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

Tsallis entropy induced metrics and CAT(k) spaces

Generalizing the group structure of the Euclidean space, we construct a Riemannian metric on the deformed set \ $\mathbb{R}^n_q$ \ induced by the Tsallis entropy composition property. We show that the Tsallis entropy is a "hyperbolic analogue" of the "Euclidean" Boltzmann/Gibbs/Shannon entropy and find a geometric interpretation for the nonextensive parameter $q$. We provide a geometric explanation of the uniqueness of the Tsallis entropy as reflected through its composition property, which is provided by the Abe and the Santos axioms. For two, or more, interacting systems described by the Tsallis entropy, having different values of $q$, we argue why a suitable extension of this construction is provided by the Cartan/Alexandrov/Toponogov metric spaces with a uniform negative curvature upper bound.Comment: 23 pages, Standard LaTeX2e, Accepted for publication in Physica

Quantum realism: axiomatization and quantification

The emergence of an objective reality in line with the laws of the microscopic world has been the focus of longstanding debates. Recent approaches seem to have reached a consensus at least with respect to one aspect, namely, that the encoding of information about a given observable in a physical degree of freedom is a necessary condition for such observable to become an element of the physical reality. Taking this as a fundamental premise and inspired by quantum information theory, here we build an axiomatization for quantum realism -- a notion of realism compatible with quantum theory. Our strategy consists of listing some physically-motivated principles able to characterize quantum realism in a ``metric'' independent manner. We introduce some criteria defining monotones and measures of realism and then search for potential candidates within some celebrated information theories -- those induced by the von Neumann, R\'enyi, and Tsallis entropies. We explicitly construct some classes of entropic quantifiers that are shown to satisfy (almost all of) the proposed axioms and hence can be taken as faithful estimates for the degree of reality (or definiteness) of a given physical observable. Hopefully, our framework may offer a formal ground for further discussions on foundational aspects of quantum mechanics.Comment: 15 pages, 4 figure

Clustering above Exponential Families with Tempered Exponential Measures

The link with exponential families has allowed $k$-means clustering to be generalized to a wide variety of data generating distributions in exponential families and clustering distortions among Bregman divergences. Getting the framework to work above exponential families is important to lift roadblocks like the lack of robustness of some population minimizers carved in their axiomatization. Current generalisations of exponential families like $q$-exponential families or even deformed exponential families fail at achieving the goal. In this paper, we provide a new attempt at getting the complete framework, grounded in a new generalisation of exponential families that we introduce, tempered exponential measures (TEM). TEMs keep the maximum entropy axiomatization framework of $q$-exponential families, but instead of normalizing the measure, normalize a dual called a co-distribution. Numerous interesting properties arise for clustering such as improved and controllable robustness for population minimizers, that keep a simple analytic form

Modular Theory, Non-Commutative Geometry and Quantum Gravity

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields

Risk measures and economic capital for (re)insurers

This contribution relates to the use of risk measures for determining (re)insurers’ economic capital requirements. Alternative sets of properties of risk measures are discussed. Furthermore, methods for constructing risk measures via indifference arguments, representation results and re-weighting of probability distributions are presented. It is shown how these different approaches relate to popular risk measures, such as VaR, Expected Shortfall, distortion risk measures and the exponential premium principle. The problem of allocating aggregate economic capital to sub-portfolios (e.g. insurers’ lines of business) is then considered, with particular emphasis on marginal-cost-type methods. The relationship between insurance pricing and capital allocation is briefly discussed, based on concepts such as the opportunity and frictional costs of capital and the impact of the potential of default on insurance rates

