A new measure of dependence: Integrated $R^2$

TL;DR

This paper introduces a new dependence measure called integrated R^2 that quantifies the relationship between variables, extending to conditional dependence, with a simple estimator and a variable selection method that outperforms existing techniques.

Contribution

The paper proposes a novel, interpretable dependence measure with a computationally efficient estimator and a consistent, tuning-free variable selection procedure.

Findings

The measure equals zero for independence and one when Y is a function of X.

The estimator has complexity comparable to classical correlation coefficients.

The method shows strong empirical performance and outperforms existing approaches.

Abstract

We introduce a novel measure of dependence that captures the extent to which a random variable $Y$ is determined by a random vector $X$. The measure equals zero precisely when $Y$ and $X$ are independent, and it attains one exactly when $Y$ is almost surely a measurable function of $X$. We further extend this framework to define a measure of conditional dependence between $Y$ and $X$ given $Z$. We propose a simple and interpretable estimator with computational complexity comparable to classical correlation coefficients, including those of Pearson, Spearman, and Chatterjee. Leveraging this dependence measure, we develop a tuning-free, model-agnostic variable selection procedure and establish its consistency under appropriate sparsity conditions. Extensive experiments on synthetic and real datasets highlight the strong empirical performance of our methodology and demonstrate substantial gains over existing approaches.