English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper studies convergence of empirical measures smoothed by a Gaussiankernel. Specifically, consider approximating $P\ast\mathcal{N}_\sigma$, for$\mathcal{N}_\sigma\triangleq\mathcal{N}(0,\sigma^2 \mathrm{I}_d)$, by$\hat{P}_n\ast\mathcal{N}_\sigma$, where $\hat{P}_n$ is the empirical measure,under different statistical distances. The convergence is examined in terms ofthe Wasserstein distance, total variation (TV), Kullback-Leibler (KL)divergence, and $\chi^2$-divergence. We show that the approximation error underthe 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 theKL divergence, squared 2-Wasserstein distance ($\mathsf{W}_2^2$), and$\chi^2$-divergence, the convergence rate is $e^{O(d)}n^{-1}$, but only if $P$achieves finite input-output $\chi^2$ mutual information across the additivewhite Gaussian noise channel. If the latter condition is not met, the ratechanges to $\omega(n^{-1})$ for the KL divergence and $\mathsf{W}_2^2$, whilethe $\chi^2$-divergence becomes infinite - a curious dichotomy. As a mainapplication we consider estimating the differential entropy$h(P\ast\mathcal{N}_\sigma)$ in the high-dimensional regime. The distribution$P$ is unknown but $n$ i.i.d samples from it are available. We first show thatany good estimator of $h(P\ast\mathcal{N}_\sigma)$ must have sample complexitythat is exponential in $d$. Using the empirical approximation results we thenshow that the absolute-error risk of the plug-in estimator converges at theparametric rate $e^{O(d)}n^{-\frac{1}{2}}$, thus establishing the minimaxrate-optimality of the plug-in. Numerical results that demonstrate asignificant empirical superiority of the plug-in approach to general-purposedifferential entropy estimators are provided.

Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation