago.

the sizes and angular separations of these CMB fluctuations give an unique picture of the universe immediately after the Big Bang and have a large implication on our understanding of primordial cosmology.


Thus, we can upper bound where (10) We will bound ρ momentarily.

for now, we turn to bounding the first term of (9) using the Cauchy-Schwartz generalization as follows: ṽt Let Ãt = 1 3 I +b t ṽt ṽ t . Note that the inequality above shows that 3 Ãt A t . using Lemma 3, we get To bound the latter quantity note that | Ã0 | = |I| = 1, and that Lemma 7 (proved in the full version of the paper), we show that Q ≤ Q + R 2 .


We start by choosing a matrix regularization weight Λ such that Now consider the generalized inverse covariance problem (3) with this specified Λ.

by construction, the assumption in Theorem 2 holds for Λ, so we can decompose this problem into k sub-problems; for each cluster c ∈ {1, . . . , k}, the subproblem has the following form: where S (c) is the sample covariance matrix of cluster c. Θ is the optimal solution of problem (3) with Λ as the regularization parameter.


Especially in the maximization case, even a naive sampling algorithm for approximate inference is not scalable: n sampling rounds need to be carried out for each node to estimate the influence, which results in an overall O(n|V||E|) algorithm.

our goal is to design a scalable algorithm which can perform influence estimation and maximization in this regime of networks with millions of nodes.


when the label is queried, we produce the importance weighted labeled example (X, Y, 1/p).


Note that the hard balanced cut is similar to the normalized 1-cut but achieves smaller cut at the prize of a larger maximal component.

the example nicely shows how the different balance criterion influence the final partition.


Most of the computational approaches of Optimal Transport use the Kantorovich relaxation of the problem to learn a probabilistic coupling γ but do not address the problem of learning the underlying transport map T linked to the original Monge problem.

it lowers the potential usage of such methods in contexts where out-of-samples computations are mandatory.


We can reduce this to m + n if we store the gradients f i (x s ), but this is not practical in most applications.

sVRG requires many more gradient evaluations than classic SG iterations of memory-based methods like SAG.


Motivated by the advantages of the particle filtering approach, we are currently working on a real time implementation of the particle filter based tempo tracker for eventual automatic' accompaniment generation such as an adaptive drum machine.

the music is quantized such that it can be typeset in a notation program.


Further, we will restrict all the non-zero edge weights ( θi,j ) in the graph classes to be the same, set equal to λ > 0.

for a given G(V, E), we have θ = λ1 E for some λ > 0.


Theorem 3.2 Scoring rule (z, η) is (strictly) proper to estimate η from z if and only if the equivalent loss where l(η) is a vector with components i (η) = (b i , η) and b i is the i-th element in Z (according to some arbitrary ordering), is (strictly) proper.

proof The proof is straightforward by noting that the expected loss can be expressed as arg min η E z { (z, η)} = arg min η E y { ˜ (y, η)} and, thus, is (strictly) proper with respect to y iff ˜ is (strictly) proper.


This will allow us to see how each process contributes to the accuracy of the estimator.

in this model, (The power spectrum is then shaped as exp(−f 2 l 2 i /2).)


In their method, a group of images is fed as an input to the network, which merges their inner representation in a deeper level of the network, along with an attention model added to each image, and producing a single prediction.

noisy labels may appear in groups with correctly labeled examples, thus diminishing their impact.


Although the curve feature is modified due to the existence of the impulse inputs, we observe that its peak is located at a similar value of ∆τ .


If this posterior probability is low, then the resampling of particles performed after processing each word provides another point at which particles representing the disfavored interpretation could be deleted.

total parse failure at the disambiguator will become more likely the greater the length of the preceding ambiguous region.


It is not straightforward to encode the calibration constraint if the probabilities are assumed to be continuous, and post-processing calibration methods [31,35] would break equal-cost constraints by modifying classifier scores.

we look to achieve the calibrated Equalized Odds relaxation by post-processing existing calibrated classifiers.


For any £ and any 4 g ¤ ¦u ¦u u ¦¤ t4 Y e m , define the matrix by: .

the eigenvalues of are all non-negative, which implies that ¢ is positive definite [1].


Since it is a capacity, it must be measured behaviourally, using tasks that we know to demand reasoning about the ways in which entities are related.

since we cannot formally prove that some series of computations will necessarily result in improved relational reasoning, we must rely on empirical verification that some computations may or may not be correlated with improved relational reasoning.


It is probable that such an ordering is close to optimal, but such orderings may not always be obvious, and the effects of a poor ordering on the resources required for policy generation can be extreme.

to characterize the reordering methods discussed in Section 3.3, we start with initially randomly shuffled orders and compare the sizes of the final value diagrams with those found using the intuitive order.


We prove non-ergodic convergence of the expected objective value sequence under diminishing non-summable stepsizes, where the stepsizes are required to be "almost" square summable.

the convergence requirements are almost equal to SGD.


The computation time of the algorithm is upper bounded by parameters of the algorithm, and constant with respect to the size of the state space.

this algorithm is suitable for MCs with large state spaces.


H is adequate for G if ∃M ∈ H such that M is adequate for G. An adequate model may be partial or incomplete in that it may not include every possible transition in the transition function T . However, the model is sufficient to produce a sound plan with respect to (T, g) for every goal g in the desired language.

the more limited the goal language, the more incomplete the adequate model can be.


A key to computation speedup is that the covariance matrices Σ k | 3 k=1 should be low-rank, since the features and tasks are enforced to be correlated for multi-task learning.

the inverses of Σ k | 3 k=1 do not exist in general and we have to compute the generalized inverses using eigendecomposition.


The BA algorithm solves a convex optimization problem that is based on Corollary 3 for loss L SB .

since the last three terms of the right-hand side of the bound of the corollary do not depend on ↵, this suggests to select ↵ as the solution of min ↵2  the following is the objective function of our optimization problem: The problem min ↵ 0 F (↵) is a convex optimization problem, which we solve via projected coordinate descent.


In this context, the bootstrap procedure obviously gives rise to a (considerable) bias towards too complex models.

too many edges are present in the learned graph structure, and the confidence in the presence of edges is overestimated.


Our algorithms work with large databases in supervised and semi-supervised settings and scale well in the alphabet size and the number of allowed mismatches.

the proposed algorithms can be readily applied to other challenging problems in sequence analysis and mining.


(Note: All convolutions in BDC-LSTM use the same kernel size as indicated in the layers.)

to predict the probability map of a 100×100 region, we need the 126×126 region centered at the same position as the input.


In the RBM between layers − 1 and , P (g ) is defined in terms on the parameters of that RBM, whereas in the DBN P (g ) is defined in terms of the parameters of the upper layers.

q(g |g −1 ) of the RBM does not correspond to P (g |g −1 ) in the DBN, except when that RBM is the top layer of the DBN.


Phase 2 // We know that there exists r i such that ∀x ∈ S i , x ∈ C i ⇔ d(x, µ i ) < r i .

// r i can be found by simple binary search S i = Sorted({S i }) // Sorts elements of {x : x ∈ S i } in increasing order of d(x, µ p ).


In all the results we report on, we use a one-back response history, i.e., m = 2.

the SIMPLE models are REG(2, 0), LUT(2, 0), and REG⊕LUT(2, 0), the ORACLE and CRF models are REG(2, 1), LUT(2, 1), and REG⊕LUT(2, 1).


In each step of the AdaBoost, a weak learner G s (•) is obtained on the weighted examples and a weight α s is assigned to it.

adaBoost iteratively builds S s=1 α s G s (•).


In the spirit of full generality, we allow H α to be any probability distribution with support greater than a set value α; particularly, it could be nonzero at arbitrarily large values.

could be interpreted as the gross contamination level.


If ¥ , then the benchmark is recovered, while ¥ ¢ allows deviations from the benchmark within the bounds 2 ¦¡ .

the active risk parameter analysis the risk sensitivity of the portfolios with respect to the quality of the forecast models.


From some perspectives, this is a little counterintuitive: not only must the machine gradually place more predictive weight on whichever principal's prior is a better predictor, but it must reward that principal by attending more to their utility function as well.

a machine implementing a Pareto optimal policy can be viewed as a kind of bet-settling device.


We thus use Mv to denote the local game matrix for the player identified with vertex V. Note that our definitions are entirely representational, and alter nothing about the underlying game theory.

every graphical game has a Nash equilibrium.


Denote (and indeed of the order of 0(1/ v'ii) even for low-dimensional problems).

as in the previous section, we estimate the coordinate j that maximizes the inner product I (x;, r(w)) I, which in tum can be approximated using approximate nearest neighbor search.


( 17), sigsoftmax has the following relation: where S is a hyperplane of S with k 15) and ( 18), we can see that the range of sigsoftmax includes the range of softmax.

we have {f s (z)|z ∈ S} ⊆ {f (z)|z ∈ S}.


Therefore, it is not necessary for a worker to send g k (as well as u k ).

we use an old value ĝk to approximate g k to further avoid computing g k . coordinate k will be skipped in the KKT filter if |ĝ k | ≤ λ − δ, where δ ∈ [0, λ] controls how aggressive the filtering is.


Consider that a data center is interested in the data sequence, {X(t)} ∞ t=1 , which cannot be observed directly.

the data center deploys a bunch of L sensors which each independently encodes its noisy observation of the sequence, {Y i (t)} ∞ t=1 , without sharing any information, i.e., the sensors are not permitted to communicate and decide what to send to the data center beforehand.


Instead, the loss can be minimized by scaling the norm of any linear predictor that separates the data to infinity.

any sequence of predictors β (t) (say, from an optimization algorithm) that asymptotically minimizes the loss in eq.


In this paper, we focus on recommending likable items.

instead of maximizing the expected reward r (T ) , we aim to maximize the expected number of likable items recommended up to time T : where X ut is the indicator random variable for whether the item recommended to user u at time t is likable, i.e., X ut = +1 if p uπut > 1/2 and X ut = 0 otherwise.


