Toward Understanding Convolutional Neural Networks from Volterra Convolution Perspective
Tenghui Li, Guoxu Zhou, Yuning Qiu, Qibin Zhao
TL;DR
The paper tackles the challenge of interpreting complex CNN architectures by proposing a unified Volterra convolution framework, where networks are approximated by a finite-term Volterra series $\sum_{n=0}^{N} H_n * x^n$. It demonstrates that most CNNs can be represented or closely approximated within this formalism, enabling analysis via a smaller set of proxy kernels rather than the full, intricate architectures. It introduces two practical kernel-inference approaches—direct calculation for white-box networks and a hacking network for black-box scenarios—and validates them on simple two- and three-layer structures, showing close matches between true and proxy kernels. The work further explores how these proxies can be used for perturbation analysis and potential adversarial applications, including a MNIST demonstration, and suggests future directions for higher-order kernels and dynamic inputs with open-source code release.
Abstract
We make an attempt to understanding convolutional neural network by exploring the relationship between (deep) convolutional neural networks and Volterra convolutions. We propose a novel approach to explain and study the overall characteristics of neural networks without being disturbed by the horribly complex architectures. Specifically, we attempt to convert the basic structures of a convolutional neural network (CNN) and their combinations to the form of Volterra convolutions. The results show that most of convolutional neural networks can be approximated in the form of Volterra convolution, where the approximated proxy kernels preserve the characteristics of the original network. Analyzing these proxy kernels may give valuable insight about the original network. Base on this setup, we presented methods to approximating the order-zero and order-one proxy kernels, and verified the correctness and effectiveness of our results.