Autodifferentiable Voronoi Tessellation: A Breakthrough in Computational Geometry
In the field of computational geometry, the Voronoi tessellation is an essential technique with a wide range of applications across various scientific disciplines. This method involves dividing a given space into distinct regions based on the proximity to a set of specified points. However, the Voronoi tessellation has long posed a challenge in optimization tasks due to its non-differentiable nature.
Fortunately, a breakthrough has now been achieved with the development of an autodifferentiable method for the 2D Voronoi tessellation. This innovative approach allows for the construction of the tessellation while enabling the computation of gradients using the backpropagation algorithm. As a result, the construction process becomes end-to-end differentiable, opening up new possibilities for seamless integration into larger computational pipelines.
In this article, we delve into the implementation details of this groundbreaking method for autodifferentiation of the Voronoi tessellation. We explore how gradients can be seamlessly passed through the construction process, providing a fully differentiable framework for extracting Voronoi geometrical parameters. This represents a significant advancement in computational geometry, as prior methods lacked the ability to obtain such parameters in a differentiable manner.
Furthermore, we showcase several important applications made possible by this autodifferentiable realization of the Voronoi tessellation. From optimization tasks in computer science to pattern recognition in image analysis, the versatility of this method is truly remarkable. By integrating autodifferentiation into the Voronoi tessellation, researchers and practitioners have gained a powerful tool that enhances their ability to solve complex problems in diverse scientific domains.
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.