  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 and CIFAR-100.
  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 in those
representations. 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 with provable guarantees based on the multiplicative weights update
method to provide upper bounds on the above matrix norm. Our algorithmic
guarantees improve upon the state of the art for this problem, and may be of
independent interest.
