Deep Convolutional Neural Networks Based on Semi-Discrete Frames

TL;DR

This paper extends Mallat's theory of scattering networks by allowing diverse semi-discrete frames in each layer, enhancing feature extraction capabilities and providing broader invariance and stability results.

Contribution

It introduces a generalized framework for deep convolutional networks using different semi-discrete frames, broadening the types of features extracted and improving theoretical guarantees.

Findings

Proves translation-invariance for the generalized feature extractor.

Establishes deformation stability for a wider class of deformations.

Removes technical conditions from wavelet-based feature extractors.

Abstract

Deep convolutional neural networks have led to breakthrough results in practical feature extraction applications. The mathematical analysis of these networks was pioneered by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on identical semi-discrete wavelet frames in each network layer, and proved translation-invariance as well as deformation stability of the resulting feature extractor. The purpose of this paper is to develop Mallat's theory further by allowing for different and, most importantly, general semi-discrete frames (such as, e.g., Gabor frames, wavelets, curvelets, shearlets, ridgelets) in distinct network layers. This allows to extract wider classes of features than point singularities resolved by the wavelet transform. Our generalized feature extractor is proven to be translation-invariant, and we develop deformation stability results for a larger class of deformations than those considered by Mallat. For Mallat's wavelet-based feature extractor, we get rid of a number of technical conditions. The mathematical engine behind our results is continuous frame theory, which allows us to completely detach the invariance and deformation stability proofs from the particular algebraic structure of the underlying frames.