Sign-random-projection locality-sensitive hashing (SRP-LSH) is a probabilistic dimension reduction method which provides an unbiased estimate of angular similarity, yet suffers from the large variance of its estimation. In this work, we propose the Super-Bit locality-sensitive hashing (SBLSH). It is easy to implement, which orthogonalizes the random projection vectors in batches, and it is theoretically guaranteed that SBLSH also provides an unbiased estimate of angular similarity, yet with a smaller variance when the angle to estimate is within $(0,\pi/2]$. The extensive experiments on real data well validate that given the same length of binary code, SBLSH may achieve significant mean squared error reduction in estimating pairwise angular similarity. Moreover, SBLSH shows the superiority over SRP-LSH in approximate nearest neighbor (ANN) retrieval experiments.

We describe an approach to incorporating Bayesian priors in the maxq framework for hierarchical reinforcement learning (HRL). We define priors on the primitive environment model and on task pseudo-rewards. Since models for composite tasks can be complex, we use a mixed model-based/model-free learning approach to find an optimal hierarchical policy. We show empirically that (i) our approach results in improved convergence over non-Bayesian baselines, given sensible priors, (ii) task hierarchies and Bayesian priors can be complementary sources of information, and using both sources is better than either alone, (iii) taking advantage of the structural decomposition induced by the task hierarchy significantly reduces the computational cost of Bayesian reinforcement learning and (iv) in this framework, task pseudo-rewards can be learned instead of being manually specified, leading to automatic learning of hierarchically optimal rather than recursively optimal policies.

We present a probabilistic formulation of max-margin matrix factorization and build accordingly a nonparametric Bayesian model which automatically resolves the unknown number of latent factors. Our work demonstrates a successful example thatintegrates Bayesian nonparametrics and max-margin learning, which are conventionally two separate paradigms and enjoy complementary advantages. We develop an efficient variational algorithm for posterior inference, and our extensive empiricalstudies on large-scale MovieLens and EachMovie data sets appear to justify the aforementioned dual advantages.

Large scale $\ell_1$-regularized loss minimization problems arise in numerous applications such as compressed sensing and high dimensional supervised learning, including classification and regression problems. High performance algorithms and implementations are critical to efficiently solving these problems. Building upon previous work on coordinate descent algorithms for $\ell_1$ regularized problems, we introduce a novel family of algorithms called block-greedy coordinate descent that includes, as special cases, several existing algorithms such as SCD, Greedy CD, Shotgun, and Thread-greedy. We give a unified convergence analysis for the family of block-greedy algorithms. The analysis suggests that block-greedy coordinate descent can better exploit parallelism if features are clustered so that the maximum inner product between features in different blocks is small. Our theoretical convergence analysis is supported with experimental results using data from diverse real-world applications. We hope that algorithmic approaches and convergence analysis we provide will not only advance the field, but will also encourage researchers to systematically explore the design space of algorithms for solving large-scale $\ell_1$-regularization problems.

Given pairwise dissimilarities between data points, we consider the problem of finding a subset of data points called representatives or exemplars that can efficiently describe the data collection. We formulate the problem as a row-sparsity regularized trace minimization problem which can be solved efficiently using convex programming. The solution of the proposed optimization program finds the representatives and the probability that each data point is associated to each one of the representatives. We obtain the range of the regularization parameter for which the solution of the proposed optimization program changes from selecting one representative to selecting all data points as the representatives. When data points are distributed around multiple clusters according to the dissimilarities, we show that the data in each cluster select only representatives from that cluster. Unlike metric-based methods, our algorithm does not require that the pairwise dissimilarities be metrics and can be applied to dissimilarities that are asymmetric or violate the triangle inequality. We demonstrate the effectiveness of the proposed algorithm on synthetic data as well as real-world datasets of images and text.

