Phi-Entropic Measures of Correlation

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 $ρ(X, Y)$, has the tensorization property if $ρ(X^n, Y^n)=ρ(X, Y)$ 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 $Φ$-ribbon, defined in this paper for any function $Φ$ from a class of convex functions ($Φ$-ribbon reduces to HC~ribbon and the maximal correlation for special choices of $Φ$). Any $Φ$-ribbon is shown to be a measures of correlation with the tensorization property. We show that the $Φ$-ribbon also characterizes the $Φ$-strong data processing inequality constant introduced by Raginsky. We further study the $Φ$-ribbon for the choice of $Φ(t)=t^2$ and introduce an equivalent characterization of this ribbon.