Near-Linear Time Algorithm for the Chamfer Distance

Ainesh Bakshi, Piotr Indyk, Rajesh Jayaram, Sandeep Silwal, Erik Waingarten

Keywords: chamfer distance, earth mover distance, high dimensional data analysis, nearest neighbor search, high dimensional data, high-dimensional geometry, sublinear algorithms, point clouds, theory

TL;DR: We give a near linear time algorithm for computing the Chamfer distance, a common relaxation of the Earth Mover (Optimal Transport) distance, improving upon the naive quadratic runtime.

Abstract:

For any two point sets $A,B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $\texttt{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, where $d_X$ is the underlying distance measure (e.g., the Euclidean or Manhattan distance). The Chamfer distance is a popular measure of dissimilarity between point clouds, used in many machine learning, computer vision, and graphics applications, and admits a straightforward $O(d n^2)$-time brute force algorithm. Further, Chamfer distance is often used as a proxy for the more computationally demanding Earth-Mover (Optimal Transport) Distance. However, the \emph{quadratic} dependence on $n$ in the running time makes the naive approach intractable for large datasets.

We overcome this bottleneck and present the first $(1+\epsilon)$-approximate algorithm for estimating Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\epsilon^2)$ and is implementable. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets. We believe that our algorithm will open new avenues for analyzing large high-dimensional point clouds. We also give evidence that if the goal is to report a $(1+\epsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

Submission Number: 3549