2 It is often taken for granted in the optical flow literature, because in the limit where two images are sampled infinitesimally close in time, there are no occluded regions, and one can focus solely on motion discontinuities.

most variational motion estimation approaches provide an estimate of a dense flow field at each location on the image domain, including occluded regions.


In instances where theoretical prediction error is available, is finite, and d = 500, the relative error between empirical and theoretical prediction error for ŷpcr and ŷbc ranges from 8.60%-33.81%; for d = 5000, it ranges from 1.72%-4.86%.

the accuracy of the theoretical prediction error formulas tends to improve as d increases, as one would expect.


Motivated by this fact, the first intuition of our algorithm is that in order to minimize disaggregation error, we can discriminatively optimize the bases B 1:k that such performing the optimization (2) produces activations that are as close to A ⋆ 1:k as possible.

of course, changing the bases B 1:k to optimize this criterion would also change the resulting optimal coefficients A ⋆ 1:k . the second intuition of our method is that the bases used in the optimization (2) need not be the same as the bases used to reconstruct the signals.


With its derivations deferred to Section 2, LAG resembles (2), given by where each ∇L m ( θk m ) is either ∇L m (θ k ), when θk m = θ k , or an outdated gradient that has been computed using an old copy θk m ̸ = θ k . Instead of requesting fresh gradient from every worker in (2), the twist is to obtain ∇ k by refining the previous aggregated gradient ∇ k−1 ; that is, using only the new gradients from the selected workers in M k , while reusing the outdated gradients from the rest of workers.

with where is the difference between two evaluations of ∇L m at the current iterate θ k and the old copy θk−1 m . If ∇ k−1 is stored in the server, this simple modification scales down the per-iteration communication rounds from GD's M to LAG's |M k |.


The rejections could be incorporated into the importance weights (which would require estimating the corresponding partition function, or assuming it to be constant across rounds), but as these rejections do not depend on the data x o , we interpret them as modifying the prior: from an initially specified prior p(θ), we obtain a modified prior excluding those parameters which likely will lead to nonsensical simulations.

the predictive model ĝ(θ) does not only lead to more efficient inference (especially in strongly under-constrained scenarios), but is also useful in identifying an effective prior -the space of parameters deemed plausible a priori intersected with the space of parameters for which the simulator is well-behaved.


By assumption, the data (y i , x i ) are multivariate normal.

e V (y This suggests studying linear prediction rules of the form ŷ(x new ) = x T new β, for some estimator β of β.


This new coarsened graph can then be used as input to another GNN layer, and this whole process can be repeated L times, generating a model with L GNN layers that operate on a series of coarser and coarser versions of the input graph (Figure 1).

our goal is to learn how to cluster or pool together nodes using the output of a GNN, so that we can use this coarsened graph as input to another GNN layer.


g c gives g c = (h s + h t )/3 at the minimum, and the regularizer reduces to ).

eA can be thought to be minimizing the sum of empirical source error on h s , empirical target error on h t and this regularizer.


However, superpixel segmentation methods in general produce a fixed number of superpixels from an image, and groups of nearby superpixels may belong to the same proto-object due to the intended over-segmentation.

we extract superpixels as image fragments for pre-processing, and subsequently merge similar superpixels into proto-objects.


If the softmax output for the provided label is below a threshold, truncated L q loss becomes a constant.

the loss gradient is zero for that sample, and it does not contribute to learning dynamics.


In our case, given the current labeling f , and a particular α ∈ 1 . . . K, each segment has the following three options: (1a) a segment with the non-zero label can retain it (1b) a segment with zero label can change it to the current non-zero label of its ancestor (if any), (2) label f i can be changed to α, (3) label f i can be changed to 0 (or kept at 0 if already there).

each step results in the 2-class inference task, where U and V potentials of the 2-class inference are induced by the U and V potentials of the multi-label problem (in fact, some boundary terms then become asymmetric if one of the adjacent segments have the current label α.


However, this solution is not practical since bigram LSA assigns zero probability to unseen bigrams.


Second, because the x's are so much higher dimensional than the y's in our applications of interest, the partitions would be dominated by the x's, unless we chose a partitioning strategy that emphasized the y's.

our strategy mitigates this difficulty (while certainly introducing others).


• The interesting distributions the system shall simulate belong to a small class of distributions.

the model does not need to approximate all distributions.


Due to the loss of phase information, signal reconstruction from magnitude-only measurements can be quite challenging.

despite a variety of proposed methods and analysis frameworks, phase retrieval still faces fundamental theoretical and algorithmic challenges.


We propose a novel prior formulation, the structured spike and slab prior, which allows to incorporate a priori knowledge of the sparsity pattern by imposing a spatial Gaussian process on the spike and slab probabilities.

prior information on the structure of the sparsity pattern can be encoded using generic covariance functions.


When one minimizes FH(P) with respect to P under the constraint of normalization L x P(X) = 1, one obtains PH.

computing marginals of PH such as PH(Xi) or PH(Xi, Xj) involves sums over all states, which is intractable for large n. one needs tractable approximations to PH.


We show that similar to Fourier analysis, spectral clustering methods are global in nature.

even with a location dependent σ(x) as in [10], these methods typically fail to simultaneously identify clusters at different scales.


Therefore, we will calculate analytically the change of the synaptic weight by correlating the derivative of the membrane potential with the activity of the NMDA channel.

for this calculation we utilise biophysical variables of the physiological cell.


In recurrent neural network (RNN) language modeling, given a corpus of tokens Y = (Y  This theorem shows that the length of the hidden vector in the output layer determines the representational power of RNN with softmax.

in language modeling, the rank of A can be extremely high since contexts can vary and vocabulary size M is much larger than d. the softmax can be the bottleneck of the representational power.


The accuracy of this approach improves with the number of constraints considered, but performance is fastest with fewer constraints.

because ascending the likelihood function requires evaluating the likelihood at many different points, we can make this ascent process much quicker by applying a version of the coarse-to-fine idea.


By the Eckart-Young theorem X k = XV k V T k , where V k ∈ R d×k is the matrix whose columns are the top-k right singular vectors of X (see, for example, e-Chapter 9 of [14]).

the optimal linear encoder is H opt = V k , and the top-k PCA-features are Z pca = XV k . Since its early beginings, [19], PCA has evolved into a classic tool for data analysis.


In particular, observe that for particular values of T , Similarly, for the Jacobian of the layer transform, .

depending on the output of the transform gates, a highway layer can smoothly vary its behavior between that of H and that of a layer which simply passes its inputs through.


Given the fact that an MFP can be reduced to an SOCP [7], we can reduce Problem 3 to the following SOCP: Problem 5.

we obtain the following result:  In this section, we discuss the properties of the proposed MTL method and the relation to existing methods.


Since the boundaries are Lipschitz, the expected volume that is incorrectly assigned to any decision set is > c 1 n −1/d , where c 1 > 0 is a constant.

if the expected excess risk at a point that is incorrectly assigned to a decision set can be greater than a constant c 2 > 0, then the overall expected excess risk is > c 1 c 2 n −1/d .


Suppose that a single machine (m = 1) receives n i.i.d.

observations X i according to P . Since the packing entropy is lower bounded as log M Θ (δ) ≥ log(1/δ), the lower bound (6) implies setting B = 1 2 log n yields the lower bound M ind (θ, P([0, 1]), B) ≥ e −2 4n .


Since the truncated SVD requires s to be set beforehand, we need to efficiently estimate the number of singular values such that must meet the condition of σ i ( Ŵ (τ ) hh ) > 2 − δ.

we compute upper bounds of the singular values that meet the condition on the basis of the following lemma: Lemma 6.


Since the SDP approximation TSVM [14] has very high time complexity O(n 6.5 ), which is difficult to process data sets with hundreds of examples.

it is only evaluated on the smaller data sets, i.e., "IBM-s" and "Course-s".


We can add any unit clause of the form ∀p i ∈ ∆ pi , q j ∈ ∆ qj : S(p i , q j ) or ∀p i ∈ ∆ pi , q j ∈ ∆ qj : ¬S(p i , q j ) to the theory α(S) in Lemma 1 and the theory is still domain-liftable.

theories in S 2 FO 2 and S 2 RU are domain-liftable.


There is no reason to limit the applicability of this approach to supervised learning.

for example, adaptive density estimation using negative log-likelihood as a loss can be regularized and the solution path be tracked using our algorithm.


Equivalently, y is an eigenvector of K µ ⋆ + λI with eigenvalue λ, which is equivalent to y ∈ null(K µ ⋆ ).

since the product of PDS functions is also PDS, (*) must be non-negative.


2.3: in general, the mixing matrix may be unknown, or, even if it is known, it may depend on the observation, x.

we need a procedure to design losses that are proper for a wide family of mixing matrices.


The set T is a hyperparameter of our method, on which we elaborate in Section 6.

the self-labeled set S T is defined as for any x ∈ S, j is the label of T j (x).


As Hopfield networks are equipped with a Lyapunov function, convergence is guaranteed.

in the limit of many weak connections per neuron, Hopfield networks exactly implement a continuous-time variant of belief propagation starting from message initialisations that prevent from running into convergence problems.


. . , v T ∈ X c , we will show how to construct distinct vectors v 1 , .

. . , v T ∈ X c such that we will have As our construction will work for any choice of distinct vectors v 1 , .


The equality above is due to our Gaussian approximation of p( θ| x t+1 , r t+1 ).

we need to minimize E rt+1| xt+1 log det[C t+1 ] with respect to x t+1 .


There are many families of copulas with different dependence shapes and different numbers of parameters, e.g.

the multivariate Clayton copula family with a scalar parameter α: for a given realization r, which can represent the counts of two neurons, we can set , where F Xi can be arbitrary univariate cdf's.


In the meantime, the residual block benefits quick convergence while ASPP aggregates multiscale features to enhance the capability of shape recovery.

we successfully obtain the semantic completed scene volume where C denotes the number of categories we predict.


As input for our models, we consider a collection of I documents, each document i having J i event mentions.

each event mention is characterized by L feature types, FT, and each feature type is represented by a finite number of feature values, f v. we can represent the observable properties of an event mention, em, as a vector of pairs (FT 1 : f v 1i ), .


The main interest of the first and third points is that we obtain a final estimator of θ 0 which is at least provably consistent: indeed, although our criteria are obtained from an assumption of independence, the generalization performance result also holds for "weakly" dependent observations and thus ensures the consistency of our approach.

as opposed to most previous active learning heuristics, our estimator will always converge (in probability) to the ML estimator.


