At this scale any manual processing of the obtained data is infeasible.

automatic spike sorting for dense MEAs has enjoyed significant recent attention [15,9,51,24,36,20,33,12].


2 e(IF + aA) < 1 ~ 'it: 11 + aAi l < 1 ~ a < H' (5) This completes the proof if all eigenvalues of A have a negative real part.

the convergence analysis can be carried out separately for the two iterations.


For a RevNet, the residual functions each must be recomputed during the backward pass.

the number of operations required for reversible backprop is approximately 4N , or roughly 33% more than ordinary backprop.


( 11), we have From above equation and ( 8) with x = y k , we get On the other hand, from the definition of the estimate sequence and (8), from these two equations, we have Since g is Lipschitz smooth, we bound f (x k+1 ) as follows: Using ( 16), ( 17), (12), and x k+1 − y k = −η(v k + ξ k ) we have where for the first inequality we used Here, we give the following where for the first inequality we used ) and convexity of g and h. Thus we have By taking expectation with respect to the history of random variables I 1 , .


Taking the high probability statements of Lemma 3 and Lemma 4 with /2 2 [0, 1/e] tells us that with probability at least 1 , kw which is required for Lemma 4 to hold.

if we set the algorithm parameter ✏ as indicated in the statement of theorem, we have that with probability at least 1 T 1/2 ↵ for all t 2 {T ↵ + 1, . . . , T } that a t = 1 and v t p t  ✏, which follows from the same argument used for Lemma 6.


Therefore, we introduce an additional tuning parameter µ > 0 and look for minimizing the criterion If we make the change of variables φ 1 = Mβ/σ, φ 2 = Nβ/σ and ρ = 1/σ, we get a convex function for which the first-order conditions [20] take the form Thus, any minimizer of (2) should satisfy these conditions.

to simplify the problem of optimization we propose to replace minimization of (2) by the minimization of the weighted 1norm j ωj |(Mβ) j | subject to some constraints that are as close as possible to (3)(4)(5).


This is because, if not, then, using Theorem 6, in each of the good major cycles t * + 1, t * + 2, . . . t * + t , err(x) falls additively by > ε 4 /8Q 2 and thus err(x t * +t ) < err(x t * ) − ε 2 ≤ 0, which is a contradiction.

in O(Q 2 /ε 2 ) good major cycles, the algorithm obtains an x = x t * * with ∆(x) ≤ ε 2 , proving Theorem 4.


The remaining terms correspond to the comparisons between a non-winner arm and the remaining arms, and is bounded by O K 2 log log T . As demonstrated in [6], L C is relatively small compared to K, and can be viewed as a constant.

the total regret In particular, this asymptotic trend can be easily seen for Condorcet dueling bandits where L C = 0.


By definition, an execution graph G for an execution contains a vertex if its label is a substring of the execution's trace t. Likewise, an edge is contained if an aggregation of the vertex labels is a substring of t. It follows 1 that the predictive distribution can be reformulated as in Equation 2to predict the probability of seeing the code position s = s n after a fragment of preceding statements f = s 1 . . . s n−1 using the trace representation of an execution.

it is not neccessary to represent execution graphs G explicitly.


Since we marginalize out α in the inference as written in (4), the observation noise becomes a Student-t distribution.

the proposed model is robust to single-view anomalies.


We observe that M-Walk outperforms the strong baseline ConvE by 4.6-10.9% in samples that require 2 or 3 steps, while it is nearly on par for paths of length one.

m-Walk does better at reasoning over longer paths than ConvE.


It is unreasonable to expect that our semidefinite relaxation is always tight.

we can sometimes further refine the projection p relax obtained by the RELAX algorithm by using it as a starting point in the original non-convex optimization.


As listed in Table 6, in second run, the relative improvements of the target model is incremental, compared to its improvement in the first run.

when learning a model using CL, with a sufficiently strong reference, the improvement is usually saturated in the first run, and there is no need, in terms of overall performance, to replace the reference multiple times.


( 1) is thus equivalent to min Using the change of variable βv = v∈V , this implies that given the optimal η (and associated ζ), β corresponds to the solution of the regular supervised learning problem with kernel matrix K = w∈V ζ w K w , where K w is n × n the kernel matrix associated with kernel k w . Moreover, the solution is then β w = ζ w n i=1 α i Φ w (x i ), where α ∈ R n are the dual parameters associated with the single kernel learning problem.

the solution is entirely determined by α ∈ R n and η ∈ R V (and its corresponding ζ ∈ R V ).


In this paper, we restrict our discussion of PSRs to linear PSRs, in which the function f t is a linear function of the tests in Q.

p(t|h) = p(Q|h)m t , where m t is a column vector of weights.


Here we use the standard squared exponential (SE) kernel.

x ∼ N (µ1, Σ), where Σ is the positive definite covariance matrix defined by For notational convenience, we define the hyperparameter set θ = [µ; γ; κ; σ 2 f ; σ 2 v ].


an incremental approach, which efficiently updates an existing solution given a new training point, can be devised.


Third, large-scale recordings are required to detect many pairs of cells and estimate their functional interactions; electrode arrays with fewer channels may not suffice.

large-scale, high-density electrophysiological recordings are uniquely well suited to the task of identifying cell types.


The DM function h is a fixed, unknown black-box.

learning good { ŶM , s} involves learning functions f and g which can adapt to h -the goal is to make Ŷ a good predictor of Y .


For example, two images containing one common object may also have other visually incompatible objects, which makes these two images less similar in terms of the B2B distance.

instead of measuring the similarities between bags, in this paper we approach MIL from a new perspective using the Class-to-Bag (C2B) distance, which assesses the relationships between the classes and the bags.


This, however, would be incorrect, since the primitive P.ratio is composed of P.perimeter and P.intensity, which makes λ 2 and λ 3 related.

along with looking at the primitives that each HF takes as input, it is also essential to model how these primitives are composed.


In the full version of the paper, we show that the expected cost function is (Hn 2 k log n)-Lipschitz.

the best approximate break-point has approximately optimal expected cost.


( 5) can be viewed as a data labelling problem, where each value x j or y j takes a label in [1, N ], with a monotonicity constraint.

let us now consider the graph (V, E) where nodes V = {v t } t=1...2M represent the list of all x j and y j in a sorted order and the edges E = {(v s , v t )} connect all pairs (x j , y j ).


( 5), we subtract the constant component, E z s j , from the signal, since it does not capture the variation of input samples.


It then follows that the condition number of the design matrix κ 2k ≤ (2 + C2 C1 )σ max (Σ)/σ min (Σ).

we can set the parameter Note that the above quantities depend only on the covariance matrix.


Actually, in this case, training input densities that approximately minimize Ï ÄË and ÇÄË were found by Â Ï ÄË and Â ÇÄË .

the difference of the errors is caused by the difference of WLS and OLS: WLS generally has larger variance than OLS.


Suppose the initial values µ x , µ y satisfy µ x = ϕ X L1(fµ y ) .

then, at every iteration t ≥ 1 of EGT, the corresponding solution z t = [x t ; y t ] satisfies x t ∈ X , y t ∈ Y, the excessive gap condition is maintained, and EGT has a convergence rate of O( 1 T ) [24].


In fact, we show that for sparse binary data, scoring all items in a large data set can be accomplished using a single sparse matrix-vector multiplication.

we get a very fast and practical Bayesian algorithm without needing to resort to approximate inference.


It is easy to see that if x is outside of the support of the data, then the metric tensor is large.

geodesics are "pulled" towards the data where the metric is small.


In the inference phase, features are extracted for auxiliary tasks and are used to modify the base feature map in the same way, so that the predictions for the primary task explicitly take this information into account, without needing any annotations.

rOCK uses auxiliary supervision as privileged information.


When a reward is collected from a state, all the policies which could reach that state are affected.

it is useful to share the information between the policies.


More explicitly, , and ARE (β SD ; β RP ) := Whenever the ARE is less than 1, our procedure is considered to have greater asymptotic power than the competing test-with our advantage being greater for smaller values of the ARE.

we seek sufficient conditions in Theorems 2 and 3 for ensuring that the ARE is small.


Note that these geometrybased methods require high-resolution images at high frame rates.

we sample high-resolution images (1024 × 1024 vs. 224 × 224 for our policies) at 5× the frame rate (145 vs. 30 images for our policy for a trajectory of length 30) along the reference trajectory along with ground truth poses.


We now recall that the individual rectangles of the class "m x n" differ only in their positions within the 4 x 4 grid.

the cardinality of the class "m x n" is equal to the number ofunique ways N m x n in which an m x n or n x m rectangle can be selected from a 4 x 4 grid, where Thus, the probability associated with an individual rectangle of class "m x n" is P N (m x n) .


For any i ∈ Z \ Y g , the Cholesky decomposition of L Yg∪{i} can be derived as where row vector c i and scalar d i ≥ 0 satisfies In addition, according to Equ.


Pair-based models with all-to-all interaction [4, 5, 6, 7, 8, 9, 10] yield under Poisson stimulation (protocol P2) a total weight change that is linear in presynaptic and postsynaptic frequencies.

as a function of postsynaptic frequency we always find a straight line with a slope that depends on the integral of the STDP function [5,7].


However, pharamacological studies which block the NMDA receptors impair a variety of brain processes, suggesting that NMDA receptors also playa role in shaping the dynamic activity of neural networks [10,3,8,11,2].

we wanted to examine the role of NMDA receptors in post-synaptic integration.


Finite history windows increase coverage, by reducing the number of possible histories to a finite scale.

finite-history window provide some generalization power.


The results show that the parameters trained on a single noise level cannot handle different levels of noises well.

in this paper, we aim to train a single model for recovering different levels of corruption, which are different noise levels in the task of image denoising and different scaling parameters in image super-resolution.


Let I n denote an indicator vector of length K that records which signatures appear at node n. For the root node, all K signatures are present, so we have I n = [1, . . . , 1].

following the typical random forest protocol [2], the training instances are recursively split according to a randomized test; it compares one dimension of the signature against a threshold t, then propagates each one to the left child l or right child r depending on the outcome, yielding indicator vectors I l and I r . Specifically, if during training we must choose two things at each node: the query attribute m and the threshold t, represented jointly as the split (m, t).


So if we consider a binary adjacency matrix on these nodes based on P, a number of columns (or rows) will indeed have all missing values.

the only information we have about these nodes are their incident noisy links in the multi-resolution network.


