Particle gradient descent model for point process generation

TL;DR

This paper introduces a gradient descent-based statistical model for stationary ergodic point processes that efficiently captures complex multi-scale geometries from a single observed realization.

Contribution

It proposes a novel model using gradient descent and wavelet-based moments to generate point processes with complex geometries, enabling fast sampling and detailed multi-scale interaction modeling.

Findings

Effective modeling of complex geometric structures

Fast sampling of new point configurations

Model validation via spectral and topological analysis

Abstract

This paper presents a statistical model for stationary ergodic point processes, estimated from a single realization observed in a square window. With existing approaches in stochastic geometry, it is very difficult to model processes with complex geometries formed by a large number of particles. Inspired by recent works on gradient descent algorithms for sampling maximum-entropy models, we describe a model that allows for fast sampling of new configurations reproducing the statistics of the given observation. Starting from an initial random configuration, its particles are moved according to the gradient of an energy, in order to match a set of prescribed moments (functionals). Our moments are defined via a phase harmonic operator on the wavelet transform of point patterns. They allow one to capture multi-scale interactions between the particles, while controlling explicitly the number of moments by the scales of the structures to model. We present numerical experiments on point processes with various geometric structures, and assess the quality of the model by spectral and topological data analysis.