Compared to the initialization, transitivity relations are learned and the approach can generalize from an exemplar to more related samples.

the similarity matrix becomes more structured (cf.


Indeed, a formalization would likely already solve the problem, yielding an algorithm to maximize it.

the arguments in this section are mostly qualitative.


While many different interpretations are possible for the notion of long-lived, here we consider agents that have to solve a sequence of tasks over a continuous lifetime.

our problem is closely related to that of transfer learning in sequential decision-making, which can be thought of as a problem faced by agents that have to solve a set of tasks.


To make the resulting optimal control computations numerically tractable, future observations need to be assumed to coincide with their most likely values (see e.g., [19,5]).

at every time step, the agent solves a constrained nonlinear optimization problem min which returns an optimal sequence of controls u 0:N −1 minimizing the objective function J.


It is also applicable to RNNs since the similar formats of activation functions.

we introduce a constant sloping factor in the activation functions to guide the activation distribution concentrating to the two poles as much as possible.


The stopping condition grants rf (A t ) 6 2 Ker(A > ).


The small oscillations in the rotation angle and in the horizontal optical angle are due to reaction delay and uncertainty; they are also predicted by GOAC and LOT.

in this well-studied case, the model produces an optimal policy that exhibits behavior which is fully in accordance with the heuristics.


If the width is decreased more structure is to be expected.

for ¥ © ( $ © , two more minima appear, as seen in Fig.


However, for Web-2, because our implementation for testing ordinal classification required too much memory for M = 2000, we did not obtain the final test NDCG scores; the partial results indicated that ordinal classification did not improve classification for this dataset.

we compute the p-values using classification results for Web-2.


Note that during normal play, we do not have access to y t . We can however utilize past games for training as replaying them in StarCraft provides both o t and y t . We note that humans generally make forward predictions on a high level at a long time-scale -humans will not make fine predictions but instead predict a general composition of units over the whole game map.

we propose to model a low-resolution game state by considering accumulated counts of units by type, downsampled onto a coarse spatial grid.


At any unsolved frontier node n, draw one sample from q(x >n |x ≤n ) in (7).

16: end function importance weights resulting from our two-step sampling can enjoy the same type of bounds described in (9).


Since most state-of-the-art trackers included in our empirical comparison use only grayscale images, we have converted all the sampled images to grayscale (but our method can also use the color images directly if necessary).

each image is represented by a vector of 1024 dimensions corresponding to 1024 pixels.


. . , v T ) is simply a permuted version of the vector z t (v 1 , .

. . , v T ) which means that their j th largest elements must agree: and the lemma follows as a consequence .


Detailed balance condition becomes For example, the well-known Metropolis-Hasting algorithm has The analogous equations for PEIL on the lattice are, In applying the acceptance formula, each component is treated independently.

moves can be accepted on some components and not on others.


In particular, our approach differs from other batch methods, which need to wait for all data to be received before starting computation, and require multiple passes over the data.

our approach is space efficient, and hence can handle the data-rich scenario with a large, increasing number of samples.


The xent-variance penalty can be further expanded as: where ent(•) denotes the entropy.

v x combines the KL-divergence penalty with an entropy penalty, which has been shown to perform well in a semi-supervised setting [7,14].


On the other hand, a logistic and probit approach both 0 and 1 at the same rate, and therefore cannot model the sparsity/skewness of the label matrix like the Bernoulli-Poisson link.

in contrast to multilabel learning models based on logistic/probit likelihood function or standard loss functions such as the hinge-loss [25,14] for the binary labels, our proposed model provides better robustness against label imbalance.


From the definition of Φ, this problem resembles a traditional k-means clustering problem with distance function d(x i , x i ) = (b i −b i ) 2 .

one could use one of several clustering algorithms to solve it.


Given a gradient at step t, i , y We assume that only one data instance of one party changes in neighbouring datasets D and D . Hence the sensitivity bound, ∇J(θ, D) − ∇J(θ, D ) ≤ 2G mn (1) .


Consequently, we can equate the output current of the bump circuit with the probability of the input under a distribution parameterized by mean µ: ( ) In addition, increasing the similarity between V in and µ is equivalent to increasing P(V in |µ).

the adaptive bump circuit adapts to maximize the likelihood of the present input under the circuit's probability distribution.


Since we are interested in optimizing performance rather than evaluating it, we evaluate the posterior distribution of the gradient of η B (θ).

for the mean we have in BPG we cast the problem of estimating the gradient of the expected return in the form of Eq.


Given the restrictions imposed on µ and σ, one can show that the normalizing quantity Z = 1 0 N (x; µ, σ 2 ) is always greater than one-third.

to sample from the posterior we simply draw from the original, closed form posterior distribution, discarding (and re-sampling) if µ ∈ [0, 1], σ ∈ [0, 1], or with probability 1 − (3Z) −1 .


Therefore, the maximum value is observed in every 2 k−1 ≤ 2 sd samples.

a direct application of LearnBool yields poly(n, 2 k−1 ) time complexity, which improves the O(n d ) bound for small s and d. Improving further, we provide a more efficient algorithm tailored for the hypergraph sketching problem, which makes use of the unique sign property and some other properties of the cut function.


For this neuron, incorporating non-Poisson renewal properties into a model with spike history dependent intensity (light gray) provides only a modest (<1%) increase in cross-validation performance.

in addition to being more tractable for estimation, it appears that the generalized linear modeling framework captures spike-train dependencies more accurately than a non-Poisson renewal process (at least for this neuron).


Once sketches are constructed, the estimation procedure will be the same regardless whether the original data are dynamic or static.

we will use static data to verify some estimators of CRS.


This guarantees that linear constraints of P(X N ) converge to the quadratic constraint of P, and hence the two problems match.

now since S ⊆ T N for all N , it is easy to see that S ⊆ lim N →∞ T N . To prove the converse, let t 0 ∈ lim N →∞ T N but t 0 ∈ S. d(t 0 , S) > 0.


( 2), this parameter w, while achieving low distortion, could potentially be in a neighborhood of parameters with high distortion.

we can optimize a more robust objective that takes into account perturbations.


Let A be any (deterministic) clustering algorithm which, given the full sample X m+u , can cluster this sample into any desired number of clusters.

we use A to cluster X m+u into 2, 3 . . . , c clusters where c ≤ m. the algorithm generates a collection of partitions of X m+u into τ = 2, 3, . . . , c clusters, where each partition is denoted by C τ .


In contrast, the DEPM indicated its best performance with the largest truncation level (T = 128).

we confirmed that the uniqueness of the optimal solution in the DEPM was considerably important in achieving good generalization ability and link prediction accuracy.


The following result has an alternate interpretation for λ: Result: The sparsegen formulation (Eq.1) is equivalent to the following, when γ = 1 1−λ (where The above result says that scaling g(z) by γ = 1 1−λ is equivalent to applying the negative L-2 norm with λ coefficient when considering projection of g(z) onto the simplex.

we can write: This equivalence helps us borrow results from sparsemax to establish various properties for sparsegen.


On the other hand, the approximately convex function fw (x) for f w (x) we consider will be such that fw (x) = G(x) except in a very small cone around w. Picking w at random, no algorithm with small number of queries will, with high probability, every query a point in this cone.

the algorithm will proceed as if the function is G(x) and fail to optimize fw .


then, the following bounds hold: where z k globally converges to z ?


This theorem shows that we can use OCSVMs to estimate high-density regions in the input space while bounding the number of examples in X lying outside these regions.

by setting ν = 1−α, we can use OCSVMs to estimate regions approximating C(α).


We assume that all these assignments are equi-probable (have the same weight) in the proposal.

to compute the (un-normalized) probability of a sample w.r.t M, we first convert the assignments on a specific sample, Ŝ(t) into one of the equi-probable assignments s by randomly choosing one of the assignments.


When K < M , the subproblem with respect to s has solution s = (B B + σ 2 Φ) −1 x.

lower dimensional random vector s has less variables among which each variable is a linear combination of the original variables of x with the combination weights stored in W = (B B + σ 2 Φ) −1 .


For |M | steps and i = 1, 2, . . . , k, perf i (M , π 2k 1 ) = perf i+k (M ′ , π2k k+1 ).

this yields an NEXP NP algorithm to decide the competitive agents problem.


A typical iterative minimization process based on the alternating direction method of multipliers (ADMM) [15,16] can be written explicitly as However, minimizing the augmented Lagrangian function with respect to M i , L and E using ADMM is expensive in practice, and moreover, the global convergence can not be guaranteed.

we propose to solve the above problem by taking one proximal gradient step.


What is common to current linear and non-linear methods is that they are often used to model the instantaneous relationship between brain signals and particular behavioral parameters.

they do not account for the temporal evolution of movement parameters, and can also not directly provide uncertainty in their predictions.


Nevertheless, in our FESL problem, on rounds T 1 + 1, . . . , T 1 + T 2 , w 2,t cannot satisfy the requirement in the beginning due to insufficient training data and w 1,t may become worse when more and more data come causing a cumulation of recovered error.

it may not be appropriate to combine the two models all the time, whereas dynamically selecting the best single may be a better choice.


Then, f i ∈ L 2 (P n ), which we equip with the inner product and it exists whenever β > 1 2 (n − 1).

h β is positive for all β defined by ( 18).


Using those directly as input to our model would require the model to learn to map each keypoint to a position on the human body.


The distribution of X = w T u follows the distribution of the f i , which is approximately exponential.

the projection of all inputs onto the weight vector has an approximately exponential distribution.


(3) Third, the lack of sufficient evidence or an insignificant improvement of the metric may make it undesirable from a practical or financial perspective to replace the default.

when a company runs hundreds to thousands of A/B tests within a year, ideally the number of statistically insignificant changes that it made should be small relative to the total number of changes made.


Instead, it computes a sufficient statistic during each update, x and P , which consists of the maximum likelihood estimate of r and b, and the covariance matrix of this estimate.

we can run this algorithm online as we learn, and its speed does not deteriorate over time.


The proof in [13] is based on the term αη k φ(s, a) dominating the sum when α → ∞.

this type of a greedy update is a special case of a natural gradient update in which the step-size approaches infinity.


