Central and Non-central Limit Theorems arising from the Scattering Transform and its Neural Activation Generalization
Abstract
Motivated by analyzing complicated and non-stationary time series, we study a generalization of the scattering transform (ST) that includes broad neural activation functions, which is called neural activation ST (NAST). On the whole, NAST is a transform that comprises a sequence of ``neural processing units'', each of which applies a high pass filter to the input from the previous layer followed by a composition with a nonlinear function as the output to the next neuron. Here, the nonlinear function models how a neuron gets excited by the input signal. In addition to showing properties like non-expansion, horizontal translational invariability and insensitivity to local deformation, the statistical properties of the second order NAST of a Gaussian process with various dependence and (non-)stationarity structure and its interaction with the chosen high pass filters and activation functions are explored and central limit theorem (CLT) and non-CLT results are provided. Numerical simulations are also provided. The results explain how NAST processes complicated and non-stationary time series, and pave a way towards statistical inference based on NAST under the non-null case.
Cite
@article{arxiv.2011.10801,
title = {Central and Non-central Limit Theorems arising from the Scattering Transform and its Neural Activation Generalization},
author = {Gi-Ren Liu and Yuan-Chung Sheu and Hau-Tieng Wu},
journal= {arXiv preprint arXiv:2011.10801},
year = {2020}
}