Isometric Signal Processing under Information Geometric Framework

Abstract

Information geometry is the study of the intrinsic geometric properties of manifolds consisting of a probability distribution and provides a deeper understanding of statistical inference. Based on this discipline, this letter reports on the influence of the signal processing on the geometric structure of the statistical manifold in terms of estimation issues. This letter defines the intrinsic parameter submanifold, which reflects the essential geometric characteristics of the estimation issues. Moreover, the intrinsic parameter submanifold is proven to be a tighter one after signal processing. In addition, the necessary and sufficient condition of invariant signal processing of the geometric structure, i.e., isometric signal processing, is given. Specifically, considering the processing with the linear form, the construction method of linear isometric signal processing is proposed, and its properties are presented in this letter.

Information geometry for Fermi-Dirac and Bose-Einstein quantum statistics

“…Here we are going to focus on useful forms to express FRIM (7) for Gibbs distributions. In (2) Gibbs distribution were defined in terms of the LMs λ however, as they come from (1) they could also be parametrized in terms the expected values A and the two set of parameters are related by (6). This is not more than a change of variables that can be characterized using…”

“…In this geometric description, the distances are obtained from the Fisher-Rao information metric (FRIM) [3,4] and represent the distinguishability between neighbouring probability distributions. This finds application in a large number of information science disciplines including machine learning [5], signal processing [6] as well as quantum information [7] and statistical physics. More specifically, the role played by geometric techniques in physics cannot be underestimated [8].…”