For SNR estimation, the input signal is transformed into spectro-temporal input features, which are neurophysiologically-motivated: experiments on amplitude modulation processing in higher stages of the auditory system in mammals show that modulations are represented in "periodotopical" gradients, which are almost orthogonal to the tonotopical organization of center frequencies [5].

both spectral and temporal information is represented in two-dimensional maps.


Local regions on the surface are rarely distinctive enough to determine the correct correspondence, whether because of noise in the scans, or because of symmetries in the object shape.

the set of candidate correspondences to a given point is usually large.


Then, let m be the cardinality of Θ, we have m ≤ 4N .


Under this arrangement, M can be factorized as M = AB T , where A is a m × 4 camera matrix and B is a n × 4 structure matrix with the last column of B, an all-one vector.

m is a rank 4 matrix with a special structure for the last column of B.


w(x) is an estimate of the importance w(x) which is non-negative by definition.

it is natural to impose w(x) ≥ 0 for all x ∈ D, which can be achieved by restricting α ≥ 0 for = 1, 2, . . . , b.


Also, the support is distributed evenly across the xi 's (4.1).

we show that even large elements are not recovered.


Furthermore, we can check the label w assigns to x j by which is a function of the Grahm matrix.

sampling from V ∩ T boils down to the problem of sampling from convex bodies, where instead of sampling a vector directly we sample the coefficients of the orthonormal basis t 1 , .


We know from Theorem 2.1 that Ψ n ≤ Ψ n−1 + β n − Z n , where β n is non-stochastic and Z n is a quantity of positive expected value.

in expectation, and modulo a small additive term, Ψ n decreases monotonically.


Since f (V ) has full rank, the solution of this problem corresponds to a global optima of , and hence to a global optima of the original optimization problem.

for sufficiently large networks, finding global optima is generally easy, and they are in a sense ubiquitous.


A necessary condition for identification is that the total number of spikes generated in response to all N trials is larger than Proof: The equivalent representation of the t-transform in equations ( 4) and (7) implies that the decoding of the stimulus u m nm (in Theorem 1) and the identification of the filter projections Ph m nm encountered here are dual problems.

the receptive field identification problem is equivalent to a neural encoding problem: the projections Ph m nm , m = 1, ..., M , are encoded with an mTEM comprised of N neurons and receptive fields u im nm , i = 1, ..., N , m = 1, ..., M . The algorithm for finding the coefficients h m l1...ln m is analogous to the one for u m l1...ln m in Theorem 1.


Note that s X and a X are subsets of φ X , and are already calculated as part of trajectory optimization.

computing optimal feedback gains comes at very little additional cost.


The zero rows provide no information on which columns of Q T X or U T X are extreme rays or extreme points.

we can restrict ourselves to finding the extreme columns of R and ΣV T . These matrices are n × n, and we have significantly reduced the dimension of the problem.


In the same reason, ones set γ = 1− exp(−α) for the EXP penalty.


Moreover, it is onto the interior of M, denoted by M o .

for any mean parameter µ 2 M o , there exists a canonical parameter ✓(µ) 2 ⌦ such that E ✓(µ) [ (X)] = µ.


the whole image), the notion of density is not well-defined at a pixel level.

given a set of dotted annotation there is no trivial answer to the question: what should be the ground truth density for this training example.


(2010) attained an O(n −1/2 ) convergence rate with bandit feedback, which Shamir (2013) showed is unimprovable.

in the single-player case, the bound (5.3) is off by n 1/6 and coincides with the bound of Agarwal et al.


Because the triangular Cholesky factor changes smoothly with the matrix square root, this modification does not change the behavior of the CMA-ES in terms of required objective function evaluations as verified empirically.

the described algorithm can and should replace the standard CMA-ES if updating and storing the covariance matrix matters.


Property P1 ensures that such a J, if exists, is unique.

when the condition is satisfied, the only permutations that contain edge e i are σ j , j ∈ J.


Let κ(Σ) be the condition number of the covariance matrix Σ, i.e., the ratio of the maximum and minimum eigenvalues; refer to Section 2 for other relevant definitions. .

there exist two covariance matrices Σ and Σ that are good entrywise approximations: but the corresponding distributions are as far as they can get, i.e., given a black-box algorithm that provides an entrywise ε-approximation to the true covariance matrix Σ, if ε ≥ 2 κ(Σ)+1 , it might output Σ, which does not approximate Σ in KL divergence or total variation distance.


Generally speaking, user U i gives item For example, if the ground-truth score of item I j is 4.8 for user U i , then user U i makes a noisy rating by giving I j score 5, yet makes an attack by giving I j score −3.

we assume that Z F ≤ δ, where δ is a small parameter.


As described previously, a spherical Gaussian is the only SSD with independent components.

a natural solution for eliminating the dependencies in a non-Gaussian spherically symmetric x wht is to transform it to a spherical Gaussian.


From the form of the truncated ΓP, thanks to the additive property of independent gamma random variables, the total sum of λ k over countably infinite atoms follows a gamma distribution as ∞ k=1 λ k ∼ Gamma(γ 0 , c 0 ), wherein the intensity function of the ΓP has a finite expectation as E[ .

the ΓP has a regularization mechanism that automatically shrinks the number of atoms according to given observations.


In general, if c " 4 " 1 C we can conclude that " 1 arg min " 4 C c min " 1 .

calculating the constants shows " 1 arg min " 4 3 min " 1 , yielding Theorem 4.3.


Alternatively, many studies compute the oscillatory activity in specific frequency bands by filtering and squaring the signal prior to averaging.

changes in oscillatory activity are termed event related synchronization or desynchronization (ERS/ERD).


The regularities mentioned above reflect different aspects of the learning problem, and are not directly comparable in general.

it is appealing to develop an algorithm that adapts to the smaller regularity of the problem.


Another alternative is to resort to synthetic imagery which would provide point correspondences by construction, but this has the downside of making it harder to generalize to larger appearance variations encountered in real image pairs we wish to handle.

it is desirable to be able to train directly from pairs of real images, and using as little annotation as possible.


As such, different users may be interested in very different partitions.

different clustering methods detect distinct types of structure, often yielding radically different results on the same data.


This priming mechanism yields priors on average that match the presentation probabilities in the task, e.g., .17 and .83 for the two responses in the 17:83 condition of the Jones and Braver experiment.

when we report results for overall error rate and reaction time in a condition, we make the assumption of rationality that the model's priors correspond to the true priors of the environment.


We give closed-form expressions of the likelihood gain and the weights of newly added features under contrastive divergence approximation [23] (other type of inference can be done similarly).

the CVI process can be very efficient, only adding small overhead to the training of a regular treeRMN.


, which is also the optimal solution to (1) and thus the model parameter.

we further note that the equivalent min-max problem ( 5) is now in a form that sums over T = the empirical average has been brought out of the logarithmic loss and we are ready to apply stochastic gradient methods.


However, even when θ is a feasible point, the unique optima of these sub-functions may not be feasible when combined.

the feasibility function g(θ) of Theorem 3 must be utilized in this case.


On the contrary, our proposed REFUEL algorithm can acquire 1.31 positive symptoms.

this experiment indicates that our algorithm is beneficial for sparse feature exploration.


We observe that in higher dimension error fluctuations for ADMM increases noticeably (see Figure 1).

projections of REASON 1 play an important role in denoising and obtaining good accuracy.


We have Let i be the total weights at time t. For each t, similarly to (3), we have For each i such that err Di (g (t) ) ≤ 12 , by Lemma 5, know that Pr[i ∈ Z (t) ] ≤ 1 100 .

if we take the expectation over the randomness of WEAKTEST at time t, we have, When χ (t) = 0, similarly to the proof of Theorem 3, we have Pr i∼p (t) (•) err Di (g (t) ) > 12 ≤ .1, and Combining ( 7), (8), and ( 9), we have (when Together with (6), we have E W (t+1) W (0) , .


In round 0, pattern nodes are initialized by a pattern x, sampled from dataset X , perturbed by external errors z, i.e., x(0) = x + z.

for cluster we have x ( ) (0) = x( ) + z ( ) , where z ( ) is the realization of errors on subpattern x ( ) .


Briefly, CG ensures that z after iteration k is a maximizer of q over a (Krylow) subspace of dimension k. For details about CG and many other approximate linear solvers, see [15].

z "converges" to p (the unconstrained maximizer of q) after n steps, but intermediate z can be used as approximate solutions.


Moreover, the regime of operation of the policy (the points achieved by the policy in the 2-D plane ) can vary a lot depending on the choice of cost function even though the same weights are used.

the cost should be chosen based on the disease.


At each iteration, the computational cost is dominated by the update of Z (−n) (n = n), which can be obtained by where (−n, −n) denotes every mode except n and n, which greatly reduces the computation.

when n = n, the updates of Z (−n) , e, Z (−n) e and Diag(Z  We provide convergence analysis for our stagewise algorithm in this section.


This construction introduces a single edge in the graph each step, although it may be a duplicate of an edge that already exists.

this technique generates multigraphs one edge at a time.


Our setup in (12) introduces a projection operator before Ψ γ a γ in the basis (11) and therefore it can be viewed as the unique hybrid parametrization, which confines the function modeled by γ to be orthogonal to the span the β basis.

there is no correlation between optimizing a γ and a β , making the problem more well-conditioned.


By Hoeffding's inequality and union bounds, we have that with a probability of at least It can be easily seen that ŷi = y i for all i ∈ J.

for all x such that κ(x, U(S a (t))) ∈ J, h nn S (x) = h nn Sa(t) (x), and hence err( The second term is at most ν(t), and it remains to bound the first term, on the condition that E holds.


In contrast the smoothness constant L 1 under 1 norm depends on the absolute value of the entries in ); no saving is achieved.

for scheme (b), the bilinear function L 2 1 saves a factor of n compared with L 2 2 .


we have x ≤ R, ∀x ∈ ∂φ(0), where R is a finite radius constant.


From Lemma 2, , and thus Ũ = ∅, which conflicts with the assumption of Ũ = ∅.

there is no direction that can decrease f 1 ( Ũ(∞) ), i.e.


We would like to note that this objective function is not convex due to the concave entropic term in the definition of .

the learning procedure described in the sequel converges to a local minimum or a saddle point of O.


However, more usefully, we can relax the integer constraint to obtain a semidefinite program that yields soft cluster assignments which approximately maximize the margin.

one can obtain soft clusterings efficiently using widely available software.


A Bayesian network is a triple (X, G, θ), where G = (N, A) is a directed acyclic graph (DAG) whose nodes are in one-to-one correspondence with variables in X, and θ = {θ i (x i , x Gi )} is a set of numerical parameters specifying (conditional) probabilities θ i (x i , x Gi ) = Pr(x i |x Gi ), for every node i in G, value x i of X i and assignment x Gi to the parents G i of X i in G. The structure G of the network represents a set of stochastic independence assessments among variables in X called graphical Markov conditions: every variable X i is conditionally independent of its nondescendant nonparents given its parents.

a Bayesian network uniquely defines a joint probability distribution over X as the product of its parameters.


Recall that the sample complexity of learning tree models is Ω(log p) [23].

the complexity of learning sparse Erdős-Rényi random graphs is akin to learning trees in certain parameter regimes.


Even though graph isomorphism is relatively fast in practice, a theory may have exponentially many symmetries.

breaking all the symmetries is often impractical, though partial symmetrybreaking is still useful.


Recall that the loss function is defined as follows: weight of the edge ((n + 1) a , ⊥) is y t , weight of the edge ((n + 1) b , ⊥) is 1 − y t ; other edges have loss 0.

the loss suffered by the policy π S is 1 if PAR S (x t ) = y t and 0 otherwise.


Moreover, 0:+ grows as the statistical dependence between A and B diminishes.

given a domain with a range of degrees of statistical dependences, the number of edges in the learned graph increases monotonically with growing scale parameter 0: when each variable has at most one parent (i. e., in the class of trees or forests).


In practice, however, we do not know which of the two events is correct.

the ideal choice in either case can be expressed using indicators as follows: Taking the expected value of A ij (X) with respect to p(x|C i (x) ∨ C j (x)) yields This results in a single likelihood correction S ij (x) = Lij (x) + α ij that is obtained simply by integrating in Equation 5 with respect to the average distribution over class c i and c j , i.e., With these assumptions, and supposing the zero set is Lebesgue measurable with measure µ, we get: where As can be seen, α ij is composed of two factors, the integral and the 1/Ψ(P i , P j ) factor.


However, the RTRBM does not allow EM-like learning because the equation ∇log is not meaningful.

to be precise, the gradient ∇log is not, in general, a continuous function of W . the gradient has to be computed differently.


Here, we minimize a hinge loss, which is defined on the joint prediction rule, while MLE minimizes the log-likelihood loss log p(s d |g d , θ), which does not depend on the latent object model.

our approach can be expected to achieve a closer dependence between the conditional scene model and the latent object model.


Neural action potentials have a finite spatial extent [6].

the spikes can be divided into distinct groups based on the geometry of the MEA and the local position of each neuron, and each group is then processed independently.


2 merely contains matrix multiplication in the fraction.

as long as the TVI operators are necessary in the task, no matter where it is needed in the architecture and the type of layer before or after it, the interpolation layer can be seamlessly plugged in.


A group's mixing weights are randomly chosen from w 1 = [0.33, 0.33, 0.33] or w 2 = [0.84, 0.08, 0.08].

a group is normal if its points are sampled from these three Gaussians, and their mixing weights are close to either w 1 or w 2 .


As defined in (5), our regularizer has a parameter, β i , associated with each training point, which can be used to encode the information of the marginal or input distribution, P (x), by setting where λ is a tradeoff or regularization parameter.

the use of β i would make the regularization take effect only in dense regions although our experiments reported were carried out by setting β i = 1 2 ; i.e., we were using a weak assumption that data are scattered uniformly throughout the whole space.


We specify the upper bound by assuming a situation when the generator is perfect.

we replayed actual past data paired with the predicted targets from the old solver network.


( 5): learning will quickly compensate for the approximations and/or sub-optimality of A and B obtained by matrices V and Λ returned by Algorithm 1.

these are the matrices we have used for the experiments described in next section.


The U-Tree algorithm automatically builds a state-feature representation starting from one that makes no distinctions between different observation vectors.

no specification of state-feature dependencies is necessary prior to learning.


With probability Pr(d L ) • Φ(0 | µ t , σ t ), the hidden state x is less than 0, making A R an incorrect decision and resulting in a penalty and earns a reward of R P . The inverse is true for A L . When A S is selected, the animal simply receives an observation at a cost of R S . Computing the value function defined in ( 5) involves an expectation with respect to future belief.

we need to compute the transition probabilities over belief states, T (b t+1 |b t , a), for each action.


In order to get an expression for the weight change in a specific time step t k we write the probabilities occurring in (7) as products over individual time bins, i.e., P ( ), according to the chain rule of information theory [5].

we rewrite L 12 as a sum over the contributions of the individual time bins, The weight change ∆ wk 1j is then proportional to the gradient of this expression with respect to the weights w 1j , i.e., ∆ wk 1j = α(∂∆L k 12 /∂w 1j ), with some learning rate α > 0.


However, these assumptions involve data, which are sampled from certain statistical distribution.

we need to verify that these assumptions hold with high probability under mild data generation process of (i.e., GLM) in high dimensions in the following lemma.


By the union bound, this event happens with probability 1 − tδ = 1 − δ/4.

consider the set of players i / ∈ I (r) ∪ G r . These are the players for whom the classifier of the round performed well but FASTTEST made a mistake and did not include them in the set G r . By linearity of expectation: the expected value of the potential function in round r conditioned on its value in the previous round is bounded by 1) .


This permits the affinity metric to efficiently pull together parameters θ m and θ n . However, for mixture models, there is the presence of hidden variables h in addition to observed variables.

we replace all the marginals p(x|θ n ) = h p(x, h|θ n ).


Theorem 1 provides an upper bound on the number of measurements that is equal to a constant factor times the statistical dimension of the descent cone, d(T L λ (x 0 )).

even though our analysis is not exact in this section, it leads us to the important observation that our proposed method is order-wise optimal in terms of the required sample complexity for perfect signal reconstruction.


Examples of such POMDPs are Utile Suffix Memory [14], infinite generalized policy representation [13], and infinite POMDPs [3].

an agent's beliefs about other agents are not part of the solutions of POMDPs.


However, it is highly likely that some sensors might fail to send the measurements correctly and sometimes report totally irrelevant measurements.

it is more accurate to employ the observation model (3) than model (1).


When reaction time and system noise become sufficiently large, the agent fails to intercept the ball (upper right grayed out area).

seemingly substantially different behaviors can be explained by means of a single model.


[3], we employed a learning rate schedule which decayed according to γ/d where d is set to the number of unique "real" datapoints seen so far.

our learning rate schedule converges to γ/N in the limit, where we have defined N = |D|.


Specifically, v(•) is called a monotone operator [21] if the following holds (with equality if and only if x = x): If v(•) is monotone, the game admits a unique Nash equilibrium x * which (per the property of a Nash equilibrium) satisfies v(x * ), x − x * ≤ 0.

if v(•) is a monotone operator, it follows that v(x), x − x * ≤ v(x * ), x − x * ≤ 0, where equality is achieved if and only if x = x * .


However, obtaining sublinear regret is generally impossible in this case.

competitive analysis studies much weaker performance metrics such as the competitive ratio, making it orthogonal to our work.


However, training-time requirements of sophisticated machine learning algorithms can overstrain the hardware of real world robots.

in most cases, ad hoc methods, hard coding of expert knowledge, and hand-tuning of parameters, or similar approaches were preferred over the use of learning algorithms on the robot.


This implies that a large spread of the prior forces the Bayes factor to favor the null model.

as in the case of the Zellner g-prior [4], Bartlett's paradox arises for the Silverman g-prior.


Bounding the expected variance by the empirical variance is generally tighter than bounding it by the empirical loss.

the PAC-Bayes-Empirical-Bernstein bound is generally tighter than the PAC-Bayes-kl bound, although the exact comparison also depends on the divergence between the posterior and the prior and the sample size.


. . , ξ N of ξ are given and that they corresponds to sensors measurements.

the distributionally robust counterpart of ( 8) is of the form of ADRO (6).


How the noise affects the firing probability is also a function of the parameters of the neuron model.

it is important to take into account the refinements of simple neuron models in terms of subthreshold resonance [3,4], spike-triggered adaptation [5,6] and non-linear spike initiation [7,5].


Thus, if Σ −1 i has strictly positive eigenvalues, so do Σ −1 i+1 and Σ i+1 .

σ i and Υ i are indeed PSD non-singular, as assumed above.


The proof of Theorem 3 relies on two facts.

first, under H 0 , Σq = Σp i.e., the plug-in estimate of Σ p . under H 0 , the null distribution approximated with Σq is asymptotically correct, following the convergence of Σp to Σ p . Second, the rejection threshold obtained from the approximated null distribution is asymptotically constant.


. . , σ ′ p where, for all j = 1, . . . , p, σ ′ j has the same support as σ j .

we note that unlike in the symmetric case, the signs of the off diagonal entries can no longer be determined using a cycle basis of induced cycles, since such a basis may contain only cycles which have no contribution to the principal minors of K. In the same example as above, the only induced cycles of G K are triangles, and any cycle basis should contain at least three cycles.


This circumvents the problem of kernel design, as no parametric form of kernel function is required here.

this form of learning a covariance matrix is also suited for problems with complex covariance structure (e.g.


( 9) and write, The squared norm, • 2 is a convex function in its vector argument and thus ∆ t is concave, which yields the following lower bound on ∆ t , The SimPerc algorithm sets µ t j to be 1/|M t | for all j ∈ M t and to be 0 otherwise.

furthermore, for all j ∈ M t , α t j is set to C. the right hand-side of Eq.


Figure 1(a) shows the initial discretization of our example domain into 12 large cells together with the start coordinates of the agent (A) and the goal region (cell containing ¢¡ ¤£ ¦¥ ¨ § ).

it can always attempt to move towards the center of each adjacent cell (that is, cell that its current cell shares a border line with).


Therefore, the spikes can be divided into distinct groups based on the geometry of the MEA and the local position of each neuron, and each group is then processed independently.

each group can be processed in parallel, allowing for high data throughput.


The values for each patch were normalized to values between a and 255, with a cone excitation corresponding to a value of 128.

the R, G, and B components of each pixel represent the relative excitations of L, M, and S cones, respectively.


Then the rest of id T = 4 nodes are changed to have id T = 3 by Line 15 of Algorithm 1 and to be further processed (even though they could be both parents at the same time) with neighbor nodes' causal identities.

step 2 of Algorithm 1 makes all variable pairs with id T = 4 to become identified either as parents, children, or with id T = 3 after taking some neighbors' MBs into consideration.


