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

Akira Kamatsuka, Koki Kazama, Takahiro Yoshida

TL;DR

This study presents a variational characterization of H-mutual information via statistical decision theory and proposes an alternating optimization algorithm for computing $H$-capacity.

Abstract

$H$-mutual information ($H$-MI) is a wide class of information leakage measures, where $H=(η, F)$ is a pair of monotonically increasing function $η$ 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 $α$, $g$-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.

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

TL;DR

This study presents a variational characterization of H-mutual information via statistical decision theory and proposes an alternating optimization algorithm for computing -capacity.

Abstract

-mutual information (-MI) is a wide class of information leakage measures, where is a pair of monotonically increasing function and a concave function , which is a generalization of Shannon entropy. -MI is defined as the difference between the generalized entropy and its conditional version, including Shannon mutual information (MI), Arimoto MI of order , -leakage, and expected value of sample information. This study presents a variational characterization of -MI via statistical decision theory. Based on the characterization, we propose an alternating optimization algorithm for computing -capacity.
Paper Structure (10 sections, 9 theorems, 36 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 10 sections, 9 theorems, 36 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Organization of the Paper
  4. Preliminaries
  5. Notations
  6. Statistical Decision Theory and Scoring Rules
  7. $H$-Mutual information ($H$-MI) 95052069064819
  8. Variational Characterization of $H$-MI
  9. Application: Deriving Algorithm For Computing $H$-Capacity
  10. Conclusion

Key Result

Proposition 1

The minimal Bayes risk is given by with the optimal decision rule $\delta^{\text{*}}\colon \mathcal{Y}\to \mathcal{A}$ given by Similarly, the maximal Bayes expected gain and the optimal decision rule $\delta^{\text{*}}\colon \mathcal{Y}\to \mathcal{A}$ are given by

Figures (1)

  • Figure 1: System model of the statistical decision theory

Theorems & Definitions (34)

  • Proposition 1: GVK027440176, alma991009366059705251
  • Remark 1
  • Example 1
  • Example 2
  • Remark 2
  • Remark 3
  • Example 3
  • Definition 1
  • Example 4
  • Definition 2: 9505206
  • ...and 24 more