On Dependence Measures Based on $Φ$-Divergence, $Φ$-Entropy, and Their Matrix Forms

Abstract

We study two information measures associated with a convex function $Φ$: the $Φ$-mutual information $\IPhi$ and the max-$Φ$-mutual information $\wIPhi$. The measure $I_Φ$ is naturally induced by $Φ$-divergence and $Φ$-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 $Φ$, 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 $Φ$-entropy and the matrix $Φ$-mutual information. We propose the matrix $Φ$-ribbons, prove tensorization and data processing properties. As applications, we derive monotonicity of matrix $Φ$-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.