k-Sliced Mutual Information: A Quantitative Study of Scalability with Dimension

TL;DR

This paper investigates the scalability of k-Sliced Mutual Information (k-SMI) in high-dimensional settings, providing theoretical bounds, estimation methods, and asymptotic analysis to understand its dependence on ambient dimension.

Contribution

It introduces a comprehensive theoretical framework for k-SMI, deriving error bounds, convergence rates, and asymptotic properties, extending the understanding of sliced mutual information's scalability.

Findings

Derived sharp bounds on Monte Carlo estimation error for k-SMI.

Established optimal convergence rates for neural estimation of k-SMI.

Provided Gaussian approximation results for population k-SMI in high dimensions.

Abstract

Sliced mutual information (SMI) is defined as an average of mutual information (MI) terms between one-dimensional random projections of the random variables. It serves as a surrogate measure of dependence to classic MI that preserves many of its properties but is more scalable to high dimensions. However, a quantitative characterization of how SMI itself and estimation rates thereof depend on the ambient dimension, which is crucial to the understanding of scalability, remain obscure. This work provides a multifaceted account of the dependence of SMI on dimension, under a broader framework termed $k$-SMI, which considers projections to $k$-dimensional subspaces. Using a new result on the continuity of differential entropy in the 2-Wasserstein metric, we derive sharp bounds on the error of Monte Carlo (MC)-based estimates of $k$-SMI, with explicit dependence on $k$ and the ambient dimension, revealing their interplay with the number of samples. We then combine the MC integrator with the neural estimation framework to provide an end-to-end $k$-SMI estimator, for which optimal convergence rates are established. We also explore asymptotics of the population $k$-SMI as dimension grows, providing Gaussian approximation results with a residual that decays under appropriate moment bounds. All our results trivially apply to SMI by setting $k=1$. Our theory is validated with numerical experiments and is applied to sliced InfoGAN, which altogether provide a comprehensive quantitative account of the scalability question of $k$-SMI, including SMI as a special case when $k=1$.