Abstract:We study two information measures associated with a convex function $\Phi$: the $\Phi$-mutual information $\IPhi$ and the max-$\Phi$-mutual information $\wIPhi$. The measure $I_\Phi$ is naturally induced by $\Phi$-divergence and $\Phi$-entropy and admits a chain-rule-compatible conditional form, but it lacks several Shannon-type identities. To address this, we introduce and analyze $\wIPhi$, a variational dependence measure obtained by optimizing over auxiliary random variables from $\IPhi$.
We establish basic calculus rules for both quantities, including chain-rule identities, data-processing-type inequalities, deterministic function rules and Markov-chain characterizations. For suitable classes of $\Phi$, we obtain closed-form expressions for $\wIPhi$ and identify conditions under which KL divergence is the unique case with full Shannon-type behavior.
We extend the above notions to the matrix case, with the existing definitions of the matrix $\Phi$-entropy and the matrix $\Phi$-mutual information. We propose the matrix $\Phi$-ribbons, prove tensorization and data processing properties. As applications, we derive monotonicity of matrix $\Phi$-ribbons for wirings of no-signaling boxes, study associated SDPI constants, and obtain ribbon regions from partial independence structures and non-Shannon-type information inequalities.
From: Chenyu Wang [view email]
[v1]
Sat, 8 Feb 2025 06:37:04 UTC (290 KB)
[v2]
Fri, 11 Sep 2026 10:00:18 UTC (554 KB)
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