Fully Dynamic Algorithms for Chamfer Distance

TL;DR

This paper introduces the first fully dynamic algorithms for approximating Chamfer distance in point clouds under  and  norms, enabling efficient updates for evolving datasets with practical applications.

Contribution

It presents novel dynamic algorithms for maintaining approximate Chamfer distances under  and  norms, reducing the problem to approximate nearest neighbor search.

Findings

Achieves -approximation with ilde{O}(rac{1}{\u03b5}^d) update time.

Achieves (1+)-approximation with ilde{O}(d n^{\u03b5^2} \u03b5^{-4}) update time.

Demonstrates competitive performance on real-world datasets.

Abstract

We study the problem of computing Chamfer distance in the fully dynamic setting, where two set of points $A, B \subset \mathbb{R}^{d}$, each of size up to $n$, dynamically evolve through point insertions or deletions and the goal is to efficiently maintain an approximation to $\mathrm{dist}_{\mathrm{CH}}(A,B) = \sum_{a \in A} \min_{b \in B} \textrm{dist}(a,b)$, where $\textrm{dist}$ is a distance measure. Chamfer distance is a widely used dissimilarity metric for point clouds, with many practical applications that require repeated evaluation on dynamically changing datasets, e.g., when used as a loss function in machine learning. In this paper, we present the first dynamic algorithm for maintaining an approximation of the Chamfer distance under the $\ell_p$ norm for $p \in \{1,2 \}$. Our algorithm reduces to approximate nearest neighbor (ANN) search with little overhead. Plugging in standard ANN bounds, we obtain $(1+\epsilon)$-approximation in $\tilde{O}(\epsilon^{-d})$ update time and $O(1/\epsilon)$-approximation in $\tilde{O}(d n^{\epsilon^2} \epsilon^{-4})$ update time. We evaluate our method on real-world datasets and demonstrate that it performs competitively against natural baselines.