Chernoff Information between Gaussian Trees

TL;DR

This paper systematically studies Chernoff information between Gaussian trees, revealing how it relates to their structure, and introduces methods for simplification and dimensionality reduction based on graph properties.

Contribution

It establishes a relationship between Chernoff information and covariance matrix eigenvalues, and proposes graph operations and reduction techniques to analyze and simplify Gaussian tree comparisons.

Findings

Chernoff information relates to generalized eigenvalues of covariance matrices.

Grafting operations can reduce complex Gaussian trees to simpler 3-node trees.

Multiple grafting operations do not necessarily increase Chernoff information.

Abstract

In this paper, we aim to provide a systematic study of the relationship between Chernoff information and topological, as well as algebraic properties of the corresponding Gaussian tree graphs for the underlying graphical testing problems. We first show the relationship between Chernoff information and generalized eigenvalues of the associated covariance matrices. It is then proved that Chernoff information between two Gaussian trees sharing certain local subtree structures can be transformed into that of two smaller trees. Under our proposed grafting operations, bottleneck Gaussian trees, namely, Gaussian trees connected by one such operation, can thus be simplified into two 3-node Gaussian trees, whose topologies and edge weights are subject to the specifics of the operation. Thereafter, we provide a thorough study about how Chernoff information changes when small differences are accumulated into bigger ones via concatenated grafting operations. It is shown that the two Gaussian trees connected by more than one grafting operation may not have bigger Chernoff information than that of one grafting operation unless these grafting operations are separate and independent. At the end, we propose an optimal linear dimensional reduction method related to generalized eigenvalues.