Besides using a somewhat different hide-andseek construction as mentioned earlier, it also relies on the simple but powerful observation that any b-memory online protocol is also a (b, κ, m/κ ) protocol for arbitrary κ.

we only need to prove the theorem for (b, κ, m/κ ) for some κ (chosen to equal d(d − 1) in our case) to automatically get the same result for b-memory protocols.


For an optimal policy and an unbiased estimator, this parameter gradient must, by definition, map to the zero policy gradient, i.e., to ∇ π J(θ k ) = 0.

an optimal policy at a fixed point is defined solely by the zero policy gradient, making the policy equal to π(0), which is the uniformly stochastic policy.


This property allows us to solve for the value function exactly using a single pass dynamic program.

our first contribution is a complete solution to the optimal sequential market making problem within a Gaussian update framework.


a' ¢ HC by where 7 ' ( TS denotes the inward branching factor of point ' ( TS .

a simple induction proof can be used to show that © 6 G ©' ¢ DC e . the new G -weighting scheme provides an efficient unbiased method for implementing GIS that does not use any free parameters.


In information theory, the Kraft-McMillan inequality states that, ignoring rounding issues, for every uniquely decodable code with a code length function λ, there is a probability distribution ω such that λ i = − log ω i for all symbols i = 1, . . . , n, and vice versa.

the log loss can be interpreted as the code length assigned to the observed outcome.


Rejection sampling might look tempting, but unfortunately does not work well in high dimensions.

we will instead opt for a Markov Chain Monte Carlo approach.


We have where the Stein matching set F * is the linear span of {P x k(x, Equivalently, f (x) = P x φ(x) + b and φ is in the linear span of {k(x, Extending Lemma 3.1, one can readily see that the SVGD fixed points can approximate the expectation of functions that are close to F * .

specifically, let F * be the neighborhood of F * , that is, the SVGD approximation can be viewed as prioritizing the functions within, or close to, F * .


Difficult to Generalize Proof of Lemma 3.2.

we compute the gradient and Hessian of g(x), rg(x) = Mx kxk 2 x, r 2 g(x) = 2xx > M + kxk 2 • I .a critical point x satisfies rg(x) = Mx kxk 2 x = 0, and thus it must be an eigenvector of M and kxk 2 is the corresponding eigenvalue.


Finally, we replace the ConvBlocks with the residual blocks [10], which we discussed in Section 4.

the best CB513 accuracy achieved by the ResNet-like models is 71.0% ('ResNet' in Table 2), and is worse than the models in Fig.


In our FaceSync application, we have more dimensions (pixels in the image) than examples (video frames) .

our covariance matrices are singular, making their inversionwhich we do as part of canonical correlation -problematic.


p∈P d(p, C) and p∈P d 2 (p, C)), where d(p, C) = min c∈C d(p, c) is the minimum distance from p to a center in C. For all the 3 objectives, given a set C ⊆ P of k centers, the best set P can be derived from P by removing the z points p ∈ P with the largest d(p, C).

we shall only use a set C of k centers to denote a solution to a (k, z)-center/median/means instance.


Initialization: From the model design, we can expect that instances in the same region have the same label.

it is reasonable to initialize α to have each region prefer only one label, that is, each αk has one element with large value and all others with small values.


( 8) maintains the stability of the fixed point h * = 0.

our approach prevents the gradient explosion caused by the bifurcation of the fixed point h * .


However, in such a 'VQ-PQ' style approach, the norms of the residuals exhibit significant variance.

the proposed multiscale approach provides significant gains for massive search even when the original data is fully normalized.


A formal way of enforcing robustness is to require that the distribution of f (X; θ f ) conditional on Z (and possibly Y ) be invariant with the nuisance parameter Z.

we wish to find a function f such that for all z, z ∈ Z and all values s ∈ S of f (X; θ f ).


In addition, our probabilistic model allows dealing with missing data and, in the supervised dimensionality reduction case, can incorporate additional unlabeled data one may have access to, making our CCA algorithm work in a semi-supervised setting.

apart from being a general, nonparametric, fully Bayesian solution to the CCA problem, our framework can be readily applied for learning useful predictive features from labeled (or partially labeled) data in the context of learning a set of related tasks.


The necessary condition (9) shows a dependence on the minimum-to-average ratio MAR instead of just the average power through SNR.

the bound shows the negative effects of relatively small components.


In this subsection we first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems.

we establish a result that a class of rank minimization problems have closed form solutions, which will be used to develop penalty decomposition methods in Subsection 2.2.


It turns out that for MLNs, both steps can be computationally expensive.


In this case, the SOM demonstrated the best performance, and the SA algorithm also had the nearly equal performance.

comparing with the other soft-max algorithms, the SA algorithm has nearly the best clustering performance.


We do so by recognizing and accounting for the potential confounding in the data and require that the learned policy improve upon a baseline no matter the direction of confounding.

we calibrate personalized decision policies to address sensitivity to realistic violations of the unconfoundedness assumption.


Critically, it does not require either re-running MaxEnt or recomputing the Hessian, since the optimization is always done around the optimal single-task parameters, ↵ (t) .

eLIRL can pay a small cost to do this optimization when it is faced with performing on a previous task, but it gains potentially improved performance on that task by benefiting from up-to-date knowledge in L, as shown in our results.


The additional two diseases that cannot be learned share all but two children with each other.

for these two latent variables, singly-coupled triplets exist but singly-coupled quartets do not.


However, efficient algorithm based on sorting and balanced binary trees has been developed to calculate the Kendall's tau statistic with a computational complexity O(n log n) [4].

the incurred computational burden is negligible.


The binomial CDF yields the probability of making at most τ max q mistakes total, and the (1 − γ) term above specifies the probability of recovering the true class if a maximum of this many mistakes were made.

there are three probabilistic events required for the semantic output code classifier to predict a novel class and the total (joint) probability of these events is: P (there are p feature predictors with true error ≤ τ max q /p) • P (at most τ max q mistakes made | there are p feature predictors with true error ≤ τ max q /p ) • P (recovering true class | at most τ max q mistakes made) and since τ max q = G −1 q (γ), the total probability is given by: In summary, given desired error parameters (1−γ) and (1−δ) for the two classifier stages, Equation 2 provides the total probability of correctly predicting a novel class.


In the following time step t, if M receives an action A t ≤ m (i.e., a feature acquisition action) from the agent and the time step t is less than the maximum length T , the sample model generates the next state S t+1 and the reward R t+1 by The next state S t+1 contains the positive and negative symptoms in S t , and also one additional feature x At (i.e., the A th t position of x) acquired by the action A t . If M receives a feature acquisition action at the last step of an episode (t = T ), the next state S t+1 and the reward R t+1 are where S ⊥ is the terminal state.

the agent is punished if it asks too many questions.


However, in practice the dependencies between patches are not the same everywhere, and our expected value for (d i − d j ) 2 may depend on the features of the local patches.

to improve accuracy we extend the model to capture the "variance" term σ 2 2rs in the denominator of the second term as a linear function of the patches i and j's relative depth features y ijs (discussed in Section 3.2).


Let x π be the set of examples selected by π.

the Gibbs error of a non-adaptive policy π is simply the optimal non-adaptive policy selects a set S of examples maximizing its Gibbs error, which is defined by p0 g (S In general, the Gibbs error of a distribution P is , where the summation is over elements in the support of P . The Gibbs error is a special case of the Tsallis entropy used in nonextensive statistical mechanics [4] and is known to be monotone submodular [7].


algorithm 1 run on V VT (or equivalently, just V) can be used to find the conic combination weights M ≈ M. Since M being the product of T with a diagonal matrix where T has unit column sum, we can extract T = M D−1 , where D is a diagonal matrix with Dii = M(:, i) 1 .


Moreover, for a fixed ⇡ p , the constraints and objective depend only on inner products between a small number of points: the 4 attack points, the class means µ (on the clean data), and the model ✓.

we can solve for the optimal attack locations with a semidefinite program on a 7 ⇥ 7 matrix.


For this general definition of diversity, the marginal gain is Since h is concave, the gain h 1 + G i ∩ S − h G i ∩ S decreases as S becomes larger.

the marginal gain of an item a is proportional to how rare each group G i a is in the list S. In each step of the greedy algorithm, we maximize r(y) + λd(y|S).


We use the same observations as used in prior work which includes joint positions, joint velocities, and for some tasks, information related to contacts.

the policy mapping is a t ∼ N (W s t + b, σ), and the goal is to learn W , b, and σ.


Rule R6 implies that we have to recurse over more than one program, hence their repeated application can mean we have to solve number of programs that is exponential in the size of program.

we can only guarantee a pseudo-polynomial expression if we use this rule.


Since there are two opposing sides, it is hard to measure performance directly.

we compare by letting (agents trained by) BC play against (adversaries trained by) other methods, and vice versa.


More specifically, given a general bounded convex body K and a continuous function F that is monotone, smooth, and (weakly) DR-submodular we show that • All stationary points of a DR-submodular function F over K provide a 1 2 approximation to the global maximum.

projected gradient methods with sufficiently small step sizes (a.k.a.


from now on we will systematically omit the +1 as well as admit non-integer codelengths (Grünwald, 2007).


Since xc is not an optimal point of f , we have f (x new ) < f (x c ).

now consider the two possible cases: (i) if xnew is consistent with the given active set and sign vector, updating x := xnew strictly decreases the objective; (ii) if xnew is not consistent with the given active set and sign vector, let xd be the first zero-crossing point (where any coefficient changes its sign) on a line segment from xc to xnew , then clearly xc = xd , and f (x d ) < f (x c ) by convexity of f , thus we finally have f ( 4 the discrete line search described in Step 3 ensures a decrease in the objective value.


With no transformations, drawing the new features in the noisy-OR tIBP model is equivalent to drawing the new features in the normal noisy-OR IBP model.

we can use the same sampling step for K new n as [10].


That is, for neurons with general non-montonic activation functions, or for neurons with monotonic activation functions whose output magnitude violates the aforementioned bounds, there are example networks that can efficiently compute any real-valued function.


Unlike binary classification tasks, negative examples are not available to estimate the optimality of the solution.

we employed a common practice [1] of using a fixed width, divided by the number of features.


