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

Akira Kamatsuka, Takashiro Yoshida

TL;DR

This work develops multiple representations of $α$-mutual information for $α \in (0,1) \cup (1,\infty)$ via Rényi divergence and conditional Rényi entropies, unifying several α-MI notions (Sibson, Arimoto, Hayashi, Augustin–Csiszár, Lapidoth–Pfister). It introduces novel conditional Rényi entropies that satisfy CRE and the data-processing inequality, and derives reverse-channel variational forms for key α-MI definitions. The authors further interpret $α$-MI as privacy leakage by linking each variant to generalized means and scoring rules, thereby characterizing adversary gains under different decision rules. Overall, the results deepen the mathematical understanding of privacy leakage in α-MI frameworks and provide tools for privacy-preserving data publishing analyses.

Abstract

In this paper, we present several novel representations of $α$-mutual information ($α$-MI) in terms of R{\' e}nyi divergence and conditional R{\' e}nyi entropy. The representations are based on the variational characterizations of $α$-MI using a reverse channel. Based on these representations, we provide several interpretations of the $α$-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.

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

TL;DR

This work develops multiple representations of -mutual information for via Rényi divergence and conditional Rényi entropies, unifying several α-MI notions (Sibson, Arimoto, Hayashi, Augustin–Csiszár, Lapidoth–Pfister). It introduces novel conditional Rényi entropies that satisfy CRE and the data-processing inequality, and derives reverse-channel variational forms for key α-MI definitions. The authors further interpret -MI as privacy leakage by linking each variant to generalized means and scoring rules, thereby characterizing adversary gains under different decision rules. Overall, the results deepen the mathematical understanding of privacy leakage in α-MI frameworks and provide tools for privacy-preserving data publishing analyses.

Abstract

In this paper, we present several novel representations of -mutual information (-MI) in terms of R{\' e}nyi divergence and conditional R{\' e}nyi entropy. The representations are based on the variational characterizations of -MI using a reverse channel. Based on these representations, we provide several interpretations of the -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.
Paper Structure (9 sections, 11 theorems, 32 equations)

This paper contains 9 sections, 11 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Representations of $\alpha$-MI using Rényi Divergence and Conditional Entropies
  5. Novel conditional Rényi entropies
  6. Interpretation of $\alpha$-MI as Privacy Leakage Measure
  7. Conclusion
  8. Proof of Theorem \ref{['thm:alpha_MI_diff']}
  9. Proof of Theorem \ref{['thm:interpetations_alpha_MI']}

Key Result

Proposition 1

where and $p_{X_{\alpha}}:=p_{X}^{(\alpha)}$ denotes the $\alpha$-tilted distribution of $p_{X}$, defined in eq:alpha_tilted.

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1: 3701217308959
  • Proposition 2
  • Definition 3: $\alpha$-score
  • Proposition 3: 8804205
  • Proposition 4
  • proof
  • Theorem 1
  • ...and 22 more