🤖 AI Summary
α-Mutual Information (α-MI) suffers from semantic ambiguity and insufficient theoretical grounding in quantifying privacy leakage.
Method: We systematically reconstruct its theoretical foundation by (i) introducing a novel conditional Rényi entropy satisfying the “conditional entropy reduction” property and the data processing inequality; (ii) establishing multiple equivalent characterizations of α-MI—via inverse-channel variational representation—in terms of Rényi divergence and the new conditional Rényi entropy; and (iii) unifying these as privacy leakage measures tailored to generalized means and gain functions.
Contribution/Results: First, we establish α-MI as a rigorous, semantically interpretable privacy leakage metric. Second, we derive a mathematically well-defined conditional Rényi entropy with transparent privacy semantics. Third, our framework provides finer-grained leakage assessment tools applicable to differential privacy, the information bottleneck, and related settings—enhancing both theoretical coherence and practical utility.
📝 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.