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 \(T_p^{(\sigma)}(\mu,\nu)=W_p(\mu*\gamma_\sigma,\nu*\gamma_\sigma)^p\) on \(\R^d\). For fixed smoothing and finite polynomial moments \(M_{q_\mu}(\mu)<\infty\), \(M_{q_\nu}(\nu)<\infty\), with \(q_\mu,q_\nu>p\), we establish upper bounds in probability of order \(\rho_{q_\mu,p,d}(m)+\rho_{q_\nu,p,d}(n)\). Here \(\rho_{q,p,d}(N)=N^{-(q-p)/(q+d)}\) for \(p d+2p\). This order also holds in expectation under \(q_\mu,q_\nu\ge2p\). When the smoothed population distance is positive, the cost bound yields this rate for the distance itself. For \(p>1\) and \(q_\mu,q_\nu>d+2p\), we also derive a first-order expansion, a separated two-sample central limit theorem, and a sample-splitting variance estimator.