In more realistic scenarios where there are inhomogeneities and variability (as is the case for biology and silicon) p is largely independent of n [9].

an efficient strategy for implementing long lasting memories in VLSI networks of spiking neurons is to use a large number of synapses with only two stable states (i.e.


and the encoding functions i (θ) are of the finite-dimensional form considered above, i (θ) = K T i Φ(θ), then the log f [b i + i (ϑ)] term in the definition of ∆ ϑ (r) simplifies: in this case, all information about θ is captured by the sufficient statistic If we again let dt = 1/n, then we find that ∆(r) converges to a finite-dimensional Gaussian random vector with mean and covariance again, if the filters K i are modeled as independent draws from some fixed distribution, then the above normalized sums converge to their expectations, by the LLN.

as in the intermediate-SNR regime, we see that inference can be dramatically simplified in this short-time setting.


The total density of cells in the optimal solution computed by the model also varies with the firing rate budget.

our model makes a specific numerical prediction relating total cell density to the ratio of midget to parasol cells.


As an example, consider the family of WFSTs T a , a 2 ⌃, with a single state Q = I = F = {0} and with T a defined by self-loop transitions with all input labels b 2 ⌃ with the same output label a, and with uniform weights.


We do this by maintaining the set INCONS of all the locally inconsistent states that are not in OPEN (lines 12-13, Figure 3).

the union of INCONS and OPEN is exactly the set of all locally inconsistent states, and can be used as a starting point for inconsistency propagation before each new search iteration.


In this section, we will show that the function −D(•, •) is monotone and consistent.

by Rizzi's result, it follows that we can find a 2-clustering {S 1 , S 2 } = {S 1 , S \ S 1 } that minimizes −D(S 1 , S 2 ), and hence maximizes D(S 1 , S 2 ).


Let us take two arbitrary vectors u and v from the s-Voronoi cell R x .

it means that s(x, v) > s(w, v) ∀w ∈ X \ {x} (8) Hence, due to linearity vector tu Corollary 1.


4, the performance of LDLFs can be improved by using more trees, but the improvement becomes increasingly smaller and smaller.

using much larger ensembles does not yield a big improvement (On Movie, the number of trees K = 100: K-L = 0.070 vs K = 20: K-L = 0.071).


However, the joint probability distribution P(x, y) is often not known in practice.

in order to utilize the MAP rule, one should first estimate P(x, y) using the training data instances.


Let Assumption 1 hold, and suppose that π * is a sequentially robust policy w.r.t.

s Ps , where to find the sequentially robust policy, we need only to solve the sequentially robust action.


Here we take a different approach and, following [31], we interpret an image depicting a digit as an histogram and normalize it to sum up to 1.

y is the unit simplex in R 128 (16 × 16 images) and we adopt the Hellinger distance to measure distances on Y.


Our function classes therefore require us to understand the specific form of the weighted VCG payment rule and its interaction with the parameter space.

our context and techniques diverge from those in [Morgenstern and Roughgarden, 2016].


This is due to the term 1/(Kγ) in the bounds of Theorem 2 dominates both the bias and the MSE as expected since K is fixed.

we observe a drop in the bias and the MSE as we increase γ up to ≈ 8 × 10 −5 , and then they gradually increase along with γ.


We also consider the problem of maximizing the global gain from a set of initial states chosen according to some probability distribution P(x).

accordingly, we define the global gain of a policy 11"": ]JL(7f) == Ex ]JL(x; 7f)P(x) and the global loss LIL of using some approximate policy 7r instead of the optimal one nIL (4) knowing the pdf over all parameters J-l we can define the expected global loss L == EJL [LIL].


It is clear that in multiple electrode recordings the phase trajectories between events are not independent, since they have to start where the previous trajectory ended.

our model violates the observation independence assumption of standard Hidden Markov Models (HMMs).


( 12) is precisely a sub-gradient of F i .

we can write By optimality of (α * , γ * ) we have Because of the max operation over α, γ, we have 13) we arrive at Hence, F ′ (K 1 , .


To learn from unlabeled data, we observe that a Bayesian approach provides us with a predictive posterior distribution-i.e., we are able to quantify predictive uncertainty.

we regularize the posterior by adding an unsupervised loss term that minimizes the predictive variance at unlabeled data points: where n and m are the numbers of labeled and unlabeled training examples, α is a hyperparameter controlling the trade-off between supervised and unsupervised components, and θ represents the model parameters.


The kernel is intrinsically non-stationary, i.e., the level of similarity between any pair of samples depends on all the available samples, whether labeled or unlabeled, thus promoting semi-supervised learning.

this approach is significantly simpler and more efficient than the alternating-optimization that is used in previous co-regularization implementations.


However, observe that evaluating a point on a related task can never reveal more information than evaluating the same point on the task of interest.

the above strategy would never choose to evaluate a related task.


As we aim to learn the local structure of flow warp error at motion boundaries, it is more effective to penalize the structure at the scale of local patches instead of the whole image.

we use the PatchGAN [17] architecture as our discriminator.


Apparently yes, because different classifier chains involve different classifiers trained on different training sets.

to reduce the influence of the label order, Read et al.


The Markovian relaxation process after β iterations, namely M β , can be represented as: .

a particle undertaking a random walk with an initial distribution p 0 acquires after β steps a distribution p β given by: p β = M β p 0 .


We define the max-margin majority voting (M 3 V) as solving the constrained optimization problem to estimate the true labels y and weights η: And in practice, the worker labels are often linearly inseparable by a single hyperplane.

we relax the hard constraints by introducing non-negative slack variables {ξ i } M i=1 , one for each task, and define the soft-margin max-margin majority voting as inf ξi≥0,η,y , where c is a positive regularization parameter and − ξ i is the soft-margin for task t i .


Let Ψ ∈ H n,3 be the hypothesis Ψ(x) = sign ( ψ, x ).


The high speed of human sensory perception [1] is puzzling given its inherent computational complexity: sensory inputs are noisy and ambiguous, and therefore do not uniquely determine the state of the environment for the observer, which makes perception akin to a statistical inference problem.

the brain must represent and compute with complex and often high-dimensional probability distributions over relevant environmental variables.


However for each new constraint we may need to design a specifically tailored algorithm.

the most common approach to impose consistency is usage of PN-consistency potentials [9].


In addition to having a more adaptive representation, the main benefit of SVDGP is that the computation of an unbiased gradient requires only linear complexity in M α , as shown below (see Appendix Bfor details).

the time complexity to compute ∇ a KL[q||p] can be reduced to O(N m M α ) if we sample over the columns of K α with a mini-batch of size N m . By contrast, the time complexity to compute ∇ B KL[q||p] will always be Θ(M 3 β ) and cannot be further reduced, regardless of the parametrization of B.


Nevertheless, nonwitnesses are usually labeled positive since the consequent margin will be larger.


In this simple example, we compare the BQ and MC estimates of the gradient (for a fixed set of policy parameters) using the same samples.

our simple bandit problem has a single state and A = R. each path ξ i consists of a single action a i .


However, the Zonotope abstraction is inherently not exact for non-linear activation functions such as ReLU, Sigmoid, and Tanh.

approximation is needed, which creates a tradeoff between the cost of computation and precision.


Since θ 1 > θ 0 , and binomial distributions have a monotone likelihood ratio (MLR) in X, larger values of X give more evidence for θ 1 over θ 0 .

φ(x) should be increasing in x as much as possible, subject to (2)-(5).


( 10) is a generalization of the process introduced in [26].

the following properties are simple extensions of the properties derived in [26].


price, season) aggravates the sparsity issue.

it is particularly important for RSs to have access to large data sets to perform accurate elicitation [12].


when X is an orthonormal matrix (especially X = I p ), the equality holds.


In this local privacy framework, we establish sharp upper and lower bounds on the convergence rates of statistical estimation procedures.

we exhibit a precise tradeoff between the amount of privacy the data preserves and the utility, measured by convergence rate, of any statistical estimator.


Since most attributes take numerical values in this dataset, the fullgrained histogram has quite a lot of bins.

the data-parallel algorithm which communicates full-grained histogram is quite slow, even slower than the sequential algorithm.


It has been observed that in face recognition the variations between the images of the same face due to illumination and viewing direction are almost always larger than image variations due to the changes in face identity [1].

while the PCA projections are optimal in a correlation sense (or for reconstruction" from a low dimensional subspace), these eigenvectors or bases may be suboptimal from the classification viewpoint.


Function ConsistentFinder(•): Let D st be the set of all directed paths from variable X s to variable X t . For any directed path d 2 D st , we define the weight of d as w d := ⇧ (u,v)2d b vu where b vu are coefficients in (1).

by this definition, it can be seen that the entry (t, the entries of matrix A are multivariate polynomials of entries of B.


In this section, we discuss a third method related to ŷpcr and ŷbc , which relies on information that is typically not available in practice.

this method is usually non-implementable; however, we believe it is useful for comparative purposes.


( 1), we deduce that if the d-th feature of x is not to be used, it is sufficient and necessary for the d-th diagonal element of M be zero.

analogous to the use of 1 -norm by the popular LASSO technique [34], we add the 1norm of M 's diagonal elements to the large margin metric learning criterion (M ) in eq.


at time steps 1, . . . , T . Recall that the noise trader sells back every bundle z t she purchases.

her total payoff is the sum over t of the difference in price at which she buys z t and price at which she sells it.


Then w 1 and w 2 are the i th columns of U and V respectively, and v T 1 v 2 is the i th diagonal element of S if v 1 and v 2 are the i th principal vectors of S 1 and S 2 .

geometrically, this theorem says that after projection on the plane (P ) defined by any one of the principal vector pairs between subspaces S 1 and S 2 , both the entire subspaces collapse to just two lines such that points from S 1 lie along one line while points from S 2 lie along the second line.


In summary, the salient features of our work are • Unknown system dynamics: We assume that the transition kernel and reward function of the MDP is unknown.

we cannot exactly evaluate the expectation required in standard dynamic programming algorithms (e.g., value/policy iteration).


We found that among various EM type algorithms, this formulation is the most revealing for our problems because it clearly relates the second projection to the scenario where an FVS F is both observed and known (Section 4.1.1).

we are able to compute the second projection exactly even though the graph structure is unknown (which allows any tree structure among the observed nodes).


Note that four of these examples are sufficiently hard to be unsolved by a variable elimination-based exact solver, even with several orders of magnitude more computational resources (200GB memory, 24 hour time limit).

dIS provides excellent anytime behavior; in particular, (N l =1, N d =10) seems to work well, perhaps because expanding the search tree is slightly faster than drawing a sample (since the tree depth is less than the number of variables).


According to their results, most errors occurs among the four indoor categories (830 images), namely, bedroom (174 images), living room (289 images), kitchen (151 images), and office (216 images).


