A Higher Precision Algorithm for Computing the $1$-Wasserstein Distance

We consider the problem of computing the $1$-Wasserstein distance $\mathcal{W}(\mu,\nu)$ between two $d$-dimensional discrete distributions $\mu$ and $\nu$ whose support lie within the unit hypercube. There are several algorithms that estimate $\mathcal{W}(\mu,\nu)$ within an additive error of $\varepsilon$. However, when $\mathcal{W}(\mu,\nu)$ is small, the additive error $\varepsilon$ dominates, leading to noisy results. Consider any additive approximation algorithm with execution time $T(n,\varepsilon)$. We propose an algorithm that runs in $O(T(n,\varepsilon/d) \log n)$ time and boosts the accuracy of estimating $\mathcal{W}(\mu,\nu)$ from $\varepsilon$ to an expected additive error of $\min\{\varepsilon, (d\log_{\sqrt{d}/\varepsilon} n)\mathcal{W}(\mu,\nu)\}$. For the special case where every point in the support of $\mu$ and $\nu$ has a mass of $1/n$ (also called the Euclidean Bipartite Matching problem), we describe an algorithm to boost the accuracy of any additive approximation algorithm from $\varepsilon$ to an expected additive error of $\min\{\varepsilon, (d\log\log n)\mathcal{W}(\mu,\nu)\}$ in $O(T(n, \varepsilon/d)\log\log n)$ time.