Optimizing Computational-Statistical Runtime for Wasserstein Distance Estimation

• URL: http://arxiv.org/abs/2605.20122v1
• Date: Tue, 19 May 2026 17:10:47 GMT
• Title: Optimizing Computational-Statistical Runtime for Wasserstein Distance Estimation
• Abstract summary: Squaredstein distance is a frequently used tool to measure discrepancy between probability distributions.<n>We develop a Sample-$ computational paradigm where we introduce a regular cartesian grid sketch of the samples.<n>We approximate $W(P,Q)$ within $$ error in $-max(2,fracd+1+o(1)1+)$ time for $0 1$ Hlder smooth distributions.
• Score: 8.375876290025586
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Squared Wasserstein distance is a frequently used tool to measure discrepancy between probability distributions. This distance is typically computed between empirical measures of size $n$ from two underlying random samples. Unfortunately, even in lower dimensional Euclidean space problems $\left( d \in \{2,3\} \right)$, algorithms for Wasserstein distance computation with approximate or exact precision guarantees scale poorly in the runtime as a function of $n$ and the desired precision. In response, we consider the computational-statistical runtime, where the goal is to estimate from samples the Wasserstein distance between potentially smooth measures up to $ε$-additive error in expectation with respect to the sampling; we allow $O(1)$ computational cost for collecting a sample. Towards this, we develop a Sample-Sketch-Solve paradigm where we introduce a regular cartesian grid sketch of the samples. We show that (especially under $α$-Hölder smooth distributions) this can compress the data without increasing asymptotic error, and also regularizes the structure which enables faster exact algorithms. Ultimately, we approximate $W_2^2(P,Q)$ within $ε$ error in $ε^{-\max(2,\frac{d+1+o(1)}{1+α})}$ time for $0 < α< 1$ Hölder smooth distributions $P,Q$ on $(0,1)^{d}$; an optimal $Θ(ε^{-2})$ for $α> 1/2$ when $d=2$ and nearly optimal as $α\to 1$ when $d = 3$.

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