We formulate em algorithm in the framework of Bregman divergence, which is a general problem setting of information geometry. That is, we address the minimization problem of the Bregman divergence between an exponential subfamily and a mixture subfamily in a Bregman divergence system. Then, we show the convergence and its speed under several conditions. We apply this algorithm to rate distortion and its variants including the quantum setting, and show the usefulness of our general algorithm. In fact, existing applications of Arimoto-Blahut algorithm to rate distortion theory make the optimization of the weighted sum of the mutual information and the cost function by using the Lagrange multiplier. However, in the rate distortion theory, it is needed to minimize the mutual information under the constant constraint for the cost function. Our algorithm directly solves this minimization. In addition, we have numerically checked the convergence speed of our algorithm in the classical case of rate distortion problem.
The problem to maximize the information divergence from an exponential family is generalized to the setting of Bregman divergences and suitably defined Bregman families.
A class of distortions termed functional Bregman divergences is defined, which includes squared error and relative entropy. A functional Bregman divergence acts on functions or distributions, and generalizes the standard Bregman divergence for vectors and a previous pointwise Bregman divergence that was defined for functions. A recently published result showed that the mean minimizes the expected Bregman divergence. The new functional definition enables the extension of this result to the continuous case to show that the mean minimizes the expected functional Bregman divergence over a set of f…
Sony Computer Science Laboratories Inc.（索尼计算机科学实验室公司）；Sony Computer Science Laboratories（索尼计算机科学实验室）
We first introduce the class of strictly quasiconvex and strictly quasiconcave Jensen divergences which are oriented (asymmetric) distances, and study some of their properties. We then define the strictly quasiconvex Bregman divergences as the limit case of scaled and skewed quasiconvex Jensen divergences, and report a simple closed-form formula which shows that these divergences are only pseudo-divergences at countably many inflection points of the generators. To remedy this problem, we propose the $δ$-averaged quasiconvex Bregman divergences which integrate the pseudo-divergences over a smal…
In this paper, we introduce new classes of divergences by extending the definitions of the Bregman divergence and the skew Jensen divergence. These new divergence classes (g-Bregman divergence and skew g-Jensen divergence) satisfy some properties similar to the Bregman or skew Jensen divergence. We show these g-divergences include divergences which belong to a class of f-divergence (the Hellinger distance, the chi-square divergence and the alpha-divergence in addition to the Kullback-Leibler divergence). Moreover, we derive an inequality between the g-Bregman divergence and the skew g-Jensen d…
A variational principle for the rate distortion (RD) theory with Bregman divergences is formulated within the ambit of the generalized (nonextensive) statistics of Tsallis. The Tsallis-Bregman RD lower bound is established. Alternate minimization schemes for the generalized Bregman RD (GBRD) theory are derived. A computational strategy to implement the GBRD model is presented. The efficacy of the GBRD model is exemplified with the aid of numerical simulations.
École Polytechnique（巴黎综合理工学院）；Sony Computer Science Laboratories Inc（索尼计算机科学研究所）；Data61；Australian National University（澳大利亚国立大学）；University of Sydney（悉尼大学）
Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized Bregman divergences as limit cases of skewed Jensen divergences. In particular, we report explicit formula of these generalized Bregman divergences when considering quasi-arithmetic means. Finally, we introduce a generalization of the Bhattacharyya statistical distances based on comparative means using relative convexity.
Total Bregman divergences are a recent tweak of ordinary Bregman divergences originally motivated by applications that required invariance by rotations. They have displayed superior results compared to ordinary Bregman divergences on several clustering, computer vision, medical imaging and machine learning tasks. These preliminary results raise two important problems : First, report a complete characterization of the left and right population minimizers for this class of total Bregman divergences. Second, characterize a principled superset of total and ordinary Bregman divergences with good cl…
The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman …
There exist two different versions of the Kullback-Leibler divergence (K-Ld) in Tsallis statistics, namely the usual generalized K-Ld and the generalized Bregman K-Ld. Problems have been encountered in trying to reconcile them. A condition for consistency between these two generalized K-Ld-forms by recourse to the additive duality of Tsallis statistics is derived. It is also shown that the usual generalized K-Ld subjected to this additive duality, known as the dual generalized K-Ld, is a scaled Bregman divergence. This leads to an interesting conclusion: the dual generalized mutual information…
The generalized Kullback-Leibler divergence (K-Ld) in Tsallis statistics [constrained by the additive duality of generalized statistics (dual generalized K-Ld)] is here reconciled with the theory of Bregman divergences for expectations defined by normal averages, within a measure-theoretic framework. Specifically, it is demonstrated that the dual generalized K-Ld is a scaled Bregman divergence. The Pythagorean theorem is derived from the minimum discrimination information-principle using the dual generalized K-Ld as the measure of uncertainty, with constraints defined by normal averages. The m…
We investigate connections between information-theoretic and estimation-theoretic quantities in vector Poisson channel models. In particular, we generalize the gradient of mutual information with respect to key system parameters from the scalar to the vector Poisson channel model. We also propose, as another contribution, a generalization of the classical Bregman divergence that offers a means to encapsulate under a unifying framework the gradient of mutual information results for scalar and vector Poisson and Gaussian channel models. The so-called generalized Bregman divergence is also shown …
