On constant factor approximation for earth mover distance over doubling metrics

TL;DR

This paper presents efficient algorithms and sketching schemes for approximating earth mover distance in metric spaces with bounded doubling dimension, achieving near-linear query times and controlled approximation factors.

Contribution

It introduces a data structure and encoding scheme for fast, approximate earth mover distance computations over doubling metrics, with provable efficiency and approximation guarantees.

Findings

Preprocessing time is near-quadratic in the size of the metric space.

Query time for EMD approximation is nearly linear in the size of the space.

The methods achieve an approximation factor proportional to the doubling dimension.

Abstract

Given a metric space $(X,d_X)$, the earth mover distance between two distributions over $X$ is defined as the minimum cost of a bipartite matching between the two distributions. The doubling dimension of a metric $(X, d_X)$ is the smallest value $\alpha$ such that every ball in $X$ can be covered by $2^\alpha$ ball of half the radius. We study efficient algorithms for approximating earth mover distance over metrics with bounded doubling dimension. Given a metric $(X, d_X)$, with $|X| = n$, we can use $\tilde O(n^2)$ preprocessing time to create a data structure of size $\tilde O(n^{1 + \e})$, such that subsequently queried EMDs can be $O(\alpha_X/\e)$-approximated in $\tilde O(n)$ time. We also show a weaker form of sketching scheme, which we call "encoding scheme". Given $(X, d_X)$, by using $\tilde O(n^2)$ preprocessing time, every subsequent distribution $\mu$ over $X$ can be encoded into $F(\mu)$ in $\tilde O(n^{1 + \e})$ time. Given $F(\mu)$ and $F(\nu)$, the EMD between $\mu$ and $\nu$ can be $O(\alpha_X/\e)$-approximated in $\tilde O(n^\e)$ time.