A Method for Auto-Differentiation of the Voronoi Tessellation

TL;DR

This paper introduces the first end-to-end differentiable method for 2D Voronoi tessellation, enabling gradient-based optimization in applications requiring geometric partitioning.

Contribution

It presents a novel autodifferentiable approach for 2D Voronoi tessellation, allowing gradients to pass through the geometric construction.

Findings

Enables end-to-end differentiability of Voronoi tessellation

Provides implementation details and applications

First to offer full set of geometrical parameters in a differentiable manner

Abstract

Voronoi tessellation, also known as Voronoi diagram, is an important computational geometry technique that has applications in various scientific disciplines. It involves dividing a given space into regions based on the proximity to a set of points. Autodifferentiation is a powerful tool for solving optimization tasks. Autodifferentiation assumes constructing a computational graph that allows to compute gradients using backpropagation algorithm. However, often the Voronoi tessellation remains the only non-differentiable part of a pipeline, prohibiting end-to-end differentiation. We present the method for autodifferentiation of the 2D Voronoi tessellation. The method allows one to construct the Voronoi tessellation and pass gradients, making the construction end-to-end differentiable. We provide the implementation details and present several important applications. To the best of our knowledge this is the first autodifferentiable realization of the Voronoi tessellation providing full set of Voronoi geometrical parameters in a differentiable way.