( 2), we can see the difference is the third and fourth terms in (6), which refine the regret of comparing a suboptimal arm and its last (K − L C − 1) inferiors into O(log log T ).

to prove Theorem 1, it suffices to show the following additional lemma: Lemma 5.


Silverstein and Choi have shown that the support of ρ(λ) can be determined by the intervals between extrema of z(m ρ ) [12] and this has been used to determine the signal component of a spectrum when O(N ) equal strength symmetry-breaking directions are present [13].

since C in equation ( 1) only contains a finite number of symmetry-breaking directions then in the limit N → ∞ these will have zero measure as defined by H. in this limit the eigenvalue density would appear to be identical to the isotropic case.


Given KN fixed activation signals z n k ∈ R T , associated to signals X n ∈ R P ×T , the D-step aims to update the K spatial patterns u k ∈ R P and K temporal patterns v k ∈ R L , by solving: where The problem ( 8) is convex in each block of variables {u k } k and {v k } k , but not jointly convex.

we optimize first {u k } k , then {v k } k , using in both cases a projected gradient descent with an Armijo backtracking line-search (Wright and Nocedal, 1999) to find a good step size.


According to the analysis in [4], the success condition (regarding L 0 ) of RPCA is where µ(L 0 ) = max{µ 1 (L 0 ), µ 2 (L 0 ), µ 3 (L 0 )} and c r > 0 is some numerical constant.

rPCA will be less successful when the coherence parameters are considerably larger.


In a graph even when the edge (s, t) is removed, (s, t) being ( , d) connected ensures that the correlation between s and t is still very high (exponentially close to 1).

resolving the question of the presence/absence of the edge (s, t) would be difficult -requiring lots of samples.


Note that deterministic finite automata (DFA) with n states can be represented by a WFA with at most n states.

the results we present here can be directly applied to classification problems in Σ .


Then, for any > 0 and any SPD matrix A 0, the function b The result shows that if at each round t we sample u t ⇠ U (@B 1 (0)), define an SPD matrix A t and play the point y .

we can use FTRL with these smoothed gradient estimates: , at the cost of the approximation error from f t to b f t . Furthermore, the norm of these estimate gradients can be bounded.


For each g, the constraint X T g θ 2 ≤ √ n g confines θ to an ellipsoid which is closed and convex.

the feasible set of the dual problem ( 17) is the intersection of ellipsoids and thus closed and convex.


If we assume that the advantages A(s, a) are distributed with N (A(s, a)|m A (s), σ 2 A (s)), all normalized advantage values Ā(s, a) have the same distribution.

the denominator of π 1 is constant for all states and we can use the term exp(τ Ā(s, a)) ∝ π 1 (a|s) directly as weighting for the regression defined in Equation (10).


One crucial requirement of a MP representation is that the parameters of a single primitive are easy to acquire from demonstrations.

to facilitate the estimation of the parameters, we will assume a Gaussian distribution for p(w; θ) = N (w|µ w , Σ w ) over the parameters w. the distribution of the state p(y t |θ) for time step t is given by ) and, thus, we can easily evaluate the mean and the variance for any time point t. As a ProMP represents multiple ways to execute an elemental movement, we also need multiple demonstrations to learn p(w; θ).


On the other hand, a lattice graph structure yields activations that are globally supported and smooth, and in this case the Laplacian eigenbasis corresponds to the Fourier transform (see Section 5.2).

the graph Laplacian eigenbasis provides an efficient representation for patterns whose structure is governed by the graph.


Assume that the inputs and outputs follow a "true" joint distribution q(x, y).

then any transmission method with codelength L satisfies (Mackay, 2003) the gain (per data point) between the codelength L and the trivial codelength H(y) is the mutual information between inputs and outputs (Mackay, 2003).


Notice that k is incremented only when an update occurs.

at the end of trial t, subscript k counts the number of updates made so far (plus one).


This is essentially because the cortical code changes slowly over presentations.

first, the hippocampus is mandatorily required if memories are to be preserved -the forgetting curve for the normals in figure 1B is actually dominated by hippocampal forgetting.


The modified prior P would then assign mass P (h ) = P (h)/N to each noisy copy h of h. The conditional distribution P (X T | h ) is still deterministic (obtained by flipping the outcome of one of the tests).

each hypothesis h i in the original problem is now associated with a set H i of hypotheses in the modified problem instance.


In particular, we expect that arbitrarily oriented lines can only be predicted using third-order statistics.

we should need at least a second-order Wiener functional to predict lines correctly.


Since the objective value is the sum of the total cost from all point-sets {G 1 , • • • , G n }, by Markov inequality, the contribution from G l should be no more than 2 times the average cost with probability at least 1/2, i.e.,  From ( 3) and triangle inequality, if replacing each q j by p l j , the objective value becomes where ( 4) follows from triangle inequality, and (5) follows from (3).


The use of the L 1 norm in the objective function prevents overfitting, but also serves a secondary purpose of promoting sparse solutions (i.e., coefficients w containing many 0s).

we can think of L 1 regularization as performing feature selection.


In particular, by taking η = Θ(1) and T = Θ(1/λ), the algorithm attains an -accurate solution after O(log(1/ )/λ) stochastic iterations of without-replacement sampling, and O(log(1/ )) sequential passes over the data to compute gradients of F (•).

this implies that as long as 1/λ (which stands for the condition number of the problem) is smaller than O(m/ log(1/ )), the number of withoutreplacement stochastic iterations is smaller than the data size m. assuming the data is randomly shuffled, we can get a solution using only sequential data passes, without the need to reshuffle.


Its usefulness, however, is limited by the fact that the quantity of interest, mutual information (typically between stimuli and neuronal responses) is hard to compute from data [5].

although this approach has led to a relatively deep understanding of neural coding in single neurons [4], it has told us far less about populations [6,7].


It is also known that decorrelated features can lead to better generalization [8,9,10,11].

this side effect may counteract, to some extent, the regularization effect of dropout.


The following lemma gives a convenient method of computing a tight upper bound on gain( t , y ) for any supergraph t of t. For any t ⊇ t and y ∈ {±1}, the gain of t , y is bounded by µ(t) (i.e., gain( t y ) ≤ µ(t)), where µ(t) is given by where I(•) is the indicator function.

if we focus on the case y = +1, then for any t ⊇ t and y ∈ {±1}, gain( t , y ) ≤ µ(t).


However, the true hyperparameters ⌘real which generated D ought to be in a region of high posterior mass unless the prior p(⌘) concentrates most of its mass away from ⌘real .

we expect even a small number of MCMC steps to produce a plausible posterior sample.


First, note that with d = 6, the agent is unbounded, because food is never more than 6 steps away.

the agent does not benefit from adapting the reward function parameters (given that we initialize to the objective reward function).


Some activations learn to be highly linear (such as PReLU) which reduces error somewhat, but sharply non-linear functions such as sigmoid and tanh fail consistently.

despite the fact that neural networks are capable of representing functions that extrapolate, in practice we find that they fail to learn to do so.


The third and final layer learns one filter (returning the predicted contact map) with the window size 9 × 9.

the overall receptive window of the convolutional network is 15 × 15, which provides the required amount of context of the co-evolution data to predict the contacts.


During the initial training phase, to reach a minima of "width" d the weight vector w t has to travel at least a distance d, and this takes a long time -about exp(d) iterations.

to reach wide ("flat") minima we need to have the highest possible diffusion rates (which do not result in numerical instability) and a large number of training iterations.


We prove that if a surrogate loss is convex, then it cannot be calibrated with respect to any of the AP, the ERR or the PD.

if one of these metrics should be optimized, the price to pay for the computational advantage of convex losses is an inconsistent learning procedure, which may converge to non-optimal predictions as the number of examples increases.


The experimental procedures are described in detail in [ 21 ] .

after spikesorting, spike trains from 32 neurons were binned at 2 0ms resolution, and the response of a neuron to a stimulus at time t was defined to consist of the the spike-counts in the 1 0 bins between 40ms and 240ms after t. each population response Y t is a 3 20 dimensional vector.


In the regularized case, the eigenvalue zero changes to α, which is then the smallest non-zero eigenvalue of the Hessian.

the condition number depends only on the largest eigenvalue of This shows that for large regularization parameters, the condition number is close to one and convergence is fast.


• Stochasticity: The outcome of subtask execution is stochastic in our setting (for example, some objects are randomly moving).

the agent needs to consider the expected outcome when deciding which subtask to execute.


Various researchers have considered the case when the data is sampled from a submanifold embedded in the ambient Euclidean space.

learning with the low dimensional manifold structure, or specifically the intrinsic topological and geometrical properties of the data manifold, becomes a crucial problem.


However, those approaches consider each visual descriptor in the image as a separate coding problem and do not take into account the fact that descriptor coding is just an intermediate step in creating a bag of codewords representation for the whole image.

sparse coding of each visual descriptor does not guarantee sparse coding of the whole image.


2 (d), it can be seen that DQN spends most of its time wandering around the same route.

its search range covers only a small proportion of the state space.


The advantage of considering the estimating function is the ability 1) to characterize an entire set of consistent estimators, and 2) to find the optimal estimator with the minimum parameter estimation variance from the set of estimators [23,29].

by applying this to WLPS, we can characterize an entire set of estimators, which maximizes the expected reward without identifying the state transition distribution, and find the optimal estimator with the minimum estimation variance.


Similarly, define êg (γ) := E g (1/γ); this function is concave in γ as it is a pointwise infimum (indexed by x) of functions linear in γ [see e.g., §3.2.3 in 6].

its derivative ê g (γ) = 1 2 P g 1/γ (x − γ −1 y) − x 2 = q g (1/γ), is a decreasing function of γ.


this sequence of prior works appears to have reached the limit of what is possible based on improving the bound on δ η t . Unfortunately, however, if the data are not adversarial, then even second-order bounds do not guarantee the best possible tuning of η for the data at hand.


The global indicator function for the satisfaction of these constraints can be written N k=1 f k (s 1 , .

