Topology reduction in deep convolutional feature extraction networks

Abstract

Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and 10,000s 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 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 − ε) · 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.

Energy Propagation in Deep Convolutional Neural Networks

“…The conditions we impose on the mother and father wavelet ψ, φ in i) are satisfied, e.g., by analytic Meyer wavelets [22,Section 3.3.5], and those on the prototype function g and low-pass filter φ in ii) by B-splines [45, Section 1]. Moreover, as shown in [44,Theorem 3.1], the exponential energy decay results in ( 28) and ( 31) can be generalized to O(a −N ) with arbitrary decay factor a > 1 realized through suitable choice of the mother wavelet or the Weyl-Heisenberg prototype function.…”

“…For input signals that exhibit mild spectral decay, the number of "operationally significant nodes" will, however, be finite in practice. For a treatment of this aspect as well as results on depth-width tradeoffs, the interested reader is referred to [44].…”

“…where ( 45) and ( 46) are thanks to |a + b| 2 ≤ 2|a| 2 + 2|b| 2 , for a, b ∈ C. Substituting ( 45) and ( 46) into (44) and noting that…”