Neural Approximation of Generalized Voronoi Diagrams

TL;DR

VoroFields is a neural framework that efficiently approximates generalized Voronoi diagrams in low-dimensional spaces using a continuous, differentiable surrogate model.

Contribution

It introduces a hierarchical neural-field approach that implicitly encodes Voronoi diagrams without explicit combinatorial construction.

Findings

Accurately recovers Voronoi cells and boundaries across various site types and metrics.

Reduces complexity through hierarchical refinement near transition boundaries.

Demonstrates effectiveness in low-dimensional geometric domains.

Abstract

We introduce VoroFields, a hierarchical neural-field framework for approximating generalized Voronoi diagrams of finite geometric site sets in low-dimensional domains under arbitrary evaluable point-to-site distances. Instead of constructing the diagram combinatorially, VoroFields learns a continuous, differentiable surrogate whose maximizer structure induces the partition implicitly. The Voronoi cells correspond to maximizer regions of the field, with boundaries defined by equal responses between competing sites. A hierarchical decomposition reduces the combinatorial complexity by refining only near envelope transition strata. Experiments across site families and metrics demonstrate accurate recovery of cells and boundary geometry without shape-specific constructions.