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 works extends the original SMI definition to k-SMI, which considers projections to k-dimensional subspaces, and provides a multifaceted account on its dependence on dimension. 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. 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.