We tested the VHEM algorithm on hierarchical motion clustering, where each of the input HMMs to be clustered is estimated on a sequence of motion capture data from the Motion Capture dataset (http://mocap.cs.cmu.edu/ ).

we start from K 1 = 56 motion examples from 8 different classes ("jump", "run", 'jog'", "walk 1" and "walk 2" which are from two different subjects, "basket", "soccer", "sit"), and learn a HMM for each of them, forming the first level of the hierarchy.


Interestingly, we note that the separation requires using these recent techniques from robust statistics to detect it, even at the learned representation level.

one could consider computing weaker statistics, such as 2 norms of the representations or correlations with a random vector, in an attempt to separate the clean and poisoned sub-populations.


A rigorous perturbation analysis of our model reveals that the dynamical filter properties of STDP make weight changes sensitively dependent on combinations of specific features of pre-and postsynaptic signals.

our analysis indicates that both cortical as well as hippocampal STDP is most susceptible for modulations in the theta frequency range.


The bound depends only on the distribution of B and holds for any polyhedral uncertainty set U .


Similarly to the matrix factorization approaches [2,3,13], automatic dimensionality selection is invoked via hyperparameters c A,i and c B,i , which enforce sparsity in the columns and rows of A and B, respectively.

when a particular set of variances c A,i , c B,i assume very small values, the posteriors of the i th column of A and i th row of B will be concentrated around zero, such that the effective number of principal directions in AB will be reduced.


The dynamics (3) define the behavior of the solution X t = U t U t and using the chain rule we can verify that Ẋt = Ut , where r t = A(X t ) − y is a vector of the residual.

even though the dynamics are defined in terms of specific factorization X t = U t U t , they are actually independent of the factorization and can be equivalently characterized as We can now define the limit point X ∞ (X init ) := lim t→∞ X t for the factorized gradient flow (4) initialized at X 0 = X init .


the ('Toyota', 'Kia') and the ('Bush', 'Cheney') tuples, rather than generally related words.

for this purpose we design a similarity graph G(V, E) with words represented as vertices V and similarity edge weights φ uv between vertices u, v ∈ V whenever u is related to v. the magnitude of φ uv can denote the similarity between words u and v. In the following we denote by y kv the topic dependent smoothing coefficients for a given word v and topic k. We impose the smoother where log p(β) is given up to an additive constant and y v allows for multiplicative topic-unspecific corrections.


In Section 4, we investigate the performance, scalability, and convergence of the sampling algorithm on the probabilistic inference problem of collective classification on a set of Wikipedia documents.

we show that the standard deviation of the marginal density function can serve as a strong indicator for the "confidence" in the classification prediction, thereby demonstrating a useful qualitative aspect of marginals over continuous MRFs.


We consider a setting where the regression model is used to guide future decisions.

after computing regression coefficients, each time we observe feature vectors x 1 , .


In a first experiment, we demonstrate the ability of our approach to solve such sequential problems, adjusting movement durations between sub goals in a principled manner, and show that it improves upon the standard modelling approach.

we apply AICO-T to the two via point problems illustrated in Fig.


We can fit the parameters of this model using the observed data and evaluate counterfactual fairness by sampling from it.

we will generate samples from the model given either the observed race and sex, or counterfactual race and sex variables.


(B) The same model can also be used more generally to fit multivariate binary data with given pairwise correlations, if x is drawn from a Gaussian distribution.

we will show that the resulting distribution closely resembles the binary maximum entropy models known as Ising models or Boltzmann machines which have recently become popular for the analysis of spike train recordings from retinal ganglion cell responses [13,12].


Because ∇ 2 E is indefinite, the posed stationary point is a saddle point with E = 1 2 (ON − off) 2 (≈ 1.219), as desired.

the target value for ON is modified to break symmetry in data.


Hence, to implement gradients wrt θ, we also constitute a reparametrization of p θ (x|z).

we consider samples x θ (z, ξ) with ξ ∼ N (0, I).


We consider a generalization of the one-bit quantized regression problem, where we seek to recover the regression coefficient β * ∈ R p from one-bit measurements.

suppose that X is a random vector in R p and Y is a binary random variable taking values in {−1, 1}.


Although each entry x ij ∈ X is independent given Y and Z, the PCSA model captures the dependencies along rows as well as columns in the joint X.

assuming D 1 is the data dimensionality while D 2 is the sample size, the samples (column vectors) in X is dependent from each other by PC-SA.


To see this, let us denote the algorithm from Theorem 2.1 as A.

its inputs are the database S and the set of candidate centers Y , and the differential privacy guarantee is only with respect to the database S. for every fixed set Y , the algorithm A Y (S) = A(S, Y ) is differentially private as a function of S. Known composition theorems for differential privacy [13] show that for every differentially private algorithm B that takes a database S and outputs a set of centers Y , we have that the composition A(S, B(S)) satisfies differential privacy.


We have presented a novel spatiotemporal ResNet architecture for video-based action recognition.

our approach is the first to combine two-stream with residual networks and to show the great advantage that results.


Fortunately, a simple power method [10] can be used to efficiently compute a maximum eigenvector solution to the inner problem by only performing matrix-vector multiplications on the individual factors of the two low rank matrices making up T (α), meaning the inner problem can be solved without ever forming a large n × n matrix T (α).

if X is n × m each inner iteration requires at most O(nm) computation.


Moreover, if X is strictly stationary, then the joint distribution of the aligned sequences (X, Y ) is strictly stationary.

the conditional transition probabilities P t (x, i, j) x have the following form: a lower bound for P t (x, i, j) is given by and the matrix of transition probabilities P t (x), with the (i, j)-th component given by P t (x, i, j), satisfies φ(P t (x)) = φ(M (x t )).


In the setting where each class is represented by multiple ellipsoids, the goal of learning is to ensure that each example is closer to its "target" ellipsoid than the ellipsoids from all other classes.

for a labeled example (x n , y n , m n ), the constraint in eq.


It is easy to see that any set such that S ⊃ M will have zero risk, with M being the "smallest" such set (with respect to set containment.)

the above error measure does not introduce an explicit penalty on the "size" of S n : enlarging any given S n can never increase the learning error.


Although such random matrices do not have the computational advantages of fast transforms or the optimalitypreservation property of tight frames, their eigenvalue behavior can be characterized analytically.

using the existing results on the eigenvalue scaling of large i.i.d.


This statement follows from Lemma 1 of Ghosal and van der Vaart (2007) and holds upon the satisfaction of the conditions The key observation behind Corollary 5.1 is that the posterior does not overshoot in terms of the width and sparsity, rewarding only small networks that are sparse.

the posterior concentrates on networks with up to the optimal number s n of links.


We propose a method that eliminates actions by utilizing an elimination signal; a specific form of an auxiliary reward (Jaderberg et al., 2016), which incorporates domain-specific knowledge in text games.

it provides the agent with immediate feedback regarding taken actions that are not optimal.


The new conditions generalize this to k > 2.

define C eor to be the multiclass extension of C bin : any cost-matrix in C eor should put the least cost on the correct label, i.e., the rows of the cost-matrices should come from the set c ∈ R k : ∀l, c(1) ≤ c(l) .


has a unique answer, then v has order r − 1 convergence to R h up to sign.

if the following conditions are met: (1) There exists a coordinate random variable s i of s such that G(s i ) = 0.


For the embeddings, we try 3 initialization schemes and choose the best one based on validation error.

these schemes are: (1) all embeddings are drawn from the Gaussian prior implied by the regularizer; (2) the embeddings are initialized from a global embedding; (3) the context vectors are initialized from a global embedding and held constant, while the embeddings vectors are drawn randomly from the prior.


As noted in Section I, there are many other decomposition methods.

the primal decomposition method is a useful machinery for problem with coupled variables [31]; and when some of the variables are fixed, the rest of the problem may decouple into several subproblems.


We further show that coupling the scheme with random rotation can significantly improve communication further.

for " = O(1), we provide an algorithm achieving the same privacy and error tradeoff as that of the Gaussian mechanism with communication per round of distributed SGD.


Note if dim H = ∞, with probability one f is not in H, but fortunately H is large enough for us to approximate the sampled functions.

it can be shown that the posterior GP(m |y , k |y ) in GPR has a dual RKHS representation in the same RKHS as the prior GP [2].


3 Masked Autoregressive Flow Consider an autoregressive model whose conditionals are parameterized as single Gaussians.

the i th conditional is given by In the above, f µi and f αi are unconstrained scalar functions that compute the mean and log standard deviation of the i th conditional given all previous variables.


On the other hand the proposed Bayesian model provides a posterior distribution over the unknown class labels as well as other latent variables and can be used for active learning.

measures such as uncertainty or information gain can be used to guide the selective sampling procedure for active learning.


A state is called locally inconsistent every time its g-value is decreased (line 09, Figure 1) and until the next time the state is expanded.

suppose that state s is the best predecessor for some state s : that is, g(s ) = min s ∈pred(s ) (g(s )+c(s , s )) = g(s)+c(s, s ).


In this work, we propose a more integrated region-based approach that combines multi-class image segmentation with object detection.

we propose a hierarchical model that reasons simultaneously about pixels, regions and objects in the image, rather than scanning arbitrary windows.


Further, suppose µ (j) 4 < 1 for all j 1 and that where C 1 , C 2 the following finite constants: and A .

if M 0 = 1 and 4  1 and µ (j) is Gaussian for all j 1, then we can take Var[M d ] is exponential in P d j=1 1/n j and if The proof of Theorem 5 is deferred to the Supplementary Material.


We present two architectural recipes in the context of multi-stage progressive encoders and empirically demonstrate their importance on compression performance.

we show that: (a) predicting the original image data from residuals in a multi-stage progressive architecture facilitates learning and leads to improved performance at approximating the original content and (b) learning to inpaint (from neighboring image pixels) before performing compression reduces the amount of information that must be stored to achieve a high-quality approximation.


Then, we can treat the input bag-of-words as a |x|-trials multinomial sample from that distribution and use as the reconstruction loss its negative log-likelihood: We now must ensure that the decoder can compute p( x = x i |φ(x)) efficiently from φ(x).

we'd like to avoid a procedure scaling linearly with the vocabulary size V , since V will be very large in practice.


