Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings

This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional space, 2-Sinkhorn divergence strictly weaker than exact 2-Wasserstein distance. Specifically, a sequence centered converges if corresponding covariance operators converge Hilbert–Schmidt norm. contrast to previous known distance only trace class In reproducing kernel (RKHS) setting, Gaussian–Sinkhorn divergence, which Sinkhorn between defined RKHS, defines semi-metric set Borel probability Polish given characteristic space. With norm convergence, we obtain dimension-independent rates same order as Maximum Mean Discrepancy. These apply particular Euclidean spaces. The complexity while dimension-dependent, exponentially faster worst case scenario literature.