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We study the behavior of the Wasserstein-$2$ distance between discretemeasures $\mu$ and $\nu$ in $\mathbb{R}^d$ when both measures are smoothed bysmall amounts of Gaussian noise. This procedure, known as Gaussian-smoothedoptimal transport, has recently attracted attention as a statisticallyattractive alternative to the unregularized Wasserstein distance. We giveprecise bounds on the approximation properties of this proposal in the smallnoise regime, and establish the existence of a phase transition: we show that,if the optimal transport plan from $\mu$ to $\nu$ is unique and a perfectmatching, there exists a critical threshold such that the difference between$W_2(\mu, \nu)$ and the Gaussian-smoothed OT distance $W_2(\mu \ast\mathcal{N}_\sigma, \nu\ast \mathcal{N}_\sigma)$ scales like $\exp(-c/\sigma^2)$ for $\sigma$ below the threshold, and scales like $\sigma$ aboveit. These results establish that for $\sigma$ sufficiently small, the smoothedWasserstein distance approximates the unregularized distance exponentiallywell.

Asymptotics of smoothed Wasserstein distances in the small noise regime