The relative magnitudes of the two brightness values may change, but the sign of the inequality does not.

the ordinal relationship between the average brightnesses of the <left-eye, forehead> pair is invariant under lighting changes.


Additionally, they offer a number of advantages including providing fairer clustering and clustering which has better long-term behavior for each user.

they ensure that every user is guaranteed to get good service (on average).


Because MRI and PET data are highdimensional, regularization techniques are needed for effective variable selection, such as the L1regularization technique [25]- [30] and the L2/L1-regularization technique [31].

l2/L1-regularization has been used for variable selection jointly on multiple related datasets, also known as multitask feature selection [31], which has a similar nature to our problem.


In this section, we analyze the asymptotic behavior for the coverage risk and prove the consistency for estimating the coverage risk by the proposed method.

we derive the asymptotic properties for the density ridges.


Finally, we also conduct a preliminary exploration into the question of how robust the learning of intrinsic rewards is to the sparsity of extrinsic rewards.

we used the delayed versions of the Mujoco domains, where the extrinsic reward is made sparse by accumulating the reward for N = 10, 20, 40 time steps before providing it to the agent.


The indices for the positive and negative samples are denoted by P and N respectively.

if i ∈ P and j ∈ N then x i belongs to positive class and x j belongs to the negative class.


This term penalizes the divergence of intrinsic features of aligned points in the embedding space M. In unrolling the manifold to the mutual embedding space, the local neighborhood relationship of either manifold is not expected to destroyed.

the local geometry of either manifold should be well preserved to avoid information loss.


We will make an assumption on cyclic graphs that restricts the strength of the feedback.

let a cycle of length ⌘ be given by (m 1 , .


We complete the picture and show that k 2 attributes are in fact necessary to obtain arbitrarily low error.

with only one attribute per example, there is an information-theoretic limit on the accuracy attainable by any regression algorithm.


Here, we use amortization to control the number of parameters of the S-EFE model.

we set the per-group embeddings ρ (s) v to be the output of a deterministic function of the global embedding vectors, ρ We use a different function f s (•) for each group s, and we parameterize them using neural networks, similarly to other works on amortized inference (Korattikara et al., 2015;Kingma and Welling, 2014;Rezende et al., 2014;Mnih and Gregor, 2014).


We work within a time-frequency representation (a spectrogram), and exploit the sparsity of speech signals in this representation.

although two speakers might speak simultaneously, there is relatively little overlap in the time-frequency plane if the speakers are different [5,4].


Then, with D W (•; x, s) denoting the distribution of W (x, s), if w n min (x, s) ≥ 0 and zero otherwise.

the EI boils down to integrating the distribution function of W (x, s) between 0 (since W is positive) and w n min (x, s).


Instead, the space is folded depending on the orientations and shifts encoded in the input weights W and biases b and on the nonlinear activation function used at each hidden layer.

this means that the sizes and orientations of identified input-space regions may differ from each other.


In addition to giving new formulations, and new properties of the solutions to these robust problems, we focus on the inherent importance of robustness, and its ability to prove from scratch important properties such as sparseness, and asymptotic consistency of Lasso in the statistical learning context.

our main contributions in this paper are as follows.


Our work shows that merely knowing whether the loss vector positively or negatively correlates with another vector is sufficient to achieve improved regret bound, when the set is uniformly convex.

rather than having access to an approximate value of c t , we only need to have access to a halfspace that classifies c t correctly with a margin.


Individuals cannot be tracked by others, and all group members can be seen together as one group.

θ c could be interpreted as cooperativeness of the group on average.


In general, given no presumption about the nature of matrix entries, it is virtually impossible to restore L 0 as the missing entries can be of arbitrary values.

some assumptions are necessary for solving Problem 1.1.


Let T denote the set of all such policies of nature.

a policy τ ∈ T of nature is a sequence of transition matrices that may be played by it in response to the actions of the agent.


This is because that, by trying to preserve neighborhood structure in the embedding, the LPP algorithm implicitly emphasizes the natural clusters in the data.

it makes the neighboring points in the ambient space nearer in the reduced representation space, and faraway points in the ambient space farther in the reduced representation space.


Then with probability at least 1 − N 2 e −Ω(d) over the random choice of training set, f maps the entire inner sphere to the same point, and the entire outer sphere to some other point, except for a γ-fraction of both spheres, where γ = N e −Ω(d) .


Instead of following the greedy search strategy described above, we can extend the framework for generating diverse captions for a given image, via beam search or probabilistic sampling.

we can retain multiple ordered pairs at each step and multiple connecting sequences for each retained pair.


If we condition on the event {Y ∈ A e } for some e, then as soon as we have calculated the region R Y ⊆ R + , Theorem 1 will allow us to perform inference on the quantity of interest P Le,g µ 2 by evaluating the expressions (4) and (5).

we are testing whether µ is significantly correlated with the group X g , after controlling for all the other selected groups, S(Y )\g = S e \g.


Based on this, we propose to use a principled approach using the likelihood ratio test to measure the similarity between data points under any arbitrary parameterized distribution, which includes the previously adopted Gaussian and exponential family distributions as special cases.

we prove that for the heavy-tailed distribution we proposed, the corresponding similarity measure leads to a distance metric, theoretically justifying its use as a similarity measurement between image patches.


The special case when ψ is a constant function is particularly illuminating, so we state and show it directly using concepts from convex optimization.

if there were no cost regularization at all, the recovered policy will exactly match the expert's occupancy measure.


As a result, many standard nonlinear kernels on X can be used to define nonlinear kernels on P as long as the kernel evaluation depends entirely on the inner product µ P , µ Q H , e.g., K(P, Q) = ( µ P , µ Q H + c) d . Although requiring more computational effort, their practical use is simple and flexible.


Let φ(x i ) denote the mapped counterpart of x i in kernel space, and then In light of the maximum scatter difference criterion [26], we take into account the ordinal label information, and propose a novel criterion named Ordinal Scatter Discrepancy (OSD) to achieve the desired data distribution after projecting Φ(X) to a low-dimensional subspace.

oSD minimizes the within-class scatter, and meanwhile maximize the noncontiguous-class scatter matrix.


Therefore, a simple way to fix the problem is to adjust the parameter β.

at the end of each iteration t, we obtain the true labels y S for the selected instances S, and compare them with our guessed labels ŷS indicated by µ * .


(8) If we further have η(t) ≥ r (t), then the conditions of Theorem 2 and Theorem 3 are satisfied, which guarantee that L r is a Lyapunov function and that the energy E r decreases.

such a heuristic would preserve the convergence rate r(t) by Theorem 2.


As predicted from the theoretical analysis, our adaptive algorithm improves significantly over its non-adaptive version.

for the left panel, the non-adaptive algorithm's error scaling depends on smallest λ i that is 1/16 while for the adaptive algorithm it scales with λ = 1/7.


To tackle such issue, we train another network to predict the trajectory individually as an alternative.

we use MotionNet [51] as the basis due to its lightweightness.


We have demonstrated the competitive results of the new loss on a number of datasets.

the Histogram loss outperformed other losses for the person re-identification problem on CUHK03 and Market-1501 datasets.


We let m A denote the number of examples from each class that suffice for A to construct a ranking function as described above.

if A is provided with a sample of draws from D η such that each class, positive and negative, has at least m A points in the sample with that true label, then algorithm A outputs a γ-advantage weak ranking function.


In this section, we show that our metric covers the existing metrics defined in the previous works [12,6,25].

we describe the relation to the metric via subspace angles and Martin's metric in Subsection 4.1 and the one to the Binet-Chaucy metric for dynamical systems in Subsection 4.2 as the special cases of our metric.


Treating the decoder in a Bayesian manner leads to several advantages.

in the small-data regime, a probabilistic decoder will be advantageous to leverage prior assumptions and avoid over-fitting.


These bounds are based on our Submultiplicative Glivenko-Cantelli Bound.

assume that E V ∼µ [V 1+θ ] ≤ C for some θ > 0 and C ≥ 1.


In recent years, researchers [I, 2] studying the statistical properties of such estimators have started discovering commonalities among proofs and analyses and have proposed unified frameworks that take advantage of such commonalities.

using a single theorem, they are able to rederive a wide range of known results on high-dimensional consisteney and error bounds for the various regularized and constrained estimators.


We minimize the reconstruction error of dual swap output with respect to the original input, and write the dual swap loss L d as follows: The dual swap reconstruction minimization here provides a unique form of self-supervision.

by swapping random parts back and forth, we encourage the element-wise separability and modularity of the obtained encodings, which further helps the encoder to learn disentangled representations under the guidance of limited weak labels.


c (•) does have the necessary characteristics of a proper loss: summaries S + with large V-ROUGE score are mapped to small values and summaries S − with small V-ROUGE score are mapped to large values.

submodularity means r(S) , so complement rouge is a submodular upper bound of the ideal supermodular loss.


For computational tractability, we also truncated our posterior representation to K topics.

we assumed that q(z id > K) = 0 for every i and d. A consequence is that observations have no effect on πk and φ k for all k > K, and these parameters can be exactly marginalized out.


To model the joint indicator variables, we define Such Bayesian approach favors simple model structures.

the priors on A k and B k enforce a sparse parametrization since, during learning, many α k ij and β k j get close to infinity whereby (the posterior distribution of) A k ij and B k j get close to zero (see [11] for an analysis of this pruning effect).


We derive lower bounds on the problem of correctly identifying two clusters under the assumption that the clusters are balanced.


In fact, little is known about inference in RUMs beyond P-L.

we are not aware of either an analytical solution or an efficient algorithm for MLE inference for one of the most natural models proposed by Thurstone [26], where each X j is normally distributed.


Such a mixed binary-continuous problem arises in Bayesian model selection with a spike-and-slab prior and we illustrate our technique by focusing on this case.

we show how to sample from the posterior of linear and probit regression models with spike-and-slab priors, while also imposing truncations in the parameter space (e.g., positivity).


These two chains are started at the same point θ 0 ∈ R d , and are run accordingly to (2) but the chain with the smallest step size is run twice more time than the other one.

these two discretizations are run until the same time horizon K k=1 γ k , where K is the number of iterations.


