Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization

We propose a class of multiplicative algorithms for Nonnegative Matrix Factorization (NMF) which are robust with respect to noise and outliers. To achieve this, we formulate a new family generalized divergences referred to as the Alpha-Beta-divergences (AB-divergences), which are parameterized by the two tuning parameters, alpha and beta, and smoothly connect the fundamental Alpha-, Beta- and Gamma-divergences. By adjusting these tuning parameters, we show that a wide range of standard and new divergences can be obtained. The corresponding learning algorithms for NMF are shown to integrate and generalize many existing ones, including the Lee-Seung, ISRA (Image Space Reconstruction Algorithm), EMML (Expectation Maximization Maximum Likelihood), Alpha-NMF, and Beta-NMF. Owing to more degrees of freedom in tuning the parameters, the proposed family of AB-multiplicative NMF algorithms is shown to improve robustness with respect to noise and outliers. The analysis illuminates the links of between AB-divergence and other divergences, especially Gamma- and Itakura-Saito divergences

Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT

Crossing symmetry (CS) is the main tool in the bootstrap program applied to CFT. This consists in an equality which imposes restrictions on the CFT data of a model, i.e., the OPE coefficients and the conformal dimensions. Reflection positivity (RP) has also played a role in this program, since this condition is what leads to the unitary bound and reality of the OPE coefficients. In this paper, we show that RP can still reveal more information, explaining how RP itself can capture an important part of the restrictions imposed by the full CS equality. In order to do that, we use a connection used by us in a previous work between RP and positive definiteness of a function of a single variable. This allows us to write constraints on the OPE coefficients in a concise way. These constraints are encoded in the conditions that certain functions of the cross-ratio will be positive defined and in particular completely monotonic. We will consider how the bounding of scalar conformal dimensions and OPE coefficients arise in this RP based approach. We will illustrate the conceptual and practical value of this view trough examples of general CFT models in d-dimensions.Fil: Lanosa, Leandro Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Leston, Mauricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; ArgentinaFil: Passaglia, Mario. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentin

A perturbative approach to the reconstruction of the eigenvalue spectrum of a normal covariance matrix from a spherically truncated counterpart

In this paper we propose a perturbative method for the reconstruction of the covariance matrix of a multinormal distribution, under the assumption that the only available information amounts to the covariance matrix of a spherically truncated counterpart of the same distribution. We expand the relevant equations up to the fourth perturbative order and discuss the analytic properties of the first few perturbative terms. We finally compare the proposed approach with an exact iterative algorithm (presented in Palombi et al. (2017)) in the hypothesis that the spherically truncated covariance matrix is estimated from samples of various sizes.Comment: 39 pages, 7 figures. v2: version accepted for publication in J. Comp. Appl. Mat

Two-component galaxies with flat rotation curve

Dynamical properties of two-component galaxy models whose stellar density distribution is described by a gamma-model while the total density distribution has a pure r^(-2) profile, are presented. The orbital structure of the stellar component is described by Osipkov-Merritt anisotropy, while the dark matter halo is isotropic. After a description of minimum halo models, the positivity of the phase-space density (the model consistency) is investigated, and necessary and sufficient conditions for consistency are obtained analytically as a function of the stellar inner density slope gamma and anisotropy radius. The explicit phase-space distribution function is recovered for integer values of gamma, and it is shown that while models with gamma>4/17 are consistent when the anisotropy radius is larger than a critical value (dependent on gamma), the gamma=0 models are unphysical even in the fully isotropic case. The Jeans equations for the stellar component are then solved analytically; in addition, the projected velocity dispersion at the center and at large radii are also obtained analytically for generic values of the anisotropy radius, and it is found that they are given by remarkably simple expressions. The presented models, even though highly idealized, can be useful as starting point for more advanced modeling of the mass distribution of elliptical galaxies in studies combining stellar dynamics and gravitational lensing.Comment: 11 pages, 5 figures, accepted by MNRA

Big data, computational science, economics, finance, marketing, management, and psychology: connections

The paper provides a review of the literature that connects Big Data, Computational Science, Economics, Finance, Marketing, Management, and Psychology, and discusses some research that is related to the seven disciplines. Academics could develop theoretical models and subsequent econometric and statistical models to estimate the parameters in the associated models, as well as conduct simulation to examine whether the estimators in their theories on estimation and hypothesis testing have good size and high power. Thereafter, academics and practitioners could apply theory to analyse some interesting issues in the seven disciplines and cognate areas

Nuclear and Quark Matter at High Temperature

We review important ideas on nuclear and quark matter description on the basis of high- temperature field theory concepts, like resummation, dimensional reduction, interaction scale separation and spectral function modification in media. Statistical and thermodynamical concepts are spotted in the light of these methods concentrating on the - partially still open - problems of the hadronization process.Comment: Review intended for EPJ A Topical Issu

On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling

Documento depositado en el repositorio arXiv.org. Versión: arXiv:1205.0482v6 [stat.CO]In this work we investigate the relationship among three classical sampling techniques: the inverse of density (Khintchine's theorem), the transformed rejection (TR) and the generalized ratio of uniforms (GRoU). Given a monotonic probability density function (PDF), we show that the transformed area obtained using the generalized ratio of uniforms method can be found equivalently by applying the transformed rejection sampling approach to the inverse function of the target density. Then we provide an extension of the classical inverse of density idea, showing that it is completely equivalent to the GRoU method for monotonic densities. Although we concentrate on monotonic probability density functions (PDFs), we also discuss how the results presented here can be extended to any non-monotonic PDF that can be decomposed into a collection of intervals where it is monotonically increasing or decreasing. In this general case, we show the connections with transformations of certain random variables and the generalized inverse PDF with the GRoU technique. Finally, we also introduce a GRoU technique to handle unbounded target densities