As the dimensionality of the data increases, the tests require a sufficient number of data points in order to be reliable.

although the data may be of arbitrary dimensionality, it is important to apply unimodality tests on one-dimensional data values.


Some examples of these types of data are functional magnetic resonance imaging (fMRI) (Grosenick et al., 2008) and near-infrared spectra (NIR) (Xiaobo et al., 2010).

it is important to develop methods for automatic induction that take into account the functional structure of the data (infinite dimension, high redundancy, etc.)


We work with time series data using a frame-by-frame analysis with N -point frames.


Let the gradient (8) vanish, then the primal iterates x k+1 can be exactly determined by which requires inverting an n × n positive definite matrix I + A T A, or equivalently, inverting an m × m positive definite matrix I + AA T via the following Sherman-Morrison-Woodbury identity, ) One basic fact is that we only need to invert such a matrix once and then use this cached factorization in subsequent iterations.

there are several cases for which the above factorization can be efficiently calculated: (i) Factorization has a closed-form expression.


Using velocity trajectories to make predictions about future states also requires that people know the inertia term.

the degree to which trajectories are used to track is a question about the inertia parameter (λ) that best matches human performance.


To get around this issue, we observe that an approximate solution for TAS lies in the row space of W T and therefore, an approximately optimal solution for AS lies in the row space of W T . We then set up the following regression problem: min X AS − XW T 2 F . Note, this regression problem is still too big to be solved in sublinear time.

we sketch it by sampling columns of AS according to the leverage scores of W T to set up a smaller regression problem.


Note that complete HOUDINI programs do not have explicit variable names.

hOUDINI follows the point-free style of functional programming [5].


Taking the expected value, it may be further simplified to: where N is the number of parameters.

canceling σ results in: for small noises and constant σ the SNR and the parameter number have a simple inverse relationship.


Consider the matrix G defined by By the Gershgorin circle theorem, G is positive definite.


Determining the rank r is the key to the whole problem, after which, the parameters D, E, F, G can be computed easily [17,20].

finding a low order system is equivalent to minimizing the rank of the Hankel matrix above, while remaining consistent with the observations.


In particular, we obtain a stochastic hill climbing algorithm by accepting a new sample only if its posterior probability improves upon that of the previous sample   (line 15).

each iteration of the stochastic hill climber requires more time, but always improves the solution.


Instead, as suggested by our analysis, we require that 1/ ln(1/ρ i ) is a uniform grid on [0, 1/(ln 1/ρ max )].

we add HOO instances in batches such that ρ i = ρ max N/i .


There is a closed lower dimensional set E ⊂ L such that the function τ * is locally constant on the complement of E. Theorem 1 implies that, with probability 1, if the model estimate m and the true model m are sufficiently close, then the actions recommended by the exactly optimal policies for both models are identical.

the impact of estimation error on the exactly optimal policy is minimal.


Firstly, the basis functions are no longer deterministic, but they are instead drawn from different GPs, so an extra layer of flexibility is added to the model.

a posterior distribution over the basis functions of the generalized linear model can be inferred from data.


A simple example is p loc (•|v) ∼ N (v, σ 2 loc ).

we consider the probability family: v∈V where p(•|v) = p atlas (•|v)p loc (•|v).


When the reference frame for an image is ambiguous, one factor that influences the inferred reference frame is the orientation of other images it is grouped with, especially when those images are identifiable in any orientation.

if we ask people to solve an arithmetic problem, where the operator × is paired with the numbers 5 aligned with the top-down axes of the page (Figure 2 (a)), they should respond 25, the result of multiplication.


( 24) the decoding variance in the three input conditions, which are (because of the symmetry, only the results for the network 1 are shown) where Var(z 1 |c vi ), Var(z 1 |c ve ) and Var(z 1 |c b ) denote, respectively, the decoding errors when only the visual cue, only the vestibular cue and both cues are presented.

it is straightforward to check that in the weak input limit, the coupled CANNs implements Bayesian inference perfectly (compare Eq.


upper-right corner in Flickr and Gplus).

in addition to competitive and parsimonious performance, the RBP-RM also produces intuitively interpretable and meaningful partitions (Figure 4).


For a given x, we assume that once it pays the cost to acquire a feature, its value can be efficiently cached; its subsequent use does not incur additional cost.

the cost of utilizing a particular prediction model, denoted by c(f z , x), is computed as the sum of the acquisition cost of unique features required by f z . Oracle Gating: Consider a general gating likelihood function q(z|x) with z ∈ {0, 1}, that outputs the likelihood of sending the input x to a prediction model, f z .


( 1) is equivalent to optimizing its logarithmic form log E s . To efficiently solve this difficult optimization problem, we can instead optimize the lower bound of log E s as a surrogate energy, by applying Jensen's inequality: With f s (z) = z −s , s > 0, we observe that E log becomes sE 0 = s i =j log( ŵi − ŵj −1 ), which is identical to the logarithmic hyperspherical energy E 0 up to a multiplicative factor s. minimizing E 0 can also be viewed as a relaxation of minimizing E s for s > 0.


Although there were no explicitly shared primitives in this dataset, the generative model was still able to model the training labels more accurately with knowledge of how heuristics used primitives, which affects the relative false positive and false negative rates.

the generative model significantly improved over the independent model.


An important quantity in this analysis is the level of non-redundancy between the hidden variables h and g. Let If Figure 1(a) is the correct induced topology among {z 1 , z 2 , z 3 , z 4 }, then the smaller ρ is, the greater the gap between ).

ρ also governs how small the ∆ i,j need to be for the quartet test to return a correct pairing; this is quantified in Lemma 4.


Ceteris paribus, subjects will prefer a larger quantity of money to a smaller quantity of money.

while the psychological de-sirability pointer could assign preferences to monetary labels capriciously (as an infant who prefers the drawings on a $1 bill to those on a $100 bill might), to model desirability behavior corresponding to knowledgeable use of currency, we constrain it to follow arithmetic ordering such that, where the notation x m denotes an item (currency token) x associated with the money label m. Then, Equation 3 reduces to, where max(M ) ≤ m * , since the contribution to p(r|x, m, c) for all larger m terms, is set to zero by the arithmetic ordering condition; the p(x|m) term binds x to all the m s it has been seen with before.


Additionally, for activations not close to unit variance, we have proved an upper and lower bound on the variance mapping.

sNNs do not face vanishing and exploding gradient problems.


On the other hand, we found that for a small sample size (small training set), each path in T represents a unique CPDAG.

b-RAI has a virtue of inherently identifying the number of unique CPDAGs required to capture the distribution.


Micro-averaging: Micro-averaged multilabel performance metrics micro are defined by averaging over both labels and examples.

let: c TN(f ) and c FN(f ) are defined similarly, then the micro-averaged multilabel performance metrics are given by: ) for micro-averaging, one applies a binary performance metric to the confusion matrix defined by the averaged quantities described in (5).


Optimizing I action with respect to the policy parameters θ is a bit trickier, however, because the expectation above is with respect to a distribution that depends on θ.

the gradient of I action with Algorithm 1 Action information regularized REINFORCE with value baseline.


The filtered inputs to the two spiking models A and B are then N A * F L * S and N B * F R * S. These will be identical for any sound S whenever N A * F L = N B * F R , implying that the two neurons fire synchronously.

for each location indicated by its filter pair (F L , F R ), we define the synchrony pattern as the set of binaural pairs of neurons (A, B) such that N A * F L = N B * F R . This pattern is location-specific and independent of the source signal S. the identity of the synchrony pattern induced by a binaural stimulus indicates the location of the source.


Besides, every user's need is different and time varying [16].

user judgments indicating the relevance of an image retrieved for a query are utilized to achieve better retrieval performance in the revised ranking of images [17].


Moreover, complete coverage holds: We say there is an angular separation if cos(b i , b k ) ≤ 0 for any i, k = 1, . . . , K (i.e., the angles for all pairs are at least π/2), then ω ∈ 1 − r Rmax , 1 = ∅.

under angular separation, the range ω that allows for full coverage is nonempty independently of K. Our result is in agreement with that of Nguyen (2015), whose result suggested that topic simplex B can be consistently estimated without knowing K, provided there is a minimum edge length a min > 0.


Here, m represents the number of objects that parameters X and Y may take.

2 evidence Γ m must contain precisely m 2 literals.


In the large sample limit, independent bootstrap samples will yield similar CI-test results.

all the paths in T will represent the same single CPDAG.


But this process can be very costly, as the number of required comparison is proportional to 1/ 2 , even for strongly suboptimal arms.

our algorithm, UnchainedBandits, when combined with decoys, first produces an "-approximation b P of P using a peeling approach and direct comparisons before refining it into P by using Algorithm 3 together with decoys.


However, unlike other bandit problems, the argument inside the KL divergence is a product of parameters θ i κ l : While d KL (•, θ i κ l ) is convex to θ i κ l , it is not convex to the parameter space {θ i }, {κ l }.

finding a set of parameters that minimizes i,l d KL (µ i,l , θ i κ l ) is non-convex, which makes PBMU difficult.


( 3) is the information geometric part [4], in which D is a Bregman divergence.

the f -GAN problem can be equivalent to a geometric optimization problem [4], like for the Wasserstein GAN and its variants [6].


It can be shown that , where f i and f j are the ith and jth rows of F respectively.

each row of F can be interpreted as a C-dimensional embedding of the corresponding node in the network.


Since the ι is given in each world, the distribution over the attributes can be represented using a Bayesian network as described in Section 3.

the probability of a state is simply the product pf P (n), P (ι), and the probability of the hidden attributes of the network.


Throughout this pap er including exp eriments, we adopt the o rigi nal defi ni t ion of the F isher kernel from [6] .

3Notice t h at p-l (t) = log tlog(l -t ) by setting A Theoretical Analysis In this section , we compare the TOP kernel with the plugin es timate in terms of p erformance measures .


( 6) with ((x i , y i ), µ) + λ 2 x i AΣ i A x i .

our online algorithm minimizes the following bound on Eq.


However their approach is limited by the fact that the MI-prompting objective is used only during testing time, while the training procedure remains the same as the standard maximum likelihood training.

during training the model is not explicitly specified for maximizing pertinent information.


Now, recall the term which dominated our analysis of Q i under H 1 , ( ⇤ ) > ⌃ 2:d ⇤ , which was equal to We replace the covariances by their rescaled versions to obtain: For the spiked covariance model, rescaling variances to one amount to rescaling with D 1 = 1 /(1+✓).

we see that our signal strength is affected only by constant factor (assuming ✓  1).


