The Bregman Variational Dual-Tree Framework

Graph-based methods provide a powerful tool set for many non-parametric frameworks in Machine Learning. In general, the memory and computational complexity of these methods is quadratic in the number of examples in the data which makes them quickly infeasible for moderate to large scale datasets. A significant effort to find more efficient solutions to the problem has been made in the literature. One of the state-of-the-art methods that has been recently introduced is the Variational Dual-Tree (VDT) framework. Despite some of its unique features, VDT is currently restricted only to Euclidean spaces where the Euclidean distance quantifies the similarity. In this paper, we extend the VDT framework beyond the Euclidean distance to more general Bregman divergences that include the Euclidean distance as a special case. By exploiting the properties of the general Bregman divergence, we show how the new framework can maintain all the pivotal features of the VDT framework and yet significantly improve its performance in non-Euclidean domains. We apply the proposed framework to different text categorization problems and demonstrate its benefits over the original VDT.Comment: Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI2013

Learning to Approximate a Bregman Divergence

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman divergence from supervision, and we provide a well-principled approach to analyzing such approximations. We develop a formulation and algorithm for learning arbitrary Bregman divergences based on approximating their underlying convex generating function via a piecewise linear function. We provide theoretical approximation bounds using our parameterization and show that the generalization error $O_p(m^{-1/2})$ for metric learning using our framework matches the known generalization error in the strictly less general Mahalanobis metric learning setting. We further demonstrate empirically that our method performs well in comparison to existing metric learning methods, particularly for clustering and ranking problems.Comment: 19 pages, 4 figure

Approximate Bregman near neighbors in sublinear time: beyond the triangle inequality

pre-printBregman divergences are important distance measures that are used extensively in data-driven applications such as computer vision, text mining, and speech processing, and are a key focus of interest in machine learning. Answering nearest neighbor (NN) queries under these measures is very important in these applications and has been the subject of extensive study, but is problematic because these distance measures lack metric properties like symmetry and the triangle inequality. In this paper, we present the first provably approximate nearest-neighbor (ANN) algorithms. These process queries in O(logn) time for Bregman divergences in fixed dimensional spaces. We also obtain polylogn bounds for a more abstract class of distance measures (containing Bregman divergences) which satisfy certain structural properties . Both of these bounds apply to both the regular asymmetric Bregman divergences as well as their symmetrized versions. To do so, we develop two geometric properties vital to our analysis: a reverse triangle inequality (RTI) and a relaxed triangle inequality called m-defectiveness where m is a domain-dependent parameter. Bregman divergences satisfy the RTI but not m-defectiveness. However, we show that the square root of a Bregman divergence does satisfy m-defectiveness. This allows us to then utilize both properties in an efficient search data structure that follows the general two-stage paradigm of a ring-tree decomposition followed by a quad tree search used in previous near-neighbor algorithms for Euclidean space and spaces of bounded doubling dimension. Our first algorithm resolves a query for a d-dimensional (1+e)-ANN in O ( logne )O(d) time and O (nlogd-1 n) space and holds for generic m-defective distance measures satisfying a RTI. Our second algorithm is more specific in analysis to the Bregman divergences and uses a further structural constant, the maximum ratio of second derivatives over each dimension of our domain (c0). This allows us to locate a (1+e)-ANN in O(logn) time and O(n) space, where there is a further (c0)d factor in the big-Oh for the query time

A Smoothed Dual Approach for Variational Wasserstein Problems

Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by Carlier et al. (2014) can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals