2022/05/04 by Adam Block, Block, Adam, Zeyu Jia +5 · 1 citation
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Information Theory (cs.IT) #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2205.02128
openalex publication_date 2022/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider an empirical measure ℙn induced by n iid samples from a d-dimensional K-subgaussian distribution ℙ and let γ= N(0,σ2 Id) be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance W2(ℙn * 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= 1\over2. For K>σ in dimension d=1 we show that the rate is slower and is given by α= (σ2 + K2)2\over 4 (σ4 + K4) < 1/2. This resolves several open problems in [GGNWP20], and in particular precisely identifies the amount of smoothing σ needed to obtain a parametric rate. In addition, for any d-dimensional K-subgaussian distribution ℙ, we also establish that DKL(ℙn * γ‖ℙ*γ) has rate O(1/n) for K<σ but only slows down to O((log n)d+1\over n) for K>σ. The surprising difference of the behavior of W22 and KL implies the failure of T2-transportation inequality when σ< K. Consequently, it follows that for K>σ the log-Sobolev inequality (LSI) for the Gaussian mixture ℙ * N(0, σ2) cannot hold. This closes an open problem in [WW+16], who established the LSI under the condition K<σ and asked if their bound can be improved.