Analysis of time-frequency scattering transforms

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.

References in corpus (3)

• Understanding Deep Convolutional Networks
• Rotationally invariant time-frequency scattering transforms
• Energy Propagation in Deep Convolutional Neural Networks

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• The Modern Mathematics of Deep Learning
• SSN: Stockwell Scattering Network for SAR Image Change Detection
• Geometric Scattering on Measure Spaces
• Rotationally invariant time-frequency scattering transforms
• Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
• On Generalizations of the Nonwindowed Scattering Transform
• On Lipschitz Bounds of General Convolutional Neural Networks
• Energy Propagation in Deep Convolutional Neural Networks
• Three-Dimensional Fourier Scattering Transform and Classification of Hyperspectral Images
• Energy Propagation in Scattering Convolution Networks Can Be Arbitrarily Slow
• Central and Non-central Limit Theorems arising from the Scattering Transform and its Neural Activation Generalization