Rate of convergence of the smoothed empirical Wasserstein distance

05/04/2022

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by   Adam Block, et al.

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Consider an empirical measure ℙ_n induced by n iid samples from a d-dimensional K-subgaussian distribution ℙ and let γ = 𝒩(0,σ^2 I_d) be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance W_2(ℙ_n * γ, ℙ*γ) = n^-α + o(1) with * being the convolution of measures. For K<σ and in any dimension d≥ 1 we show that α = 12. For K>σ in dimension d=1 we show that the rate is slower and is given by α = (σ^2 + K^2)^2 4 (σ^4 + K^4) < 1/2. This resolves several open problems in <cit.>, and in particular precisely identifies the amount of smoothing σ needed to obtain a parametric rate. In addition, we also establish that D_KL(ℙ_n * γℙ*γ) has rate O(1/n) for K<σ but only slows down to O((log n)^d+1 n) for K>σ. The surprising difference of the behavior of W_2^2 and KL implies the failure of T_2-transportation inequality when σ < K. Consequently, the requirement K<σ is necessary for validity of the log-Sobolev inequality (LSI) for the Gaussian mixture ℙ * 𝒩(0, σ^2), closing an open problem in <cit.>, who established the LSI under precisely this condition.