Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses

We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1162 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Geodesics on the manifold of multivariate generalized Gaussian distributions with an application to multicomponent texture discrimination

We consider the Rao geodesic distance (GD) based on the Fisher information as a similarity measure on the manifold of zero-mean multivariate generalized Gaussian distributions (MGGD). The MGGD is shown to be an adequate model for the heavy-tailed wavelet statistics in multicomponent images, such as color or multispectral images. We discuss the estimation of MGGD parameters using various methods. We apply the GD between MGGDs to color texture discrimination in several classification experiments, taking into account the correlation structure between the spectral bands in the wavelet domain. We compare the performance, both in terms of texture discrimination capability and computational load, of the GD and the Kullback-Leibler divergence (KLD). Likewise, both uni- and multivariate generalized Gaussian models are evaluated, characterized by a fixed or a variable shape parameter. The modeling of the interband correlation significantly improves classification efficiency, while the GD is shown to consistently outperform the KLD as a similarity measure

Multivariate texture discrimination based on geodesics to class centroids on a generalized Gaussian Manifold

A texture discrimination scheme is proposed wherein probability distributions are deployed on a probabilistic manifold for modeling the wavelet statistics of images. We consider the Rao geodesic distance (GD) to the class centroid for texture discrimination in various classification experiments. We compare the performance of GD to class centroid with the Euclidean distance in a similar context, both in terms of accuracy and computational complexity. Also, we compare our proposed classification scheme with the k-nearest neighbor algorithm. Univariate and multivariate Gaussian and Laplace distributions, as well as generalized Gaussian distributions with variable shape parameter are each evaluated as a statistical model for the wavelet coefficients. The GD to the centroid outperforms the Euclidean distance and yields superior discrimination compared to the k-nearest neighbor approach

The Burbea-Rao and Bhattacharyya centroids

We study the centroid with respect to the class of information-theoretic Burbea-Rao divergences that generalize the celebrated Jensen-Shannon divergence by measuring the non-negative Jensen difference induced by a strictly convex and differentiable function. Although those Burbea-Rao divergences are symmetric by construction, they are not metric since they fail to satisfy the triangle inequality. We first explain how a particular symmetrization of Bregman divergences called Jensen-Bregman distances yields exactly those Burbea-Rao divergences. We then proceed by defining skew Burbea-Rao divergences, and show that skew Burbea-Rao divergences amount in limit cases to compute Bregman divergences. We then prove that Burbea-Rao centroids are unique, and can be arbitrarily finely approximated by a generic iterative concave-convex optimization algorithm with guaranteed convergence property. In the second part of the paper, we consider the Bhattacharyya distance that is commonly used to measure overlapping degree of probability distributions. We show that Bhattacharyya distances on members of the same statistical exponential family amount to calculate a Burbea-Rao divergence in disguise. Thus we get an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods found in the literature that were limited to univariate or "diagonal" multivariate Gaussians. To illustrate the performance of our Bhattacharyya/Burbea-Rao centroid algorithm, we present experimental performance results for $k$-means and hierarchical clustering methods of Gaussian mixture models.Comment: 13 page

On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means

The Jensen-Shannon divergence is a renown bounded symmetrization of the unbounded Kullback-Leibler divergence which measures the total Kullback-Leibler divergence to the average mixture distribution. However the Jensen-Shannon divergence between Gaussian distributions is not available in closed-form. To bypass this problem, we present a generalization of the Jensen-Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using generalized statistical mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen-Shannon divergence between probability densities of the same exponential family, and (ii) the geometric JS-symmetrization of the reverse Kullback-Leibler divergence. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen-Shannon divergence between scale Cauchy distributions. We also define generalized Jensen-Shannon divergences between matrices (e.g., quantum Jensen-Shannon divergences) and consider clustering with respect to these novel Jensen-Shannon divergences.Comment: 30 page

kk-MLE: A fast algorithm for learning statistical mixture models

We describe $k$-MLE, a fast and efficient local search algorithm for learning finite statistical mixtures of exponential families such as Gaussian mixture models. Mixture models are traditionally learned using the expectation-maximization (EM) soft clustering technique that monotonically increases the incomplete (expected complete) likelihood. Given prescribed mixture weights, the hard clustering $k$-MLE algorithm iteratively assigns data to the most likely weighted component and update the component models using Maximum Likelihood Estimators (MLEs). Using the duality between exponential families and Bregman divergences, we prove that the local convergence of the complete likelihood of $k$-MLE follows directly from the convergence of a dual additively weighted Bregman hard clustering. The inner loop of $k$-MLE can be implemented using any $k$-means heuristic like the celebrated Lloyd's batched or Hartigan's greedy swap updates. We then show how to update the mixture weights by minimizing a cross-entropy criterion that implies to update weights by taking the relative proportion of cluster points, and reiterate the mixture parameter update and mixture weight update processes until convergence. Hard EM is interpreted as a special case of $k$-MLE when both the component update and the weight update are performed successively in the inner loop. To initialize $k$-MLE, we propose $k$-MLE++, a careful initialization of $k$-MLE guaranteeing probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201

A unified framework for information integration based on information geometry

We propose a unified theoretical framework for quantifying spatio-temporal interactions in a stochastic dynamical system based on information geometry. In the proposed framework, the degree of interactions is quantified by the divergence between the actual probability distribution of the system and a constrained probability distribution where the interactions of interest are disconnected. This framework provides novel geometric interpretations of various information theoretic measures of interactions, such as mutual information, transfer entropy, and stochastic interaction in terms of how interactions are disconnected. The framework therefore provides an intuitive understanding of the relationships between the various quantities. By extending the concept of transfer entropy, we propose a novel measure of integrated information which measures causal interactions between parts of a system. Integrated information quantifies the extent to which the whole is more than the sum of the parts and can be potentially used as a biological measure of the levels of consciousness

Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, April 2015

A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada, and Director of the Centre for Mathematical and Statistical Sciences, India. He has published over 300 research papers and more than 25 books on topics in mathematics, statistics, physics, astrophysics, chemistry, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, President of the Mathematical Society of India, and a Member of the International Statistical Institute. He is the founder of the Canadian Journal of Statistics and the Statistical Society of Canada. He is instrumental in the implementation of the United Nations Basic Space Science Initiative. The paper is an attempt to capture the broad spectrum of scientific endeavors of Professor A.M. Mathai at the occasion of his anniversary.Comment: 21 pages, LaTe