In the limit as a approaches 1, estimation of θ is based upon word tokens.

when a → 1, all terms in the sum over (z, ℓ) go to zero, except that in which every word token has its own table.


Or, by iteratively proposing a ξ t taking into account the last state h t−1 proposed and the generative deterministic mechanism F g . The latter allows the inference network to peek at states h t that F g could generate from h t−1 before proposing an actual target h t . This allows the inference model to track a multi-modal F g without need for F q to match its expressiveness.

this might offer the possibility to learn multi-modal generative models, without the need to employ complex multi-modal distributions in the inference model.


In other words, when the 0-1 loss over the training dataset is 0, then the AP loss is also 0.

instead of optimizing the AP loss over all the samples, we adopt a two-stage approximate strategy.


We believe that continued progress on the ALE environment that preserves these advantages will extend to broad advances in other domains with significant perception and policy selection challenges.

our immediate goal in the work reported here is to build even better performing general-purpose Atari Game playing agents.


Hence, an intervention that blocked the WNT5A protein from activating its receptor could substantially reduce the ability of WNT5A to induce a metastatic phenotype.

we consider the following immediate reward function in belief space: R(s, = 50 128 i=1 s(i) δ x i (1)=0 − 10||a|| 1 . actions are available for controlling the system: A = {[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0]}.


. . , N K and drawing µ from an infinite-dimensional Dirichlet distribution over the positive integers: where α > 0 is a fixed concentration parameter and µ (0) = (µ m ≥ 0 for all m. The probability of C N conditioned on N and µ is Posterior inference for the NBD model involves alternating between (a) sampling C N from P (C N | N, a, q, µ) using the chaperones algorithm and (b) sampling µ from its conditional posterior: where L m is the number of clusters of size m in C N . Although µ is an infinite-dimensional vector, only the first N elements affect P (C N | a, q, µ).

it is sufficient to sample the (N + 1)-dimensional vector (µ 1 , .


As a result, we get F (x s ) − Φ(u s ) ≤ 2( + s ).

we can use the duality gap F (x s ) − Φ(u s ) as a certificate to monitor the progress of optimization.


However, at each temperature the same standard EM updates are used.

the method does not provide strong guarantees on the global optimality of the resulting solution.


The discriminator is designed to discriminate whether the photos are retouched by an expert or by our own method: D β ( Sf ) = φ(P T (s 0 , A)).


A vertex u is then free to purchase an edge to a vertex v at grid distance d = δ(u, v) at a cost of d α for α > 0.

longer-distance edges now have higher cost rather than lower probability, but again in a power law form.


For simplicity, we first assume that L is a deterministic learning algorithm, so that in each of the m trials, its choice of action is some deterministic function of the outcomes of the earlier trials.

in each of the m trials, L chooses a vector of actions a ∈ A N , and receives the global reward signal R(s, a) ).


In essence, the noisy label ỹ is flipped from the groundtruth label y by an unknown noise transition matrix.

current techniques tend to improve the accuracy of classifier (x → y) by estimating the noise transition matrix (y → ỹ).


Then (16) becomes min q KL[q||q t ] in order to project q t back to subspace parametrization specified by M basis functions.

(16) can also be viewed as performing stochastic natural gradient ascent with a KL divergence projection.


For a HST metric, the two complexity measures coincide as its packing and covering numbers are the same.

for a HST metric ∆ T we will simply denote the complexity of (K, ∆ T ) as C(T ).


Therefore, graph data often takes the form illustrated in Figure 1a: the nodes are divided into a core and a fringe, and we only see edges that involve a member of the core.

the core set is a vertex cover of the underlying graph-since a vertex cover, by definition, is a vertex set incident to all edges.


In the extreme case when µ = ∞, the second term in ( 7) can be shown to be the generalization bound using the weighted samples in the source domain only (the target data will not be effective in this case), and the last term equals to 2λ α,β + d H∆H (D α,β , D T ).

effective transfer is possible in this case only if the divergence is small.


The above equation then requires which implies that where .

taking union bounds on all matrix entries, we have P (7) does not hold < (p + 5p 2 )e −Cn + 2p 2 e −Cn/ν < (7 + 1 n )p 2 e −Cn/ν 2 , where the second inequality is due to the fact that p > n and ν > 1.


in order to find the nearest node to q, the algorithm has to find the direct node on each branch, and then compare them to find the NN.


For example, as discussed earlier, the analysis of [14] shows that when SNR • MAR → ∞, lasso requires m > 2k log(n − k) + k + 1 (14) for reliable recovery.

like thresholding, lasso does not achieve a scaling better than m = O(k log(n − k)), even at infinite SNR.


In other words, all Z ∈ Ω 1 , forms the set of extreme points of Γ 1 .

proof: It is easy to check that any convex combination i θ i Z i , such that Z i ∈ Ω 1 , resides in Γ 1 , with the following two facts: (1) a convex combination of PSD matrices is still a PSD matrix; (2) all eigenvalues of Z must satisfy: .


According to [21], for each node-wise regression, the complexity is O(nd min{n, d}).

for each vertex j, the computational complexity in Step 2 is the total computational complexity of our Algorithm 1 is , where O means up to a log term.


employed is the cosine of the angle between the kernel matrices where each matrix is "flattened" into a vector of dimension m 2 .

this definition implies that the alignment is bounded above by 1 and can attain this value iff the two kernel matrices are identical.


The number of iterations will still be the same, but now, at each iteration, each node should transmit cs bits to its log(s) neighbors.


( 1), a certain feature can be shared by some tasks but not all the tasks.

the proposed formulation can leverage the common features shared among tasks.


Here we use average pooling over adjacent, disjoint 3-by-3 spatial blocks.

applied to the 27-by-27-by-K 1 representation from layer 1, this yields a 9-by-9-by-K1 pooled representation.


. . , X p |Xp | in which the variables X p could have been sampled.

the joint proposal distribution is given by Q Hence, computing the joint proposal Q(x p ) becomes easy given a deterministic selection policy ⇡.


However, to train directly on compressed data without decompressing, we must maintain the feature in sorted order throughout the duration of training, a prerequisite for RLE.

to compute all splits for a given depth t, we introduce a data structure to record the most recent splits at depth t − 1.


We must consider all possible patterns of k i 's for label ζ i .

the largest value of k i is m. for fixed q, we can bound the number of possibilities by m q .


The rest is obtained by backpropagation.

the derivative of the loss with respect to the network's output is: expanding the right hand side and denoting C = 1/ √ 2π we have: Equation 9 provides a simple analytical formula for computing the gradient of Houdini with respect to its input, hence an efficient way to obtain the gradient with respect to the input of the network x by backpropagation.


Furthermore, since X ∞ (αI) converges to X, the scalars , where λ k (•) is defined as the eigenvalue corresponding to eigenvector v k and not necessarily the k th largest eigenvalue.

for all k such that λ k ( X) > 0, by the continuity of log, we have Defining ν(β) = s∞(β) /β, we conclude that for all k such that λ Thus, for every ∈ (0, 1], for sufficiently large β we have shown that lim β→∞ A * (ν(β)) I and lim β→∞ A * (ν(β)) X = X establishing the optimality of X for (5).


At iteration t of the update rule we make a call to the black box ✓(x t ) giving a -approximation to the backward mapping ✓(x t ), compute x t s ✓(x t ), and then project onto [q✏, 1) p . Proposition 4.6 ensures that x t+1 2 M 1 .

the update x t+1 = P (x t s ✓(x t )) is the same as x t+1 = P M1 (x t s ✓(x t )).


Once a CAP TCHA is broken (i.e., computer programs can successfully pass the test), bots can impersonate humans and gain access to services that they should not.

it is important for CAP TCHAs to be secure.


For these experiments the increased confidence comes at the cost of not measuring the full contact map.

the integration of multi-resolution noisy Hi-C data with high-confidence Capture-C data is not only an interesting problem in the context of general network densoising but it is also biologically relevant.


To summarize, from the small enclosing subgraphs extracted around links, we are able to accurately calculate first and second-order heuristics, and approximate a wide range of high-order heuristics with small errors.

given adequate feature learning ability of the model used, learning from such enclosing subgraphs is expected to achieve performance at least as good as a wide range of heuristics.


Support vector machines are arguably amongst the most successful classification algorithms of all times, however their multi-class extensions such as one-vs-all [4] or clever refinements of the loss functions [10,21] all require at least one weight vector per class label.

the MTL adaptation of svm [15] requires all tasks to share an identical set of labels (or require side-information about task dependencies) for meaningful tranfer of knowledge.


However, proper initialization of MIL remains elusive in general, as it often requires engineering experience with the individual problem structure.

it is still of broad interest to develop an initializationindependent formulation for MIL.


First, the large model and training set sizes make it computationally expensive to train these models.

it would be desirable to ensure that the model's performance is as good as it can be, given a particular budget of training data and training time.


We can further improve on this method by using trusted data to train the corrected classifier.

we apply no correction on examples from the trusted set encountered during training.


The main challenge is that maximizing F corresponds to a nonconvex optimization problem (as the function F is not concave), and a priori it is not clear why gradient methods should yield a competitive ratio.

studying gradient methods for such nonconvex problems poses new challenges: Do (stochastic) gradient methods converge to a stationary point?


Intuitively speaking, such q init already encodes certain preferences towards a few preconditioned shift truncations of the ground truth.

we randomly choose an index i and set the initialization point as For a generic kernel a 0 ∈ S k−1 , A T A is close to a diagonal matrix, as the magnitudes of off-diagonal entries are bounded by column incoherence µ.


These differences yield an important consequence: VDMGP can infer automatically the latent dimensionality K of data, but SPGP-DR is unable to, since increasing K is never going to decrease its likelihood.

vDMGP follows Occam's razor on the number of latent dimensions K. We will assess VDMGP on real-world datasets.


For instance, X can be a set of documents, each of which is represented as a vector of word-features where f i is the frequency of the ith word (in some fixed word enumeration).

we represent each element as a distribution over its features, and are interested in a partition of the data based on these feature conditional distributions.


In non-optimized Matlab code, ML took 0.57 seconds per epoch while CML took 1.27 seconds due to extra bound computations.

efficiency is close to EM for practical problems.