Moreover, Alice and Bob have common knowledge of all these facts.

now, consider the following Pareto optimal policy that favors Alice (principal 1) when o 1 is red, and Bob (principal 2) when o 1 is green: (none, all) This policy can be viewed intuitively as a bet between Alice and Bob about the value of o 1 , and is highly appealing to both principals: π is more appealing to both Alice and Bob than an agreement to deterministically split the cake (half, half).


As both strategies proposed by [1] are consistent with all the assumptions that we have made in section 3.4 when applied to RNNs, BPTT- Chen's √ t algorithm requires to remember √ t hidden states and √ t internal RNN states (excluding input hidden states), while the recursive approach requires to remember at least log 2 t hidden states.

the model does not allow for a fine-grained control over memory usage and rather saves some memory.


( 9) at iteration t, the item r t (w) we added can be shown to be the delta over w * t , that is, r t (w) = I(w = w * t ); in addition, we have q t (w) = 1 t t s=1 I(w = w * s ) when the step size is taken to be β t = 1 t . Therefore, this Frank-Wolfe update can be naturally interpreted as greedily inserting a hidden unit into the current model p(v | q t , ϑ t ).

if we update the temperature parameter as α t ← t, according to Proposition 3.1, we can directly transform our model p(v | q t , ϑ t ) to a regular RBM after each Frank-Wolfe step, which enables the convenient blocked Gibbs sampling for inference.


A Gaussian graphical model [18] is then employed by representing different genes as nodes and the regulation between genes as edges in the graph.

two genes are conditionally independent given the others if and only if the corresponding entry of the precision matrix of the multivariate normal distribution is zero.


We can see the GAN objective, as a special instance of (7) by identifying each terms in the expectations of ( 7) and (8).

choosing the last nonlinearity in the discriminator as the sigmoid D ω (x) = 1/(1 + e −Vω(x) ), corresponds to output activation function is g f (v) = − log(1 + e −v ); see Table 2.


We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the original data matrix, then one can recover a near optimal solution to the optimization problem by using the sketch.

we use this approach to obtain sparse principal components and show that for m data points in n dimensions, O( −2 k max{m, n}) elements gives an -additive approximation to the sparse PCA problem ( k is the stable rank of the data matrix).


By pipelining schedule, the master machines do not become a bottleneck even with a large number of workers.

the pipelined strategy does not occur any parallelization errors if parameters x for push can be ordered in a manner that does not depend on their actual values (e.g.


This reflects the non-zero cue strength between items that were presented as part of the same training pair.

the model without contextual retrieval has correctly learned the relationships described by the edges of the graph.


A particular disadvantage that is our focus here is the fact that the principal components are usually linear combinations of all variables.

all weights in the linear combination (known as loadings), are typically non-zero.


For example, the gradient restart in [17] is only studied for unconstrained quadratic problems, and the speed restart in [20] is only studied for unconstrained strongly convex problems.

it is not guaranteed (to our knowledge) that these heuristics preserve the original convergence rate of the non-restarted method, when the objective function is not strongly convex.


Under these additional distributional assumptions in (6), what form would the CRF distribution in Theorem 1 take?

what would be the form of the observation functions ✓(X)?


If β ≤ 1/ √ 2, then there exists a sequence of vectors {v 0 (n)} n≥0 , such that almost surely such that the following happens.

for any sequence of vectors {v 0 (n)} n≥0 , we have, almost surely in the high-dimensional limit, the maximum likelihood estimator is positively correlated with the signal v 0 (n) if and only if Explicit (although non-elementary) expressions for R + (β) can be computed, along with the limit value of the risk R(v ML ; v 0 (n)) for sequences of vectors {v 0 (n)} n≥1 whose entries empirical distribution converges.


Our method exhibits several robustness properties.

the proof of Theorem 2.2 suggests that as the number of tests are increased the performance also improves.


If for all pairs {z i , z j } ⊂ Z and all s ∈ , and if SpectralQuartetTest returns a pairing {{z i , z j }, {z i , z j }}, then {{z i , z j }, {z i , z j }} = {{z 1 , z 2 }, {z 3 , z 4 }}.

the spectral quartet test never returns an incorrect pairing as long as the singular values of E[z i z j ] lie in an interval of length 2∆ i,j around the singular values of Σi,j .


We use differential equations based approaches to provide some physics insights into analyzing the dynamics of popular optimization algorithms in machine learning.

we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a natural connection of optimization algorithms to physical systems.


Observation: [17] If p(x) is separable, and similarity between datapoints is defined as e − xi−xj 2 / 2 then the eigenfunctions of the continuous weighted Laplacian, L p have an outer product form.

if Φ i (x) is an eigenfunction of the weighted Laplacian defined on R 1 with eigenvalue λ i then Φ Specifically for a case of a uniform distribution on [a, b] the eigenfunctions of the onedimensional Laplacian L p are extremely well studied objects in mathematics.


Restricting this search space amounts to constraining the class of variational distributions that we consider.

we aim to constrain the aggregated variational posterior [2]: Here p data (X) denotes the empirical distribution.


Gibbs sampling is an MCMC algorithm which systematically explores the sample space by repeatedly drawing samples from the conditional distributions of the individual hidden dimensions.

the transition probability from the current sample to a new candidate sample is given by p(s new h | s current \h ).


More recently, several works seek to achieve a better convergence time than O(1/") under weaker assumptions than Lipschitz gradient and strong convexity of the dual function.

building upon the recent progress on the gradient type methods for optimization with Hölder continuous gradient (e.g.


We define the cumulative loss of the BMA algorithm after T rounds to be Importantly, note that even though the algorithm we consider is a Bayesian one, our theoretical results do not assume that the data comes from any particular probabilistic model.


In particular, SGD can only select the minima with s  1/⌘ whereas GD does not have this requirement.

for the toy example discussed above, at x = 1, we have s = 1.8 > 1/0.7, so as predicted SGD with ⌘ = 0.7 will escape and finally converge to minimum x = 0, where s = 0 < 1/0.7.


The objective of the algorithm is to sequentially select the instances to query for label information and output a classifier ĥ that is close to h * while making as few queries as possible.

we consider a non-parametric setting as in [6,17] where the hypothesis space is the smooth boundary fragment class the decision boundaries of classifiers in this class are epigraph of smooth functions (see Figure 1 for example).


Instead, we consider the conditions that lead to a feature receiving zero weight.

we show that (i) as a direct result of feature-wise independence of the uncertainty set, a slight change of a feature that was originally assigned zero weight still gets zero weight (Theorem 4); (ii) using Theorem 4, we show that "nearly" orthogonal features get zero weight (Corollary 1); and (iii) "nearly" linearly dependent features get zero weight (Theorem 5).


random perturbations are applied for every assignment x ∈ X.

when the random perturbations follow the Gumbel distribution (cf.


The online version of EXIT mitigates this by aggregating all datasets generated so far at each iteration.

instead of training πi on D i , we train it on D = ∪ j≤i D j .


In general, larger values of p, as well as larger differences between p and q, require smaller training sets for a desired level of accuracy.

learning progresses faster where it is more useful.


In this section, we introduce a proximal-point method based on Kullback-Leibler (KL) proximal function and establish its relation to the existing approaches based on natural gradients [1,5,6].

for conditionally-conjugate exponential-family models, we show that each iteration of our proximal-point approach is equivalent to a step along the natural gradient.


Additionally, overloading and AD are well established techniques that have been applied to machine learning, and even to applicationspecific programming languages for machine learning, e.g., LUSH [12] and DYNA [4].

dYNA applies a nonstandard interpretation for ∧ and ∨ as a semiring (× and +, + and max, .


Reject option: In the initial stage of the algorithm, we let the transduction step use the reject R as an additional label (besides the class labels) to label the unsupervised target points.

our transduction algorithm can decide to not label (reject) some of the points so that they will not be used for adaptation.


. . , i n } where i 's are elements of [n].

a top-k list has k elements that are strictly ordered and ranked above the remaining n − k elements that are incomparable to each other.


The idea in [22] is to directly use the bits from b-bit minwise hashing to construct hash tables.

we hash the data points using k random permutations and store each hash value using b bits.


Then Sparse Proximal SAGA converges geometrically in expectation with a rate factor of at least ρ = 1 5 min{ 1 n , a 1 κ }.

for x t obtained after t updates, we have the following bound: Remark.


Therefore, when the optimal value function cannot easily be reduced to a low-dimensional representation, we argue to apply a complementary form of regularisation on the target side.

we propose to replace the optimal value function as target for training with an alternative value function that is easier to learn, but still yields a reasonable-but generally not optimal-policy, when acting greedily with respect to it.


All these problems can be reduced to each other.

the MVC, MC, and SAT problems can all be represented as instances of the MIS problem, as reviewed in the supplementary material.


We first derive two equations and obtain two nonlinear operators for geodesically convex optimization instead of the linear extrapolation step in Euclidean space.

we analyze the global convergence properties of our accelerated method for geodesically strongly-convex problems, which show that our method improves the convergence rate from O((1−µ/L) k ) to O((1− µ/L) k ).


Let S := {λ/(λ − 1) | λ is a real negative eigenvalue of T * } ∪ {0}.

then the following holds: • For all r ∈ (0, 1] \ S, P (r) := (rI + (1 − r)T * ) −1 T * is well-defined and P (r) is a stochastic matrix if and only if r ≥ r * .


Since the expected length of message Y i is bounded by B i , Shannon's source coding theorem [6] implies that H(Y i ) ≤ B i .

inequality (18) establishes a link between the initial distribution (17) and the number of bits used to transmit information, that is, We can now apply the quantitative data processing inequality (19) in the bound (16).


Compared to handcrafted methods [25,17,36,7], most CNN-based methods have better performance.

methods that use STN-based feature transformations, namely UCN-ST [5] and CAT-FCSS [19], show improved ability to deal with geometric variations.


To this end, we construct a lightweight function to gather feature responses over large neighbourhoods and use the resulting contextual information to modulate original responses of the neighbourhood elements.

we define a gather operator ξ G which aggregates neuron responses over a given spatial extent, and an excite operator ξ E which takes in both the aggregates and the original input to produce a new tensor with the same dimensions of the original input.


When recovering the data, DNA storage systems receive clusters that consist of p-noisy copies of the centers.

