Fast Algorithms for a New Relaxation of Optimal Transport

TL;DR

This paper introduces a new family of optimal transport objectives parameterized by , which interpolate between Earth Mover's distance and faster computable metrics, enabling efficient approximation algorithms in high-dimensional spaces.

Contribution

It proposes a novel class of optimal transport objectives  that allow for faster algorithms when  > 1, bridging the gap between computational efficiency and distance accuracy.

Findings

Provides an algorithm for  > 1 with near-linear runtime.

Achieves additive -approximation of the new metric.

Demonstrates faster computation compared to traditional Earth Mover's distance.

Abstract

We introduce a new class of objectives for optimal transport computations of datasets in high-dimensional Euclidean spaces. The new objectives are parametrized by $\rho \geq 1$, and provide a metric space $\mathcal{R}_{\rho}(\cdot, \cdot)$ for discrete probability distributions in $\mathbb{R}^d$. As $\rho$ approaches $1$, the metric approaches the Earth Mover's distance, but for $\rho$ larger than (but close to) $1$, admits significantly faster algorithms. Namely, for distributions $\mu$ and $\nu$ supported on $n$ and $m$ vectors in $\mathbb{R}^d$ of norm at most $r$ and any $\epsilon > 0$, we give an algorithm which outputs an additive $\epsilon r$-approximation to $\mathcal{R}_{\rho}(\mu, \nu)$ in time $(n+m) \cdot \mathrm{poly}((nm)^{(\rho-1)/\rho} \cdot 2^{\rho / (\rho-1)} / \epsilon)$.