On the Monotone Measure of Correlation

TL;DR

This paper revisits and generalizes the concordant monotone correlation (CMC), establishing its properties, connections to other measures, and analyzing its computational complexity with an exact exponential-time algorithm.

Contribution

It introduces new properties of CMC, relates it to rank correlations and the FKG inequality, and provides an exact algorithm for its computation.

Findings

CMC captures various rank-based correlations like Kendall tau.

CMC satisfies data processing and tensorization properties.

An exponential-time algorithm guarantees exact CMC computation.

Abstract

Based on the notion of maximal correlation, Kimeldorf, May and Sampson (1980) introduce a measure of correlation between two random variables, called the "concordant monotone correlation" (CMC). We revisit, generalize and prove new properties of this measure of correlation. It is shown that CMC captures various types of correlation detected in measures of rank correlation like the Kendall tau correlation. We show that the CMC satisfies the data processing and tensorization properties (that make ordinary maximal correlation applicable to problems in information theory). Furthermore, CMC is shown to be intimately related to the FKG inequality. Furthermore, a combinatorical application of CMC is given for which we do not know of another method to derive its result. Finally, we study the problem of the complexity of the computation of the CMC, which is a non-convex optimization problem with local maximas. We give a simple but exponential-time algorithm that is guaranteed to output the exact value of the generalized CMC.