Non-classicalities in quantum walks and an axiomatic approach to quantum realism

Orientador: Prof. Dr. Renato Moreira AngeloTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Física. Defesa : Curitiba, 26/05/2022Inclui referências: p. 125-137Resumo: No cerne das estranhezas da mecânica quântica estão a superposição de estados e a complementariedade de Bohr, noções conflitantes com a nossa percepção de realidade física macroscópica. Recentemente, uma hipótese de realismo foi formulada assumindo que a mecânica quântica constitui uma teoria física completa. Esta hipótese parte de uma ideia que é compartilhada também por defensores do Darwinismo Quântico: de que a codificação de informação sobre um dado observável em um grau de liberdade físico é uma condição necessária para que tal observável se torne um elemento de realidade física. Nesta tese, nós exploramos tal proposta de realismo dentro da teoria quântica em duas partes. Na Parte I nós estudamos um sistema físico conhecido como caminhadas quânticas e analisamos como se dá a emergência de realidade física objetiva de observáveis de spins durante a evolução de diversas não-classicalidades entre os subsistemas, a citar, não-localidade de Bell, direcionamento quântico, emaranhamento, discórdia quântica, irrealismo e não-localidade baseada em realismo. Motivados por esta análise, nós buscamos, na Parte II, nos aprofundar ainda mais no conceito de realismo dentro da mecânica quântica. Tomando a ideia de fluxo de informação do sistema para o ambiente como condição necessária para a emergência de realidade física, nós construímos uma axiomatização para o aqui chamado realismo quântico-em oposição ao realismo clássico. Nossa estratégia consiste em listar alguns princípios motivados fisicamente que sejam capazes de caracterizar o realismo quântico de maneira independente de "métrica". Introduzimos alguns critérios que definem monótonas e medidas de realidade e, em seguida, procuramos potenciais candidatos dentro de algumas teorias da informação célebres (entropias de von Neumann, Rényi e Tsallis) e também por medidas geométricas (distâncias do traço, Hilbert-Schmidt, Bures e Hellinger). Construímos explicitamente algumas classes de quantificadores entrópicos e geométricos, entre os quais que alguns satisfazem todos os axiomas propostos e, portanto, podem ser tomados como estimativas fiéis para o grau de realidade (ou definidade) de um dado observável físico. Nós esperamos que nossa estrutura possa oferecer uma base formal para futuras discussões sobre aspectos fundamentais da mecânica quântica.Abstract: At the heart of the strangeness of quantum mechanics are the superposition of states and Bohr's complementarity, notions that are in conflict with our perception of macroscopic physical reality. Recently, a realism hypothesis has been formulated assuming that quantum mechanics constitutes a complete physical theory. This hypothesis starts from an idea that is also shared by supporters of Quantum Darwinism: that the encoding of information about a given observable in a physical degree of freedom is a necessary condition for such an observable to become an element of physical reality. In this thesis, we explore such a proposal of realism within quantum theory into two parts. In Part I we study a physical system known as quantum walks and analyze how the emergence of objective physical reality of spin observables occurs during the evolution of several non-classicalities between subsystems, namely, Bell nonlocality, quantum steering, entanglement, quantum discord, irrealism, and realism-based nonlocality. Motivated by this analysis, we seek, in Part II, to get even further into the concept of realism within quantum mechanics. Taking the idea of information flow from the system to the environment as a necessary condition for the emergence of physical reality, we build an axiomatization for the here called quantum realism-as opposed to classical realism. Our strategy is to list some physically motivated principles that are capable of characterizing quantum realism in a "metric" independent way. We introduce some criteria that define monotones and measures of reality and then we look for potential candidates within some famous information theories (von Neumann, Rényi and Tsallis entropies) and also by geometric measures (trace, Hilbert- Schmidt, Bures, and Hellinger distances). We explicitly build some classes of entropic and geometric quantifiers, among which some satisfy all the proposed axioms and, therefore, can be taken as faithful estimates for the degree of reality (or definiteness) of a given physical observable. We hope that our framework can provide a formal basis for future discussions of fundamental aspects of quantum mechanics

Calculus, heat flow and curvature-dimension bounds in metric measure spaces

The theory of curvature-dimension bounds for nonsmooth spaces has several motivations: the study of functional and geometric inequalities in structures which arc very far from being Euclidean, therefore with new non-Riemannian tools, the description of the \u201cclosure\u201d of classes of Riemannian manifolds under suitable geometric constraints, the stability of analytic and geometric properties of spaces (e.g. to prove rigidity results). Even though these goals may occasionally be in conflict, in the last few years we have seen spectacular developments in all these directions, and my text is meant both as a survey and as an introduction to this quickly developing research field