Neural Approximation of Generalized Voronoi Diagrams

Panagiotis Rigas; George Ioannakis; Ioannis Emiris

cs.CGMar 27, 2026

Neural Approximation of Generalized Voronoi Diagrams

Authors: Panagiotis Rigas, George Ioannakis, Ioannis Emiris

Organizations: Department of Informatics & Telecommunications, National and Kapodistrian University of Athens, Greece · Archimedes, Athena Research Center, Greece · Institute for Language and Speech Processing, Athena Research Center, Greece

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.

Figures & tables

Figure 1: Hierarchical VoroFields construction. Local partitioning, fat-bisector sampling, and recursive refinement of Voronoi strata. Final regions are obtained via tree parsing.

Panel of Figure 2 — Figure 2: Arrangement of distance functions for 1D segments, its lower envelope, and the projected Voronoi stratification.

Dataset	Layers	Dim	Metric	Top-1 (%)	Top-2 (%)
Squares	5	$\mathbb{R}^{2}$	$L_{\infty}$	$93.46\pm 1.75$	$97.53\pm 0.29$
Cuboids	7	$\mathbb{R}^{3}$	$L_{\infty}$	$89.40\pm 1.84$	$97.46\pm 0.83$
Ellipses	5	$\mathbb{R}^{2}$	$L_{2}$	$94.73\pm 1.01$	$98.46\pm 0.56$
Segments	5	$\mathbb{R}^{2}$	$L_{2}$	$94.47\pm 1.21$	$97.93\pm 0.49$
Segments	7	$\mathbb{R}^{3}$	$L_{2}$	$93.72\pm 1.83$	$97.59\pm 1.09$
NC-Polygons	7	$\mathbb{R}^{2}$	$L_{2}$	$92.66\pm 1.73$	$97.20\pm 0.37$

Table 1: Leaf-level Top-

m

accuracy (%) across shape families (

n=10

k sites). Layers denote hierarchy depth until leaf size

\tau

$\ell_{\max}/\text{bbox}$	10%	20%	30%	40%	50%
Baseline	91.93	84.20	68.46	71.13	72.80
Increased $\varepsilon$	94.33	89.34	78.53	75.53	76.80

Table 2: Top-1 accuracy (%) for 2D segments under increasing relative length. Enlarging

\varepsilon

improves boundary coverage.

Neural Surrogate Modeling

Neural surrogate modeling uses learned models to approximate costly simulations, enabling faster prediction, design, control, and inference.

See all 410 papers

Geometry-Aware Probabilistic Circuits via Voronoi Tessellations
Mar 12, 2026Sahil Sidheekh, Sriraam NatarajanProbabilistic Circuits Density Estimation
The University of Texas at Dallas, Richardson, TX, USA.
Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples
May 8, 2026Yicen Li, Ruiyang Hong, Anastasis Kratsios +2Hierarchical Classification Clustering
Department of Mathematics and Statistics, McMaster University, Canada · Vector Institute, Canada · University of Cambridge, United Kingdom
Scalable Sparse Autoregressive Variational Inference for Non-Gaussian Spatial Fields
Oct 9, 2026Paul F. V. Wiemann, Matthias KatzfussVariational Inference
Department of Statistics, The Ohio State University, Columbus OH 43210, USA · Department of Statistics, University of Wisconsin–Madison, Madison WI 53706, USA

Mar 12, 2026J/K move · Enter open · S save

Geometry-Aware Probabilistic Circuits via Voronoi Tessellations

Sahil Sidheekh, Sriraam Natarajan

The University of Texas at Dallas, Richardson, TX, USA.

Probabilistic Circuits Density Estimation

Open paper AskPDF arXiv

Probabilistic circuits (PCs) enable exact and tractable inference but employ data independent mixture weights that limit their ability to capture local geometry of the data manifold. We propose Voronoi tessellations (VT) as a natural way to incorporate geometric structure directly into the sum nodes of a PC. However, naïvely introducing such structure breaks tractability. We formalize this incompatibility and develop two complementary solutions: (1) an approximate inference framework that provides guaranteed lower and upper bounds for inference, and (2) a structural condition for VT under which exact tractable inference is recovered. Finally, we introduce a differentiable relaxation for VT that enables gradient-based learning and empirically validate the resulting approach on standard density estimation tasks.

Abstract

Neural Surrogate Modeling

Explore similar work

Geometry-Aware Probabilistic Circuits via Voronoi Tessellations

Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples

Scalable Sparse Autoregressive Variational Inference for Non-Gaussian Spatial Fields