Limit Distribution for Smooth Total Variation and $\chi^2$-Divergence in High Dimensions

Abstract

Statistical divergences are ubiquitous in machine learning as tools for measuring discrepancy between probability distributions. As these applications inherently rely on approximating distributions from samples, we consider empirical approximation under two popular $f$-divergences: the total variation (TV) distance and the $\chi^2$-divergence. To circumvent the sensitivity of these divergences to support mismatch, the framework of Gaussian smoothing is adopted. We study the limit distributions of $\sqrt{n}\delta_{\mathsf{TV}}(P_n\ast\mathcal{N},P\ast\mathcal{N})$ and $n\chi^2(P_n\ast\mathcal{N}\|P\ast\mathcal{N})$, where $P_n$ is the empirical measure based on $n$ independently and identically distributed (i.i.d.) observations from $P$, $\mathcal{N}_\sigma:=\mathcal{N}(0,\sigma^2\mathrm{I}_d)$, and $\ast$ stands for convolution. In arbitrary dimension, the limit distributions are characterized in terms of Gaussian process on $\mathbb{R}^d$ with covariance operator that depends on $P$ and the isotropic Gaussian density of parameter $\sigma$. This, in turn, implies optimality of the $n^{-1/2}$ expected value convergence rates recently derived for $\delta_{\mathsf{TV}}(P_n\ast\mathcal{N},P\ast\mathcal{N})$ and $\chi^2(P_n\ast\mathcal{N}\|P\ast\mathcal{N})$. These strong statistical guarantees promote empirical approximation under Gaussian smoothing as a potent framework for learning and inference based on high-dimensional data.