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


Moreover, the simplicity of our hard thresholding operator allows us to easily extend it to more complicated sparsity structures.

we show that the methods we propose can be generalized to the sparse overlapping group setting, and to hierarchies of (overlapping) groups.


There is no statistical difference between our results FESS and those based on FPS [3].

the poor performance of [8] is explained by the under-training of HMMs [6].


This has a negative side-effect on the advancement of AD systems.

many time series AD systems' accuracy is being misrepresented, because point-based Precision and Recall are being used to measure their effectiveness for range-based anomalies.


Nevertheless, the reduction reveals new possibilities for training deep models that have not been previously considered.

we discover that regret matching with simple initialization can offer competitive training performance compared to state-of-the-art deep learning heuristics while providing sparser solutions.


The features from earlier layers in φ (see Figure 5) are centered and l 2 -normalized before computing cycleconsistency.

we consider both (a) the 2-way cycle-consistency, P φ , between the test video and the first training video, and (b) the 3-way cycle-consistency, P 3 φ , between the test video and the first two training videos.


However, if we generalize the model slightly, then it can support efficient search.

when we construct a long-range link (v, w) out of v, we do not choose w uniformly at random; rather, we choose it with probability proportional to d −α , where d is the lattice distance from v to w. In this way, the long-range links become correlated to the geometry of the lattice.


The acceleration slows down after the seen/total ratio reaches 0.4.

when there is a limited budget, our approach enables the agent to learn from 40% of all possible (ε, τ ) pairs and yet generalize well across the tasks and environments.


Other researchers have proposed strategies for combining probabilistic models [11], and some of these methods may account well for our data.

the product of experts approach [12] should lead to predictions that are qualitatively similar to the predictions of our combined model.


It also suffices to suppose that H approximates P with exponent β for all β < β, and that H has complexity exponent p for all p > p. Now, it is shown in [10] that the RKHS H has an approximation exponent β = 1 if and only if H contains a minimizer of the l-risk.

if H has approximation exponent β for all β < 1 but not for β = 1 then H does not contain such a minimizer but Theorem 2.3 gives the same result as for β = 1.


Since this inequality is attained whenever S = U Diag(✓)U , where U are the eigenvectors of W > W , we see that kW k cl = k (W )k ⇥ , that is, the cluster norm coincides with the spectral box-norm.

we see that the spectral k-support norm is a special case of the cluster norm, where we let a tend to zero, b = 1 and c = k. Moreover, the methods to compute the norm and its proximity operator described in the following section can directly be applied to the cluster norm.


The relations between PLST and CPLST when coupled with M5P are similar to the ones when coupled with linear regression.

in the yeast, scene, and emotions, CPLST outperforms PLST.


We focus on the network architecture as in the previous section.

let F α,B (a) := n j=1 α j S(a, b j ) be the output of a neural net with activation function S, where: (1) a is an input vector from the set an n-dimensional real-valued vector.


The ML estimate of the sparsity pattern is where |J| denotes the cardinality of J.

the ML estimate is the set of k indices such that the subspace spanned by the corresponding columns of A contain the maximum signal energy of y.


Suppose that the resulting chain {s t } is ergodic.

it is possible to go from every state to every other state (not necessarily in one move).


Using the recent optimal convergence analysis of Oja's algorithm [7], one can conclude that after T 1 = Θ log r σ ηλ iterations, where λ = max{0, −λ min (∇ 2 f (x 0 ))}, then we not only have that x t+1 − x 0 2 is blown up, but also it aligns well with the minimum eigenvector of ∇ 2 f (x 0 ).

if λ ≥ δ, then the algorithm must stop before T . Finally, one has to carefully argue that the error does not blow up in this iterative process.


For the sake of comparison, we adhered to the experimental protocols described in [28], so that the performance of our method is directly comparable.

4 train/test splits were suggested in [28]: 1) 'maximal': train on frames 600:5:1400 (in Matlab notation) 2) 'downscale': train on frames 1205:5:1600 (the most crowded) 3) 'upscale': train on frames 805:5:1100 (the least crowded) 4) 'minimal': train on frames 640:80:1360 (10 frames).


In the discrete case, our method optimizes the same variational objective of Titsias [15], but is a significant improvement over greedy forward selection using the variational objective as selection criteria, or some other criteria.

given the cost of evaluating the variational objective on all training points, Titsias [15] evaluates the objective function on a small random subset of candidates at each iteration, and then select the best element from the subset.


( 5), with probability at least 1 2 (1 ), PONSS using θ ≥ and T = 2eBnk 2 log 2k finds a subset X with |X| ≤ k and Proof.

let J max denote the maximum value of j ∈ [0, k] such that in P , there exists a solution x with |x| ≤ j and f (x) Since the final selected solution x from P has the largest F value (i.e., line 19 of Algorithm 3), we have the desired approximation bound is reached.


Note also that, while κ f itself is an upper bound of κf (X), often one does have an upper bound on κf (X) if one knows the function class of f (for example, say concave over modular).

an immediate corollary is that the class of concave over modular functions f (X) = k i=1 λ i [w i (X)] a , λ i ≥ 0, for a ∈ (0, 1) can be learnt within a factor of min{ Next, we apply our results to the minimization of submodular functions under constraints.


Our discrete log-concavity tester improves on previous work in terms of both sample and time complexity.

[2] gave a log-concavity tester with sample complexity O √ n/ 2 + 1/ 5 , while [11] obtained a tester with sample complexity Õ √ n/ 7/2 .


Then, if we let F = T −1 Λ T be the eigendecomposition of F , then where b j is the j-th column of T −1 , can be used as an alternative to the empirical Ritz vector φj .

we have the following theorem: Theorem 2.


It is easy to see that the closer to 0 the µ i 's are, the closer G S (x; ξ) and N Y −1 ∇ S f (x; ξ) will be.

lim µi→0,∀i Applying the asynchronous parallelism, we propose a generic asynchronous stochastic algorithm in Algorithm 1.


In the bandit case, the same result holds for ε-approximations of Nash equilibria if we introduce an exploration factor ε > 0 that guarantees that action choice probabilities never fall below ε.

if the algorithm is run with a suitably decreasing exploration factor, the sequence of play converges to a bona fide Nash equilibrium with probability 1.


First, the optimal solution is in the span of the training instances.

furthermore, from its definition it is clear that w r l = 0 only for the closest prototypes to x l . w r l = 0 if and only if µ r is either the closest prototype to x l which have the same label as x l , or the closest prototype to x l with alternative label.


By the law of large numbers for empirical distributions, the sequence of empirical distributions µ m,0 = 1 m m i=1 δ ui converges (almost surely, for W 2 ) to µ 0 .

our proof of Theorem 2.6 gives an alternative proof of the existence claim in Proposition 2.5 (the latter remains necessary for the uniqueness of the limit).


In order to lower bound the minimax risk of a given class P, we use a finer granularity than the P d classes described in Section 5.

let P p be the permutation class of distributions consisting of a single distribution p and all of its permutations.


We consider a composite convex optimization problem associated with regularized empirical risk minimization, which often arises in machine learning.

our goal is to minimize the sum of finite smooth convex functions and a relatively simple (possibly) non-differentiable convex function by using first order methods in mini-batch settings.


In experiment, we demonstrate the superiority of our proposed N -pair-mc loss to the triplet loss as well as other competing metric learning objectives on visual recognition, verification, and retrieval tasks.

we report much improved recognition and verification performance on our finegrained car and flower recognition datasets.


Third, two new algorithms for ordinal embedding are proposed and evaluated in experiments.

ordinal embedding aims to represent items as points in R d so that the distances between items agree as well as possible with a given set of ordinal comparisons such as item i is closer to item j than to item k. the goal is to find a geometric representation of data that is faithful to comparative similarity judgments.


Let Ψ(S) denote the minimum attainable p-value corresponding to the contingency table of feature combination Z S . By definition, p S ≥ Ψ(S), therefore Ψ(S) > δ implies that feature combination Z S can never be deemed significantly associated, and hence it cannot cause a false positive.

feature subsets S for which Ψ(S) > δ are irrelevant as far as the FWER is concerned.


We point out that many of the theorems proven in this paper depend on the value of β(f ), which is defined as min u:q(u)>0 p(u).

if f occasionally outputs a very rare label, then β(f ) is tiny and much of our analysis breaks down.


Furthermore, the cost function c is generalized to a subadditive function.

we note that two examples given above can be rephrased using this problem (see Section 4 for details).


Assuming a i ≥ β for all i, we have Q t ≤ 1/β, and setting η = β(log d + 1)/(2T ) guarantees that the regret of SLEEPINGCAT against the best fixed policy is bounded as We now turn our attention to the problem of learning with semi-bandit feedback where the learner only gets to observe the losses associated with its own decision.

we assume that the learner observes all components i of the loss vector such that V t,i = 1.


When the training distribution is sufficiently close (top row), the modes "snap" together and the mean of the two modes is assigned high probability.

objects with 35% red are the most likely to be generated, even though they never appeared in the training set.


We also compare against the state-of-the-art methods in budgeted learning (iv) GREEDYMISER [22]: it is a modification of gradient boosted regression tree [8] to incorporate feature cost.

each weak learner (a low-depth decision tree) is built to minimize squared loss with respect to current gradient at the training examples plus feature acquisition cost.


We now wish to generalize the results to the population of New York City (NYC), but we find the distribution P (x, y, z) in LA to be different from the one in NYC (call the latter P * (x, y, z)).

the average age in NYC is significantly higher than that in LA.


We assume we are given a sequence of T observations, which we stack to create a sequence of samples from the F n |h t 's: To learn the graph structure, we make the approximation of ignoring the dynamical component of the model.

we treat each f t as an observation, and try to estimate the density of the resulting unordered set, ignoring the t subscripts (we appeal to density estimation because many good algorithms have been developed for structure induction).


The LP solution is very sparse since the optimal solutions for the linear program lie on the corners of the simplex in the positive orthant spanned by the constraints.

it lacks a means to control the sparseness of the coefficients α α α.


The heuristic computes the error for a given validation image at level k in the pyramid as L k (I k ) = min {zj } ||G k (z j , u(I k+1 )) − h k || 2 where {z j } is a large set of noise vectors, drawn from p noise (z).

the heuristic is asking, are any of the generated residual images close to the ground truth?


