A Hybrid Scattering Transform for Signals with Isolated Singularities

TL;DR

This paper introduces a two-layer hybrid scattering transform combining wavelet and Gabor filters to analyze signals with isolated singularities, enhancing understanding of filter design in neural networks.

Contribution

It proposes a novel hybrid scattering transform that characterizes signals with isolated singularities and demonstrates the synthesis of sparse signals from Gabor measurements.

Findings

Characterizes signals with isolated singularities using the hybrid transform.

Shows Gabor measurements can synthesize sparse signals.

Provides insights into filter choices for neural network layers.

Abstract

The scattering transform is a wavelet-based model of Convolutional Neural Networks originally introduced by S. Mallat. Mallat's analysis shows that this network has desirable stability and invariance guarantees and therefore helps explain the observation that the filters learned by early layers of a Convolutional Neural Network typically resemble wavelets. Our aim is to understand what sort of filters should be used in the later layers of the network. Towards this end, we propose a two-layer hybrid scattering transform. In our first layer, we convolve the input signal with a wavelet filter transform to promote sparsity, and, in the second layer, we convolve with a Gabor filter to leverage the sparsity created by the first layer. We show that these measurements characterize information about signals with isolated singularities. We also show that the Gabor measurements used in the second layer can be used to synthesize sparse signals such as those produced by the first layer.