Analysis of time-frequency scattering transforms

TL;DR

This paper develops a novel time-frequency scattering transform combining Gabor representations with Mallat's framework, providing theoretical insights and an efficient algorithm for feature extraction.

Contribution

It introduces uniform covering frames including Gabor systems, proves energy propagation and decay properties of the Fourier scattering transform, and presents a fast algorithm.

Findings

Energy propagates along frequency decreasing paths.

Energy decays exponentially with depth.

The fast Fourier scattering transform performs efficiently.

Abstract

In this paper we address the problem of constructing a feature extractor which combines Mallat's scattering transform framework with time-frequency (Gabor) representations. To do this, we introduce a class of frames, called uniform covering frames, which includes a variety of semi-discrete Gabor systems. Incorporating a uniform covering frame with a neural network structure yields the Fourier scattering transform $\mathcal{S}_\mathcal{F}$ and the truncated Fourier scattering transform. We prove that $\mathcal{S}_\mathcal{F}$ propagates energy along frequency decreasing paths and its energy decays exponentially as a function of the depth. These quantitative estimates are fundamental in showing that $\mathcal{S}_\mathcal{F}$ satisfies the typical scattering transform properties, and in controlling the information loss due to width and depth truncation. We introduce the fast Fourier scattering transform algorithm, and illustrate the algorithm's performance. The time-frequency covering techniques developed in this paper are flexible and give insight into the analysis of scattering transforms.