Now, it is easy to check that d ρ (x, x ) := ρ(x) − ρ(x ) H defines a metric that generates ρ −1 (τ V ) and, since ρ is isometric with respect to this new metric, we can conclude that (X, d ρ ) is a compact metric space.

we are back in the situation of Theorem 2.2, and hence k X is universal with respect to the space C(X, d ρ ) of functions X → R that are continuous with respect to d ρ .


However, as the size of the test candidates n gets bigger, which is happening nowadays, the number of required pairwise labeling grows exponentially O(n 2 ) leading to the unfeasible implementation.

there is an urgent need to reduce the number of pairwise comparisons, that is, selecting part of the pairs but without loosing the aggregation accuracy.


From Lemma 2, the near optimal auxiliary function ρ * can be regarded as minimizing the mean squared residuals to zero between R(s, a) and the estimator R ρ (s, a; ω) ∇ θ ln{d θ (s) ι π(a|s;θ)} ω + ρ(s, s +1 ).

the meaning of this near optimality of g * ι,θ (s, a; ω) is interpreted as a near minimization of the Euclidean distance between r and its approximator R ρ * (s, a; ω), so that ρ * works to reduce the distance of the regressand r and the subspace of the regressor ∇ θ ln{d θ (s) ι π(a|s;θ)} of the M-estimator ω.


It is clear that the sparse recovery version of ( 1) is highly non-convex, and possibly has multiple local minima [22].

as is typical in modern non-convex methods [13,23,57] we use an spectral technique to obtain a good initial estimate.


This paper lies at the intersection between the subfields of object detection and multiple instance learning.

we discuss the relationship with previous work in each subfield separately.


Substitution of this bound into Theorem 2 yields the PAC-Bayes-Empirical-Bernstein inequality given in Theorem 4.

the PAC-Bayes-Empirical-Bernstein inequality is based on two subsequent applications of the PAC-Bayesian bounding technique.


The dynamics of such collective computations can be seen in the coordinated activity of all of the neurons within the local network; although each individual such neuron may reflect this coordinated component only noisily.

we hope to identify the computationallyrelevant network dynamics by purely statistical, unsupervised means-capturing the shared evolu-tion through latent-variable state-space models [1,2,3,4,5,6,7,8].


In particular, a discrete time drift diffusion process can be derived by subtracting the likelihoods of Eq.

( 5): the change in y within a trial, in which the true stimulus is constant, is Gaussian: δy ∼ N (2r/δt, 4r 2 σ 2 /δt).


Since LR only depends on β * , x i , we can always scale the samples x i to make the magnitude of β * less than 1.

without loss of generality, we assume that β * ∈ S p−1 .


To analyze the computational complexity of each SGD step, we note that Hamming distance between g (defined in (10)) and h = sgn(W x) is bounded above by the depth of the tree d. This is because only those elements of g corresponding to the path to a selected leaf can differ from sgn(W x).

for SGD the expression ( g − h) x T needed for Line 6 of Alg 1 can be computed in O(dp), if we know which bits of h and g differ.


From a Bayesian perspective, optimization of either problem can be viewed, after a exp[−(•)] transformation, as a challenging MAP estimation task with a quadratic likelihood function and a prior that is both improper and discontinuous.

unfortunately, an exhaustive search for the optimal representation requires the solution of up to m n linear systems of size n × n, a prohibitively expensive procedure for even modest values of m and n. in practical situations there is a need for approximate methods that efficiently solve (1) or (2) with high probability.


The arguments above easily generalize to the multiblock case.

when learning , by writing E ij the matrix with a 1 in position (i, j) and zeros elsewhere, we update L k as From the above updates it is not transparent whether the Kronecker product saves us any computation.


However, these still take up memory that is worst-case linear in the length of the input sequence.

they are not applicable to reinforcement learning.


If the trial participants are ignorant of their group assignment, it is assumed that they have indeed been blinded.

ρ 1 heavily relies on the proportion of the non-decisive participants.


In particular, two reads inside of a cluster will have edit distance O(pm), since they are p-noisy copies of the same center.

any two reads in different clusters will be far apart in edit distance whenever p c ind is a small enough constant.


The where and what modules are connected by two links: first, the input to the unsupervised path of the what module is the generated layout from the where module; second, we apply the same affine transformation parameters to the generated shape, so that it can be inserted to the same location as being predicted in the where module.

the final output is obtained as follows: where Â and s are generated from the generator of the where and the what modules, respectively: The STN is able to make a global connection between x from the input end, and x in the output end.


When many such maximizing sets exist, we assume that the latent options are one particular set from amongst these choices; for discussion, the particular choice does not matter, but it is important to have a single set.

our goal is to recover the termination sets of the latent options from the sampled subgoal data; these can be used to construct a library of options that approximate the latent options and have the following desirable properties: • Recall: The termination sets of the library options should contain a maximal portion of the termination sets of the latent options.


Whereas previous methods single out a single "best" correspondence across images, in a Bayesian framework we are interested in characterizing our knowledge about the unknowns conditioned on the data only, averaging over all possible correspondences.

we are interested in the posterior distribution P(OIU), where 0 collects the unknown model parameters mi and Xj.


With respect to the learning rule, the pseudo-inverse is close to SVM rule in the regime we work, so we believe that is a good tradeoff between realism and tractability.

although the unavoidable simplicity of our model, we believe its analysis yields important insights which will likely carry over to the more realistic domains of deep networks studied in ML and neuroscience.


It is not hard to see that problem ( 16) is a special case of the general rank minimization problem (2) with f (X) ≡ 0, ν = 1, Ω = m×n , and X = {X ∈ m×n : ij = M ij , (i, j) ∈ Θ}.

the PD method proposed in Subsection 2.2 for problem (2) can be suitably applied to (16).


Also, since the projections are not too accurate in this case, we can get away with fewer iterations.

there is a tradeoff between estimation error ✏ and model selection error .


For f t and the fringe nodes on the context path below it, we apply the same updates while treating fringe nodes normally.

the recursion in (6) does not carry over to fringe nodes, but otherwise all nodes update their values of β in the same manner.


While traditional PAC-Bayes analyzes distributions directly on H, we instead analyze distributions on Θ.

instead of applying the loss directly to a random hypothesis, we apply it to the output of a learning algorithm, whose inputs are a dataset and a random hyperparameter instantiation.


After epiphany, the total number of queries is no more than that of running CAL from scratch; this number is at most M CAL .

the total query complexity is at most M + M CAL + 3 β ln 4 δ .


For θ = 0 the Frank copula corresponds to independence.

the usual χ 2 independence test is a special case of our linear correlation test.


For learning the model parameters, we used three deterministic exploration policies, each corresponding to always taking one of the three actions.

we could use the more efficient version of the algorithm described in the second paragraph of Section 4.1 and at the end of Section 5.


With the weight matrix S defined in ( 5), we have L r = 1 1+Fr .

proof From observations 1,2,3, we see that L r = 1 1+Fr .


For example, performing elementwise operations takes nearly the same amount of time on the GPU whether operating on tens or on thousands of elements, and multiplying a few hundred different vectors by the same matrix is significantly slower than executing a single (equivalent) matrix-matrix product using an optimized GEMM implementation on either a GPU or a CPU.

careful grouping of operations into batches that can execute efficiently in parallel is crucial for making the most of available hardware resources.


We have shown that for typcial structure in real world data, strong eigendirections adversely affect shrinkage by driving the shrinkage strength to zero.

finally, we have proposed an algorithm which automatically restricts shrinkage to the orthogonal complement of the strongest eigendirections if appropriate.


Linear interpolation of the edge weights between the t i is used to smoothly add the new edges and gradually delete the ones to be deleted.

almost always, there are 105 edges in the graph and 10 edges have weights that are varying smoothly (Figure 1).


That is, in the case of c → ∞, the objective in (1) is a monotonically decreasing function of a.

the optimal strategy for a finite pool is to choose the negative instance with the smallest x •1 value and the positive instance with the largest x •1 value.


The distortion measure is so far defined on a symbol-by-symbol basis.

we extend the definition to sequences: The distortion between sequences J , Ĵ ∈ J M is defined by the distortion for a sequence is the average distortion per symbol of the elements of the sequence.


The feature map φ(x) : X − → H projects x into a typically high-dimensional (possibly infinite) feature space such that linear models in the feature space can model the target variable y effectively.

gP can equivalently be formulated as where w is assigned a multivariate Gaussian prior distribution and marginalized.


However, neither pointwise submodularity implies adaptive submodularity nor vice versa.

our assumptions in this paper can be applied to a different class of utilities than those in [2].


However, the complexity of learning depends upon the number of sets, which we are denoting by m, and would be trivial if m = |X |.

we would like to identify concepts which admit m ⌧ |X |.


In order to obtain a low spectral radius of 1m +o:K this interval should lie symmetrically around zero, which is equivalent to ),min = -),max' This yields 0:* = 1 >'=in l ~I >'=ax l < k with H = IAmaxl.

0:* leads to convergence according to Theorem 1.


In particular, if G µ (B) is the optimal value of some perturbed density restricted to B, then the event that the optima over ⌦ is contained in B is equivalent to the event that G )) is a probability measure proportional to µ(B) [12].

we can use the Gumbel process for a continuous measure µ(B) = R x2B exp( (x)) on R d to model a perturbed density function where the optimum is distributed / exp( (x)).


Given the framework of partition-based CTBNs, we need to provide a way to determine useful partitions.

we introduce partition-based CTBN learning using regression tree modifications in place of CTBN learning using graph operators of adding, reversing, and deleting edges.


We can now formulate the problem of learning the parameters of the decision DAG as a joint minimization of the objective over the split parameters {θ i } and the child assignments {l i }, {r i }.

the task of learning the current level of a DAG can be written as: Maximizing the Information Gain.


For example, [33] discusses training sets with 10 11 items and 10 9 distinct features, requiring novel algorithmic approaches and architectures.

there has been a renewed emphasis on scaling up machine learning techniques by using massively parallel architectures; however, methods relying solely on parallelism can be expensive (both with regards to hardware requirements and energy costs) and often induce significant additional communication and data distribution overhead.


− Upper bound on Rademacher complexity: Though existing Rademacher complexity bounds [8,22] of neural networks can be used for input data with bounded support, direct applications of those bounds to the unbounded sub-Gaussian input data yield order-level loose bounds.

we develop a tighter bound on the Rademacher complexity that exploits the sub-Gaussianity of the input variables.