We assume that for every random vector ξ, F (x, ξ) is a convex and continuous function in x ∈ X .


(c) For any F UHPR (T, δ) with F UHPR (T, δ) → ∞ as T → ∞, there is an algorithm that satisfies that uniform high-probability regret bound on some MDP but makes infinitely many mistakes for any sufficiently small accuracy level ε > 0 for that MDP.

a high-probability regret bound (uniform or not) cannot be converted to a finite (ε, δ)-PAC bound.


Two neurons A and B responding to two different sides (left and right), with receptive fields N A and N B , transform the signals N A * F L * S and N B * F R * S into spike trains.

synchrony between A and B occurs whenever N A * F L = N B * F R , i.e., for a specific set of filter pairs (F L , F R ).


Since the loss is always non-negative for both A and B, (3) implies that loss(A, AW W T ) > 0.


Given the training data points {(x i , ŷi )} N i=1 , a natural candidate for the center of the Wasserstein ball is the empirical distribution PN = 1 N N i=1 δ (xi,ŷi) , where δ (xi,ŷi) denotes the Dirac point measure at (x i , ŷi ).

we henceforth examine the distributionally robust optimization problem equipped with a Wasserstein ambiguity set.


However, the off-diagonal elements of the Hessian are often dominated by the diagonal.

we will use the diagonal approximation: where λ sets regularization strength and is related to (but not exactly equal to) 1 2 σ 2 in equation ( 6).


For node i, with probability the random walk enters the cycle, with probability 1/2 the random walk chooses any neighbor in the clique; and with probability 1/2 − the random walk stays at node i.

we can show that the expected return time to node i is E i [T i ] scales linearly in n and k. Suppose we observe only the local neighborhood of constant size r around node i.


A drug is labeled positive if any of its constituent conformations has the binding ability greater than the threshold, otherwise negative.

each sample (a drug) is a bag of instances (its constituent conformations).


32.3.2]-this is not true in general if C is not compact.

under the assumptions of Theorem 2, there is a point Y ∈ M such that X = Y Y is an optimal extreme point of (SDP); then, of course, Y itself is optimal for (P).


As a result, on event E, the regret of UGapEc cannot be bigger than , because then it contradicts the stopping condition of the algorithm, i.e., B J( n+1) ( n + 1) < .


using only semi-metric allows us to consider a larger class of functions.


If T is odd then for every odd t ∈ T : Pr(P (t)) = exp(w)Pr(¬P (t)).


As a comparison, note that the corresponding ODE for gradient descent with updates whose finite difference scheme has the characteristic equation det(λ − (1 − ∆t∇ 2 f )) = 0.

to guarantee −I n 1 − ∆t∇ 2 f I n in worst case analysis, one can only choose ∆t ≤ 2/L for a fixed step size, which is much smaller than the step size 2/ √ L for (3.1) when ∇f is very variable, i.e., L is large.


So the expected number of positive sampled points would be less than ( 1 2 − ǫ)(2 log n/ǫ 2 ) = (log n/ǫ 2 − 2 log n/ǫ).

by Chernoff bounds, the chance of the sample having a majority of positive-predicting functions would be at most e −(2 log n/ǫ) 2 /2(log n/ǫ 2 −2 log n/ǫ) = e −2 log n/(1−ǫ) = n −2/(1−ǫ) .


Furthermore, the mean and the variance remain in the domain Ω, that is g(Ω) ⊆ Ω, where When iteratively applying the mapping g, each point within Ω converges to this fixed point.

we consider activations of a neural network to be normalized, if both their mean and their variance across samples are within predefined intervals.


This simple form of "recurrence" also failed to improve ImageNet performance.

non-trivial recurrent connectivity is likely needed to explain the visually-evoked responses observed here.


Similarly, if E <~ Yo the map will not spread different inputs.

this formulation of the problem does not have a satisfactory solution.


one can pre-compute the eigenvectors once, and only update the eigenvalues as a function of θ k . This will make the computation efficient both in time and memory.


⌃, with a single state Q = I = F = {0} and with T ' defined by self-loop transitions with input label a 2 ⌃ and output '(a), all weights uniform.


The first considers the behavior of EM when the dimension d = d n may grow with the sample size n. Our proof of Theorem 7 reveals that the parameter error of the t-th iterate (in Euclidean norm) is of the order d/n as t → ∞.

we conjecture that the theorem still holds as long as d n = o(n).


As observed in Table 1, preprocessing images with BASS or using a learned hash code through the AE leads to much better performance on Gravitar, Montezuma's Revenge and Venture.

a static or adaptive preprocessing step can be important for a good hash function.


This guarantees that on any finite interval [t 0 , T ), the functions η(t) and w(t)/W (t) are bounded.

−η(t)∇f (X) and w(t) W (t) (∇ψ * (Z) − X) are Lipschitz functions of (X, Z), uniformly in t ∈ [t 0 , T ).


The complexity of inference in graphical models is typically exponential in some parameter of the graph, such as the size of the largest clique.

it is often required to find a subgraphical model that has lower complexity (smaller clique size) without introducing a large error in inference results.


If training data is sparse-as is often the case in practice-it is impossible to estimate this distribution reliably.

we lack essential information to evaluate (1) in the first place.


This means that the model complexity does not affect the asymptotic variance.

overfitting would hardly occur in the estimation of the relative PE divergence even when complex models are used.


From Theorem 1, we know that comparing other arms with their superiors will also result in Θ(log T ) regret.

although the current D-TS + algorithm performs well in most practical scenarios, one may further improve its performance by taking these additional comparisons into account in R (1) i (t).


1 It differs from the standard (induction) regression estimation problem by the fact that the learning algorithm is given the unlabeled test examples beforehand.

it may exploit that information and achieve a better result than via the standard induction.


The proposed reconstruction method assumes linear summation of synaptic inputs (both excitatory and inhibitory) and is sensitive to non-linearity of synaptic integration.

it is most useful for studying connections onto neurons, in which synaptic integration is close to linear.


For example, in the wireless industry, domestic monthly churn rates are 2-3% of the customer base.

service providers are highly motivated to identify subscribers who are dissatisfied with their service and offer them incentives to prevent churn.


In this case, the perturbation matrices in the lemma can be expressed as ∆ 1 = Ξ, ∆ 2 = Ξ, P = 2(1 − ξ )Ξ.

we get the expansions of the eigenvalues and eigenvectors as d k + ε χ 1k , c k + ε χ 2k , v k + εψ 1k and u k + εψ 2k , where    is satisfied, the O(ε) term χ 1k (resp.


For example, we could let DcjJ be an information divergence and qN some kernel estimate, that is, a filtered version of the empirical measure (see [4] for an independent approach along these lines).

this doesn't quite work, however, because the kernel induces an arbitrary scale; if this scale is larger than the natural scale of f and p(< V, X » for some V but not others, our estimate will be biased away from K. DcjJ and PN have to be asymptotically scale-free in some sense.


ccrypt's sensitivity to EOF inputs suggests that the problem has something to do with its interactions with standard file operations.

randomly sampling function return values may identify key operations close to the bug.


Let us write ŵt = ŵt + ŵt⊥ , where ŵt is the projection of ŵt on the rows of Π πt F , i.e.

∃α t ∈ R |S| : ŵt = F T Π T πt αt , and ŵt⊥ is orthogonal to the rows of Π πt F . Our problem now is to find a distribution on deterministic policies P that satisfies the conditions stated in Proposition 1 in addition to the feature matching conditions φ π k (s) = φk .


For a given orthonormal basis {ϕ j } of L 2 ([0, 1]), smoothness parameter β > 1/2 and radius C, the function class F β [C] is given by If we choose the trigonometric basis as our orthonormal basis, then membership in the class F β [C] corresponds to certain smoothness constraints on the derivatives of f . More precisely, for j ∈ N, consider the orthonormal basis for L 2 ([0, 1]) of trigonometric functions: Now consider a β-times almost everywhere differentiable function f for which .

definition 1 (essentially) captures densities that have Lipschitz-continuous (β − 1)th derivative.


Then h Q (x) = argmax l∈[L] A∈path(x),l(A)=l w Q (A), where path(x) denotes all nodes of T 0 containing x.

h Q (x) is computable from weights proportional to w Q (A) at every node.


This term is convex and, therefore, the problem in (7) is convex: The objective is convex (sum of linear and convex functions) and constraint z p ∈ ∇ L is affine.

one can minimize this constrained convex problem for each p by solving the Karush-Kuhn-Tucker (KKT) conditions 8 .


In this case, the probability of an output sequence only depends on the total number of zeros or ones in the sequence.

we only get t independent measurements from windows of length t, hence windows of length O(n) instead of O(log 2 n) are necessary for identifiability (also refer to Blischke [8] for more discussions on this case).


In contrast, the MacKay updates do not even guarantee cost function decrease.

both methods can become trapped at a solution such as γ = 0; a fixed point of the updates but not a stationary point or local minimum of L(γ).


In practice, however, we only know an estimated class prior π.

a classifier is selected to minimize Expanding the above risk based on (2) gives Thus, the estimated class prior affects the risk with respect to 2 π − π and 1 − π.


This is tiny compared to any vector operation involving billionscale of variables.

it is not necessary to parallelize Algorithm 3 in implementation.


More specifically, the topology preservation of similarity amongst different viewing angles and distances allows a direct translation of the visual similarity between two views to their relative location within an environment.

only a small number of place cells are required for position reconstruction.


Clearly, if we set all bids b ij to 1, the goal of the budgeted allocation part will be maximizing the cardinality.

this is a clear generalization of the bi-objective online allocation to maximize weight and cardinality, and the same hardness results hold here.


Clearly, we can not expect T 1 (x), T 2 (x) to arbitrarily small if there is sufficient overlap between p 1 and p 2 .

we will restrict ourselves to the following class of probability distributions for each individual class which has reasonably fast tail decay.


Presumably this reflects the collective modulation of wideband (or harmonic) sounds, with features that span many frequencies.

a synthesis of BSE and PAD should incorporate the notion of auditory features.


We make this intuition rigorous in Section 2.1.

the inference for y jl can be expressed by a minimax entropy program:   Now we justify the principle of choosing y jl by minimizing entropy.


More generally, we show analytically that, for data lying on symmetric manifolds, optimal solutions of objectives, from which similarity-preserving networks are derived, have localized receptive fields.

nonnegative similarity-preserving mapping (NSM) implemented by neural networks can model representations of continuous manifolds in the brain.


