Even Faster Algorithm for the Chamfer Distance

TL;DR

This paper presents an improved algorithm for approximating the Chamfer distance between point sets, significantly reducing the computational complexity and narrowing the gap to the theoretical lower bound.

Contribution

It introduces a faster near-linear time algorithm for (1+eps)-approximate Chamfer distance computation, improving upon previous methods.

Findings

Reduces running time from O(ndlog(n)/eps^2) to O(nd(loglog(n)+log(1/eps))/eps^2)

Narrowed the gap to the lower bound from O(log n) to O(loglog n)

Applicable to high-dimensional point set comparisons in machine learning and vision

Abstract

For two d-dimensional point sets A, B of size up to n, the Chamfer distance from A to B is defined as CH(A,B) = \sum_{a \in A} \min_{b \in B} \|a-b\|. The Chamfer distance is a widely used measure for quantifying dissimilarity between sets of points, used in many machine learning and computer vision applications. A recent work of Bakshi et al, NeuriPS'23, gave the first near-linear time (1+eps)-approximate algorithm, with a running time of O(ndlog(n)/eps^2). In this paper we improve the running time further, to O(nd(loglog(n)+log(1/eps))/eps^2). When eps is a constant, this reduces the gap between the upper bound and the trivial Omega(dn) lower bound significantly, from O(log n) to O(loglog n).