Thus after t cycles of uniform Newscast, the original variance φ 0 of the local estimates is reduced on the average to φ t ≤ φ 0 /(2 √ e) t . The fact that the variance drops at an exponential rate means that the nodes learn the correct average very fast.

indeed, using Chebyshev's inequality P t [|µ i − µ| ≥ ε] ≤ φ t /(nε 2 ) we see that for any ε > 0, the probability that some node makes an estimation error larger than ε is dropping exponentially fast with the number of cycles t. we can derive a bound on the number of cycles that are needed in order to guarantee with high probability that all nodes know the correct answer with some specific accuracy: With probability 1 − δ, after 0.581(log n + 2 log σ + 2 log 1 ε + log 1 δ ) cycles of uniform Newscast holds max i |µ i − µ| ≤ ε, for any ε > 0 and data variance σ 2 .


In this work, we study these theoretical properties for more standard convolutional architectures from the point of view of positive definite kernels [27].

we consider a functional space derived from a kernel for multi-dimensional signals, which admits a multilayer and convolutional structure that generalizes the construction of convolutional kernel networks (CKNs) [15,16].


It is also possible to extend the policy gradient framework to deterministic policies µ µ µ θ : S → A.

under certain conditions we can write the gradient of the objective J(θ) = E s∼p µ µ µ [R(s, a)] as: Since this theorem relies on ∇ a Q µ µ µ (s, a), it requires that the action space A (and thus the policy µ µ µ) be continuous.


Unfortunately, the problem we address is more challenging, since the spatial correlations in a 2-D grid structure create numerous dependency cycles.

our graphical model structure prevents exact inference from being feasible.


Here we consider the simple case of a uniform region prior, with a known prior probability of an outbreak P 1 .

if there is an outbreak, it is assumed to be equally likely to occur in any spatial region.


Here, p is the total number of features (or covariates) and n is the total number of observations (or the sample size).

in general, f belongs to a class F of functions from R p → R. One further assumes that the functions in F depend on at most s of the features, usually with s p. for every f ∈ F, there is some f 0 : R s → R and a subset S ⊂ [p] with |S| ≤ s such that f (z 1 , .


Assuming the graph structure is correct, from a regularization perspective we want to encourage smooth functions, to reflect our belief that labels should vary slowly over the graph.

[2] and [7] suggest a general principle for creating a semi-supervised kernel K from the graph Laplacian L: transform the eigenvalues λ into r(λ), where the spectral transformation r is a non-negative and decreasing function 1 Note that it may be that r reverses the order of the eigenvalues, so that smooth φ i 's have larger eigenvalues in K. A "soft labeling" function f = c i φ i in a kernel machine has a penalty term in the RKHS norm given by Ω(||f Since r is decreasing, a greater penality is incurred for those terms of f corresponding to eigenfunctions that are less smooth.


For simplicity, we will set h = h. The goal of training is to find the weight matrices, W , U , and V , that minimize a training objective defined over the training data (with regularization).

we assume the availability of t training examples {(x i , y i )} t i=1 , and denote the feature matrix X := (x 1 , .


In this section, we explore the potential of AGS in solving complex MMAP problems.

we evaluate the anytime performance of AGS and two natural baselines.


show that counts n are the sufficient statistic for planning.

the joint-value function of a policy π over horizon H can be computed by the expectation over counts as (Nguyen et al., 2017): Set Ω 1:H is the set of all allowed consistent count tables as: is the distribution over counts (detailed expression in appendix).


Lemma 1 Assume that all the piles are of size n i = 1 and that ǫ k ≤ ǫ p+k = ǫ p+k+1 .

then the following two inequalities cannot hold simultaneously: if the bound on |β − βk+p | is smaller than the bound on |β − βk |, then the bound on |β − βk+p+1 | must be smaller than both unless ǫ k+p+1 > ǫ k+p .


Instead, we assume that an oracle provides a random linear transformation (i.e., a sketch) of the gradient, which is the information available to drive the iterative process.

given a fixed distribution D over matrices S 2 R n⇥b (b 1 can but does not need to be fixed), and a query point x 2 R n , our oracle provides us the random linear transformation of the gradient given by ⇣(S, x) Information of this type is available/used in a variety of scenarios.


Note: Theorem 1a-1b gives consistency results for ( 9) and (10), respectively, under certain conditions on the parameters λ cn > 0, λ ds > 0, and α * > 1.

these conditions assume knowledge of the noise variance ξ 2 and spikiness ratio α sp (Θ * ).


We studied the generalization performance of some regularized multi-category classification methods.

we derived a class-size independent generalization bound for a standard formulation of multi-category support vector machines.


Similar to our setting, those early papers studied the dynamics of various online learning algorithms in high dimensions.

they show that the mean-squared error (MSE) of the estimation, together with a few other "order parameters", can be characterized as the solution of a deterministic system of coupled ordinary differential equations (ODEs) in the large system limit.


S ( to optimize (6), followed by the method of iterative projections to ensure that the constraints ( 7) and ( 8) hold.

we will repeatedly take a gradient step S ¡ 6 S 1© 2 8 E !


In addition, Net-Trim encounters explosion of memory and time on large-scale datasets and large-size parameters.

space complexity of the positive semidefinite matrix Q in quadratic constraints used in Net-Trim for optimization is O 2nm 2 l m l−1 .


The authors thank Bob Williamson and Bernhard Sch6lkopf.

we present a simple sparse greedy technique to approximate the maximum a posteriori estimate of Gaussian Processes with much improved scaling behaviour in the sample size m. computational requirements are O(n 2 m), storage is O(nm), the cost for prediction is 0 ( n) and the cost to compute confidence bounds is O(nm), where n «: m. We show how to compute a stopping criterion, give bounds on the approximation error, and show applications to large scale problems.


For a weight vector w ∈ H, we use notation w 2 H = w • w. We associate each possible input/output pair (x, y) ∈ X × Y with a feature vector f x,y ∈ H. Our classifier is characterized by a weight vector w ∈ H, with the following classification rule: Note that computational issues are ignored in this paper.

we assume that the above decision can be computed efficiently (either approximately or exactly) even when GEN(x) is large.


However, as recently shown by Lattimore and Hutter [6] for the infinite horizon setting, it is possible to obtain much tighter sample complexity results by using a more fine grained categorization.

a key idea is that in order to obtain accurate estimates of the value function of a policy from a starting state, it is sufficient to have only a loose estimate of the parameters of (s, a)-pairs that are unlikely to be visited under this policy.


These works relate the change in the payoff to the change in the action.

there has been a vast research on Lipschitz MAB with stochastic payoffs [22,29,30,21,26], where, roughly, the expected reward is Lipschitz.


To achieve this, we propose to augment the regular autoencoder proposed in Section 2 so that, from the sentence representation in a given language, a reconstruction can be attempted of the original sentence in the other language.

we now define language specific word representation matrices W x and W y , corresponding to the languages of the words in x and y respectively.


Error bounds are derived for SOSlasso and its consistency is established for squared error loss.

sOSlasso is motivated by multisubject fMRI studies in which functional activity is classified using brain voxels as features.


An initial state s 0 is a set of propositions that describes a closed world state from which the team trace O starts to be observed.

activities at time step t = 0 can be applied in the initial state s 0 .


In this paper, we propose the Dual Space Gradient Descent (DualSGD) to address the computational problem encountered in the projection and merging strategies in the budgeted approach [8,9,17,7] and the excessive number of random features in the random feature approach [15,16].

the proposed DualSGD utilizes the random-feature space as an auxiliary space to store the information of the vectors that have been discarded during the budget maintenance process.


Eight recently published approaches are selected for comparison.

mCZSL [1] directly uses region annotations to extract image features, and the rest approaches employ two popular CNN architectures to extract image features, i.e., VGGNet [17] and GoogleNet [20].


Contemporary image-to-image translation approaches leverage the powerful ability of deep neural networks to build meaningful representations.

gANs have proven to be the gold standard in achieving appealing image-to-image translation results.


it follows from the expansion property that Pr (S 1 ⊕ S 2 ) ≥ Pr (S 1 ∧ S 2 ).

therefore and so Pr (  this corresponds to both the positive and negative parts of the confident region expanding in the way given in the proof of Theorem 1, with an η fraction of the new edges going to examples of the other label.


[19], this check can be replaced by a check on whether the multilinear extension of z L−1 is correctly computed at a randomly picked point in the field, with minimal impact on the soundness error.

the verifier picks two vectors, q L−1 ∈ F log(n L ) p and r L−1 ∈ F log(b) p at random, evaluates ZL−1 (q L−1 , r L−1 ), and checks whether it was correctly computed using a specialized sum-check protocol for matrix multiplication due to Thaler [18] (described in Section 3.1).


We observe that this p 1 behaves similarly to 2( 1 4 − r π 2 ).

as r varies, EH-Hash's p 1 returns values close to twice those returned by H-Hash's p 1 (see plot illustrating this in supplementary file).


The general strategy of the proposed model is to represent the principal axis of the signal s more accurately as the signal is more degraded (by optical blur and/or sensory noise).

the two neurons come to represent the identical dimension when the degradation is sufficiently large.


Note that Thompson sampling draws actions only from the support of the posterior distribution of A * .

it never selects an action a if P (A * = a) = 0.


We modify the network pipeline so that instead of producing predictions for a single object and relationship at a pixel, a feature vector is used to produce predictions for a set of s o objects and s r relationships.

given a feature vector f from a single pixel, the network will for example output s o object class labels, s o bounding box predictions, and s o embeddings.


However, providing provable guarantees for this modified algorithm seems beyond the current techniques.

extending a much simpler analysis of the zooming algorithm [20] to arbitrary time horizon remains a challenge.


The dataset above satisfies a set of linear constraints, that has the null-space property.

for any h 2 M ⌘ span(A), the following holds with high probability: ), r = O(n/k) and a constant ↵ < 1.


The algorithm makes use of a simple space decomposition scheme based on Bregman balls [8], but deploying this decomposition for the range search problem is not straightforward.

one of the bounds required results in a non-convex program to be solved, and the other requires comparing two convex bodies.