. . , s N ), where f k is the constraint for candidate cluster x k : the joint distribution of the mixture model and data factorizes as follows: The factor graph [10] in Fig.


We assume that the cost to compute f (s) given s is significant.

care should be taken when choosing the next experiment, as picking optimum points may reduce the runtime of the algorithm by orders of magnitude.


many of the most successful annotation algorithms in the literature [7,8,9,10,11] have opted to focus on tag assignment and often fix a large number of hand-crafted features for input to their algorithms.


Although the Z cue has no vertical mirror symmetry, the similarity surfaces are nearly symmetric with respect to the vertical center line.

using a single cue, localization is only possible modulo a mirror along the vertical center.


From the definition of H − , we can choose f : For Part 3, first note that, for any φ, ψ is continuous on [0, 1] and ψ(0) = 0 by Lemma 4, part 6, and hence θ i → 0 implies ψ(θ i ) → 0.

we can replace condition (3b) by (3b') For any sequence To see that (3a) implies (3b'), let φ be classification-calibrated, and let (θ i ) be a sequence that does not converge to 0.


These caching techniques make the complexity of the inference scale linearly with the number of objects in each image, regardless of image or object size.

our algorithm benefits from the computational efficacy of matching pursuit.


Our result, however, shows that this is not true in general.

specifically, when µ = 0, Async-SGD yields an effective iterations of complexity: which is faster than that of MSGD (Liu et al., 2018): there exists fundamental difference between these two algorithms.


Add to S as many candidates from L as possible using B F . Figure 4: The main steps of the IQRP algorithm.

when the heap is full, T is the value of v associated with the l+1'th largest candidate encountered so far.


Again, this allows 1 to reason as follows: if 2 or 3 had said yes the second time, they would have been the only two muddy children.


However, the different gates values for each element of the feature maps in the spatial domain seemingly do not have the function of spatial attention.

the soft attention mechanism [15] is additionally introduced into the input of ConvLSTM in [5], in order to make ConvLSTM focus on the noticeable spatial features.


If the classifier's decisions for untrusted inputs are less reliable than the original noisy labels, then the network's utility is limited.

to obtain a reliable neural network, a large trusted dataset is necessary.


In this case, the entrywise sum of K −1 y is equal to the sum of the entries in K −1 .

we show hardness for computing the latter quantity (see Section F in the supplementary material for the proof).


This means that the membrane potential is purely driven to its threshold by input fluctuations, or a random walk in membrane potential.

the neuron's firing will be memoryless, and close to a Poisson process.


In the next two subsections, we bound Let A := {x : 0 ≤ x 1 ≤ h and h ≤ x 2 ≤ 1 − h}.

we have For x ∈ A, we have for x ∈ A we have Since p ∈ Σ κ (2, L) and 0 < x 1 ≤ h, we have For any x ∈ A, we can bound the bias term where the last inequality follows from the fact that ∂p(x) ∂x1 , ∂p(x) ∂x2 ≤ Lh, by the Hölder condition and the assumption that the density p(x) has vanishing partial derivatives on the boundary points.


Since the number of symbols in X is np we normalize by this effective block length to define the information rate r n,m per symbol as r n,m = sup p(X ) I (X ; X ) np .

we seek bounds on the capacity of this channel.


Different independent samples are obtained by repeating this process.

the equations can be interpreted as the equilibrium distribution of a dynamic system in the presence of noise which is constant during equilibration, but differs across measurements (data points).


The architecture consists of k sequentially connected RNN cells in the lower hierarchical layer and one RNN cell in the higher hierarchical layer, see Figure 1 and Section 3.

the hidden state of the lower layer is updated k times per time step, whereas the hidden state of the higher layer is updated only once per time step.


We want to "construct" new variables T , where the relations between the observed variables and these new compression variables are specified using a DAG Gin over X U T where the variables in T are leafs.

each Tj is a stochastic function of a set of variables U j = Pa~;in ~ X.


It is also intriguing to examine the internal working mechanism of tLSTM.

we visualize the memory cell which gives insight into how information is routed.


The dynamics of the system is constrained by the evolution equations.


This is undesirable, since the amount of information contained in a demonstration grows as the number of blocks increases.

we need an operation that can map variable-dimensional inputs to outputs with comparable dimensions.


The other part of the cost, O(lumn) , can be reduced by using careful pre-computation.

written in detail, we have: The crucial observation is the existence of α ij , which are independent of dimension index d. they can be pre-computed efficiently.


Using the expert-provided skills, BKT achieves a mean 11% relative improvement over the baseline model across the five datasets.

bKT with expert-provided skills is sensitive to the temporal dynamics of learning.


While it is often infeasible to store the original matrix A in the memory, it is also often infeasible to materialize all pairwise distances in A.

in applications such as online learning, databases, search engines, online recommendation systems, and online market-basket analysis, it is often more efficient if we store B ∈ R n×k in the memory and estimate any pairwise distance in A on the fly only when it is necessary.


Notice that from Lemma 3.5 we know z j+1 z j 2 W for all j 2 [k].

thus for all w 2 W 1 \ W k+1 there exists i 2 we must have h 2 + (W \ B(0, r)) ✓ W k+1 .


However, M exhibits some nice properties for any Θ: M 0, M I, tr(M ) = tr((ΘΘ ) † (ΘΘ )) = rank(ΘΘ ) = rank(Θ).

(10) Here I is the identity matrix, and we also use M 0 to encode M = M . tr(M ) provides a convenient proxy for controlling the rank of the latent representation, i.e.


When not all the members appear in an existing set, the algorithm creates a new equivalence class containing only those members that do.

as the algorithm processes more and more text, it bootstraps itself and enriches the RDS graph structure with new SPs and their accompanying equivalence sets.


For the Boltzmann distribution, the gradient of the policy with respect to the reward parameters is given by the equation where τ is the Boltzmann temperature (see [10]).

computing The value of Q t depends on the reward parameters θ t , the model, and the planning depth.


Because of the lack of complete dataset, the direct evaluation of missing values filling accuracy is impossible.

we use the mortality prediction results calculated by the same classifier but trained on different imputed datasets to determine the performance of imputation methods.


What is surprising is that a particular jigsaw piece becomes very strongly associated with a particular face part (far more so than when clustering by appearance alone).

by learning the shape of each jigsaw piece, our model has effectively identified small and large face parts of widely different shapes and aspect ratios.


In particular, we show that for multilabel metrics constructed as instance-, micro-and macroaverages, the population optimal classifier can be decomposed into binary classifiers based on the marginal instance-conditional distribution of each label, with a weak association between labels via the threshold.

our analysis extends the state of the art from a few known multilabel classification metrics such as Hamming loss, to a general framework applicable to many of the classification metrics in common use.


Hence the input vectors used to train the next layer would be x .

if the computational cost of a single inner product x i β k (e.g., logistic regression) is one, then that for T hidden layers would be about T t=1 (K t−1 + K t + 1)K t+1 /(V + 1).


Consequently, when applying the algorithm to the variational reformulation of the problem of interest (3), we are able to get an ǫ-optimal solution within at most O(1/ǫ 2 ) LMO calls.

semi-Proximal Mirror-Prox generalizes previously proposed approaches and improves upon them in special cases of problem (3); see Appendix D.2.


This is because if u ∈ B � (i 1 , r 1 ) ∩ B � (i 2 , r 2 ), then assuming r 1 ≥ r 2 (w.l.o.g.

), triangle inequality implies that d(i 1 , i 2 ) < r 1 (the strict inequality is because we defined the balls to be 'open'); in an integral solution, we will set at most one of x i1r1 and x i2r2 to 1.


We stress that while Local's neighbors are ranked based on visual similarity, our method's "neighborhood" uses visual similarity only to guide its sampling during training, then directly predicts which instances are useful.

purer visual similarity in the retrieved examples is not necessarily optimal.


This is exactly the state we would like to achieve for our two distributions of samples: bringing the two distributions into alignment without collapsing either distribution into a trivial form.

our sample-based method proposes using the Coulomb energy as an objective function to be minimized.


Each dataset was divided into 10% for testing and 90% for training.

the input to all of the algorithms tested was, for each point in the training set: (1) a vector in ℜ n (where n is different for each set) and (2) a value from 1 to 5 denoting the rank of the group to which it belongs.


Firstly, the model can be very sparse since it covers topical bigrams in O(V 2 • K) where V and K denote the vocabulary size and the number of topics.


Recall the definition that v over all unit-length vectors, it must lie in the range of because otherwise the component outside the range will not contribute.

we can choose v Consider the following decomposition: The second and the third term can be bounded by Wely's inequality (Lemma C.4) and basic properties of � A (Lemma A.3).


However ( 2) is equal to and for each n, the supremum in the limits is over a set parameterized by U (n) points, which without loss of generality we may take to be countable (due to the density and countability of rational points).

for a fixed n, the quantity in the limits is a random variable.


To extract event classes, we employed the event identifier described in [6].

this baseline will categorize events into a small number of clusters, since the event identifier is trained to predict the five event classes annotated in TimeBank [26].


= inf When Z is continuous, ρ CVaR (Z; α) is well-known to be the mean of the α-tail distribution of Z, E [ Z| Z > q α ], where q α is a (1 − α)-quantile of Z.

selecting a small α makes CVaR particularly sensitive to rare, but very high costs.


A vertex subset, V ⊆ V, induces a subgraph consisting of V together with all edges whose both ends are within V . In this paper, all subgraphs are vertex-induced.

we abuse the notation and denote both the subgraph and the vertex subset by the same symbol.


Since the function V l is convex and 2 ) is by definition the closest point to the surface V l , then w E should be inside the ball centered around ŵ with a radius of .


1] contains only a less precise variant of (4.1).)

the formulae (4.1)-(4.2) offer an a priori mechanism for selecting the sketch size k to achieve a desired error bound.


samples of a variable distributed according to Bernoulli(p i ), an unbiased estimator for p k i is their product, namely the estimator which is 1 if all the tosses come up heads, and otherwise is 0.

if we average over all t k subsets of size k, and then average over the population, we still derive an unbiased estimator.


