Fully Dynamic Algorithms for Chamfer Distance

We study the problem of computing Chamfer distance in the fully dynamic setting, where two sets 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 $dist_{\mathrm{CH}}(A,B) = \sum_{a \in A} \min_{b \in B} dist(a,b)$, where $dist$ 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.