A loss function measures the discrepancy between the true values and their estimated fits, for a given instance of data. In classification problems, a loss function is said to be proper if a minimizer of the expected loss is the true underlying probability. We show that for binary classification, the divergence associated with smooth, proper, and convex loss functions is upper bounded by the Kullback-Leibler (KL) divergence, to within a normalization constant. This implies that by minimizing the logarithmic loss associated with the KL divergence, we minimize an upper bound to any choice of los…
The paper introduces scaled Bregman distances of probability distributions which admit non-uniform contributions of observed events. They are introduced in a general form covering not only the distances of discrete and continuous stochastic observations, but also the distances of random processes and signals. It is shown that the scaled Bregman distances extend not only the classical ones studied in the previous literature, but also the information divergence and the related wider class of convex divergences of probability measures. An information processing theorem is established too, but onl…
Colorado School of Mines（科罗拉多矿业学院）；Johns Hopkins University（约翰斯·霍普金斯大学）
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothness condition, which is satisfied by any finite-order polynomial objective function. Further, this work develops new Bregman-divergence based algorithms that are guaranteed to converge to a second-order stationary point for any relatively smooth problem. In addition, the proposed…
We discuss unbiased estimation equations in a class of objective function using a monotonically increasing function $f$ and Bregman divergence. The choice of the function $f$ gives desirable properties such as robustness against outliers. In order to obtain unbiased estimation equations, analytically intractable integrals are generally required as bias correction terms. In this study, we clarify the combination of Bregman divergence, statistical model, and function $f$ in which the bias correction term vanishes. Focusing on Mahalanobis and Itakura-Saito distances, we provide a generalization o…
Information divergences are commonly used to measure the dissimilarity of two elements on a statistical manifold. Differentiable manifolds endowed with different divergences may possess different geometric properties, which can result in totally different performances in many practical applications. In this paper, we propose a total Bregman divergence-based matrix information geometry (TBD-MIG) detector and apply it to detect targets emerged into nonhomogeneous clutter. In particular, each sample data is assumed to be modeled as a Hermitian positive-definite (HPD) matrix and the clutter covari…
A general class of Bayesian lower bounds when the underlying loss function is a Bregman divergence is demonstrated. This class can be considered as an extension of the Weinstein--Weiss family of bounds for the mean squared error and relies on finding a variational characterization of Bayesian risk. The approach allows for the derivation of a version of the Cramér--Rao bound that is specific to a given Bregman divergence. The new generalization of the Cramér--Rao bound reduces to the classical one when the loss function is taken to be the Euclidean norm. The effectiveness of the new bound is ev…
Four problems related to information divergence measures defined on finite alphabets are considered. In three of the cases we consider, we illustrate a contrast which arises between the binary-alphabet and larger-alphabet settings. This is surprising in some instances, since characterizations for the larger-alphabet settings do not generalize their binary-alphabet counterparts. Specifically, we show that $f$-divergences are not the unique decomposable divergences on binary alphabets that satisfy the data processing inequality, thereby clarifying claims that have previously appeared in the lite…
It is well-known that there are a number of relations between theoretical finance theory and information theory. Some of these relations are exact and some are approximate. In this paper we will explore some of these relations and determine under which conditions the relations are exact. It turns out that portfolio theory always leads to Bregman divergences. The Bregman divergence is only proportional to information divergence in situations that are essentially equal to the type of gambling studied by Kelly. This can be related an abstract sufficiency condition.
School of Liberal Arts, Korea University of Technology and Education（韩国技术教育大学文理学院）
Exponential dispersion model is a useful framework in machine learning and statistics. Primarily, thanks to the additive structure of the model, it can be achieved without difficulty to estimate parameters including mean. However, tight conditions on cumulant function, such as analyticity, strict convexity, and steepness, reduce the class of exponential dispersion model. In this work, we present relaxed exponential dispersion model K-LED (Legendre exponential dispersion model with K cumulants). The cumulant function of the proposed model is a convex function of Legendre type having continuous …
Sony Computer Science Laboratories, Inc（索尼计算机科学实验室公司）
We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis' quadratic entropy related to the conformal flattening of the Fisher-Rao curved geometry. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, …
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 diverge…
The notion of Bregman divergence and sufficiency will be defined on general convex state spaces. It is demonstrated that only spectral sets can have a Bregman divergence that satisfies a sufficiency condition. Positive elements with trace 1 in a Jordan algebra are examples of spectral sets, and the most important example is the set of density matrices with complex entries. It is conjectured that information theoretic considerations lead directly to the notion of Jordan algebra under some regularity conditions.
Unbiased Estimating Equation on Inverse Divergence and Its Conditions
Media Center Toyohashi University of Technology Toyohashi, Aichi, Japan Email；Department of Computer Science；Engineering Toyohashi University of Technology Toyohashi, Aichi, Japan Email