First, the fast timescale tracks both q π , q π κ and not just q π .

it enables access to both the 1-step-greedy and κ-greedy policies.


As shown in Table 1, a significantly better error was achieved with 1000 basis functions (giving a gap of 3 5 ¨10 4 ).

it remains open how one can automatically choose an appropriate basis set size B.


The method is based on minimizing the expected reconstruction error E[g(h(θ)) − θ] of the task parameter θ onto the manifold M. Here h is the mapping from the manifold to the lower dimensional Euclidean space and g is the mapping from the lower dimensional Euclidean space to the manifold.

the composition g • h maps a point belonging to manifold to the manifold, using the mapping to the Euclidean space as an intermediate step.


We can show that the LSIF solution α is piecewise linear with respect to the regularization parameter λ.

the regularization path (i.e., solutions for all λ) can be computed efficiently based on the parametric optimization technique [4].


Replacing all occurrences of the hard threshold function with a continuous relaxation H(z) ⇡ (z) := z = 1 + exp z 1 however results in a reparameterizable computational graph.

we can compute low-variance gradient estimates for the relaxed model that approximate the gradient for the discrete model.


[20] that encourages prespecified groups of parameters to be simultaneously sparse.

the combinatorial problem in ( 6) can be approximated by the following convex (and thus tractable) problem: Problem 1 (Group lasso regression: GL-REG).


The linear SVM solves the following optimization problem [3,4,11]: where λ ∈ R + is the regularization parameter and l hinge the hinge loss function defined as: With a linear SVM, the feature space is the same as the input space.

when the input features are the voxels of a 3D image, each element of w opt = (w opt v ) v∈V also corresponds to a voxel.


1b (the "i" and "j" dimensions), as well as in the residue types (the "k" and "l" dimensions) at (i, j) and in its neighborhood.

a convolutional neural network [7] with convolutions over (i, j), i.e.


This is equivalent to learning only from the hardest examples resulting in association errors.

the two different regimes of ↵ favor either minimizing the overlap of the sample distributions or correcting cluster assignments sample-wise.


For FO 2 , the membership check can be done in time linear in the size of the theory, whereas for RU, only a worst-case exponential procedure is known.

fO 2 currently offers a faster membership check than RU, but as we show later, RU subsumes FO 2 .


Optimal subset recovery is NP-hard [6] and usually involves searches over all the n k possible support sets of x.

most attention has focused on approximate methods for reconstruction.


The measurement at time will be denoted ¡ £¢ , and ¤ ¢ denotes the control asserted in the time interval ¥ ¦ ¨ § © .

the data is given by Following common notation in the controls literature, we use the subscript ¢ to refer to an event at time and the superscript ¢ to denote all events leading up to time .


Then f h (x, y) > λ + δ/2 > f h (x, ŷ) but y ∈ [ŷ − τ, ŷ + τ ], a contradiction since ŷ is the maximizer of the KDE in τ radius neighborhood when restricted to X = x.

there exists y 0 ∈ M x such that y 0 ∈ [ŷ − τ, ŷ + τ ].


γ h (n) tk is calculated through a recurrent neural network using (h ) as its inputs.


Proof: By Lemmas 1 and 2, and by the continuity of matrix inverses [11], £ R is a continuous function of 1 .


Typically one step of gradient descent on data from the replay buffer is taken per action in the environment, as (with the exception of [Barth-Maron et al., 2018]) a greater reliance on replay data leads to unstable performance.

we propose that the replay buffer may frequently contain information that could significantly improve the policy of an agent but never be fully integrated into the decision making of an agent.


The approach is based on the observation that the beliefs in ( 6) share the same trajectories through space and differ only in the observations they integrate.

we first generate sample trajectories through the estimation window without considering the observations, and then use importance sampling to generate the beliefs needed for the gradient estimation.


Although different sampling techniques could be used in principle [15], our goal is to model correlations between each variable X uv and some neighboring region G uv in G. Such a neighborhood should be large enough to make ∆λ 2 (u, v, G uv ) sufficiently informative with respect to the overall network, but also small enough to keep the spectral decomposition of G uv computationally tractable.

in order to sample G uv , we propose to draw V uv by performing k 'snowball waves' on G [16], using u and v as seeds, and then setting E uv to be the edge set induced by V uv in G (see Algorithm 1 for the details).


On the other hand, there is increasing evidence that these convex regularizations tend to produce too severely shrunken parameter estimates.

these convex regularizations could lead to miss-identification of the weak-effect disease-related brain regions, which unfortunately make up a large portion of the disease-related brain regions especially in early AD.


3 Nevertheless, our batch construction is designed to achieve the utmost potential of such (N +1)-tuplet loss, when using deep CNNs as embedding kernel on large scale datasets both in terms of training data and number of output classes.

the proposed N -pair-mc loss is a novel framework consisting of two indispensable components: the (N +1)-tuplet loss, as the building block loss function, and the N -pair construction, as the key to enable highly scalable training.


Our results indicate that the tester requires significantly less time than the ANN algorithm to build the ANN index, most times just a 1 /10-fraction.

it can often detect badly chosen parameters of the ANN algorithm at almost no additional cost and before the ANN index is fed into the remaining data processing pipeline.


In computer vision, an image can be decomposed into different regions, and only some delineate objects.

mIL can be used in sophisticated tasks, such as identifying the location of object parts from bounding box information in images [4].


. . , N R . For the sake of conciseness and clarity, in the following discussion we will focus on the case of R = 2 regions, the generalization offering no difficulties.

the document probability in Equation ( 4) is now replaced with: Here, x j i denotes the ith word in the jth region.


In yet other scenarios, correspondences are sought not between projections of the same 3D point in different images, but between semantic analogs across different instances within a category, such as beaks of different birds or headlights of cars.

in its most general form, the notion of visual correspondence estimation spans the range from low-level feature matching to high-level object or scene understanding.


Moreover, important datasets continue to increase in size, easily reaching millions of sequences.

the resulting algorithms need the ability to efficiently handle large alphabets and datasets as well as establish measures of similarity under complex sequence transformations in order to accurately classify the data.


The cost function being minimized represents the ℓ 0 (quasi)-norm of w (i.e., a count of the nonzero elements in w).

unfortunately, an exhaustive search for the optimal representation requires the solution of up to M N linear systems of size N × N , a prohibitively expensive procedure for even modest values of M and N . in practical situations there is a need for approximate procedures that efficiently solve (1) with high probability.


first, given any x i ∈ X d , the number of points in X d that differ from x i by exactly k bits is d k . Second, it is possible to find a linear combination (i.e.


However, the trouble is that at each step of the recursion, there may exist m > 1 candidate features that satisfy this property.

we want this coverability requirement to hold not only recursively but also for each candidate feature (at each recursive call).


Since the integration in ( 1) is in general intractable, Laplace's method [23] is employed in FAB inference for approximation.

gFIC can be derived as a surrogate function of the marginal log-likelihood.


Similar to an approach in supervised learning, e.g., [14], a multi-class problem can be converted into binary classification forms.

our presentation below focuses on the binary classification problem only; i.e.


So standard network techniques are not adequate in analyzing these networks.

they resorted to the concept of a hypergraph, and showed how the concept of network centrality can be adapted to hypergraphs.


However, our binary base kernels originate from more general quantizations: they correspond to their regions of constantness.

the resulting optimization problem is also more involves and thus calls for approximate solutions.


clusters in a DP mixture model), and do not impose an inherent ordering on the components.

in order to combine information about the components from multiple processors, the correspondence between components must first be found.


However, it is known to provide a false sense of security [30,6,2].

we require upper-bounds of the gradients.


Even for a deterministic algorithm that offers prices from a finite set P, the number of states of this MDP would be in ⌦(T |P| ), which quickly becomes intractable.

in view of the prohibitive cost of computing his actions, the model of a fully strategic buyer does not seem to be realistic.


Player i is represented by a vertex labeled i in G. We use N G (i) ~ {I, ... , n} to denote the set of neighbors of player i in G-those vertices j such that the undirected edge (i , j) appears in G. By convention, NG(i) always includes i itself.

the interpretation is that each player is in a game with only his neighbors in G. if ING(i) I = k, the matrix Mi has k indices, one for each player in NG(i) , and if x E [0, Ilk, Mi(X) denotes the payoff to i when his k neighbors (which include himself) play x.


Once a replay is done, we let the agent use a new behavior policy ⇡ 0 b to collect more episodes, add them to D, and continue to update ⇥ from the augmented D. This process repeats for multiple stages, and the model parameters learned from the previous stage will be used as the initialization for the next stage.

we can focus on learning at a single stage, which was formulated in Section 2 as one of solving the optimization problem (7).


The second criterion is because, intuitively, a node i's topological position within an enclosing subgraph can be described by its radius with respect to the two center nodes, namely (d(i, x), d(i, y)).

we let nodes on the same orbit have the same label, so that the node labels can reflect nodes' relative positions and structural importance within subgraphs.


Recently, grouped or depthwise convolution has attracted attention because it reduces the computational complexity greatly while maintaining accuracy.

it has been adopted in many new architectures [2,24,8,18,26].


Conversely, excitable neurons (such as those with low thresholds) spike early in the theta cycle.

variability in excitability translates into variability in timing.


The problem stems from the fact that in the definition of external regret, the benchmark is a function of the player's actions.

if the adversary is adaptive, or even memory-bounded for some m > 0, then, external regret does not take into account how the adversary would react had the player selected some other action.


(3.7) Here α = β 0 , β , where β is the first leading eigenvector of M. Note that by (3.6) we have γ ∈ (0, 1).

the optimization error term in (3.7) decreases at a geometric rate to zero as t increases.


When computing y, the split hidden dimension is reduced out.

the results of that computation get allreduced across all processors.


Note that the IC oracle never generalizes beyond the level from which the examples are drawn, and thus, exhibits very different generalization results compared to the human participants in our experiment.

visual concept learning poses more realistic and challenging problems for computer vision studies.


