Network Maximal Correlation

TL;DR

Network Maximal Correlation (NMC) is a new multivariate measure for nonlinear associations among variables, with algorithms for discrete and Gaussian cases, and applications in graphical model inference and gene dependency analysis.

Contribution

The paper introduces NMC as a novel measure, provides algorithms for its computation, and demonstrates its applications in graphical models and biological data analysis.

Findings

NMC can be computed efficiently for discrete variables using alternating conditional expectation.

For Gaussian variables, NMC relates to the Max-Cut problem, enabling specific solutions.

NMC effectively captures nonlinear dependencies in gene expression data.

Abstract

We introduce Network Maximal Correlation (NMC) as a multivariate measure of nonlinear association among random variables. NMC is defined via an optimization that infers transformations of variables by maximizing aggregate inner products between transformed variables. For finite discrete and jointly Gaussian random variables, we characterize a solution of the NMC optimization using basis expansion of functions over appropriate basis functions. For finite discrete variables, we propose an algorithm based on alternating conditional expectation to determine NMC. Moreover we propose a distributed algorithm to compute an approximation of NMC for large and dense graphs using graph partitioning. For finite discrete variables, we show that the probability of discrepancy greater than any given level between NMC and NMC computed using empirical distributions decays exponentially fast as the sample size grows. For jointly Gaussian variables, we show that under some conditions the NMC optimization is an instance of the Max-Cut problem. We then illustrate an application of NMC in inference of graphical model for bijective functions of jointly Gaussian variables. Finally, we show NMC's utility in a data application of learning nonlinear dependencies among genes in a cancer dataset.