In addition, we also report an upper bound on the performance that can be achieved by the nearest neighbor using ground-truth similarities.

the nearest training pose for a query is identified by minimizing the average distance between their ground-truth pose annotation.


The sensory data base available probably tries to cover too many aspects of beef meat affecting to its sensory quality.

it is not possible to obtain more detailed conclusions from the polynomial model.


This explains the underfitting reasoning in [4] behind the subpar reconstruction ability of ALI.

in ALICE, we explicitly add a cycle-consistency regularization to accelerate and stabilize training.


Let us assume that Q has a density h with respect to µ.

then, the sets {h > ρ}, ρ > 0, describe the concentration of Q. to define anomalies in terms of the concentration we only have to fix a threshold level ρ > 0, so that an x ∈ X is considered to be anomalous whenever x ∈ {h ≤ ρ}.


. . , φ(x n ) ∈ R n−1 as follows: For any 1 ≤ i < j ≤ n, φ(x i ) and φ(x j ) agree on all coordinates except i to (j − 1).

so the embedding preserves the distance between these points.


From Lemma 3 below, we know that in an open-loop POMDP, two belief trajectories can only get closer to each other, as they progress.

to get a δ-cover of the space R(b 0 ) that the robot scout can reach from a given b 0 , it suffices to first compute a δ/2-cover C of the initial belief points for all possible subsequences of moves and then take the union of the δ/2-covers of the belief points traversed by the subsequences whose initial belief points lie in C. The δ-cover of R(b 0 ) then has its size bounded by O(n m (2m /δ) m (16 mn/δ) 2 m + h ), where h = log ζ (δ/4n).


The kernel of h k defined by K 2 and ( 7) is denoted by K : N × N → R and depends on β, θ.

letting E[•] denote the expectation operator, the prior model on the impulse responses is given by The noise variance σ 2 will always be estimated via a preliminary step using a low-bias ARX model, as described in [24].


In light of this, the motivation to formulate the problem of multi-step ahead prediction in the online setting is quite clear: attaining regret in this setting would imply that our algorithm's performance 6: end for is comparable with the best 1-step ahead recursive AR predictor in hindsight (even if the latter is misspecified).

our goal is to minimize the following regret term: where K denotes the set of all 1-step ahead recursive AR predictors, against which we want to compete.


The second player to act in a zero-sum game can maximize/minimize using a pure strategy (i.e., a single value assignment to all variables).

these LPs consider only the set of pure strategies of the opponent to find the first player's mixed equilibrium strategy.


For a column j with knots, IS requires O( 2 c(j) + nc(j) 2 ) -or worst-case O(n 3 ) time.

sequential implementation of IS has running time O(n 3 m).


Such an aggressive approach may be sensitive to outliers and over-fitting.

we allow some of the constraints to remain violated by introducing a tradeoff between the change to ω t and the loss attained on (X t , y t ).


Therefore, the interesting and challenging regime is where b is constant and not scaling.

finally, observe the interesting consequence that under the conditions on d, since the induced distribution π is close to π, it implies connectivity of G. the analysis of our algorithm provides an alternative proof of connectivity in an Erdos-Renyi graph (of course, by using heavy machinery!).


Notice this is the same as variational inference for unsupervised LDA: since we averaged the response variable out of the right-hand side in ( 12), what remains is the standard unsupervised LDA model for Z 1:N and θ.

given a new document, we first compute E q [Z 1:N ], the variational posterior distribution of the latent variables Z n . Then, we estimate the response with Up to this point, we have confined our attention to an unconstrained real-valued response variable.


For any t 2 [1, T ], define a distribution D t over the pairs (i, n), with n in {1, 2} where Z t is the normalization factor given by Z In order to derive an explicit formulation of the descent direction that is based on the weighted error of the classification function h j and the expected value of the rejection function r j , we use the distributions D  Using these definition, we show (see Appendix D) that the descent direction is given by This equation shows that Z 1,t and 2 p c(1 c)Z 2,t re-scale the weighted error and expected rejection.

finding the best descent direction by minimizing this equation is equivalent to finding the best scaled trade-off between the misclassification error and the average rejection cost.


Note that for this dataset we report only accuracy (recall), because, unlike traditional object detection, there are no overlapping detections and every stroke is assigned to a symbol.

a false positive always causes a false negative, so recall and precision are redundant: e.g., misclassifying one segment in a three-segment "H" makes it impossible to recognize the original "H" correctly.


For any given policy π (mapping belief state to action), the expected cost is At any time t, the observer can either choose to stop and declare one of the locations to be the target, or choose to continue and look at location λ t+1 .

the expected cost associated with stopping and declaring location i to be the target is: And the minimum expected cost for continuing sensing at location j is: The value function V (p, i), or the expected cost incurred following the optimal policy (π * ), starting with the prior belief p 0 = p and initial observation location λ 1 = i, is: Then the value function satisfies the following recursive relation (Bellman, 1952), and the action that minimizes the right hand side is the optimal action π * (p, k): This can be solved using dynamic programming, or more specifically value iteration, whereby we guess an initial value of the value function and iterate eq.


Furthermore, let the benefit of an intervention observed in a particular participant be expressed as a real number ag .

and without loss of generality, a greater x ag indicates greater benefit.


Of course, with such a large C, IRL can easily overfit when provided a finite dataset.

we will incorporate a (closed, proper) convex cost function regularizer ψ : R S×A → R into our study.


The reason for this is that the sensors are very accurate (variance=0.01).

the distributions are very peaked and we are simply tracking the mode.


observation makes sense because H prom is related to active regions for "ON" genes.


The three objectives: sparsity, reconstruction and discrimination may not always be consistent.

weighting factors are introduced to adjust the tradeoff among these objectives, as the weighting factor λ in equation (3).


The LBL model makes a linear prediction of the next word representation as where C (i) , i = 1, . . . , n − 1 are K × K context parameter matrices.

r is the predicted representation of r wn .


Clearly, for binary matrices, P(i, j) determines the PDF of the values that appear in cell (i, j).

we call P(i, j) the entry-wise PDF of entry (i, j), and P the PDF matrix.


As we will see in the next section, the MSPE of ridge regression can be bounded in a sharp way when the design matrix is approximately low rank, and there we will analyze mspe( β |X) for the pilot estimator.

when near low-rank structure is available, the only remaining issue in controlling the right side of line ( 12) is to bound the quantity The very recent work of Bobkov and Ledoux [15] provides an in-depth study of this question, and they derive a variety bounds under different tail conditions on F 0 .


Hence, we neither impose this constraint explicitly nor do so by a line search on µ k , since in practice this degrades the performance of the algorithm substantially.

the resulting iterates Y k may not be positive definite, while the iterates X k remain so.


While an interesting generalization of the usual KDE, this approach assumes that the data manifold M is known in advance, and that we have access to certain geometric quantities related to this manifold such as intrinsic distances between its points and the so-called volume density function.

this Riemannian KDE cannot be used directly in a case where the data lives on an unknown Riemannian submanifold of R N . Certain tools from existing nonlinear dimensionality reduction methods could perhaps be utilized to estimate the quantities needed in the estimator of [9], however, a more straightforward method that directly estimates the density of the data as measured in the subspace is desirable.


However, an overly large choice of λ causes numeric instability, and will decrease the accuracy of the classifier found.

to find the most accurate classifier that satisfies the fairness condition one can simply perform a grid or binary search for the smallest λ such that the condition holds.


For an index tree T , let For a node G i j , the set A i j is the sum of B t k corresponding to all its descendant nodes and itself, and the set C i j the sum excluding itself.

by the definitions of A i j , B i j , and C i j , we have which implies that P ∂φ(0 .


We note that the shrinkage operator can also be expressed by S γ (w) = w − P γB∞ (w), γ ≥ 0.

(8) problem (5) can be written more compactly as inf Remark 2.


. . , x r T ) is compatible with the underlying dynamic of X.

we introduce a dynamic potential that measures the compatibility of the sequence of representatives.


The G matrix is crucial in capturing the word correlation information from the weakly-related external source U.

to incorporate the external source into the learning of the word-correlation matrix R, we regularize R according to G by introducing an internal representation of words W = (w 1 , w 2 , .


This point d corresponds to the ray l d in 3D.

the distribution of the points on the plane maps to the distribution of the gaze ray by parameterizing the 3D ray as l d (p, v d ) = p+ v d where v d = v+d and > 0.


First, the baseline intensity γ 0 captures users' inherent and long-term preferences to items, regardless of the history.

second, the triggering kernel γ(t, t i ) quantifies how the influence from each past event evolves over time, which makes the intensity function depend on the history T . a Hawkes process is essentially a conditional Poisson process [14] in the sense that conditioned on the history T , the Hawkes process is a Poisson process formed by the superposition of a background homogeneous Poisson process with the intensity γ 0 and a set of inhomogeneous Poisson processes with the intensity γ(t, t i ).


However, the model in ( 4) is only applicable to a single kernel, and the network needs to be retrained when the blur kernel changes.

this increases the complexity and difficulty for practical applications as blur kernels are of a great variety.


If we use the time complexity from [17], then finding the span of the top k singular vectors of an n × m matrix takes O(nmk) time.

if ϑ is a constant, both simple-EVD and c-EVD need O(nαr) time, whereas, Alt-Min-RPCA needs O(nαr 2 ) time per iteration [17].


Deep networks can be naturally specified as differentiable programs, and functional programs can compactly represent popular deep architectures [26].

symbolic search through a space of differentiable functional programs is particularly appealing, because it can at the same time select both which pretrained neural library functions should be reused, and also what deep architecture should be used to combine them.


To do so, note first that h f errs on an example (x, y) iff y h f (x) ≤ 0.

the exp-loss function, defined as exp(−y h f (x)), is a smooth upper bound of the zero-one error, which equals to 1 if y h f (x) ≤ 0 and to 0 otherwise.


Further for the Pascal dataset we do not have any initial depth estimate.

we start with a depth map where each point in the space is given same constant depth value.


But in more general scenarios, the inputs are likely to revert back to f 0 before another transition into f 1 , and so on.

we need at least two populations of detectors, one that detects the onset (f 0 to f 1 ), and one that detects the offset (f 1 to f 0 ).


One particular area where we feel these ideas might prove very useful is in modeling relationships between objects within an image.

in this paper we propose, in an homage to Bush, the Visual Memex Model, as a first step towards operationalizing the direct modeling of associations between visual objects, and compare it with more standard tools for the same task.


Inspired by recent progress in 3D face analysis [12,13], we introduce a dense correspondence field to bind the 2D and 3D surface spaces.

the prerequisite of our warping procedure is decomposed into two well-constrained problems: dense correspondence field estimation and facial texture map recovering.