SGD explores the space of parameters in small and noisy steps, and that noise is averaged out by the stochastic gradient contributions accumulated in each weight.

it is important to maintain sufficient resolution for these accumulators, which at first glance suggests that high precision is absolutely required.


Thus, we can compute the expected distance E(D j |v j ) of an image as: This represents the expected number of modifications in v from 1 to 0 (α) or from 0 to 1 (β).

over all images, we can define the expected distance E(D|v) as where α i , β i ∈ [0, 1] and ∝ rank represents that the proportionality is only related to the relative ranking of the image memorability scores, and M is the total number of images.


In other words, the image parts which most describe the domain will remain unchanged, which makes it straightforward for the discriminator D T to detect the image as a fake.

the only way to find an equilibrium between generator F S→T , attention map A S , and discriminator D T is for A S to focus on the objects or areas that the corresponding discriminator thinks are the most descriptive within its domain (i.e., the horses).


Denote by m f , v f , m g and v g the means and variances of Q old for f (x n ) and g(x * n ), respectively.

then, EP only requires five quadratures to update each γ n . One to compute log Z n and four extras to compute its derivatives with respect to m f , v f , m g and v g . After convergence, Q can be used to approximate predictive distributions and the normalisation constant Z can be maximised to find good values for the model's hyper-parameters.


Here, one can see that there are only s = √ n correct matches, which have a large number of hyperedges among them, whereas very few hyperedges may be present for other combinations.

the partition model has two groups of size √ n and (n − √ n), respectively.


In continuous domains with many variables, however, value function approximation may require hundreds of even thousands of features to represent the overall task's value function, and updates are usually linear time.

"lightweight" options that use fewer features than needed to solve the overall problem are desirable in high-dimensional domains, or when we may wish to create many skills.


This process is repeated until there are no more features.

we obtain an ordered sequence of features.


We achieve this by minimizing the cross-entropy for sgm(r(x; φ)) across the training instances: The observation 2 says that z \t = 0 → z t = 1 given y.

simply ignoring the case z \t = 0 and fixing q(z t |x, y; φ) = Ber(z t ; 1) is a close approximation of p(z t |x, y), especially under mean-field assumption (see the Appendix A for justification).


The second tradeoff, "remembering" versus "forgetting," is captured by restarting Exp3 and forgetting any acquired information every ∆ T pulls.

old information that may slow down the adaptation to the changing environment is being discarded.


Their sentiment can however be inferred from the utterances themselves.

to drive our autoencoder-based recommendation module we build a model that takes as input both the dialogue and a movie name, and predicts for that movie the answers to the associated movie dialogue form.


Model order selection: Because of the sparsity penalties, the solutions are relatively insensitive to model order (P, Q).

these are typically set to high values and the effective model order is controlled through the sparsity hyper-parameters.


As size of the space grows, the probability that two graphs will contain similar sub-structures becomes very small.

a graph becomes similar to itself but not to any other graph in the training data.


During testing, which takes place on one long trajectory, this feature would be much less informative.

the tree models were trained on fewer, longer trajectories (of length 40).


We can vary our λ on Liver to recover the same accuracy and solution as Shogun with a further decrease in our training time.

another very positive thing is that SMO-MKL appears to be relatively stable across a large operating range of p. The code is, in most of the cases as expected, fastest when p = 2 and gets slower as one increases or decreases p. Interestingly though, the algorithm doesn't appear to be significantly slower for other values of p. it is hoped that SMO-MKL can be used to learn sparse kernel combinations as well as non-sparse ones.


In fact, the weak derivative can be used in place of the derivative, as the proof of the reduction (see also [13]) to final form uses integration by parts only.

just the existence of a weak derivative for f suffices.


For a diagonal covariance matrix, strong generalization along a particular dimension would be produced if the covariance matrix has a high variance along that dimension, but low variances along the remaining dimensions.

this model can learn to strongly generalize along one dimension, but people often make strong generalizations along multiple dimensions [5], such as in Equation 2 when r < 1.


The graphical representation of of VCC-FMM does not involve variable ancestors (parents of parents).

the marginal probabilities p(Π = π) are simply the parameters of the multinomial distribution of the parent variable.


Specifically, we want to pick the policy that leads to the largest possible region of attraction according to Theorem 2.

this requires that for each discrete state in X τ the corresponding state-action pair under the policy must be in the set D n . we optimize the policy according to The region of attraction that corresponds to the optimized policy π n according to (3) is given by V(c n ), see Fig.


However, since Kolmogorov complexity is uncomputable, this does not solve the causal discovery problem in practice.

other notions of complexity need to be considered.


To explain, we first note that optimizing the second term of R y w.r.t G actually enforces the structure information to be fully persevered in y, because C is asked to recover the structure y from G(y, z), which is generated conditioned on y, regardless of the uncertainty of z (as z is marginalized out during sampling).

minimizing R y indicates the following constraint: min C,G E y∼p(y) p c (y|G(y, z 1 )), p c (y|G(y, z 2 )) , ∀z 1 , z 2 ∼ p(z), where a, b is some distance function between a and b (e.g.


The total posterior probability of an outbreak, on the other hand, will still be higher for outbreak than non-outbreak days, so the "tot" methods can perform well on OTC as well as ED data.

our main result is that the Bayes tot method, which infers baselines from past counts and uses total posterior probability of an outbreak to decide when to sound the alarm, consistently outperforms the frequentist method for both ED and OTC datasets.


Theorem 2 shows that the log-softmax has at most d + 1 linearly independent output vectors, even if the various inputs are applied to the model.

if the vectors of true log probabilities logP * (y|x) have more than d + 1 linearly independent vectors, the softmax-based model cannot completely represent the true probabilities.


The algorithm assumes the observed values y d are sorted.

without loss of generality, assume that y 1 < y 2 < .


We show that Foveated Feature Congestion (FFC) clutter scores (r(44) = −0.82 ± 0.04, p < 0.0001) correlate better with target detection (hit rate) than regular Feature Congestion (r(44) = −0.19 ± 0.13, p = 0.0774) in forced fixation search; and we extend foveation to other clutter models showing stronger correlations in all cases.

our model allows us to enrich clutter perception research by computing fixation specific clutter maps.


Our method, on the other hand, splits the coordinates among the different principal vectors, and therefore its input is the number of principal vectors, or parts, rather than the size of each part.

the natural way to use our algorithm is to search for all principal vectors together.


Our approach corresponds to maximizing mutual information between the empirical distribution on the inputs and the induced label distribution, regularized by a complexity penalty.

we call our approach Regularized Information Maximization (RIM).


In fact, observe that by equation 10, the bound for a variable-disjoint cycle graph with a maximum cycle size of l is at least (l−2) l , independently of the number of cycles.

if the minimum size of a cycle is 20, the quality for a fixed point is guaranteed to be at least 90%.


By using the CV likelihood framework we are guaranteed to (asymptotically) perform as well as the best model in the parametrized family.

the question arises whether the family of densities defined by a Gaussian model in a kernel-induced feature space is "rich enough" such that no systematic errors occur.


The symmetrized Sherali-Adams polytopes for an uprooted hypergraph ∇G(V + , E + ) (as given in Definition 2), is: Definition 16.

for any i ∈ V + , and any integer r ≥ 2 such that max E∈E + |E| ≤ r ≤ |V + |, we define the symmetrized Sherali-Adams polytope of order r uprooted at i to be for each collection of measures over hyperedges in L i r (∇G), there exist corresponding flippinginvariant, locally consistent measures on sets of size r which contain i (and their subsets).


In sharp contrast, we have found that restarts of the backfitting MCMC algorithm give remarkably similar results even in difficult problems.

we run one long chain rather than multiple starts.


. . , h T −1 } is the set of hypotheses output by A when provided with {(X t , y t )} T t=1 .


This quantity is maximized when these two distributions are equal.

if the true p Y follows the change of variable model (1), the recovered g converges to the true f −1 0 in the sense that they both describe a change of variable from the prior distribution p X to the distribution p Y .


denotes taking the 'real part' of the argument.)

the resulting generative model can also be written as: each pair of basis functions A R i ,A I i forms a 2-dimensional subspace and is controlled by an amplitude a i and phase φ i that determine the position within each subspace.


Let r i = y πi y i and Since y and y ′′ disagree on exactly |k − k ′′ | locations (and similarly for y π and y ′′ π ), the number of locations where r and r ′′ disagree is therefore at most 2|k − k ′′ |.

for any r ′′ and any h ∈ H, We observe that where z i = y i − y ′′ i .


Recall that the algorithm specifies α := λ 3 and generates a sequence of evenly-spaced indicies b t i := nα(i For every i ∈ 0..  10), ( 11) and Lemma 1, there exists some N 1 such that for all n ≥ N 1 we have, Since λ ≤ λ min , we have α ∈ (0, λ min /3].

for every t = 1, 2 and every change point π k , k ∈ 1..κ we have Define µ min := min (t,i)∈I ζ(t, i).


Independent of the value of discretization granularity s, at any round of data collection, each output bit b i is still sampled according to the distribution given by formula (1).

the algorithm above provides the same accuracy guarantees as given in Theorem 1.


Note that most connections in this circuit are local, just like in a biological neural circuit.

we see that minimizing total wire length tends to generate biology-like circuit structures.


And with s = jω, our synthesized BM design (passive) is where the two first-order systems are both low-pass filters (LPFs), with time constants τ 1 and τ 2 , respectively; b is a gain factor.

i in can be expressed in terms of I mem as: Comparing this expression with the design target (Equation 3) yields the circuit analogs: Note that the mass M (x) is a constant (i.e., 1), which was also the case in our mathematical model simulation.


The QP can partially be solved analytically with respect to α: where ρ is the Huber's loss otherwise.

the on-manifold solution of (11) corresponds to the robust projection by the Huber's loss.


Recall that λ ě λ max yields βpλ,Ωq " 0, in which case ρ :" y ´X βpλ,Ωq " y is the optimal residual and y{λ max is the dual solution.

as for getting λ max " Ω D pX J yq, the scaling computation in (9) requires a dual norm evaluation.