Under the hypothesis of Lemma 1, if f T (λu) ≤ λ 2 f T (u), and satisfies (2), then for any u ∈ S, and any λ > 0 we have where L = t∈M∪U η t t (u), and In particular, choosing the optimal λ, we obtain The intuition and motivation behind this Corollary is that a classification algorithm should be independent of the particular scaling of the hyperplane.

w t and αw t (with α > 0) make exactly the same predictions, because only the sign of the prediction matters.


The polyhedral convex set P k ⊃ D used in the preceding section is updated in each iteration, i.e., a sequence P 0 , P 1 , • • • is constructed such that P 0 ⊃ P 1 ⊃ • • • ⊃ D. The update from P k to P k+1 (k = 0, 1, . . .) is done in a way which is standard for pure outer approximation methods [12].

a certain linear inequality l k (x, t) ≤ 0 is added to the constraint set defining P k , i.e., we set The function l k (x, t) is constructed as follows.


While our analysis aimed at achieving unbiased transport under asymptotic conditions, additional considerations need to be taken into account when dealing with finite samples.

when sample sizes vary significantly across studies, statistical power considerations need to be invoked along with bias considerations.


Our hardness proof applies to the potentially simpler problem where only the eigenvalues are desired.

we show that computing the sum of the eigenvalues (i.e., the trace of the matrix) is hard.


These methods can be grouped as: EM with generative mixture models, bootstrapping methods (Self-training, Cotraining and the Yarowsky Algorithm), discriminative models (Transductive Support Vector Machines (TSVM) [2]) and data based methods, including Manifold Regularization [1], Information Regularization [17], and Low Density Separation(LDS) [11].

tSVM extends the maximum margin principle of SVM to unlabeled data.


Unfortunately, there are hardness results showing that even with improper learning, and even if the data is generated exactly from a small, depth-2 neural network, there are no efficient algorithms which can find a predictor that performs well on test data.

[15] and [12] have shown this in the case of learning intersections of halfspaces, using cryptographic and average case complexity assumptions.


Therefore, it is important to reduce CK whenever P is large.

reducing the number of units in the first convolutional layers has more impact than in the later ones.


In this paper, we provide a rigorous experimental analysis of various multiple object class learning strategies within a generative hierarchical framework.

we propose, evaluate and compare three important types of multi-class learning: 1.)


(1) into multiple (easy) subproblems, one for each factor.

agreement constraints between overlapping subproblem maximizers are then defined, and the resulting program takes the following form, 2 min (2) Here, 'r 2 p' (similarly, 'c 2 r') represents parent-child containment relationships, often represented as a region graph [Wainwright and Jordan, 2008], and are Lagrange multipliers for the agreement constraints, defined for every region r, assignment x r , and parent p : r 2 p. these constraints enforce that the maximizing assignment in a parent region p agrees with the maximizing assignment in the child region r on the values of the variables in r (which are also in p due to containment).


From an application perspective, while, the domain of natural images has received a lot of interest, the domain of diagram analysis is not very well studied.

[12] analyzed graphs and finite automata sketches, [17] studied food-web diagrams and [40] studied geometry diagrams.


We also became aware of a parallel work [27], which considers private graphon estimation (including SBMs).

for the logarithmic degree regime, [27] obtains a (non-efficient) procedure to estimate parameters of graphons in an appropriate version of the L 2 norm.


TS has been preliminarily considered for dueling bandits [3,21].

recent work [3] proposes a Relative Confidence Sampling (RCS) algorithm that combines TS with RUCB [4] for Condorcet dueling bandits.


We propose a novel generative model that is able to reason jointly about the 3D scene layout as well as the 3D location and orientation of objects in the scene.

we infer the scene topology, geometry as well as traffic activities from a short video sequence acquired with a single camera mounted on a moving car.


In this feedback model, the learner gets to observe the loss she suffered in every time step.

the learner does not receive any information about losses of actions she did not choose.


This assumption relates to the behavior of the denoiser g 1 (r 1 , γ 1 ) in the case when the input is of the form, r 1 = x 0 + z.

the input is the true signal with a Gaussian noise perturbation.


A quantum state vector is a complex unit vector whose components have magnitudes that are equal to the square-roots of the probabilities.

if ψ is a quantum state vector and p is the corresponding probability distribution then where the unit column vector ψ is called the quantum state vector which sits in the vector space C n , • is the Hadamard (pointwise) product and † is the complex conjugate transpose.


Center bias has been much discussed in the saliency literature, and as such, we include results in this section that apply a different strategy for considering center bias.

in the following center bias appears more directly as a factor that influences the relative weights assigned to a likelihood estimate defined by local pixels.


Using this result, it is trivial to sharpen the computational aspects of Theorem 1 in special cases.

we can generate a √ n-preserving dictionary with only d n eff ( √ n) elements instead of the √ n required by uniform sampling.


Claim 18 implies that the tails of w E E are reasonably thick.

we show that there must be a threshold T > 0 satisfying the desired property in Step 9 of our algorithm.


Batchnorm works on minibatches in stochastic gradient training, and uses the mean and variance of the minibatch to normalize the activations.

consider a particular node in the deep network, producing a scalar value for each input example.


Consider a map in the form of an undirected graph G and let X be a set of binary variables, which are in one-to-one correspondence with the edges of graph G. For example, the graph in Figure 2 will lead to 12 binary variables, one for each edge in the graph.

a variable instantiation x will then be interpreted as a set of edges in graph G. instantiation x includes edge e iff the edge variable is set to true in instantiation x.


Although seemingly unrelated at first glance, the layers of a deep neural network (DNN) can be viewed as iterations of some algorithm that have been unfolded into a network structure [9,11].

iterative thresholding approaches such as IHT mentioned above typically involve an update rule comprised of a fixed, linear filter followed by a non-linear activation function that promotes sparsity.


Learning structure and global mixtures: To simultaneously learn global mixture labels m n and edge structures associated with each mixture E m , we use an EM algorithm for learning mixtures of trees [20,15].

meila and Jordan [20] describe an EM algorithm that iterates between inferring distributions over tree mixture assignments (the E-step) and estimating the tree structure (the M-step).


We know that m k differs from all previous x s in at least one location.

m k must differ from x s * in at least one location.


The tiering problem has the appealing property [19] that the solutions for increasing λ form a nested subset.

relaxing capacity constraints never demotes but only promotes pages.


It is convenient to define Λ = 1/λ so that small Λ corresponds to a more sparse graph.

λ = 0 corresponds to the empty graph with no edges.


To make the bandit version of FTAL -differentially private, we replace the value v t = t τ =1 p β f t (w † t + βu t )u t with a private approximation v † t computed using the tree-based sum protocol.

at each time step t we output See Algorithm 3 (Appendix E.1) for details.


from K k=1 w zmk δ k , and therefore there is nothing in the model that encourages the image features, x ml and x ml ′ , which are associated with the same image-feature atom θ * j , to be assigned to the same object k. To address this limitation, we add a clustering step within each of the images; this is similar to the structure of the hierarchical Dirichlet process (HDP) [21].

consider the following augmented model: where v m ∼ Stick T (α v ), and G k is as defined in (3).


However, it has not been seriously studied how to use CCA for LSDR because W z does not contain orthogonal rows.

unlike PLST, round-based decoding cannot be used and it remains to be an ongoing research issue for designing a suitable decoding scheme with CCA [18].


This suggests an interesting hypothesis why choosing a binary code with very high temporal precision might be advantageous even if the signal of interest by itself does not change at that time scale: the sensory inputmay sometimes be too noisy, so that repeated, independent samples from the signal of interest may sometimes lead to neuronal firing and sometimes not.

a binary code at the short time scale is useful independent from the correlation time of the signal to be encoded, if uncertainties have to be taken into account, because any surplus available amount of temporal precision is maximally used for uncertainty representation in a self-adjusting manner.


We observe significant speed-ups brought by our algorithm, which is almost 50 times faster than the convex ones.

when the dimension d scales up to several thousands, the computation of SVD in PPA and ADMM takes enormous time and therefore the computational time of them increases dramatically.


Specifically, suppose that after each iteration t, each player i observes a partial payoff function ũi t : S i → R describing their payoff as a function of only their own action, s i , holding the action played by the other player fixed.


Even with a small, sampled, set of template orbits, not including the actual orbit of x, we can still compute an invariant signature.

observe that when g is unitary gx, τ k = x, g −1 τ k . the inner product of the transformed image with a template is the same as the inner product of the image with a transformed template.


However, the graphical structure of latent factors (i.e., conditional dependence/independence) is not considered in these works.

the sparse structure of the latent factor network is often preferred but has been never been explicitly explored in the learning process [2,8,23].


The infinite Tucker decomposition (InfTucker) generalizes the Tucker model to infinite feature space via a tensor-variate Gaussian process (TGP) [19].

in a probabilistic framework, we assign a standard normal prior over each element of the core tensor W, and then marginalize out W to obtain the probability of the tensor given the latent factors: p(M|U (1) , .


In a recent paper, Chambolle and Pock [11] proposed a primal-dual algorithm (PDA) for the following convex-concave saddle-point problem: ρ and h are possibly non-smooth convex functions, f is a convex function and has a Lipschitz continuous gradient defined on dom ρ with constant L, and T is a linear map.

briefly, given x 0 , y 0 and algorithm parameters ν x , ν y > 0, PDA consists of two proximal-gradient steps: where D x and D y are Bregman distance functions corresponding to some continuously differentiable strongly convex ψ x and ψ y such that dom ψ x ⊃ dom ρ and dom ψ y ⊃ dom h. and D y is defined similarly.


However, these works exclusively focused on using coding at the computation level, i.e., certain linear computational steps are performed in a coded manner, and explicit encoding/decoding operations are performed at each step.

[13] used MDS-coded distributed matrix multiplication and [7] focused on breaking up large dot products into shorter dot products, and perform redundant copies of the short dot products to provide resilience against stragglers.


The two phase configurations (peaks-add and peaks-subtract) were used to test the multiple channel hypothesis: if combined stimuli are decomposed and processed in separate channels, their perceived speeds should be independent of the phase configuration.

the difference in overall contrast of the two configurations should not affect perceived speed (Fig 3).


The trace norm 1 X Σ is the sum of the singular values of X.

the characterization in terms of the singular value decomposition allows us to characterize low trace norm matrices as the convex hull of bounded-norm rank-one matrices: the trace norm is a convex function, and the set of bounded trace norm matrices is a convex set.


