Gabor Scattering Moments for Sparse Signals and Poisson Processes

Michael Perlmutter, Department of Mathematics, Michigan State University, USA

We present a unified machine learning model for sparse signal analysis in both the deterministic and the statistical setting. 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 decouples the roles of scale and frequency. In the deterministic setting, this will allow us to establish a compressive-sensing type result where, under mild assumptions, we can completely recover a sparse signal (up to translations and reflections) with a sufficient number of measurements. In the statistical setting, we will assume that our sparse signal can be modeled as compound Poisson noise and show that our measurements allow us to estimate the Poisson arrival rate λ as well as the first and second moments of the arrival values A_{i .}