Φ-Entropic Measures of Correlation

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 (Xn, Yn) 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.