For example, when assuming additive noise, it may happen that neither of the two directions provides a sufficiently good fit to the data and hence no decision can be made.

we would like to drop this kind of assumptions that limit the model class.


The total number of loops required for convergence of EP is typically independent of N , and can be upper bounded by a small constant, say 10, making the entire inference process O(8N 3 ) = O(N 3 ).

our algorithm has the same limiting time complexity as i.i.d.


Direct_node is too conservative in variable selection and hence it tends to include many noisy variables.

it is not surprised to see its power is large.


a valid interactive communication protocol implies the rank-one condition but not all rank-one protocols are valid interactive communication schemes.

the above is a relaxation with larger feasible set of protocols, but it turns out that the optimal solution of the above optimization problem is the randomized response, which is a valid (non-interactive) communication protocol.


We note that although DDPG trains a deterministic policy µ φ , its behavior policy, which is used to collect experience during training is augmented with Gaussian (or Ornstein-Uhlenbeck) noise [25].

actions are collected as a t ∼ N (µ φ (s t ), σ) for fixed standard deviation σ, which we will shorten as a t ∼ µ φ (s t ).


On the other hand, multiplying the matrix by k x largely cancels this effect, since k x is effectively a row of a kernel matrix similar to K , and will likely concentrate most weight on the high eigenvalue eigenvectors.

ultimately, it is not K itself, but the relationship between the eigenvectors of K and the data vector y that determines which part of the spectrum of K the result of GP regression is most sensitive to.


( 2) uniquely determines the feature mapping φ.

kernel selection (learning) is one of the central issues in kernel methods.


We assume the emission distribution is also normally-distributed, Pr(s t |x) = N (s t |x, σ 2 e ), which, from Bayes' rule, results in a piecewise normal posterior distribution where and the running average st = t t =1 s t /t.

the posterior distribution depends only on s and t, the two sufficient statistics of the sequence s 1:t . For the case of a piecewise uniform prior, the variance σ 2 t = σ 2 e t , which decreases inversely in proportion to elapsed time.


In other words, there exists a z ∈ dom(h) such that for any X * ∈ X , we have A(X * ) = z.

we can express X as where v * > −∞ is the optimal value of (1).


Consequences of such errors can be disastrous and even potentially fatal as shown by the recent Tesla autopilot crashes.

there is an urgent need for formal analysis systems that can rigorously check neural networks for violations of different safety properties such as robustness against adversarial perturbations within a certain L-norm of a given image.


Since principal neurons should be able to transmit fast changes in the stimuli, we assume that the time constant of the principal cells is small compared to that of the interneurons.

we can assume that the first equation reaches equilibrium instantaneously: where we defined !


the well-established nonlinear exogenous (NX) model, the input space consists of discretized values of the so-called manipulated variables [3].

x k at time k can be given as where (u k ) k , u k ∈ R d2 , represents the discretized manipulated trajectory.


In contrast, the and £ responses are much slower.

if $ £ ¡ ¤£ $£ ¡ ¤¦ ¦£ ¡ £ V( moves at & ¨¢ upwards as in maneuver III.1, the helicopter will tend to track the ¦ -component of the trajectory much more quickly, so that it accelerates into a climb steeper than & ¢ , resulting in a "bowed-out" trajectory.


OOPS, however, is searching in program space instead of raw solution space.

in principle it should be able to solve arbitrary instances by discovering the problem's elegant recursive solution: given and three pegs G % ¨ § ¨© (source peg, auxiliary peg, destination peg), define procedure The -th task is to solve all Hanoi instances up to instance .


Here we replace the exact computation of A i • X by employing the l 2 -sampling technique used in [15] in order to estimate this quantity by viewing only a single entry of the matrix A i (line 9 ).

an important property of this sampling procedure is that if we can estimate the product A i • X with constant variance, which is important for our analysis.


On the other hand, the most favourable case (in terms of relative efficiency) seems to be when µ 2 = • • • = µ K : then and T * = 2( √ K − 1 + 1) 2 /∆ 2 , leading to lim sup while Lai and Robbins' lower bound yields 2(K − 1)/∆.

the difference grows with K as 2 √ K − 1 log(T )/∆ , but the relative difference decreases.


We show that prequential coding on top of standard learning, yields much better codelengths than variational inference, correlating better with test set performance.

despite their many parameters, deep learning models do compress the data well, even when accounting for the cost of describing the model.


However, instead of a global image transformation, each keypoint in the image can undergo an independent transformation.

we propose a convolutional version to generate the transformed activations, called the convolutional spatial transformer.


Let Ĥ(W ) denote the piecewise continuous function achieved when the discrete problem is solved approximately using submodular optimization, then we have Ĥ(W ) ↵ • H(W ) for every W , where ↵ 2 [0, 1] is the approximation factor.


Moreover compared to Newton's method, all these stochastic variants replace the coefficient of the leading term, i.e., O(nd), with some lower order terms that only depend on d and condition numbers (assuming nnz(A) ≈ nd).

one should expect these algorithms to perform well when n d and the problem is moderately conditioned.


Similar to the membrane potentials dynamics, we can distinguish two situations: either there were supra-threshold neurons at the previous moment of time or not.

we have a model with parameters α, u 0 , τ 1 , τ 2 and N . Our main focus will be on the influence of α on the cumulative dynamics of the network.


Note that the practical value of T is usually proportional to the number of involved nodes (or workers).

the total number of workers and the upper bound of the staleness are treated as the same in the following discussion and this notation T is abused for simplicity.


As we have solved for Y * in closed form in Equation ( 8), we can plug this result into Equation ( 1) to obtain a formulation of the problem of optimizing Π * that does not require a simultaneous optimization over Y .

under these conditions, Equation ( 1) is equivalent to By evaluating the first order conditions on Equation (10), we can see that the relaxed solution, Π * , to Equation (10) must lie in the principal subspace of H n M H n . 1 for the problem of simultaneously optimizing the structure matrix, Y 0, and the partition matrix, one can use the same spectral relaxation as in Equation ( 4), and use the resulting partition matrix to solve for the optimal assignment for Y using Equation (8).


If the true gradients are g(θ, w) = ∇ w L D and h(θ, w) = ∇ θ L G , then we can define g(θ, w) = g(θ, w) + M (w) and h(θ, w) = h(θ, w) + M (θ) with random variables M (w) and M (θ) .

the gradients g θ, w and h θ, w are stochastic approximations to the true gradients.


We can see this by observing that membrane potentials after learning can be interpreted as representation errors (projected onto the read-out kernels).

learning acts to minimise the magnitude of membrane potentials.


Goal predicates are named by prepending a 'g' to the corresponding world predicate.

the MDP states are sets of world and goal facts involving some or all objects in O.


Consider the shortest path distance between arbitrary pairs of nodes i and j (that are not eliminated.)

because the corrupted nodes are removed, the whole path is around C. Also, by the uniform density assumption, walking on the shortest path in G is equivalent to walking on C with "uniform speed", i.e., each edge on the path will map to an approximately fixed distance on C. the shortest path distance scales with the latent distance, i.e., (d − 1) c , which implies Theorem 2.2 (cf.


Small values of the transition weights in the k th row of Π therefore prevent component k from being excited by the other components and by itself.

specifically, because the shape parameter for the gamma prior over θ (t) k involves a linear combination of θ (t−1) and the transition weights in the k th row of Π, small transition weights will result in a small shape parameter, shrinking θ (t) k . the component weights play a critical role in the PGDS by enabling it to automatically turn off any unneeded capacity and avoid overfitting.


In this case we attempt to make each ofTA and TB sufficient to separate A from B.

on one hand we attempt to compress, and on the other hand we attempt to make T A and T B as informative about each other as possible.


Moreover, previous approaches in this vein have relied (either explicitly or implicitly) on independence assumptions amongst the time-frequency coefficients; see, e.g., [4].

with the aim of improving upon this popular class of audio noise reduction techniques, we have used these approaches as a starting point from which to proceed with a fully Bayesian analysis.


In this work, we only consider gradient Lipschitz without assuming Hessian Lipschitz or Hessian smooth.

our result is not directly comparable to the methods in this category.


It is easy to see that any algorithm that has a positive expected payoff on some sequence necessarily loses on another sequence.

we are concerned with finding a prediction strategy that has exponentially small loss bounds but also has low regret against a number of given prediction strategies.


The minimax-optimal binwidth n is known to be of the form O n −1/(2+d) , assuming a training data of size n from a metric space of unknown dimension d, and unknown Lipschitz target function f . This setting of n would then yield a minimax error rate of O n −2/(2+d) .

the dimension d is the most important problem variable entering the rate (and the tuning of n ), while other problem variables such as the Lipschitz properties of f are less crucial in comparison.


EMD f is the maximum possible EMD for each of the three image features; it is well defined in this situation such that the largest difference between intensities is black to white, hues that are 180 • apart, and a horizontal gradient against a vertical gradient.


3 relates the metric to the original l 2 error, but with an additional term, − 1 n 2 ij d i d j , that credits mistakes if they are in the same direction and penalizes them if they oppose.

an imperfect prediction will have lower error when its mistakes are consistent with one another.


On the other hand, our model operates under a semi-supervised learning setting, where both the labeled training and the unlabeled testing samples are de-noised simultaneously.

the geometry of the sample space is better modeled on the low-dimensional subspace, by interweaving both labeled training and unlabeled testing data.


The corresponding rate loss is ∫ ti+τ ti λ(y, r π (t ′ |x, t))dt ′ .

the loss of deciding a i = −1 instead of a i = +1 is the difference in the corresponding λ-induced loss.


Here, ∆ G is the adjacency matrix such that and D∆ G [i, j] = 0, for i = j.

defining VΛV = L G as the eigen-system of the graph Laplacian, and the projection of z t onto the feature space VΛ − as zt = Λ − V zt yields: the transportation distance can be approximated by a 2 metric through a similarity transformation of the original space.


Our experimental results suggest that the quality of the embedding is sensitive to this parameter.

in the experiments we report results for the two versions of the loss: with C = 10 that is close to optimal for re-identification datasets, and with C = 25 that is close to optimal for the product and bird datasets.


The adversarial noise model (N1) is very general, only imposing an upper bound on the entry-wise noise level.

ν can be correlated with x * in some complicated unknown way.


Then, the MH ratio with the auxiliary variable is .

the intractable computation of the MH ratio is replaced by generating particles Z * and Z under θ.


