Neural Approximation of Generalized Voronoi Diagrams

2026-03-27

AI Summary

Traditional methods struggle to efficiently construct generalized Voronoi diagrams for finite point sets under arbitrary metrics, particularly when confronted with complex distance functions and high combinatorial complexity. This work proposes VoroFields, a novel framework that introduces neural fields to this problem for the first time. VoroFields learns a continuous, differentiable proxy function via hierarchical neural fields, where Voronoi cells are implicitly defined by regions of maximal response, and cell boundaries emerge at locations where competing sites yield equal responses. By integrating hierarchical decomposition with adaptive regional refinement focused on envelope transition zones, the method reduces optimization complexity. Requiring no manual construction and supporting any computable distance function, VoroFields accurately recovers both cell structures and boundary geometries across diverse point configurations and metrics, demonstrating strong generality and high precision.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Learning with Manifolds

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• generalized Voronoi diagrams
• neural approximation
• point-to-site distance
• low-dimensional domains
• finite geometric site sets

Innovation

• neural fields
• generalized Voronoi diagrams
• differentiable approximation
• hierarchical decomposition
• implicit partitioning