Comparing Poisson and Gaussian channels (extended)
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Consider a pair of input distributions which after passing through a Poisson channel become ϵ-close in total variation. We show that they must necessarily then be ϵ^0.5+o(1)-close after passing through a Gaussian channel as well. In the opposite direction, we show that distributions inducing ϵ-close outputs over the Gaussian channel must induce ϵ^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).
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