Comparing Poisson and Gaussian channels (extended)

TL;DR

This paper compares the effects of Poisson and Gaussian channels on input distributions, showing how closeness in one translates to the other, and applies this to improve bounds in Poisson mixture estimation.

Contribution

It establishes quantitative relationships between distribution closeness after Poisson and Gaussian channels, and improves bounds in Poisson mixture estimation in Gaussian optimal transport.

Findings

Poisson and Gaussian channels induce similar smoothing effects.

Closeness in Poisson channels implies a specific power of closeness in Gaussian channels.

Improved the upper bound for Poisson mixture estimation from n^{-0.1} to n^{-0.25}.

Abstract

Consider a pair of input distributions which after passing through a Poisson channel become $\epsilon$-close in total variation. We show that they must necessarily then be $\epsilon^{0.5+o(1)}$-close after passing through a Gaussian channel as well. In the opposite direction, we show that distributions inducing $\epsilon$-close outputs over the Gaussian channel must induce $\epsilon^{1+o(1)}$-close outputs over the Poisson. This quantifies a well-known intuition that ''smoothing'' induced by Poissonization and Gaussian convolution are similar. As an application, we improve a recent upper bound of Han-Miao-Shen'2021 for estimating mixing distribution of a Poisson mixture in Gaussian optimal transport distance from $n^{-0.1 + o(1)}$ to $n^{-0.25 + o(1)}$.