$\Phi$ -Entropic Measures of Correlation

Abstract

A measure of correlation is said to have the tensorization property if it does not change 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 $(X^{n}, Y^{n})$ denotes $n$ 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 the HC ribbon are special cases of the new notion of $\Phi $ -ribbons, defined in this paper for a class of convex functions $\Phi $ . $\Phi $ -ribbon reduces to the HC ribbon and the maximal correlation for special choices of $\Phi $ , and is a measure of correlation with the tensorization property. We show that the $\Phi $ -ribbon also characterizes the recently introduced $\Phi $ -strong data processing inequality constant. We further study the $\Phi $ -ribbon for the choice of $\Phi (t)=t^{2}$ and introduce an equivalent characterization of this ribbon.