|Limit Distribution for Smooth Total Variation and 𝛘²-Divergence in High Dimensions
|Ziv Goldfeld, Kengo Kato, Cornell University, United States
|L.7: High-dimensional Statistics
|Statistics and Learning Theory
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|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 χ²-divergence. To circumvent the sensitivity of these divergences to support mismatch, the framework of Gaussian smoothing is adopted. We study the limit distributions of √nδTV(Pn∗Nσ,P∗Nσ) and nχ²(Pn∗Nσ‖P∗Nσ), where Pn is the empirical measure based on n independently and identically distributed (i.i.d.) observations from P, Nσ:=N(0,σ²Id), and ∗ stands for convolution. In arbitrary dimension, the limit distributions are characterized in terms of Gaussian process on Rᵈ with covariance operator that depends on P and the isotropic Gaussian density of parameter σ. This, in turn, implies optimality of then −1/2 expected value convergence rates recently derived for δTV(Pn∗Nσ,P∗Nσ) and χ²(Pn∗Nσ‖P∗Nσ). These strong statistical guarantees promote empirical approximation under Gaussian smoothing as a potent framework for learning and inference based on high-dimensional data.