Particle gradient descent model for point process generation

Antoine Brochard (1, 2) , Bartłomiej Błaszczyszyn (1) , Stéphane Mallat (3, 4, 5) , Sixin Zhang (6)

1 DYOGENE - Dynamics of Geometric Networks
2 Huawei Technologies Co., Ltd [Shenzhen]
3 DI-ENS - Département d'informatique - ENS Paris
4 Flatiron Institute
5 Collège de France - Chaire Sciences des données
6 IRIT-SC - Signal et Communications

This paper introduces a generative model for planar point processes in a square window, built upon a single realization of a stationary, ergodic point process observed in this window. Inspired by recent advances in gradient descent methods for maximum entropy models, we propose a method to generate similar point patterns by jointly moving particles of an initial Poisson configuration towards a target counting measure. The target measure is generated via a deterministic gradient descent algorithm, so as to match a set of statistics of the given, observed realization. Our statistics are estimators of the multi-scale wavelet phase harmonic covariance, recently proposed in image modeling. They allow one to capture geometric structures through multi-scale interactions between wavelet coefficients. Both our statistics and the gradient descent algorithm scale better with the number of observed points than the classical k-nearest neighbour distances previously used in generative models for point processes, based on the rejection sampling or simulated-annealing. The overall quality of our model is evaluated on point processes with various geometric structures through spectral and topological data analysis.

Domaines: Probabilités [math.PR]; Autres [stat.ML]; Théorie de l'information [cs.IT]; Apprentissage [cs.LG]

Antoine Brochard, Bartłomiej Błaszczyszyn, Stéphane Mallat, Sixin Zhang. Particle gradient descent model for point process generation. 2020. ⟨hal-02980486v1⟩