In Gibbs sampling we need the posterior conditional distribution for each indicator given all the remaining indicators and the data: where the covariance matrix depends on the parameters .

for the GP expert, we compute the above conditional density by simply evaluating the GP on the data assigned to it.


Moreover, for all cycles C of G N , π K (C) only depends on the vertex set of C, not its edge set.

in the sequel, for the ease of notation, we denote by π K (J) = π K (C) for any cycle C with vertex set J.


In forest CTBNs, the number of intensities can be exponential in the number of parameters.

the forest model has much greater potential expressivity per parameter than the other models.


From the linear modeling of the observer's displacement and the subsequent multiplicative nature of zoom, scaling should follow a Weber-Fechner law stating that subjective sensation when quantified is proportional to the logarithm of stimulus intensity.

we choose the scaling z drawn from a log-normal distribution P Z , defined in (6).


We show that the (right) singular vectors of B do not converge to the eigenvectors of K (the matrix with entries κ(x i , x j )).

it is necessary to take the product and use regularization.


Otherwise, for any i such that ↵ ⇤ i = 0 it follows that ✓ i 0, which implies  i .

for any nonzero weight ↵ ⇤ i > 0 the following holds: Squaring and summing Equation ( 2) over all the nonzero entries of ↵, we arrive at the following equation for : Algorithm 1 k ⇤ -NN Input: vector of ordered distances 2 R n , noisy labels y 1 , .


One can follow the chain of direct hypernyms all the way to the top of the tree, "entity".

we can detect a concrete sense by examining its hypernym tree to see if it contains one of the following nodes: 'article', 'instrumentality','article of clothing', 'animal', or 'body part'.


However, it should be clear that we need this restriction when considering D−eigenspaces themselves since the target space has to be unequivocally defined, otherwise convergence cannot occur.

it can happen in this special case that the reconstruction error converges while the projection space itself does not.


In biology, it is known that Abf1p acts as a global transcriptional regulator in yeast [19].

the genomic regions in telomeres would be good candidates for novel putative eQTL hotspots that regulate the expression levels of many genes.


For MapReduce algorithms this is particularly useful because the MST requires a substantially smaller space than the original graph and can be stored in one machine.

once we have the MST, we can obtain affinity or single linkage in one round.


The reason why π IG fails is that there are complementarities among tests; a set of tests can be far better than the sum of its parts.

we would like to optimize an objective function that encourages differentiation among classes, but lacks complementarities.


This would suggest working with H 1 to further limit the risk of overfitting, however, as already pointed out, H 2 is more convenient since it is a convex class.

in the next section, we will consider both hypothesis sets and introduce two distinct learning kernel algorithms, each based on one of these families.


However, the terms from resampling events contribute to the majority of the variance of the estimator.

the gradient estimator that we found most effective in practice consists only of the gradient ∇ (θ,φ) log pN (x 1:T ), the solid red arrows of Figure 1.


The objective function of the embedding problem is then given by [17].

the optimal embedding is found by solving argmin v T Dv=1 v T (D − A)v. Setting u = D 1/2 v, this is solved by finding the eigenvector corresponding to the smallest eigenvalue of L . Lemma 1 provides an efficient dimension reduction framework that summarizes the information in the entire network into few feature vectors.


However, assuming the condition number 1/λ is smaller than the data size (assuming each f i (•) is O(1) smooth, and λ is the strong convexity parameter of F (•)), only O(m log(1/ )) gradient evaluations are required to get an -accurate solution, for any .

we can get a high-accuracy solution after the equivalent of a small number of data passes.


If they are the same, they will share one set of basic attributes; if they are distinct, there will be two sets.

the possible worlds of our probability model may differ in the number of random variables, and there will be no single equivalent Bayesian network.


Theorem 1 provides a desired value h ⇤ = "/4 C, where we recall that C is the Lipschitz parameter of the optimal action-value function Q ⇤ (see Lemma 1).

we need to use a small h ⇤ if we demand a small error ", or if Q ⇤ fluctuates a lot with a large C. Theorem 1, combined with Proposition 1, immediately yield the following bound that quantify the number of samples required to obtain an "-optimal action-value function with high probability, if the sample path is generated per the uniformly random policy.


Unfortunately, this expectation generally cannot be computed efficiently, since it involves summing over exponentially many sets Y .

to use the multilinear extension in practice requires estimating its value and derivative via Monte Carlo techniques.


That is, As mentioned in Section 2, GBS is tailored to minimize L 1 (Π), and hence may not produce a good suboptimal solution for the exponential cost function with λ > 1.

we derive an extension of GBS for the problem of exponential costs.


5 Here we can show that size estimation is possible provided that is at least δ −1 times a logarithmic factor, and this bound is asymptotically tight.

our estimation methods work even when the data provides much less than a full path back to the root for each node made public.


(A* can stop search as soon as it is about to expand a goal state without actually expanding it.


= 0 and, at each iteration, to seek to update a so as to minimize the square error plus the distance from the previous value of a.

we perform the following update: where we used a notation !


In general, we observed that the performance of the tree-based method improved with smaller values for η min , with an expected increase in the computational cost.

in order to give an overall characterization of the performance of fitted Q-iteration, we report the results obtained with the extreme values of η min .


In all such cases, we could observe only the time stamp when a piece of information has been received by a particular entity, but the exact path of diffusion is missing.

it is an interesting and challenging question whether we can uncover the diffusion paths based just on the time stamps of the events.


While sharing the most layers, each branch can be considered as an independent multi-class classifier in that all of them independently learn high-level semantic representations.

taking the ensemble of all branches (classifiers) can make a stronger teacher model.


Therefore, the only possible consistent domain is the singleton {s a,i } or {u a,i } (i.e., P(M s i→a = 1, M u i→a = 0) + P(M s i→a = 0, M u i→a = 1) = 1).

either 0 or 1 can possibly be a consistent value assignment, but not both.


The idea underlying the algorithm for the personalized setting is quite intuitive: If we were to learn a classifier that is on average good for the players, then we have learned a classifier that is good for a large fraction of the players.

a large fraction of the players can be simultaneously satisfied by a single good global classifier.


By regularity condition 3a, and assuming a = 0, we have that a((∂/∂θ)f θ0 (x)) ⊤ , Z is a continuous real random variable for any x, unless Z = 0 in which case the lemma is trivial.

the conditions of Lemma D.3 apply: the two sides of Eq.


The linear embeddings are defined as follows: ( The embedding transforms W θ , W φ , and W g are all learnable and have the shape of m × l, m × l, m × m, respectively.

the proposed non-local operation can be written as where δ i (X) = j∈Si φ(X i , X j ).


To derive the minimax optimal algorithm, we compute conditional values (using similar techniques to Theorem 7), and so following Eq.

( 9) the minimax-optimal algorithm must use a minimax-optimal algorithm is simply constant-learning-rate gradient descent with learning rate 1 .


That is, there are no write-after-write races (handling these is possible, but complicates the analysis).

notice that, if we let x t denote the value of the vector x in the central store S after t writes have occurred, then since the writes are atomic, the value of x t+1 is solely dependent on the single thread that produces the write that is serialized next in S. If we let Gt denote the update function sample that is used by that thread for that write, and v t denote the cached value of x used by that write, then Our hardware model further constrains the value of ṽt : all the read elements of ṽt must have been written to S at some time before t. for some nonnegative variable τi,t , ) where e i is the ith standard basis vector.


Specifically, we set L V = V 2 /2, and calculate the gradient: We can simplify this equation considerably by observing that the contribution of the second summation is largely averaged out under a wide variety of realistic conditions 4 .

it can be neglected, and we obtain the following local learning rule: This is a Hebbian plasticity rule, whereby connectivity changes in proportion to the presynaptic firing rate ōj and post-synaptic membrane potential V i .


(iii) Motivated by (ii), we consider solutions of the following form where Then the objective value is maximized at Hence it is better off to move out probability mass from ω 1 to ω 2 and from ω 2 to ω 1 .

we conclude that there exist a least favorable distribution supported on Ω.


In general cases, as aforementioned, matrix completion is an ill-posed problem.

some assumptions are necessary for studying Problem 1.1.


GPs with SE, MA, and RQ kernels are all truly Bayesian nonparametric models -these kernels are derived from infinite basis function expansions.

as seen in Figure 1 g), h), i), these methods are completely able to capture the information in the training region; however, these kernels do not have the proper structure to reasonably extrapolate across the missing region -they simply act as smoothing filters.


the PD method proposed in Subsection 2.2 for problem (2) can be suitably applied to (17).


Thus, Ax 2 measures the strength of linear trend x in A, and A 2 measures the strongest linear trend in A.

minimizing A B 2 minimizes the strength of the strongest linear trend of A not captured by B.


Our approach to learning marginal predictors relies on the extended input pattern X = (P X , X).

we study the natural algorithm of minimizing a regularized empirical loss over a reproducing kernel Hilbert space associated with the extended input domain P X ×X .


The similarity term y T Ly is based on the idea of normalised cut (41) that encourages nearby superpixels with similar appearance to have the same label.

a similarity matrix W i is defined to represent local interactions between superpixels of same image.


Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision.

there is need for efficient optimization procedures for submodular functions, especially for minimization problems.


The analysis is constructive, and determines appropriate hyperparameter settings for the GP-UCB acquisition function.

we make a step forward to resolving the problem that, despite being used for hyperparameter tuning, BO algorithms themselves have hyperparameters.


Each term ∂Qw /∂n(i, j) is evaluated at the overall state-action counts n sa t which in turn depend on the policy and actions of other agents.

it overcomes the second limitation of the factored critic in theorem 2 where terms f w ignore policy and actions of other agents.


While interpretable, all of these models rely on aggregated features, ignoring the temporal relation among features inherent to EHR data.


However, in other problems such as network intrusion detection, we are interested in detecting the most extreme value output by the sources.

in our work we study extreme regret which measures the efficiency of an algorithm compared to the oracle policy selecting the source with the heaviest tail.


The function − log(y t k ), corresponding to the softmax with cross-entropy loss, is convex [11].