In particular, letting p = µ/ µ 1 for µ a random measure taken from either D or D , we note that with high probability p 2 = Θ(n −1/2 ) and p 3 = Θ(n −2/3 ).

m = o(n 2/3 / 4/3 ) samples are sufficient to distinguish between p being uniform and being Ω( ) far from uniform.


Theoretical results indicate that for a fixed solution quality, CFR takes a number of iterations at most quadratic in the size of the game [12,Theorem 4].

as we consider larger games, more iterations are required to obtain a fixed solution quality.


( 1), we can see that the MAP problem reduces to finding a truth assignment that maximizes the sum of weights of satisfied clauses.

any weighted satisfiability solver such as MaxWalkSAT [20] can used to solve it.


in a training set, a model specialized to only one of the two questions may have troubles to answer a question like 'what is the color of glasses?'

one has to specialize models while being aware of general knowledge shared across diverse examples.


2, thus the computational complexity for the i-th scale is O((d−2i+2) 2 cH W n/s).

the computational complexity of the entire layer can be calculated as: which is definitely smaller than that O(d 2 cH W n) of the traditional convolution operation when s > 2.


With the hyperkernels so far far we can only learn kernels that are a sum of rotationally invariant terms.

the learnt kernel will have a locally isotropic character.


Let δ > 0 and v ∼ B n , the smoothed version of a function f : R n → R is defined as follows: The next lemma of [5] ties between the gradients of f and an estimate based on samples of f : Lemma 5.

let u ∼ S n , and consider the smoothed version f defined in Equation (4), then the following applies: n δ f (x + δu)u is an unbiased estimator for the gradients of the smoothed version.


The transition matrix P [x] asymptotically approximates the spacey walk or spacey random surfer.

it is feasible to compute an eigenvector of P [x] matrix and use it with the sweep cut procedure on a generalized notion of conductance.


The measurements in Figure 4 show that these feedback filters result in cueinvariant response properties at both simple-and complex-cell levels.

our unsupervised algorithm learns cue-invariant response properties from natural image data.


If we can bring these two thresholds together, we will achieve the target CA and CR rates.

the first algorithm we propose involves training a series of classifiers, attempting to make classifier n+1 achieve better CA and CR rates by focusing its effort on examples from classifier n that lie between θ CA and θ CR ; the positive examples must be pushed above θ CR and the negative examples must be pushed below θ CA .


The functions and constraints usually have different structures.

it is useful and sometimes necessary to split and solve them separately, which is exactly the forte of ADMM.


A large class of graph-based algorithms in semi-supervised learning and clustering is based either explicitly or implicitly on the cut.

we discuss first in Section 2.1 the hypergraph cut and the corresponding approximations.In Section 2.2, we introduce in analogy to graphs, the total variation on hypergraphs as the Lovasz extension of the hypergraph cut.


This objective function corresponds to a scenario without any context φ as there is only one payoff function f that we are trying to maximize.


If we focus only on the first row of the inequality system, we obtain In fact, when the contribution from E c (q) can be ignored, then applying [16, Lemma 3] shows that v(t + 1) converges linearly if −γ 1 a 0 + γ 2 1 a 1 < 0, which is possible as a 0 > 0.

if E c (t) also converges linearly, then it is anticipated that E g (t) would do so as well.


The class predicted for each example x is ĉ x argmax pi x . Hastie and Tibshirani also prove that the pi x are in the same order as the non-iterative estimates pi x ∑ j i r i j x for each x.

the pi x are sufficient for predicting the most likely class for each example.


6 Convolutional layers can be rewritten in the same form as fully connected layers (see proof of Lemma 1 in the supplementary material).

the stability results in the next section will be formulated for the fully connected case, but apply to both.


However, the complete treatment history is not necessarily available for all patients in our clinical data set.

by restricting the prediction model for a target sample only to the data from its corresponding cluster, the model might ignore relevant information from the other clusters.


Current "tall" deep learning models, however, are not captured by this setting.

theoretically analyzing the benefit of depth has gained much attention in the recent literature [31,6,9,32,23,20,25].


Convergent iterations that compute the mean can also be used in turn to compute variances using a variety of methods [4,11].

we focus on the problem of estimating the mean at each node.


Recalling that α ≥ 0, we have that sign(v j • x) = f (x) if and only if angle from v to x lies in (π/2, π/2 + α).

since the margin condition implies α ≤ π/2 − γ as noted above, we have after conditioning on G, we have that v 1 , ..., v k are still independent and the events sign(v 50π , where we used Lemma 2 for the final inequality.


In particular, in case of white gaussian noise, the noise energy is uniformly distributed over all wavelet coefficients at various scales.

at sufficiently high SNR's, the large coefficients of the signals are only slightly distorted by the noise coefficients, and the estimation of the unmixing matrix is almost not affected by the presence of noise.


For there to be any hope to learn something about the value of a given state, this state (or its neighbors) must be visited at least once.

to study the convergence rate of the algorithm, we need a way to quantify how often ⇡ samples from different regions of the state-action space Z = X ⇥ A.


We show there are problem dependent constants a(Θ, ) and b(Θ, ) such that the rates of convergence of all possible procedures are lower bounded by a(Θ, )/ √ nI * and that there exist procedures achieving convergence rates of b(Θ, )/ √ nI * , where the ratio b(Θ, )/a(Θ, ) is upper bounded by a universal constant.

we establish and quantify explicitly the tradeoff between statistical estimation and the amount of privacy.


The number of dimensions (5) and the standard deviation of the Gaussian smoothing filter (20 ms) were cross-validated to find the best possible PSTH performance.

our evaluation is focuses on each model's ability to predict trial-to-trial co-variation in firing around the mean.


To obtain a stable transition matrix, [5] propose a soft-normalization technique which is our choice in this paper.

we are interested in the LDS tuple with the constraints, ( Equivalent Representation.


Observers can more easily track 4 of 16 objects at a higher speed than 8 of 16 objects (Figure 5), even though the stimulus presentation is identical in both cases [1].

this limitation must be a consequence of uncertainty that may be modulated by task -a flexible resource [2].


But these edges have already been oriented as uw and wv by the experiment.


This theorem gives the uniqueness property by bounding the strengths of the interactions, i.e., {|J ij |} ij∈E .

the condition does not depend on the signs of the interactions.


An appropriate context may include a larger area encompassing the local neigbourhood described by w, or the entire scene in question.

the presumed independence of the ICA decomposition means that a sparse representation allows the estimation of the n-dimensional space described by W to be derived from n one dimensional probability density functions.


On Beam Rider, our network achieves low loss after learning to segment the game's light beams, however these beams are purely visual effects and are unimportant for action selection.

the game enemies, which are much smaller in size, are ignored by the network, and the resulting learned representation is not that useful for a reinforcement learning agent.


We emphasize that this operation is performed once.

instead of NM's per-iteration cost of O(p 3 ) due to inversion, Newton-Stein method (NewSt ) requires O(p 2 ) per-iteration and a one-time cost of O(rp 2 ).


Unlike DR in manifold learning, since precision and recall are almost equivalent to continuity and injectivity (Theorem 1), we are interested in embeddings in the topological category, instead of the smooth or the Riemannian category.

our work can be considered as manifold learning from the perspective of information retrieval, which leads to the following result.


Next, we examine the threshold control scheme in the equation (6), where the threshold is controlled to maintain the mean firing rate of the network at f. q(t) in equation ( 15) is equal to the mean firing rate because q(t) x i (t) under the condition x i (t) = {0, 1}.

the threshold is adjusted to satisfy the following equation, Figure 3 shows the overlap m t (t) as a function of loading rate α with f = 0.1.


Moreover, its number of rounds is logarithmic in n and L. This is in sharp contrast with the previously best known algorithm, DISCOVER [12], where the number of rounds scales with min(m, |OP T |) 4 .

fASTCOVER not only improves exponentially over DISCOVER in terms of speed but also its number of rounds decreases as the number of available machines m increases.


When dC(t)/dt < 0, pirouette probability is increased whereas when dC(t)/dt > 0, pirouette probability is decreased.

runs down the gradient are truncated and runs up the gradient are extended, resulting in net movement toward the gradient peak.


The proteins from these experiments form a suitable data set for this analysis, as they (a) are varied in terms of structure and "difficulty", (b) have previously unknown structures, which have been subsequently made public, (c) are timestamped and (d) they have been subject to contact prediction attempts by other groups whose results are publicly available.

training on CASP9 and CASP10 data sets allowed us to avoid cross-contamination.


Finally, the IL penalties are τ j (z j + |s j /σ| 2 ) 1/2 , enforcing sparsity more strongly for smaller z j .

lanczos approximation errors lead to strengthened sparsity in subsequent ILs, but least so for sites with largest true z j .


However, both of the definitions of risk and robustness based on prediction change [32] and corrupted instance [21,20] could compute large risk and small robustness for such h. In fact, in a recent work [33] it is shown that for definitions based on corrupted input, correctness might be provably at odds with robustness in some cases.

even though all these definitions could perhaps be used to approximate the risk and robustness when we do not have access to the ground truth c ′ on the new point x ′ , in this work we separate the definition of risk and robustness from how to compute/approximate them, so we will use Definition 2.1 by default.


Notice that this is a deterministic and convex optimization, which only involves three scalar optimization variables.

the optimal α * can, in principle, be efficiently numerically computed.


Consider the cost function: The absolute value ensures that we measure both positive deviations f (x) > 0 where the prototypes underfit the density the samples, and negative deviations f (x) < 0, where the prototypes overfit the density of the samples.

we focus primarily on the magnitude of deviation, rather than its sign.


a revised distribution for entries of A as yields the same output from an estimator that only takes in the observed values.

the estimators in (2.4) and (2.5), as well as the results of Theorems 1, 2, 3, can be easily adapted to the case of partially observed graphs.


But this time there is an additional complication: When the basis is overcomplete, some of the eigenvectors will be associated with zero eigenvalues, and the logarithmic probability will be linear instead of quadratic.

in such a case we must sample a cut-exponential distribution,


Then for any h ∈ I , there is no ancestor of h in an information set in D(I, a).

π σ T +1 −i (h ) does not depend on the strategy in D(I, a).


However, instead of a generalized expectation criteria, we use the prior to generate noisy labels for examples.

mFs can complement any concept learning algorithm, not just discriminative probabilistic models as in [7].


The only other significant update is the posterior computation q(w) that is O(d 3 ), where d is the dimensionality of the feature space.

this scheme is fairly efficient and has time complexity similar to that of other non-probabilistic approaches.


