Topology Reduction in Deep Convolutional Feature Extraction Networks

TL;DR

This paper investigates how the topology of deep convolutional networks, specifically depth and width, affects their feature extraction capabilities, providing theoretical insights and design principles for resource-constrained applications.

Contribution

It generalizes energy decay results for scattering networks and offers methods to design shallow networks that retain most input signal energy.

Findings

Energy decay rate can be controlled via prototype functions.

Networks of fixed depth can capture a specified percentage of input energy.

Prototype function bandwidth influences the number of significant nodes.

Abstract

Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and $10$,$000$s of nodes. Such network sizes entail significant computational complexity due to the large number of convolutions that need to be carried out; in addition, a large number of parameters needs to be learned and stored. Very deep and wide CNNs may therefore not be well suited to applications operating under severe resource constraints as is the case, e.g., in low-power embedded and mobile platforms. This paper aims at understanding the impact of CNN topology, specifically depth and width, on the network's feature extraction capabilities. We address this question for the class of scattering networks that employ either Weyl-Heisenberg filters or wavelets, the modulus non-linearity, and no pooling. The exponential feature map energy decay results in Wiatowski et al., 2017, are generalized to $\mathcal{O}(a^{-N})$, where an arbitrary decay factor $a>1$ can be realized through suitable choice of the Weyl-Heisenberg prototype function or the mother wavelet. We then show how networks of fixed (possibly small) depth $N$ can be designed to guarantee that $((1-\varepsilon)\cdot 100)\%$ of the input signal's energy are contained in the feature vector. Based on the notion of operationally significant nodes, we characterize, partly rigorously and partly heuristically, the topology-reducing effects of (effectively) band-limited input signals, band-limited filters, and feature map symmetries. Finally, for networks based on Weyl-Heisenberg filters, we determine the prototype function bandwidth that minimizes---for fixed network depth $N$---the average number of operationally significant nodes per layer.