Differentiable Approximations for Distance Queries

2026-07-30

AI Summary

Traditional geometric distance queries are non-differentiable, posing a significant barrier to their integration into gradient-based machine learning pipelines. This work proposes a globally differentiable approximation of the Euclidean distance that leverages regularized distances, a partition of unity, and a novel approximate Voronoi diagram. The resulting smooth function achieves a (1+ε)-approximation guarantee while avoiding lifting transformations and ray casting, and exhibits asymptotically optimal bounds on both gradient and Hessian norms. The method enables end-to-end learning and attains state-of-the-art performance in terms of memory overhead and query efficiency for approximate nearest neighbor search, all while preserving theoretically optimal derivative bounds.

Abstract

The widespread use of gradient-based optimization has motivated the adaptation of various classical algorithms into differentiable solvers compatible with learning pipelines. In this paper, we investigate the enhancement of traditional geometric query problems such that the result consists of both the geometric function as well as its gradient. Specifically, we study the fundamental problem of distance queries against a set of points $P$ in $\mathbb{R}^d$, which also underlies various similarity measures for learning algorithms. The main result of this paper is a multiplicative $(1+\varepsilon)$-approximation of the Euclidean distance to $P$ which is differentiable at all points in $\mathbb{R}^d \setminus P$ with asymptotically optimal bounds on the norms of its gradient and Hessian, from a data structure with storage and query time matching state-of-the-art results for approximate nearest-neighbor searching. The approximation is realized as a regularized distance through a partition-of-unity framework, which efficiently blends multiple local approximations, over a suitably defined covering of space, into a smooth global approximation. In order to obtain the local distance approximations in a manner that facilitates blending, we develop a new approximate Voronoi diagram based on a simple point-location data structure, simplifying away both the lifting transformation and ray shooting.

Problem

• differentiable approximation
• distance queries
• Euclidean distance
• gradient-based optimization
• approximate nearest neighbor

Innovation

• differentiable distance approximation
• partition of unity
• approximate Voronoi diagram
• gradient-based optimization
• smooth geometric queries