Topology Reduction in Deep Convolutional Feature Extraction Networks
This work addresses computational efficiency for CNNs under resource constraints, such as in low-power embedded and mobile platforms, by providing theoretical insights into topology reduction, but it is incremental as it builds on prior scattering network results.
The paper tackles the problem of reducing computational complexity in deep convolutional neural networks (CNNs) by analyzing how network topology (depth and width) affects feature extraction, specifically for scattering networks with Weyl-Heisenberg filters or wavelets. It shows that networks of fixed depth can be designed to guarantee that a specified percentage (e.g., ((1-ε)·100)%) of input signal energy is contained in the feature vector, and determines optimal prototype function bandwidth to minimize operationally significant nodes per layer.
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.