A Near-Linear Time Algorithm for the Chamfer Distance

• URL: http://arxiv.org/abs/2307.03043v1
• Date: Thu, 6 Jul 2023 15:07:48 GMT
• Title: A Near-Linear Time Algorithm for the Chamfer Distance
• Authors: Ainesh Bakshi, Piotr Indyk, Rajesh Jayaram, Sandeep Silwal, Erik Waingarten
• Abstract summary: Chamfer distance is a popular measure of dissimilarity between point clouds. We present the first $(1+epsilon)$-approximate algorithm for estimating the Chamfer distance with a near-linear running time. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets.
• Score: 21.018781726524946
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\text{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, 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, the 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 the Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\varepsilon^2)$ 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 \emph{report} a $(1+\varepsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

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