Both these approaches require adding noise to every entry of the given input matrix or to every entry of the non-private output matrix.

both of these methods would incur an additive error of O(nd).


The data center then collects the L codeword sequences, z 1 , • • • , z L . Since all the L codewords are of the same length m, the combined data rate will be R = L × m/n.

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


The discussion above reveals the relationship with deformable convolution.

the trajectory convolution can be efficiently implemented similar to the way discussed in [2].


Running an additional 50 subjects would raise the power to only .17.

although we did not observe a statistically significant improvement at the inferred optimum compared to sensible alternative policies, the results are consistent with our inferred optimum being an improvement over the type of policies one might have proposed a priori.


Participants were allowed to complete as many unique trials as they wished.

a total of 20,000 AMT HITs were collected, and a total of 100,000 images were shown to the participants.


there is hope that we can reduce our our original problem of finding an optimal subgraph H ∈ H as one of finding subgraphs of H (i) ⊆ G (i) and H (j) ⊆ G (j) that are compatible, in the sense that they match up on the overlap V ij , and for which D(P G P H ) is minimized.


Given a pairs of new data points ϕ n1 and ϕ n2 , we use the fact that the learned kernel is a linear transformation kernel, along with the first result of the theorem (W = αI + ΦSΦ T ) to compute the learned kernel as: As mentioned in Section 2, many Mahalanobis metric learning methods can be viewed as a special case of (2).

a corollary of Theorem 1 is that we can constructively apply these metric learning methods in kernel space by solving their corresponding kernel learning problem, and then compute the learned metrics via (3).


All of these quantities are dimensionless, in that if either A and d or x and d are scaled together, these ratios will not change.

the results can be applied to any scaling of A, d and x, provided that the quantities SNR, MAR and SNR min are computed appropriately.


PROOF Suppose, we do not augment the flow in the complementary path P while augmenting P . In the next iteration the original algorithm of [22,23] picks P or any path with the same path length since the path length of P and P is equal before the iteration and they do not share any common edges.

by forcing P we are not modifying the course of the original algorithm and the analysis in [22,23] holds here as well.


Since the random variable ξ in Equation (3.1) can be very heavy tailed, the according elliptical distributed random vector can be heavy tailed.

as has been pointed out by various authors (Tyler, 1987;Croux et al., 2002;Han and Liu, 2013b), the leading eigenvector of the sample covariance matrix Σ can be a bad estimator in estimating u 1 = Θ 1 (Σ) under the elliptical distribution.


It is clear, however, that all properties are "indirectly" elicitable if one first elicits the distribution using a standard proper scoring rule.

in the present work, we suggest replacing this question with a more nuanced one: how elicitable are various properties?


These pooled output nonlinearities are likely a critical component of nonlinear computation across the visual hierarchy.

the SpAM framework may be particularly useful for modeling neurons or fMRI signals recorded in higher and more nonlinear stages of visual processing beyond V1.


Given such a covering, the PSOS(4) relaxation is maximize (PSOS(4)) Notice that variables ij only enter the above program if {i, j} ✓ R `for some `.

the dimension of the above optimization problem is O(r P m `=1 |R `|2 ), which is O(nr) if the regions have bounded size; this will be the case in our implementation.


( 5) is not constructed by simply summarizing sufficient statistics of data and a prior.

we derive a single-pass update for the Dirichlet parameter α using the following interpretation.


Negative values of v are ruled out, too, since they would amount to encouraging (rather than penalizing) training errors in (5).

in the primal problem (5) only v E (0,1] makes sense.


Let x k ∈ L be an example that incurs a margin violation (i.e., slack term) of ξ k . The expected contribution of x k to the SVM criterion is s(x)ξ k because x k will be drawn s(x) times on average into each re-sampled data set.

training the SVM on the re-sampled data or optimizing with re-scaled slack terms lead to identical optimization problems.


This is because in most applications of learning to rank, it is more important to do well at the top of the list than at the bottom, as users typically pay attention only to the top few items.

it is desirable to give up performance on the lower part of the list in order to gain quality at the top.


Define S : Π Θ × V × (0, 1) → {0, 1} such that can be rewritten as The interpretation of L is as follows: If the ρ-quantile of H for the current policy distribution f v is greater than the constraint threshold d, we select policies in Π Θ by their H-values in order to increase the probability of drawing feasible policies.

π θ is highly ranked if If the proportion of feasible policies is higher than ρ, we select policies that are both feasible and with large objective values, i.e., π θ is highly ranked if Intuitively, S can be considered as the indicator function of the highly-ranked or elite samples.


Note that even if ρy depends via a low-pass filter τ ρ ρy = −ρ y + Y (t) on the past spike train Y of the postsynaptic neuron, the description of the problem will turn out to be simpler if the mean firing rate is considered as a separate variable.

let us write the instantaneous weight change as The goal is now to determine the simplest functionals F and G that would be consistent with the experimental protocols P 1 and P 2 introduced above.


The maximal cliques of the graphical model are the fully connected nodes (f (x 1 ), .

. . , f (x n )) and the pairs (y i , f (x i )), i = 1, . . . , n. the joint probability of random variables f = {f (x i )} and y = {y i } is defined as p(f , y) for classification (1) and normalization factor Z (hereafter Z is defined such that the joint probability sums to 1).


Our new understanding of the deep linear models at least provides the following theoretical fact: the bad local minima would arise in a deep nonlinear model but only as an effect of adding nonlinear activations to the corresponding deep linear model.

depending on the nonlinear activation mechanism and architecture, we would be able to efficiently train deep models.


permutation matrices A, and we have to compute probabilities for all of them.

the exact filter is only applicable to problems with small N . Even if we want to keep track of K ≪ N likely permutations-as attempted by filters like the multihypothesis EKF or the particle filter-the required number of tracks K will generally have to scale exponentially with N (albeit at a slower rate).


We show that, with high probability, our private algorithm can answer up to ≈ n/V queries with a misclassification rate of ≈ V /n, which is essentially the optimal misclassification rate attainable without privacy.

answering those queries essentially comes with no cost for privacy.


Second, the state identity test requires solving the following problem over the function class G: The objective and the constraints are linear functionals of G, all empirical expectations involve polynomially many samples, and the number of constraints is |D h | which remains polynomial throughout the execution of the algorithm, as we will show in the sample complexity analysis.

the LP oracle can directly handle this optimization problem.


In real applications, a BCI with high-level FP rates could cause frustration for subjects.

in this work only the LF-ASD performance when the FP values are less than 1% were studied.


Notice that in all three datasets the one-vs-each bound gets much closer to the exact softmax cost compared to Bouchard's bound.

oVE tends to give a tighter bound despite that it does not contain any variational parameters, while BOUCHARD has N extra variational parameters, i.e.


can be combined by fusing their latent spaces, resulting in a multi-view model -for example, [14,5] model both text and network data from the same mixed-membership vectors.

our MMTM can serve as a basic modeling component for massive real-world networks with copious side information.


In this section we give an alternative definition of bisubmodularity; it will be helpful later for describing a relationship to the roof duality.

as is often done for bisubmodular functions, we will encode each half-integral value x i ∈ {0, 1, 1 2 } via two binary variables (u i , u i ) according to the following rules: 0 ↔ (0, 1) 1 ↔ (1, 0) 1 2 ↔ (0, 0) labelings in K 1/2 will be represented via labelings in the set is a set with 2n nodes.


First, we expand the original model p( w, α, τ |X, t) via while maintaining α and τ unchanged.

the data likelihood and the prior on w become Setting r = r 0 = 1, we update q(τ ) and q(α) as in the regular VB.


Suppose rank(X * ) ≤ k. Then we have f Proof.

we have the following relation similar to (2.3): From Theorem 3.4 we know that h S+1 ≤ (1 after setting X S+2 = W S+1 , we still have h S+2 ≤ 1 e h 1 + κε 2 (according to Lemma 5.1).


For finite graphs, the presence of cycles in the graph degrades the BP estimate and we show that the EP solution with a tree-structured approximation is less sensitive to the presence of such loops, and provides more accurate estimates for the marginal of each bit.

it makes the expectation propagation with a tree-structured approximation (for short we refer to this algorithm as tree-structured EP or TEP) a more practical decoding algorithm for finite length LDPC codes.


[25] and its bibliography), where higher dimensional independent random vectors are to be separated.

characteristic function-based tests [6,13] and kernel independence measures might work better for this problem.


To determine which bundles z s are sold at time t, write t = 2 j m where m is odd, and sell all bundles z s purchased during the previous 2 j−1 time steps which are not yet sold.

the noise trader will sell bundles purchased at times s = t − 1, t − 2, t − 4, t − 8, . . . , t − 2 j−1 ; in particular, when t is odd we have j = 0, so no previous bundles will be sold.


The basic and straightforward way to quantize a full precision value x to a k-bit fixed-point one is the uniform quantization (UQ) [31], which can be implemented as This quantization essentially reduces the representation precision rather than changing the dynamic range, thus x needs to be mapped into [0,1] before quantization and then recovered back to the original range.

the practical UQ is usually modified to In order to observe how UQ affects the accuracy of RNNs, we evaluate a 2-bit quantized LSTM on PTB and the results are shown in Table 1(a).


Since the feedback on playing a permutation is the relevance of top ranked item, each move x has an associated transformation matrix (vector) M x ∈ {0, 1} n , with 1 in the place of the item which is ranked at the top by x and 0 everywhere else.

m x • θ gives the relevance of item ranked at the top by x.


Hence, for datasets exhibiting multi-class structure such as Text, DIFFRAC is more able to utilize unlabelled points since it based on a multi-class clustering algorithm rather than algorithms based on binary SVMs, where multi-class extensions are currently unclear.

our experiments support the fact that semisupervised learning algorithms built on clustering algorithms augmented with labelled data acting as hints on clusters are worth for investigation and further research.


Finally, h S J chooses its outcome by randomly choosing an hypothesis in {h i : i ∈ J}, where the probability of choosing each hypothesis in this set equals (1/|J|).

the expected loss suffered by h S J on an example (x, y) is (1/|J|) i∈J ℓ(y, h i (x)).


Adopting these notations, the set C can be rewritten in terms of the probability vector p as where the system of linear equations Ap = b represents all the low order marginal constraints in B; and the notation 1 denotes the vector of all ones.

problem (2) can be reformulated as where p x,0 and p x,1 denote the elements of the vector p corresponding to the probability values P(X = x, Y = 0) and P(X = x, Y = 1), respectively.


