One or Two Components? The Scattering Transform Answers

TL;DR

This paper investigates how wavelet scattering networks can model multicomponent signals, providing criteria for component interference, visualizing similarity spaces, and analyzing the growth of scattering depth with bandwidth.

Contribution

It introduces a simple criterion for component interference, applies manifold learning to scattering coefficients, and generalizes the framework to multiple sine waves with theoretical analysis.

Findings

Renormalizing second-order nodes assesses component interference.

Manifold learning visualizes similarity spaces of signals.

Scattering depth grows logarithmically with bandwidth.

Abstract

With the aim of constructing a biologically plausible model of machine listening, we study the representation of a multicomponent stationary signal by a wavelet scattering network. First, we show that renormalizing second-order nodes by their first-order parents gives a simple numerical criterion to assess whether two neighboring components will interfere psychoacoustically. Secondly, we run a manifold learning algorithm (Isomap) on scattering coefficients to visualize the similarity space underlying parametric additive synthesis. Thirdly, we generalize the "one or two components" framework to three sine waves or more, and prove that the effective scattering depth of a Fourier series grows in logarithmic proportion to its bandwidth.