From Distance Correlation to Multiscale Graph Correlation

TL;DR

This paper introduces Multiscale Graph Correlation (MGC), a new dependence measure that generalizes distance correlation, offering improved detection of complex dependencies with strong theoretical guarantees and practical performance.

Contribution

The paper develops a unified framework for MGC, establishing its theoretical properties and demonstrating its superior ability to detect diverse dependencies over existing methods.

Findings

MGC is universally consistent for dependence testing.

MGC outperforms distance correlation in complex dependency detection.

Sample MGC is nearly unbiased and converges reliably.

Abstract

Understanding and developing a correlation measure that can detect general dependencies is not only imperative to statistics and machine learning, but also crucial to general scientific discovery in the big data age. In this paper, we establish a new framework that generalizes distance correlation --- a correlation measure that was recently proposed and shown to be universally consistent for dependence testing against all joint distributions of finite moments --- to the Multiscale Graph Correlation (MGC). By utilizing the characteristic functions and incorporating the nearest neighbor machinery, we formalize the population version of local distance correlations, define the optimal scale in a given dependency, and name the optimal local correlation as MGC. The new theoretical framework motivates a theoretically sound Sample MGC and allows a number of desirable properties to be proved, including the universal consistency, convergence and almost unbiasedness of the sample version. The advantages of MGC are illustrated via a comprehensive set of simulations with linear, nonlinear, univariate, multivariate, and noisy dependencies, where it loses almost no power in monotone dependencies while achieving better performance in general dependencies, compared to distance correlation and other popular methods.