Observe that, although there is a continuum of possible values for δ, we can limit the search to the values of feature j that are observed in the data (i.e., δ ∈ {x ij ; i = 1, ... , n}), since all other values do not lead to different configurations of S τ1 and S τ2 .

there are at most n j ≤ n unique thresholds to consider for each feature.


A large vq(n), which increases the likelihood that n.th < vq(n), may reflect a less promising node (i.e., large q-value), or many threads working on n, or both.

our strategy is an automated and dynamic way of tuning the number of threads working on solving a node n as a function of how promising that node is.


Since the eigenvalues of ∇ 2 F (t) are real and continuous with respect t, the eigenvalues of ∇ 2 F (1) must be positive reals.

we define the symmetrization of u i→j and u j→i by u i→j u j→i = β i→j β j→i .


While residual models have been successful in image analysis [14] and generation [16], they might suffer from the vanishing gradient problem in the same way as recurrent neural networks do.

in the same spirit as LSTM gates [3] and GRU gates [19] improve over pure RNNs, one can introduce convolutional LSTM and GRU operators.


Clearly, how close they can be estimated depends on the number of unlabeled examples used u, dimension d and confidence probability δ.

type-I uncertainty inherently gives rise to a perturbation size defined by 1 (u, d, δ) such that, a fixed u defines a perturbation size 1 (d, δ).


for t ≥ 2, Denote by Φ(Θ) = max w∈W w, Θ the so-called support function of W. The support function, being the maximum of linear and hence convex functions, is itself convex.


Under the condition of Theorem 4, the PNN approximation error depends on both q * and ψ The former term provides a scaling normalization for the loss function.


When a gradient is unreasonably large, the iterative step might overshoot and end up leaving this basin of attraction.

wF moving along the preceding direction might not come close to the truth unless z is already very close to x.


It also modifies w to be w+3x since '£r:Yr(w.x-br):SO Yr = 3.

the inner-product W• x increases by 311x11 2 .


Thus, our result implies a worst case −Θ(log log n) bound on the value of a game with multi-variable interventions and no latent variables.

the ability to randomize turns out to be much more helpful to the designer than the adversary.


With the key observation that the dual variables in NESTT serve as the "memory" of the past gradients, one can specialize NESTT to SAGA/SAG/IAG.

nESTT naturally generalizes these algorithms to the nonconvex nonsmooth setting.


Therefore, using the definition of the concentration coefficient C f , we have that: This results actually follows closely the one of [9, Theorem 4.2].

there remains to bound the Bellman residual Considering the following decomposition, The policy π c (the decision rule of the classifier) is greedy with respect to Qπ E = θ c μπ E . for any state-action couple (s, a) ∈ S × A we have: Taking the expectation according to ρ E and noticing that T * v π E ≥ v π E , we bound the first term: There finally remains to bound the term Let us write M ∈ R |S|×|S| the diagonal matrix defined as M = diag(χ {πc(s) =π E (s)} ).


However, under some assumption and suitable choice of B, we prove that A B m is computable in Proposition 3.6 below.

we obtain an estimation formula of A B m (see [20], [21], [7] for other results on the estimation of Banach limit).


If the energy is submodular, the MAP labeling y * = arg max y∈Y n p(y, I) can be computed exactly with a single graph cut in time c(n), where c(n) is worst-case low-order polynomial (the true complexity depends on the chosen min-cut/max-flow algorithm), but nearly linear time in practice [12,13].

submodularity reduces the complexity of an optimization over 2 n states to nearly-linear time.


In detail, they quantify the statistical error with the `2-norm and do not constrain the input of M (•) and M n (•) to be sparse.

our subsequent statistical analysis is different from theirs.


Instead of relying on the implicit lifting provided by the kernel trick, we propose explicitly mapping the data to a low-dimensional Euclidean inner product space using a randomized feature map z : R d → R D so that the inner product between a pair of transformed points approximates their kernel evaluation: Unlike the kernel's lifting φ, z is low-dimensional.

we can simply transform the input with z, and then apply fast linear learning methods to approximate the answer of the corresponding nonlinear kernel machine.


However, in the case of different motions, the blur cannot be modeled with a single kernel, and trying to deconvolve the entire image with the same kernel will cause serious artifacts.

the task of deblurring needs to involve segmentation of the image into regions with different blurs.


This would make it easier for arbitrary regions of space to share the same covariance structures; at present the areas 'controlled' by a particular expert tend to be local.

a potentially undesirable aspect of the current model is that strong clustering in input space can lead us to infer several expert components even if a single GP would do a good job of modelling the data.


We show that Quick Shift attains similar guarantees.

quick Shift, a simple and already popular procedure, can simultaneously recover the modes with segmentation tuning, provide clustering assignments to the appropriate mode, and can estimate the cluster tree of an unknown density f with the strong consistency guarantees.


Each QSDP gives rise to one SQLP and one SCSDP.

for each data set, 11 SQLPs and 11 SCSDPs are derived.


The generic case in any real world application is that the majority of training vectors contains relevant information, while single spurious vectors may be present due to noise or other disturbing factors.

in most applications the equations ( 7) or (9) will be incompatible and can only be satisfied approximately.


To address this, first note that there is no issue with the first K o components of q m and q i ; these can be merged directly since they each correspond to the K o components of q o .

the component identification problem reduces to finding the correspondence between the last K m = K m − K o components of the minibatch posterior and the last  Note that the maximum number of components after merging is K i + K m , since each of the last K m components in the minibatch posterior may correspond to new components in the intermediate posterior.


More details about these estimators are given in Appendix A.

the Curse of Horizon The importance weight w 0:T is a product of T density ratios, whose variance can grow exponentially with T . IS-based estimators have not been widely successful in longhorizon problems, let alone infinite-horizon ones where w 0:1 may not even be well-defined.


Instead, it looks for the initial prior Θ 0 such that SVGD-updates lead to minimizing the empirical loss on task-validation sets.

like other meta-learning methods, the BFA model can still suffer from overfitting despite the fact that we use a flexible Bayesian inference in the inner update.


We have z i := gap(α i ) where t ∈ [0, t] and the different elements of z are allowed to be based on different, possibly stale iterates α (t ) .

the entries of z can be continuously updated during the course of the algorithm.


The balance between the shifts in the two directions exactly corresponds to the push effect.

the model produce a push-effect graph, but it is not truly producing a push effect as was originally conceived by the experimentalists.


However, in our case, the function ψ(t) = exp(−2iϕ(t)) is discontinuous at t = 0; it changes extremely rapidly in the vicinity of t = 0, and hence its Fourier transform (appropriately defined) does not decay quickly at all.

l(a, w) is a poor target for approximation with a smooth function 1, the difference between h and ψ increases as |t| ↘ 0.


However, for non-quadratic losses, the round complexity bounds are somewhat worse, and there are no guarantees for strongly convex and smooth functions which are not self-concordant.

the question of the optimal round complexity for such functions remains open.


For example, if users must invest large amounts of time to evaluate each recommended item, overestimating essentially costs the user time.

uneven amounts of overestimation could cost one type of user more time than the other.


Under the proposed non-local framework, these methods can be represented as Here δ i (X) = j∈Ci Φ(X) j i and C i denotes the set of indices of the K i selected patches.

each row Φ(X) i has only K i non-zero entries.


However, once that set of models is decided, the designs chosen by ADO are optimal for discriminating those -and only those-models.

the designs that we found and the data we have collected in these experiments are not necessarily optimal for discriminating between, say, a power model and a logarithmic model.


Note that, since we do the filtering in the frequency domain, the size of P has, in contrast to the number of SVs, no direct influence on the computation time needed for filtering.

we have a 2500 dimensional input space and only 286 points in this space, that describe a vesicle.


we must have where the equality follows from the definition of f as a product of the f i .


In backward sampling, we sample z t,i from t = T to t = 1 from the equation below: Because of I(u < π), values of cluster indices k are limited within a finite set.

the variety of sampled z t,i will be limited a certain finite number K given U.


Back to the ecosystem protection example, since the lifespan of sensors is known to us, e.g., how long their battery will run out is a prior knowledge, we usually spread a set of new sensors before the old ones wear out.

the data stream arrives in a way as shown in Figure 1, where in period T 1 , the original set of features are valid and at the end of T 1 , period B 1 appears, where the original set of features are still accessible, but some new features are included; then in T 2 , the original set of features vanish, only the new features are valid but at the end of T 2 , period B 2 appears where newer features come.


The nodes are coloured depending on their weights, as shown in Table 2: for node t, define w t = j∈Q x j (y j,+ − y j,− ), where Q is the set of prefixes of node t (including node t itself.

the blue central nodes are the large prefixes (e.g., /8 prefixes), and the classification they output is slightly malicious; this means that an IP address without a longer matching prefix in the tree is typically classified to be malicious.


We denote k=1 α k , then the margin on the ith example (x i , y i ) can be rewritten as, The derivative of the margin on ith example with respect to α t is calculated as, Since c ≥ a i , depending on the sign of b i,t , the derivative of the margin on the ith sample (x i , y i ) is either positive or negative, which is irrelevant to the value of α t . This is also true for the second derivative of the margin.

the margin on the ith example (x i , y i ) with respect to α t is either concave when it is monotonically increasing or convex when it is monotonically decreasing.


First, cross-validation is not required to select λ, since changing λ only changes the magnitude of scores, not their sign.

for the purposes of classification, one can simply use λ = 1.


Recent cue conflict experiments have shown that the visual system's estimate of a scene parameter, as evinced in a visual percept, is often simply a weighted average of the parameter as specified by each cue separately [1]- [2].

a properly balanced cue-conflict stimulus might come to evoke the same percept as a "natural" or cue-concordant stimulus.


The above inequality is equivalent to P(T=IIV=v) P(T=I)P(v=vIT=I) where the right-most ratio is a function of V , L(V), the likelihood ratio.

the above rule is equivalent to: If L(v) > 1 -P then decide "Yes" otherwise decide "No" , , , P with p = P(T = 1).


By doing sparse PCA analysis of these text data, we hope to find interpretable principal components that can be used to summarize and explore the large corpora.

we set the target cardinality for each principal component to be 5.


For example, the false positive goal is relatively easy to meet, compared to the detection rate goal which is near 100%.

the threshold θ produced by AdaBoost must be discarded in favor of a threshold computed directly from the ensemble performance.