Approximating the Earth Mover's Distance between sets of geometric objects

TL;DR

This paper introduces polynomial-time approximation algorithms for computing the Earth Mover's Distance between various geometric object sets, providing near-optimal solutions and transport plans for continuous distributions.

Contribution

It presents the first combinatorial algorithms with provable approximation ratios for EMD between continuous geometric objects such as line segments and simplices.

Findings

Achieves (1 + ε)-approximation for point-to-geometry distributions.

Provides algorithms with small additive error for set-to-set geometric distributions.

Computes explicit transport plans alongside approximate distances.

Abstract

Given two distributions $P$ and $S$ of equal total mass, the Earth Mover's Distance measures the cost of transforming one distribution into the other, where the cost of moving a unit of mass is equal to the distance over which it is moved. We give approximation algorithms for the Earth Mover's Distance between various sets of geometric objects. We give a $(1 + \varepsilon)$-approximation when $P$ is a set of weighted points and $S$ is a set of line segments, triangles or $d$-dimensional simplices. When $P$ and $S$ are both sets of line segments, sets of triangles or sets of simplices, we give a $(1 + \varepsilon)$-approximation with a small additive term. All algorithms run in time polynomial in the size of $P$ and $S$, and actually calculate the transport plan (that is, a specification of how to move the mass), rather than just the cost. To our knowledge, these are the first combinatorial algorithms with a provable approximation ratio for the Earth Mover's Distance when the objects are continuous rather than discrete points.