On Maximal Correlation, Hypercontractivity, and the Data Processing Inequality studied by Erkip and Cover

TL;DR

This paper introduces a geometric perspective on maximal correlation and hypercontractivity, simplifies existing proofs, and corrects a false data processing inequality with a precise constant.

Contribution

It offers a new geometric characterization of maximal correlation and hypercontractivity, and corrects a previously claimed data processing inequality with an exact constant.

Findings

New geometric characterization of maximal correlation

Simplified proofs of known hypercontractivity properties

Counterexample and correction for a data processing inequality

Abstract

In this paper we provide a new geometric characterization of the Hirschfeld-Gebelein-R\'{e}nyi maximal correlation of a pair of random $(X,Y)$, as well as of the chordal slope of the nontrivial boundary of the hypercontractivity ribbon of $(X,Y)$ at infinity. The new characterizations lead to simple proofs for some of the known facts about these quantities. We also provide a counterexample to a data processing inequality claimed by Erkip and Cover, and find the correct tight constant for this kind of inequality.