Design and Analysis of Convolutional Neural Layers: A Signal Processing Perspective

Convolutional layers (CLs) are ubiquitous in contemporary deep neural network (DNN) models, commonly used for automatic feature extraction. A CL performs cross-correlation between the input to the layer and a set of learnable kernels to produce the layer output. Typically, kernel weights are randomly initialized and automatically learned during model training using the backpropagation and gradient descent algorithms to minimize a specific loss function. Modern DNN models comprise deep hierarchical stacks of CLs and pooling layers. Despite their prevalence, CLs are perceived as a magical tool for feature extraction without solid interpretations of their underlying working principles. In this work, we advance a method for designing and analyzing CLs by providing novel signal processing interpretations of the CL by exploiting the correlation and equivalent convolution functions of the layer. The proposed interpretations enable the employment of CLs to develop finite impulse response (FIR) filters, matched filters (MFs), short-time Fourier transform (STFT), discrete-time Fourier transform (DTFT), and continuous wavelet transform (CWT) algorithms. The main idea is to pre-assign the CL kernel weights to implement a specific convolution- or correlation-based DSP algorithm. Such an approach enables building self-contained DNN models in which CLs are utilized for various preprocessing and feature extractions tasks, enhancing the model portability, and cutting down the preprocessing computational cost. The proposed DSP interpretations provide an effective means to analyze and explain the operation of automatically trained CLs in the time and frequency domains by reversing the design procedures. The presented interpretations are mathematically established and experimentally validated with a comprehensive machinery fault diagnosis application example illustrating the potential of the proposed methodology.