Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments

Abstract

Gaussian smoothing has emerged as an effective technique for reducing the sample complexity of optimal transport. In this paper, we study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost \(T_p^{(σ)}(μ,ν)=W_p(μ*γ_σ,ν*γ_σ)^p\) on \(\R^d\). For fixed smoothing and finite polynomial moments \(M_{q_μ}(μ)<\infty\), \(M_{q_ν}(ν)<\infty\), with \(q_μ,q_ν>p\), we establish upper bounds in probability of order \(ρ_{q_μ,p,d}(m)+ρ_{q_ν,p,d}(n)\). Here \(ρ_{q,p,d}(N)=N^{-(q-p)/(q+d)}\) for \(p<q<d+2p\), \(N^{-1/2}\log N\) at \(q=d+2p\), and \(N^{-1/2}\) for \(q>d+2p\). This order also holds in expectation under \(q_μ,q_ν\ge2p\). When the smoothed population distance is positive, the cost bound yields this rate for the distance itself. For \(p>1\) and \(q_μ,q_ν>d+2p\), we also derive a first-order expansion, a separated two-sample central limit theorem, and a sample-splitting variance estimator.