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Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast\mathcal{N}_σ$, for $\mathcal{N}_σ\triangleq\mathcal{N}(0,σ^2 \mathrm{I}_d)$, by $\hat{P}_n\ast\mathcal{N}_σ$, where $\hat{P}_n$ is the empirical measure, under different statistical distances. The convergence is examined in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $χ^2$-divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf{W}_1$) converges at rate $e^{O(d)}n^{-\frac{1}{2}}$ in remarkable contrast to a typical $n^{-\frac{1}{d}}$ rate for unsmoothed $\mathsf{W}_1$ (and $d\ge 3$). For the KL divergence, squared 2-Wasserstein distance ($\mathsf{W}_2^2$), and $χ^2$-divergence, the convergence rate is $e^{O(d)}n^{-1}$, but only if $P$ achieves finite input-output $χ^2$ mutual information across the additive white Gaussian noise channel. If the latter condition is not met, the rate changes to $ω(n^{-1})$ for the KL divergence and $\mathsf{W}_2^2$, while the $χ^2$-divergence becomes infinite - a curious dichotomy. As a main application we consider estimating the differential entropy $h(P\ast\mathcal{N}_σ)$ in the high-dimensional regime. The distribution $P$ is unknown but $n$ i.i.d samples from it are available. We first show that any good estimator of $h(P\ast\mathcal{N}_σ)$ must have sample complexity that is exponential in $d$. Using the empirical approximation results we then show that the absolute-error risk of the plug-in estimator converges at the parametric rate $e^{O(d)}n^{-\frac{1}{2}}$, thus establishing the minimax rate-optimality of the plug-in. Numerical results that demonstrate a significant empirical superiority of the plug-in approach to general-purpose differential entropy estimators are provided.