Asymptotic Analysis of Higher-order Scattering Transform of Gaussian Processes

TL;DR

This paper provides an asymptotic analysis of the higher-order scattering transform of Gaussian processes, showing convergence to a chi-square process and quantifying the rate of convergence using advanced probabilistic techniques.

Contribution

It introduces a novel analysis of the scattering transform with quadratic nonlinearity for Gaussian processes, establishing convergence results and explicit bounds.

Findings

STQN output converges to a chi-square process with one degree of freedom.

Total variation distance between the distribution of STQN output and the chi-square process decreases exponentially.

Recursive formula and probabilistic tools are used to analyze the nonlinearity of STQN.

Abstract

We analyze the scattering transform with the quadratic nonlinearity (STQN) of Gaussian processes without depth limitation. STQN is a nonlinear transform that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the deep convolutional neural network. We prove that with a proper normalization, the output of STQN converges to a chi-square process with one degree of freedom in the finite dimensional distribution sense, and we provide a total variation distance control of this convergence at each time that converges to zero at an exponential rate. To show these, we derive a recursive formula to represent the intricate nonlinearity of STQN by a linear combination of Wiener chaos, and then apply the Malliavin calculus and Stein's method to achieve the goal.