A Variational Characterization of $H$-Mutual Information and its Application to Computing $H$-Capacity

TL;DR

This paper introduces a variational approach to characterize $H$-mutual information, a generalized measure of information leakage, and develops an algorithm to compute its capacity, broadening the understanding of information measures.

Contribution

It provides a variational characterization of $H$-mutual information and proposes an optimization algorithm for calculating $H$-capacity, extending existing methods to a broader class of information measures.

Findings

The variational characterization enables new computational techniques.

The proposed algorithm effectively computes $H$-capacity.

The framework generalizes classical mutual information concepts.

Abstract

$H$-mutual information ($H$-MI) is a wide class of information leakage measures, where $H=(\eta, F)$ is a pair of monotonically increasing function $\eta$ and a concave function $F$, which is a generalization of Shannon entropy. $H$-MI is defined as the difference between the generalized entropy $H$ and its conditional version, including Shannon mutual information (MI), Arimoto MI of order $\alpha$, $g$-leakage, and expected value of sample information. This study presents a variational characterization of $H$-MI via statistical decision theory. Based on the characterization, we propose an alternating optimization algorithm for computing $H$-capacity.