Asymptotics of smoothed Wasserstein distances in the small noise regime
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We study the behavior of the Wasserstein-2 distance between discrete measures μ and ν in ℝ^d when both measures are smoothed by small amounts of Gaussian noise. This procedure, known as Gaussian-smoothed optimal transport, has recently attracted attention as a statistically attractive alternative to the unregularized Wasserstein distance. We give precise bounds on the approximation properties of this proposal in the small noise regime, and establish the existence of a phase transition: we show that, if the optimal transport plan from μ to ν is unique and a perfect matching, there exists a critical threshold such that the difference between W_2(μ, ν) and the Gaussian-smoothed OT distance W_2(μ∗𝒩_σ, ν∗𝒩_σ) scales like exp(-c /σ^2) for σ below the threshold, and scales like σ above it. These results establish that for σ sufficiently small, the smoothed Wasserstein distance approximates the unregularized distance exponentially well.
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