  While the class of Polynomial Nets demonstrates comparable performance to
neural networks (NN), it currently has neither theoretical generalization
characterization nor robustness guarantees. To this end, we derive new
complexity bounds for the set of Coupled CP-Decomposition (CCP) and Nested
Coupled CP-decomposition (NCP) models of Polynomial Nets in terms of the
$\ell_\infty$-operator-norm and the $\ell_2$-operator norm. In addition, we
derive bounds on the Lipschitz constant for both models to establish a
theoretical certificate for their robustness. The theoretical results enable us
to propose a principled regularization scheme that we also evaluate
experimentally in six datasets and show that it improves the accuracy as well
as the robustness of the models to adversarial perturbations. We showcase how
this regularization can be combined with adversarial training, resulting in
further improvements.