Then We believe that some of the constants (including lower bound on d and exact upper bound on λ/σ) can be tightened, but the results demonstrate matching scaling behavior of clustering error with d, n and λ/σ.

we see (ignoring constants and log terms) that The result is quite intuitive: the dependence on dimension d is as expected.


the frequencies are large enough then X 2 is approximately χ 2 -distributed with df = (N −1)(M −1)−s degrees of freedom, where N is the number of rows, M is the number of columns, and s is the number of parameters in the H 0 model (1 for the Frank family).

for a given significance level α the test accepts the hypothesis H 0 that the observed frequencies are a sample from the distribution formed by the expected frequencies, if X 2 is less than the (1 − α) point of the χ 2 -distribution with df degrees of freedom.


If the learning rate is non-zero, then shaping bonuses are exactly predicted away over the course of normal learning.

even though the same bonus is provided on trial 25 as trial 1, the TD error becomes 0 since the shaping bonus is predicted away.


Inspired by [48], we are able to create a k + 4 classifier to also guarantee a high classification accuracy.

we suggest that (4) needs to be minimized together with the main classifier loss where γ strikes the balance between classification and confusion.


However, the widely-available sensors that provide electricity consumption information, namely the so-called "Smart Meters" that are already becoming ubiquitous, collect energy information only at the whole-home level and at a very low resolution (typically every hour or 15 minutes).

energy disaggregation methods that can take this wholehome data and use it to predict individual appliance usage present an algorithmic challenge where advances can have a significant impact on large-scale energy efficiency issues.


Associated with the public key P there is a (possibly probabilistic) encryption function E P and associated with the secret or private key S there is a (deterministic) decryption function D S . Informally, the system should have the following security properties: • For any n-bit x, the value of the function E P (x) can be computed in polynomial time from inputs x and P . Similarly, D S (y) can be computed efficiently given y and S. • For any n-bit input x, D S (E P (x)) = x.

decryption is the inverse of encryption.


We explicitly update the exponentially-decayed moving averages µ and σ, and optimize the rest of the model using gradient optimization, with the gradients calculated via backpropagation: These gradient equations reveal another interpretation of Batch Renormalization.

because the loss is unaffected when all x i are shifted or scaled by the same amount, the functions ({x i + t}) and ({x i • (1 + t)}) are constant in t, and computing their derivatives at t = 0 gives if we consider the m-dimensional vector ∂ ∂xi (with one element per example in the minibatch), and further consider two vectors p 0 = (1, . . . , 1) and p 1 = (x 1 , .


In this case, noise averaging causes the algorithm more greedily pursues local minimizers with lower loss values.

the distribution of iterates produced by the algorithm becomes increasingly concentrated near minimizers as the learning rate vanishes (see, e.g., the large-deviation estimates in [22]).


1 is an L 2 margin maximizing separating hyper-plane.

if this margin-maximizing hyper-plane is unique, then the solutions converge to it The condition on in the above Lemma is satisfied by many popular loss functions such as logistic, exponential, squared hinge losses.


Sounds are generated based on the AR process a i = λa .

a given pattern -a smoothed version of a 0 -slowly varies, and hence the sound deviates from periodicity, but still evokes a pitch.


However, the guarantees obtained under these weaker assumptions usually require more intricate derivations, provide weaker noise robustness, and often do not hold uniformly for all potential target signals.

to keep the exposition simple and straightforward we assume (4) which is known to hold (with high probability) for various ensembles of random matrices (e.g., Gaussian, Rademacher, partial Fourier, etc).


In this sense, we can say with the higher confidence level, decisions are made more correctly while using more resources.

the efficiencies between strategies can be compared using the accuracies as well as the average number of nearest neighbors used.


Note that ρ DS is a non-monotonic function of σ T . For a range of σ T corresponding to intervals shorter than the typical duration of visual fixation, ĉD αα is significantly larger than ĉS αα .

fixational instability strongly influences correlated activity in the model when it moves the direction of gaze within a range of a few arcmin and is correlated over a fraction of the duration of visual fixation.


In large games, this value is difficult to compute since opponents cannot share private information.

we only investigate exploitability for 2-player zero-sum games.


Specifically, for a fixed value of tr(Σ), the quantity tr(D −1 σ )/ |||R||| F , is much smaller in in the presence of correlation.

when comparing (a) with (c), and (b) with (d), we see that correlation improves the performance of our test relative to SD, as expected from the bound in Theorem 3.


If a dynamical model is constructed based on a sequence of states x 1:N , then simulations generated by the dynamical model would be reconstructing the already observed time evolution of the system for all time steps t ≤ N , and predicting the time evolution of the system for all time steps t > N . We would like to train a dynamical model that we can ultimately use to predict the future time evolution of a given system.

during training we generate the A-matrix based on only the first T /2 entries of X and Ỹ , thereby enforcing that the last T /2 entries of Ỹpred are purely predictions for how the system will evolve in time.


QM9 is a widely used benchmark for the prediction of various molecular properties in equilibrium [10,40,41].

the forces are zero by definition and do not need to be predicted.


The full extensive game is divided into subtrees and each subtree is supplied its own abstraction: Definition 3.3 Let α 0 , α 1 , ..., α p be abstractions for the game Γ and for each i ∈ N , let G i = {G i,0 , G i,1 , ..., G i,p } be a grafting partition for player i satisfying I ∩ G i,j ∈ {∅, I} for all j ∈ {0, ..., p} and I ∈ α j,I i .

each abstract information set is contained entirely in some part of the grafting partition.


Since σ2 n and = σ 2 n are linear in λ and λ, we have σ2 .

for the upper bound σ2 n , note that although it scales linearly with n S , the eigenvalues of K(1) scales with n, thus does not depend on π S even for moderate sizes n S . the lower bound is not as useful as the upper bound.


The contributions u i Y will thus be formed as a superposition of u i Y = u i G + u i N . Now, by Theorem 1, we know that G will be very close to zero for the latter coefficients, while on the other hand, the noise N will be equally distributed over all coefficients.

the kernel PCA coefficients s = u i Y will have the shape of an evenly distributed noise floor u i N from which the coefficients u i G of the relevant information protrude (see Figure 2(b) for an example).


Similarly, if the optimal r is such that there exists j ∈ I min (r) (that is r(j) = r min ) and p j ≤ 2 −Λ , then a distribution Q that is completely concentrated on j has D(Q||P ) ≥ Λ and achieves ρ(Q) = r min as in the unrestricted game.


A second, distinct inference problem is to determine the correct response on the current trial from the current stimulus and the trial history.

in our approach, control and response selection are cast as inference under uncertainty.


The weight of trip i ∈ D m,t in this prediction is given by weight where are the respective parameter values for trip i, and (l 1 , l 2 , l 3 , l 4 , l 5 ) are tunable hyperparameters.

we have 6 hyperparameters to tune: (m, l 1 , l 2 , l 3 , l 4 , l 5 ).


Furthermore, for any MDP, learning its optimal Q-function is sufficient to solve the problem.

we can safely replace the distribution over tasks with the distribution over their optimal value functions.


As a result, the dimension of the input space S was at most 401, and log-softmax output vectors are theoretically at most 402 linearly independent vectors from Theorem 2.

we confirmed that the range of log-softmax is a subset of the d + 1 dimensional vector space.


On the other hand, the irrepresentability condition for the Lasso requires only that 2|ρ| < 1, i.e., ρ ∈ (−0.5, 0.5).

in the regime |ρ| ∈ [0.2017, 0.5), the Lasso irrepresentability condition holds while our log-determinant counterpart fails.


uGM (1) with bivariate nodes is equivalent to MVB (2).


At the highest sampling rate (20 Hz), connections at the fastest timescales (50 ms) were estimated most effectively, whereas at the slowest sampling rates (0.2 Hz), only the slowest timescale connections (5 s) were estimated; intermediate sampling rates (2 Hz) estimated connections at intermediate timescales (0.5 s).

dGC estimated robustly those connections whose process timescale was closest to the sampling rate of the data.


Setting m ρ = −σ −2 (1+A) −1 +δ and expanding (10) 2 ).

as N → ∞, if the stationary points at −σ −2 (1 + A) −1 + δ exist they will define a small interval of z centred on λ u (A, σ 2 ) and so define an approximate contribution of N −1 δ(λ − λ u (A, σ 2 )) to the spectrum, in agreement with the previous calculations using replicas.


However, in practice one often assumes that a node can have at most, say, 2 or 3 parents.

it makes sense to consider settings where the input is restricted, by e.g.


In Section 3, we formed a ranking by first independently drawing permutations π p and π q , of object sets {1, . . . , p} (vegetables) and {p + 1, . . . , n} (fruits), respectively, drawing a (p, q)-interleaving (i.e., a relative ranking permutation, τ Y ∈ Ω p,q ), and shuffling to form σ = τ Y [π p π q ].

an object i ∈ {1, . . . , p} is ranked in position τ Y (π p (i)) after shuffling (and an object j ∈ {p + 1, . . . , n} is ranked in position τ Y (π q (j)) ).


That is, update r(t) − 1 completed before the Read for g t , but update r(t) completed after.

our loss (for coordinate i) is g t x r(t) , and we desire a bound on We say an update s is outstanding at time t if the Read for Update s occurs before update t, but the Update occurs after: precisely, s is outstanding at t if r(s) ≤ t < s. We let F t ≡ {s | r(s) ≤ t < s} be the set of updates outstanding at time t. We call the sum of these gradients the forward gradient sum, g fwd t ≡ s∈Ft g s . Then, ignoring constant factors and terms independent of T , we show that AdaptiveRevision has a per-coordinate bound of the form Theorem 3 gives the precise result as well as the n-dimensional version.


• More importantly, as observed in the seminal work of Kleinberg [14] in most real-world graphs there are a small number of vertices (hubs) that have much higher connectivity than most vertices.

infecting the hubs is likely to infect a large fraction of vertices.


Therefore, for all i, ∂ 2 L(Q,λ) qi < 0, and (1) is strictly concave.

since Q * is an extremum of the (strictly concave) Lagrangian function, it is the unique global maximum.


However, it is generally impractical to simply initialize the model to include all possible features: exact inference in such a model is almost invariably intractable, and approximate inference methods such as loopy belief propagation [17] are likely to give highly inaccurate estimates of the gradient, leading to poorly learned models.

we propose an algorithm schema that gradually introduces features into the model, and lets the L 1 -regularization scheme eliminate them via the optimization process.


For instance, if a radar scans 180 degrees rather than 360 degrees, then any new storm cells that appear in the unscanned areas will not be observed.

the new storm cells that will be observed will depend on the scanning action of the radar.