# Adversarial robustness via robust low rank representations

2020 Conference on Neural Information Processing (NeurIPS)

## Abstract

Adversarial robustness measures the susceptibility of a classifier to imperceptible perturbations made to the inputs at test time. In this work we highlight the benefits of natural low rank representations that often exist for real data such as images, for training neural networks with certified robustness guarantees.

Our first contribution is for certified robustness to perturbations measured in $\ell_2$ norm. We exploit low rank data representations to provide improved guarantees over state-of-the-art randomized smoothing-based approaches on standard benchmark datasets such as CIFAR-10.

Our second contribution is for the more challenging setting of certified robustness to perturbations measured in $\ell_\infty$ norm. We demonstrate empirically that natural low rank representations have inherent robustness properties that can be leveraged to provide significantly better guarantees for certified robustness to $\ell_\infty$ perturbations. Our certificate of $\ell_\infty$ robustness relies on a natural quantity involving the $\infty \to 2$ matrix operator norm associated with the representation, to translate robustness guarantees from $\ell_2$ to $\ell_\infty$ perturbations. A key technical ingredient for our certification guarantees is a fast algorithm based on the multiplicative weights update method to provide sharp upper bounds on the above matrix norm