Scattering Statistics of Generalized Spatial Poisson Point Processes

We present a machine learning model for the analysis of randomly generated discrete signals, which we model as the points of a homogeneous or inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by S. Mallat, our construction is a mathematical model of convolutional neural networks and is naturally invariant to translations and reflections. Our model replaces wavelets with Gabor-type measurements and therefore decouples the roles of scale and frequency. We show that, with suitably chosen nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes based on the first and second moments of the arrival intensity $\lambda(t)$, as well as the absolute moments of the charges associated to each point.