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

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.