Phi-Entropic Measures of Correlation

TL;DR

This paper introduces the $ ext{Phi}$-ribbon, a new class of correlation measures with tensorization properties, unifying existing measures like maximal correlation and HC ribbon, and characterizes related data processing inequalities.

Contribution

The paper defines the $ ext{Phi}$-ribbon for convex functions, unifies existing correlation measures, and links it to $ ext{Phi}$-strong data processing inequalities.

Findings

$ ext{Phi}$-ribbon generalizes maximal correlation and HC ribbon.

$ ext{Phi}$-ribbon has the tensorization property.

Characterization of $ ext{Phi}$-ribbon for $ ext{Phi}(t)=t^2$.

Abstract

A measure of correlation is said to have the tensorization property if it is unchanged when computed for i.i.d.\ copies. More precisely, a measure of correlation between two random variables $(X, Y)$ denoted by $\rho(X, Y)$, has the tensorization property if $\rho(X^n, Y^n)=\rho(X, Y)$ where $(X^n, Y^n)$ is $n$ i.i.d.\ copies of $(X, Y)$.Two well-known examples of such measures are the maximal correlation and the hypercontractivity ribbon (HC~ribbon). We show that the maximal correlation and HC ribbons are special cases of $\Phi$-ribbon, defined in this paper for any function $\Phi$ from a class of convex functions ($\Phi$-ribbon reduces to HC~ribbon and the maximal correlation for special choices of $\Phi$). Any $\Phi$-ribbon is shown to be a measures of correlation with the tensorization property. We show that the $\Phi$-ribbon also characterizes the $\Phi$-strong data processing inequality constant introduced by Raginsky. We further study the $\Phi$-ribbon for the choice of $\Phi(t)=t^2$ and introduce an equivalent characterization of this ribbon.