It is defined with respect to the following set of invertible 2 × 2 matrices G ab for {a, b} ∈ E, coined gauges: The GM, gauge transformed with respect to G = [G ab , G ba : {a, b} ∈ E], consists of factors expressed as: Here one treats independent x ab and x ba equivalently for notational convenience, and {G ab , G ba } is a conjugated pair of distinct matrices satisfying the gauge constraint G ab G ba = I, where I is the identity matrix.

then, one can prove invariance of the partition function under the transformation: GT results in the gauge transformed distribution p G (x) = 1 Z a∈V f a,G (x a ).


We allocate all positive training samples for the "model" set, while all positive test samples and 1% of negative test samples are assigned in the "evaluation" set.

we regard the positive samples as regular and the negative samples as irregular.


Sakai & Fukai [5,6] proved that Q-learning does not lead to matching behavior.

soltani-Wang's model is intrinsically incapable of reproducing matching behavior.


Thus, for low interaction strengths, the MF magnetization is maximized by focusing the external field on the nodes of highest out-degree in the network, independent of b and H. To study the structure of solutions to MF-IIM at high interaction strengths, we make the simplifying assumptions that J is strongly-connected and b ≥ 0 so that Theorem 6 guarantees a stable nonnegative steady state m. For large β and an additional external field h ∈ F H , m takes the form where d in i = j J ij is the in-degree of node i.

in the high-β limit, we have: where i * = arg min i (d in i + b i + h i ).


However, the spectral analysis of M becomes much more complicated to get for s ≥ 2, where the analysis of at least cubic equations are involved.

for the sake of brevity, we shall skip the detailed discussion here.


Examining the posterior p( , conditioned on all other parameters, we find that most settings of a and b tend to be non-informative, especially in the case of sequential learning (discussed further in Section 5).

the average sparsity level of the representation is inferred by the data itself and each sample x i has its own unique sparse representation based on the posterior, which renders much more flexibility than enforcing the same sparsity level for each sample.


( 4) starting in the nonnegative orthant remains there.

for simplicity we will consider trajectories that are confined to the nonnegative orthant x 2: 0.


However, during one update, the triplet loss only compares an example with one negative example while ignoring negative examples from the rest of the classes.

the embedding vector for an example is only guaranteed to be far from the selected negative class but not necessarily the others.


Since the symmetries that can be exploited highly depend on the amount of evidence, we encode the SRM after evidence is instantiated, via a process called shattering [2].

while a pseudotree encodes a graphical model, a schematic encodes an (SRM, evidence set) pair.


Thresholding Softmax Scores: This approach assumes that samples from a class on which the network was not trained would have probability scores distributed across all the known classes, hence making the maximum softmax score for any of the known classes low.

if the system thresholds the maximum score, it may avoid classifying such a sample as one of the known classes.


In contrast, for hard tanh networks, p(q ⇤ ) = erf(1/ p 2q ⇤ ).

one can always move along the critical line in the ( w , b ) plane towards the point (1, 0), thereby reducing q ⇤ , increasing p(q ⇤ ), and decreasing, to an arbitrarily small value, the prefactor 1 p(q ⇤ ) p(q ⇤ ) controlling the linear growth of both max and 2 JJ T . So unlike either ReLU networks, or Gaussian networks, one can achieve dynamical isometry up to depth L by choosing q ⇤ small enough so that p(q ⇤ ) ⇡ 1 1 L . In essence, this strategy increases the fraction of neurons operating in the linear regime, enabling orthogonal hard-tanh nets to mimic the successful dynamical isometry achieved by orthogonal linear nets.


Note, however, that dv/dt after convergence is small but non-zero in each of the target memories.

strictly speaking, the target patterns haven't become fixed points of the dynamics, but only slow points from which the system will eventually drift away.


For the remaining datasets, the Type I errors of the Pearson and Bootstrap approximations are close.

for larger datasets, the bootstrap is to be preferred, since it costs O(m 2 ), compared with a cost of O(m 3 ) for Pearson (due to the cost of computing ( 8)).


In addition to two-sample homogeneity test, inlier-based outlier detection, and transfer learning, density-ratios can be useful for tackling various machine learning problems, for example, multi-task learning, independence test, feature selection, causal inference, independent component analysis, dimensionality reduction, unpaired data matching, clustering, conditional density estimation, and probabilistic classification.

it would be promising to explore more applications of the proposed relative density-ratio approximator beyond two-sample homogeneity test, inlier-based outlier detection, and transfer learning.


In this case, pseudo correlations again appeared but there was a significant difference in the mean firing rates between the stimuli.

the independent model can be used to extract almost all the information.


Let C be the set of consensus decisions: then one can reformulate (3) in a decentralized way as follows: where Next, we consider the implementation of PDA in (5) to solve (11).

let Φ(x) given the initial iterates x 0 , θ 0 , λ 0 and parameters γ > 0, τ i , κ i > 0 for i ∈ N , choosing D x and D y as defined in Definition 1, and setting ν x = ν y = 1, PDA iterations given in (5) take the following form: Starting from µ 0 = λ 0 , compute for i ∈ N Using extended Moreau decomposition for proximal operators, λ k+1 can be written as Let Let B {x : , where p(x) Although x-step and θ-step of the PDA implementation in ( 12) can be computed locally at each node, computing λ k+1 requires communication among the nodes.


Because marginalization is a linear operation, there is a linear operator A such that n C = An V . Moreover, F n A is the image of F * n A under A.

the image of the Dobra basis under A is a Markov basis for .


No matter how refined the various early developments of PLS become, they are still linear projections.

in the cases where the variables of the input and output spaces are not linearly related, the challenge of the data is still poorly handled.


For example, a first-order Markov chain along the primary structure of a protein is hardly related to the causal mechanisms underlying polypeptide evolution.

the potential of combining biological modeling with kernelization in the framework of MDFE remains vastly unexplored.


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

f 0 satisfies and the result follows from Lemma 3.


The result shows that the determinant of the Hessian at the fixed points, which appears in the formula of zeta function, must satisfy a strong constraint.

in addition to the known result on the one-cycle case, we show that the LBP fixed point is unique for any unattractive connected graph with two cycles without restricting the strength of interactions.


Such an index i 2 [m] can be efficiently computed because our later lemmas shall ensure that at least (1 ↵)m indices in [m] are valid choices for i.

one can for instance guess a random index i and verify whether it is valid.


In other words, refining a partition can only reduce the margin.

refining a clustering (i.e., splitting a cluster) can only reduce the separation.


For convenience, we also include the auxiliary segment s The relation kernel computes the number of common patterns between two sentences s and t, where the set of patterns is restricted to the three types introduced in Section 3.

the kernel rK(s, t) is expressed as the sum of three sub-kernels: f bK(s, t) counting the rK(s, t) = f bK(s, t) + bK(s, t) + baK(s, t) bKi(s, t) = Ki(s b , t b , 1) number of common fore-between patterns, bK(s, t) for between patterns, and baK(s, t) for between-after patterns, as in Figure 3.


While our theorem specified an optimal setting of the regularization parameter n and ⌫, this optimal setting depended on unknown model parameters.

as is standard with high-dimensional regularized estimators, we set tuning parameters n = c p log p/n and ⌫ = c 0 p log p/n by a holdoutvalidated fashion; finding a parameter that minimizes the `2 error on an independent validation set.


It is also irreducible and aperiodic, because with ε > 0, all actions have positive probability to be chosen in a convention state.

theorem 4 in [17] applies and thus the chain has a unique stationary distribution circling around the stochastic stable conventions of {V Q 1 , V Q 2 }.


Using the fourth property in Lemma 1, we have SGN(u) ⊆ SGN(x * ).

it follows from ( 6)-( 8) that 0 ∈ ∂g λ1 λ2 (x * ).


In this simple setting, we can tune the step size optimally as η * = O( (1 + P T )/T ) and obtain an improved O( T (1 + P T )) dynamic regret bound, which matches the lower bound in Theorem 2.

when bounding the general dynamic regret, we face the following challenge: On one hand, we want the regret bound to hold for any sequence of comparators, but on the other hand, to get a tighter bound, we need to tune the step size for a specific path-length.


It can be shown [4] that KCCA is obtained by substituting where Y ∈ R k×n is the label indicator matrix.

kCCA is a special case of the proposed formulation.


We remark that T is a small integer in practice.

the number of rounds of communication is a small integer.


In particular, we can invoke [5, Theorem 2 and Section 4.3] which guarantees that for some positive absolute constants C 1 , C 2 , and C 3 if (8) holds then holds for all valid B , thereby for all valid X , with probability exceeding 1 − e −C3n .

with C = C 2 , the target matrix X would be feasible in (7).


This has the effect of destroying the fixed point attractors in each WTA stage.

persistent activity can no longer appear in a single WTA stage if there is no input.


Equally possible is that two items will have same user preference on average, and can be exchanged without hurting the optimal ranking.

existence of an unique optimal ranking, which indicates that each item will have different average user preference than every other item, is unlikely.


If i and j are two pixels in different objects, we know they will be independent, and as such uncorrelated, resulting in zero entries in the covariance matrix.

if we arrange the pixels by their object assignment, the eigenvectors of such a covariance matrix would be of the form: where v is an eigenvector of the stationary (within-object) covariance and the rest of the entries are zeros, thus eigenvectors of the covariance will be zero on one side of the occlusion mask and Fourierlike on the other side.


EM objective) arg max w y P (x, y|w) = arg max w max qy y q y log P (x, y|w) +H(q y ) where q y is a distribution over possible missing values [10].

much of the analysis we provide for joint EM also applies to marginal EM, leaving only a separate convex relaxation of the entropy term that can be conducted independently.


In addition, it is reasonable to expect that the optimal setting of α should vary across training processes and learning tasks.

it is desirable to design an approach to choose α adaptively and automatically as q changes during the training iterations, according to the distribution of the ratio p(x)/q(x).


Because bandwidth is limited and data transmission is energy-intensive, it is desirable to reduce the amount of information sent over the communication channel.

if a particular region of the visual field can be identified as having high salience, that part of the image can be selectively transmitted with high resolution and the surrounding scene can be be compressed.