Therefore, we adopt a different approach, which is based on a conjugate gradient method.

we consider an approximate time-varying linear process with the same diffusion term, that is the same system noise covariance: where g(t, x) = −A(t)x+b(t), with A(t) ∈ R d×d and b(t) ∈ R d . the approximate posterior process q(X|Σ) is restricted to be a Gaussian process [4].


In addition, VHEM-H3M allows for efficient estimation of HMM mixtures from large datasets using a hierarchical estimation procedure.

in a first stage intermediate HMM mixtures are estimated in parallel by running standard EM on small independent portions of the dataset, and the final model is estimated from the intermediate models using the VHEM algorithm.


By explicitly modeling and learning the output structure and task structure, our model also addresses the limitations of the existing models that typically assume certain specific type of output structures (e.g., tree [13]) or task structures (e.g., shared sparsity [11]).

a model with task relatedness structure based on shared sparsity on the task weight vectors may not be appropriate in many real applications where all the features are important for prediction and the true task structure is at a more higher level (e.g., weight vectors for some tasks are closer to each other compared to others).


In particular, we show that it can be written as Rg, where rows of R are "near-orthogonal" (with high probability) and have norm ||z|| 2 , and the vector g is close to Gaussian (g has independent sub-Gaussian elements), and hence the projections behave "near-independently".

g = vec(D 1 ) (vector of diagonal entries of D 1 ), and R is a function of D 2 , D 3 and z. Theorem 4.


Joint belief states maintain some of the properties that regular states have in perfect-information games.

it is possible to determine an optimal strategy in a subgame rooted at a joint belief state independently from the rest of the game.


The goal of the generator is to "fool" the discriminator by producing samples that are as close to real data as possible.

d and G are learned as where p(x) is the true data distribution, and p z (z) is usually defined to be a simple distribution, such as the standard normal distribution.


Estimation In a next experiment we aim to determine how well the overall quality of the matches is.

whether the objects matched share similar properties.


Our second result demonstrates the deviation between the empirical means of Pr ′ and β(x) Pr in feature space, given β(x) is chosen perfectly in the population sense.

this result shows that convergence of these two means will be slow if there is a large difference in the probability mass of Pr ′ and Pr (and thus the bound B on the ratio of probability masses is large).


Several models have been proposed recently to address the limitations described above [1,2,6,27].

[6] proposed using multilayer SBNs [22], to impose correlation structure across topics, while providing each document with the ability to control its topic activation probabilities, without the need of a global beta-Bernoulli process [32].


Another method is to train all three components together with a gradient reversal layer [Ganin and Lempitsky, 2015].

the encoder admits gradients from both the discriminator and the predictor, with the gradient from the discriminator negated to push the encoder in the opposite direction desired by the discriminator.


When learning a function f , we may have access to not only the output values f (x i ) for training points x i , but also the values of its j-th order derivatives with respect to the input, D j x f (x i ).

instead of the typical training set consisting of pairs {(x i , f (x i ))} N i=1 we have access to In this situation, the derivative information can easily be incorporated into training a neural network model of f by making derivatives of the neural network match the ones given by f . Considering a neural network model m parameterised with θ, one typically seeks to minimise the empirical error in relation to f according to some loss function When learning in Sobolev spaces, this is replaced with: where j are loss functions measuring error on j-th order derivatives.


In recent years, there has been a resurgence of interest in maximum entropy models in neuroscience and related fields [13,14,15].

maximum entropy models can be useful for model-based estimation of the information content of neural populations [11], as direct information-estimates do not scale well for large population sizes.


Finally, we conduct an experiment that extends the complexity from the prior models even further.

we explore the capacity of our framework to handle models with stochastic dimensionality -having the number of latent variables itself determined by a random variable, and models that can be composed of other smaller (sub-)models.


However, for the other three methods, nearly 10 out of 20 runs involve at least one extraneous variable.

we compare our results with those reported in [14].


In section 3.2, we improve upon [4], by showing that (in this generalized framework) the optimal iteration complexity of oblivious, deterministic or stochastic, finite sum algorithms with both first-order and coordinate-descent oracles cannot perform better than Ω(n + κ ln(1/ )), unless the strong convexity parameter is provided explicitly.

the richer expressiveness power of this framework allows addressing incremental gradient methods, such as Incremental Gradient Descent [6] and Incremental Aggregated Gradient [8,IAG].


These algorithms differ from the standard value function based ADP methods in how the greedy policy is computed.

at each iteration CbPI algorithms approximate the entire greedy policy as the output of a classifier, while in the standard methods, at every given state, the required action from the greedy policy is individually calculated based on the approximation of the value function of the current policy.


To determine the causal direction, we propose three decision rules.

we can determine the causal direction by directly comparing the KCDC measures for the two directions, i.e.


[9] propose a novel batch-mode active learning scheme on SVMs that exploits semi-supervised kernel learning.

a kernel function is first learned from a mixture of labeled and unlabeled examples, and then is used to effectively identify the informative and diverse instances via a min-max framework.


After performing such inference, we retain those columns of Ψ that are used in the representation of the data in D, thereby inferring D and hence M . To impose our desire that the vector of dictionary weights α is sparse, one may adjust the parameters a and b.

as discussed in [17], in the limit K → ∞, the number of elements of z i that are non-zero is a random variable drawn from Poisson(a/b).


All these problems of linear averaging can be well addressed by a KL-averaging method which averages the model (instead of the parameters) by finding a geometric center of the local models in terms of KL divergence (Merugu and Ghosh, 2003;Liu and Ihler, 2014).

it finds a model Liu and Ihler (2014) studied the theoretical properties of the KL-averaging method, and showed that it exactly recovers the global MLE, that is, θ * KL = θmle , when the distribution family is a full exponential family, and achieves an optimal asymptotic error rate (up to the second order) among all the possible combination methods of { θk }.


Note that these are very similar to Fourier features, but the cos, sin-pair corresponding to each frequency is normalised to have unit L 2 norm.

ψ(•) can be thought of as evaluations of the phase function at the selected frequencies.


We are interested in the concentration bounds of the random variables defined by where the expectation E θ * [•] is taken over the underlying true distribution (5).

by the "law of the unconscious statistician" we have {Z s,k } and {Z st,l } are zeromean random variables.


In this case, assuming that Θ = R d , the algorithm will reduce to those described in the papers [8,12].

the maximum quasi-likelihood estimator becomes the least-squares estimator and as noted earlier, the algorithm behaves identically to one which chooses the parameter optimistically within the confidence ellipsoid {θ : θ − θt Mt ≤ ρ(t)}.


The stimulus filter estimates and training log-likelihood varied considerably as a function of the approximation interval (Figure 3a,b).

the paGLM-2 estimate was highly similar to the exact ML estimate for some intervals while too large or small for other intervals.


Figure 2(b) shows that, the solutions that focus on nominal parameters (i.e., λ close to 1) achieve good performance for small α, but worse performance for large α.

if the decision maker is risk neutral, then the solutions based on nominal parameters are good.


the proposed model is an infinite mixture model, where the probability for the dth view of the nth instance is given by where θ n = {θ nj } ∞ j=1 are the mixture weights, θ nj represents the probability of choosing the jth latent vector, α is a precision parameter, N (µ, Σ) denotes the Gaussian distribution with mean µ and covariance matrix Σ, and I is the identity matrix.


Although tractable in principle, LP relaxations pose a real computational challenge.

for many applications, standard LP solvers perform poorly due to the large number of variables and constraints [33].


In the context of search engines, it has been argued (e.g., by [12,18,1]) that a user's clicking behavior provides implicit feedback of a specific form on the ranking.

since users will typically read the search results from first to last, when a user skips some links that appear earlier in the ranking, and instead clicks on a link that appears later, her action suggests that the later link was more informative or relevant.


Varying λ allows us to 'interpolate' between the two extremes of minimum dictionary cost and minimum pointer cost.

λ can be interpreted as tracing out a regularization path that allows a more flexible representation of D. To formalize our compression criterion, let S = { x i . . . be the set of all unique k-grams in D, and P = { (s, l) | s = x l . . . x l+|s|−1 } be the set of all m = |P| (potential) pointers.


In contrast, our proposed encoder HCIAE (Section 4.1) localizes the region in the image that can help reliably answer the question.

in addition to the history and the question guiding the image attention, our visual dialog encoder also reasons about the history when identifying relevant regions of the image.


Algorithm 2: CLAP: Confidence Lift And Project  We will assume that R = (R 1 , .

. . , R m ) is a covering of G (in the sense introduced in the previous section), and -without loss of generality-we will assume that the edge set is E is the maximal set of edges that is compatible with R being a covering.


A chain is said to satisfy detailed balance if and only if Intuitively this means that if we start the chain in a stationary distribution, the amount of probability that flows from i to j is equal to the one from j to i.


we are motivated by real-world scenarios where the expected reward of an arm decreases over time instances that it has been pulled.


s ≥ r or s ≥ t), because otherwise the output matrix C would be rank inefficient and the problem is degenerated.

each worker i can store two matrices Ãi ∈ F s× r m q and Bi ∈ F , computed based on arbitrary functions of A and B respectively.


To attack these computational challenges, we introduce restrictions on the mass matrix G(θ, φ) to enable efficient simulation.

we restrict G(θ, φ) to have the form where G θ and G φ are the precision matrices of r θ and r φ , respectively.


Since we primarily care about shape, the basic image-level feature representation relies only on edges, making our model invariant to appearance such as color and texture.

given a clean font image we use 16 oriented filters to detect edges, followed by local suppression that keeps at most one edge orientation active per pixel (Fig.


We also briefly explain the optimization landscape of streaming PCA problems as follows.

we consider the eigenvalue decomposition ⌃ = R⇤R > with ⇤ = diag( 1 , 2 , .


Then if In most reasonable regimes, the first term of the max in (3) will dominate the number of iterations necessary.

the number of iterations will depend linearly on the second moment bound B.


The simple potential function is defined as where λ > 0 is a hyperparameter that controls the magnitude of reward shaping.

when the state s is not the terminal state S ⊥ , ϕ(s) carries a scaled version of the number of positive symptoms that have been identified before entering state s. By the definition f (s, s ) := γϕ(s ) − ϕ(s), scaling up λ would be equivalent to scaling up f to make the auxiliary rewards larger in magnitude.


