Topological and Algebraic Properties of Chernoff Information between Gaussian Graphs

TL;DR

This paper investigates how Chernoff information distinguishes Gaussian graphs, revealing that generalized eigenvalues of covariance matrices determine it, and introduces a partial ordering for Gaussian trees to aid classification.

Contribution

It establishes the role of generalized eigenvalues in Chernoff information for Gaussian graphs and proposes a partial ordering for Gaussian trees based on this measure.

Findings

Chernoff information depends on generalized eigenvalues of covariance matrices.

Unit generalized eigenvalues do not influence Chernoff information.

Partial ordering of Gaussian trees can be derived using Chernoff information.

Abstract

In this paper, we want to find out the determining factors of Chernoff information in distinguishing a set of Gaussian graphs. We find that Chernoff information of two Gaussian graphs can be determined by the generalized eigenvalues of their covariance matrices. We find that the unit generalized eigenvalue doesn't affect Chernoff information and its corresponding dimension doesn't provide information for classification purpose. In addition, we can provide a partial ordering using Chernoff information between a series of Gaussian trees connected by independent grafting operations. With the relationship between generalized eigenvalues and Chernoff information, we can do optimal linear dimension reduction with least loss of information for classification.