Approximating mutual information of high-dimensional variables using learned representations

TL;DR

This paper introduces a new method called latent MI (LMI) approximation that leverages low-dimensional structures in high-dimensional data to accurately estimate mutual information with feasible sample sizes, outperforming existing techniques.

Contribution

The authors develop LMI, a novel approach that applies nonparametric MI estimation to learned low-dimensional representations, enabling accurate MI approximation in high-dimensional data with low intrinsic dimensionality.

Findings

LMI accurately estimates MI in variables with over 1,000 dimensions.

LMI outperforms existing techniques in high-dimensional MI estimation.

Application to biological data reveals insights into protein interactions and cell fate transitions.

Abstract

Mutual information (MI) is a general measure of statistical dependence with widespread application across the sciences. However, estimating MI between multi-dimensional variables is challenging because the number of samples necessary to converge to an accurate estimate scales unfavorably with dimensionality. In practice, existing techniques can reliably estimate MI in up to tens of dimensions, but fail in higher dimensions, where sufficient sample sizes are infeasible. Here, we explore the idea that underlying low-dimensional structure in high-dimensional data can be exploited to faithfully approximate MI in high-dimensional settings with realistic sample sizes. We develop a method that we call latent MI (LMI) approximation, which applies a nonparametric MI estimator to low-dimensional representations learned by a simple, theoretically-motivated model architecture. Using several benchmarks, we show that unlike existing techniques, LMI can approximate MI well for variables with $> 10^3$ dimensions if their dependence structure has low intrinsic dimensionality. Finally, we showcase LMI on two open problems in biology. First, we approximate MI between protein language model (pLM) representations of interacting proteins, and find that pLMs encode non-trivial information about protein-protein interactions. Second, we quantify cell fate information contained in single-cell RNA-seq (scRNA-seq) measurements of hematopoietic stem cells, and find a sharp transition during neutrophil differentiation when fate information captured by scRNA-seq increases dramatically.