Several Representations of $\alpha$-Mutual Information and Interpretations as Privacy Leakage Measures

TL;DR

This paper introduces new mathematical representations of $$-mutual information using Re9nyi divergence, offering interpretations as privacy leakage measures and proposing novel conditional Re9nyi entropies with desirable properties.

Contribution

It provides novel representations of $$-mutual information via Re9nyi divergence and introduces new conditional Re9nyi entropies with key properties.

Findings

New representations of $$-mutual information using Re9nyi divergence.

Interpretations of $$-mutual information as privacy leakage measures.

Proposed conditional Re9nyi entropies satisfying key information-theoretic properties.

Abstract

In this paper, we present several novel representations of $\alpha$-mutual information ($\alpha$-MI) in terms of R{\' e}nyi divergence and conditional R{\' e}nyi entropy. The representations are based on the variational characterizations of $\alpha$-MI using a reverse channel. Based on these representations, we provide several interpretations of the $\alpha$-MI as privacy leakage measures using generalized mean and gain functions. Further, as byproducts of the representations, we propose novel conditional R{\' e}nyi entropies that satisfy the property that conditioning reduces entropy and data-processing inequality.