The value of a policy under rewards It is expressed as the weights multiplied by the accumulated discounted feature value given a policy, which we define as MDP solver We assume that there is a RL algorithm ALG that takes an MDP as input and outputs an optimal policy and its corresponding representation in the feature space.

aLG(M ) Maxmin Learning All weights that are consistent with the agent's knowledge form a set P R . We will discuss several formal ways to define this set in Section 3.


To further validate the consistency of layout distribution between the ground truth and inserted instances by our method, we evaluate whether RGB images rendered using the inserted masks can be reasonably recognized by a state-of-the-art image recognizer, e.g., YOLOv3 [21].

given an RGB image generated by [31], we check whether the YOLOv3 detector is able to detect the inserted instance in the rendered image as shown in Figure 8 and Figure 9.


Note that this definition applies to both quality-to-rank and rank-to-rank noise models.

in PM-α noise models every pairwise comparison in the true ranking is preserved in a sample with probability at least α more than the probability of it not being preserved.


Due to the huge size of BNC, we randomly selected a maximum of 20 pseudo documents from each of the from the long BNC texts.

we conducted sentence-based random sampling as follows.


The use of difference maps speeds up the convergence of model training since the model focuses on learning the missing appearance details instead of synthesizing the target image from scratch.

the training already starts from a reasonable result.


The surrogate likelihood (which is convex), was used to initialize the univariate and clique likelihood, while the univariate likelihood was used to initialize the smooth classification error.

1 There are two redundancies, as adding a constant to a(yi, yj ) or c(yi) has no effect on p. 2 dL da(y,y ′ ) = P (i,j) dL dθ(y i =y,y j =y ′ ) , dL db(y) = P i dL dθ(y i =y) xi, and dL dc(y) = P i dL dθ(y i =y) .


. . , s K ) so that ψ(B * ) = max k s k and the sample complexity is the same.

even though there is no sharing of variables at all there is surprisingly no penalty from regularizing jointly with the 1 / 2 -norm.


Figure 4(b) shows the effect that the feature dimensionality (controlled by the codebook size of OMP-1) has on the performance of the linear and non-linear classifiers, and Table 1 provides representative numerical results.

when the codebook size is low, the assumption that we can approximate the non-linear function f as a globally linear classifier fails, and in those cases the R 2 SVM and RBF SVM clearly outperform the linear SVM.


The resulting estimator is shown to be effective for difficult estimation problems arising in Markov random field inference.

improvements are achieved over standard MCMC estimators when the distribution has multiple peaked modes.


We then define the connecting module as a classifier, which takes the left and right phrases P (l) and P (r) as input and outputs a normalized score S(P (m) j | P (l) , P (r) , I) for each j ∈ {1, . . . , L}.

we adopt a two-level LSTM model [2] to encode P (l) and P (r) respectively, as shown in Figure 3.


Since we would like to usually specify 1 or 2 keypoints, in our experiments we set the "on" probability to 0.1.

each of the 15 bird parts only had a 10% chance of acting as a conditioning variable for a given training image.


The main idea is to use the Markov property to eliminate the dependence of the confidence intervals on the unknown quantities (including π ⋆ and γ ⋆ ).

we estimate the transition probabilities from the sample path using simple frequency estimates: as a consequence of the Markov property, for each state, the frequency estimates converge at a rate that depends only on the number of visits to the state, and in particular the rate (given the visit count of the state) is independent of the mixing time of the chain.


We analyze the learning properties of the stochastic gradient method when multiple passes over the data and mini-batches are allowed.

we consider the square loss and show that for a universal step-size choice, the number of passes acts as a regularization parameter, and optimal finite sample bounds can be achieved by early-stopping.


We propose a computational definition of saliency that deviates from existing models by equating saliency to discrimination.

the salient attributes of a given visual class are defined as the features that enable best discrimination between that class and all other classes of recognition interest.


In the original model, the interactions between the letter and word layers were asymmetric.

inhibitory connections only ran from letter neurons to word neurons, and not vice versa.


In other words, Kernel PCA is a generalization of conventional PCA since different kernels can be utilized for different nonlinear projections.

we can now project the vectors in Rf to a lower dimensional space spanned by the eigenvectors weI>, Let x be a test sample whose projection is <t>(x) in Rf, then the projection of <t>(x) onto the eigenvectors weI> is the nonlinear principal components corresponding to <t>: we can extract the first q (1 ~q ~m) nonlinear principal components (Le., eigenvectors w4» using the kernel function without the expensive operation that explicitly projects the samples to a high dimensional space Rf" The first q components correspond to the first q non-increasing eigenvalues of (6).


We represent conditional distributions in the RKHS using conditional mean embeddings [18].

given observations {(x i , y i )} n i=1 of a pair of random variables (X, Y ), we construct the embeddings of the two sets of conditional distributions, {p(X|Y = y i )} n i=1 and {p(Y |X = x i )} n i=1 .


There are other consistencies we wish to enforce in our model.

we must match a source phrase to a translation phrase slot if and only if the source phrase is consistently chosen by all of its source words.


The gap between the log-likelihood and the ELBO is the Kullback-Leibler (KL) divergence between q(z) and the posterior p θ (z|x): log p θ (x) − L(q, θ; x) = D KL (q(z)||p θ (z|x)).

(6) the amortized variational inference approach introduces a conditional variational distribution, z ∼ q φ (•|x), to approximate the intractable posterior distribution.


It will therefore appear reasonable to take the length of the cut as a measure of its complexity but weight it depending on the density of the probability distribution p through which it passes.

we propose the weighted length of the boundary, represented by the contour integral along the boundary cut p(s)ds to measure the complexity of a cut.


This intriguing function of primate brain inspires us to develop a novel deep neural net, called multi-view perceptron (MVP), which can disentangle identity and view representations, and also reconstruct images under multiple views.

given a single face image of an identity under an arbitrary view, it can generate a sequence of output face images of the same identity, one at a time, under a full spectrum of viewpoints.


This paper presents a novel machine learning problem -multiple instance learning on structured data (MILSD) for incorporating additional structure information into multiple instance learning.

a general framework of MILSD is proposed for dealing with the additional structure information in different scenarios.


Here each node of the graphical model is indexed by two subscripts, i for row and j for column.

in the simplest case, the maximal cliques are we are using a cross-shaped stencil to connect vertices.


The final set of classifiers C contains stochastic classifiers, whose output only depends on the outputs of the view-specific classifiers.

associated to a set of classifiers C, there is a function For simplicity, in the rest of the paper, when the context is clear, the function x → Φ C (h 1 , ..., h V , x) will be denoted by c h1,...,hV .


Assuming a simple white Gaussian noise model, this can be achieved through the following loss While the supervised learning of analysis operator has been considered for solving denoising problems [5,17], here we address more general scenarios.

we argue that, when used along with metric learning, it is often better suited for classification tasks than its synthesis counterpart, because the non-generative nature of analysis models is more suitable for feature learning.


Consider the effect on dX (¡ a of perturbing the affinity 0 1' ) for the X t a -edge, to 0 1' ) §¦ (' ) .

we show in the Appendix that the derivative of X 1¡ fX ¦ 1' ) a wa with respect to ¦ (' ) , evaluated at ¦ 1' ) ¥ # , satisfies Here X w ) 5a are the X t a elements of eigenvector x and X 8 % %8 ) 5a are degrees of nodes X t a (Eq.1).


The present paper is inspired and draws upon recent results on selective classification [6,7,8], and can be viewed as a natural continuation of the results of [8].

we adapt the basic definitions of selectivity and the general outline of the derivation and strategy presented in [8].


Then, Proposition 2 in [19] states that that there exist constants κ l 1 and κ l 2 , independent of n, p such that with probability at least 1 − c 1 exp(−c 2 n), for some constants c 1 , c 2 > 0, if n > 4(κ l 2 /κ l 1 ) 2 ηd log(p).

with probability at least 1 − c 1 exp(−c 2 n), the loss function L satisfies RSC(k) with parameter κ l 1 provided n > 4(κ l 2 /κ l 1 ) 2 ηd log(p).


each entry t i ( ) of t B ( ) is equal to its Taylor expansion t derivative of t i ( ) evaluated at = 0.


Given two graphs G and H, let and likewise H j .

let K GG , K HH , and K GH be the kernel matrices, induced by the embedding kernel k. they are defined such that Let G be the set of graphs with attribute domains A 1 , A 2 , ..., A L . Let G and H be two graphs in G. Let m G and m H be the corresponding graph embeddings.


In this section we consider theoretical properties of a semantic output code classifier that determine its ability to recognize instances of novel classes.

we will address the question: Under what conditions will the semantic output code classifier recognize examples from classes omitted from its training set?


Note that this parameter is fairly important.

when = 1, the entropy of most random variables is too large to be discriminative.


First, it is not clear that an optimal k-clustering can be a refinement of an optimal 2-clustering.

we need to be sure that there is an optimal k-clustering in which S 1 is the union of some of the clusters, and S 2 is the union of the remaining.


(1)(left) with and Here, the residual Jacobian portion for the parameters of the integrating unit was omitted because they were merely fine-tuned with a steepest-descent type method owing to our knowledge-based design for input-partition to avoid (too many) local experts.

the spectral reflectance signals (16 inputs) were converted to the hue angle as input to the integrating unit that consists of five bell-shaped basis functions, partitioning that hue-subspace alone in a fuzzy fashion into only five color regions (red, yellow, green, blue, and violet) for five 16-16-10 MLP-experts, each of which receives all the 16 spectral signals as input [hence, Z=5; n=2,210 (CA=85; n B =1,360); mlu dn =6.5].


This paper studies the properties of the objective function introduced there.

we show that the objective function is continuously differentiable and its gradient can be explicitly computed.


It was shown in [CMM17] that if the columns of a matrix are sampled by over-approximations to their ridge leverage scores (with appropriately set ), the sample is a projection-cost preserving sketch for the original matrix.

it can be used as a surrogate in computing a low-rank approximation.


We then replace binary evidence Γ m by unary evidence Γ .

the ground literals of the binary predicate p are replaced by ground literals of the unary predicates q i and r i (see Example 4).