Residual Networks as Nonlinear Systems: Stability Analysis using Linearization

TL;DR

This paper models pre-trained ResNets as nonlinear systems and uses linearization to analyze their stability and response to perturbations, revealing insights into the behavior of residual units and the impact of network modifications.

Contribution

It introduces a stability analysis framework for ResNets using linearization and singular value decomposition, providing new understanding of perturbation propagation and network robustness.

Findings

Most singular values of residual units are close to 1.

Scaling and weight adjustments significantly affect singular value distributions.

Adversarial perturbations exhibit a dramatic increase near the network's end.

Abstract

We regard pre-trained residual networks (ResNets) as nonlinear systems and use linearization, a common method used in the qualitative analysis of nonlinear systems, to understand the behavior of the networks under small perturbations of the input images. We work with ResNet-56 and ResNet-110 trained on the CIFAR-10 data set. We linearize these networks at the level of residual units and network stages, and the singular value decomposition is used in the stability analysis of these components. It is found that most of the singular values of the linearizations of residual units are 1 and, in spite of the fact that the linearizations depend directly on the activation maps, the singular values differ only slightly for different input images. However, adjusting the scaling of the skip connection or the values of the weights in a residual unit has a significant impact on the singular value distributions. Inspection of how random and adversarial perturbations of input images propagate through the network reveals that there is a dramatic jump in the magnitude of adversarial perturbations towards the end of the final stage of the network that is not present in the case of random perturbations. We attempt to gain a better understanding of this phenomenon by projecting the perturbations onto singular vectors of the linearizations of the residual units.