This paper presents a kernel-based discriminative learning framework on probability measures. Rather than relying on large collections of vectorial training examples, our framework learns using a collection of probability distributions that have been constructed to meaningfully represent training data. By representing these probability distributions as mean embeddings in the reproducing kernel Hilbert space (RKHS), we are able to apply many standard kernel-based learning techniques in straightforward fashion. To accomplish this, we construct a generalization of the support vector machine (SVM) called a support measure machine (SMM). Our analyses of SMMs provides several insights into their relationship to traditional SVMs. Based on such insights, we propose a flexible SVM (Flex-SVM) that places different kernel functions on each training example. Experimental results on both synthetic and real-world data demonstrate the effectiveness of our proposed framework.

Keypoint matching between pairs of images using popular descriptors like SIFT or a faster variant called SURF is at the heart of many computer vision algorithms including recognition, mosaicing, and structure from motion. For real-time mobile applications, very fast but less accurate descriptors like BRIEF and related methods use a random sampling of pairwise comparisons of pixel intensities in an image patch. Here, we introduce Locally Uniform Comparison Image Descriptor (LUCID), a simple description method based on permutation distances between the ordering of intensities of RGB values between two patches. LUCID is computable in linear time with respect to patch size and does not require floating point computation. An analysis reveals an underlying issue that limits the potential of BRIEF and related approaches compared to LUCID. Experiments demonstrate that LUCID is faster than BRIEF, and its accuracy is directly comparable to SURF while being more than an order of magnitude faster.

This technical note considers the problems of blind sparse learning and inference of electrogram (EGM) signals under atrial fibrillation (AF) conditions. First of all we introduce a mathematical model for the observed signals that takes into account the multiple foci typically appearing inside the heart during AF. Then we propose a reconstruction model based on a fixed dictionary and discuss several alternatives for choosing the dictionary. In order to obtain a sparse solution that takes into account the biological restrictions of the problem, a first alternative is using LASSO regularization followed by a post-processing stage that removes low amplitude coefficients violating the refractory period characteristic of cardiac cells. As an alternative we propose a novel regularization term, called cross products LASSO (CP-LASSO), that is able to incorporate the biological constraints directly into the optimization problem. Unfortunately, the resulting problem is non-convex, but we show how it can be solved efficiently in an approximated way making use of successive convex approximations (SCA). Finally, spectral analysis is performed on the clean activation sequence obtained from the sparse learning stage in order to estimate the number of latent foci and their frequencies. Simulations on synthetic and real data are provided to validate the proposed approach.

We study the behavior of the A* search algorithm when coupled with a heuristic h satisfying (1-epsilon1)h* <= h <=(1+epsilon2)h*, where 0 <= epsilon1, epsilon2 < 1 are small constants and h* denotes the optimal cost to a solution. We prove a rigorous, general upper bound on the time complexity of A* search on trees that depends on both the accuracy of the heuristic and the distribution of solutions. Our upper bound is essentially tight in the worst case; in fact, we show nearly matching lower bounds that are attained even by non-adversarially chosen solution sets induced by a simple stochastic model. A consequence of our rigorous results is that the effective branching factor of the search will be reduced as long as epsilon1+epsilon2 < 1 and the number of near-optimal solutions in the search tree is not too large. We go on to provide an upper bound for A* search on graphs and in this context establish a bound on running time determined by the spectrum of the graph. We then experimentally explore to what extent our rigorous upper bounds predict the behavior of A* in some natural, combinatorially-rich search spaces. We begin by applying A* to solve the knapsack problem with near-accurate admissible heuristics constructed from an efficient approximation algorithm for this problem. We additionally apply our analysis of A* search for the partial Latin square problem, where we can provide quite exact analytic bounds on the number of near-optimal solutions. These results demonstrate a dramatic reduction in effective branching factor of A* when coupled with near-accurate heuristics in search spaces with suitably sparse solution sets.

Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD without such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove that after T rounds, the suboptimality of the last SGD iterate scales as O(log(T)/\sqrt{T}) for non-smooth convex objective functions, and O(log(T)/T) in the non-smooth strongly convex case. To the best of our knowledge, these are the first bounds of this kind, and almost match the minimax-optimal rates obtainable by appropriate averaging schemes. We also propose a new and simple averaging scheme, which not only attains optimal rates, but can also be easily computed on-the-fly (in contrast, the suffix averaging scheme proposed in Rakhlin et al. (2011) is not as simple to implement). Finally, we provide some experimental illustrations.