  Designing neural networks with bounded Lipschitz constant is a promising way
to obtain certifiably robust classifiers against adversarial examples. However,
the relevant progress for the important $\ell_\infty$ perturbation setting is
rather limited, and a principled understanding of how to design expressive
$\ell_\infty$ Lipschitz networks is still lacking. In this paper, we bridge the
gap by studying certified $\ell_\infty$ robustness from a novel perspective of
representing Boolean functions. We derive two fundamental impossibility results
that hold for any standard Lipschitz network: one for robust classification on
finite datasets, and the other for Lipschitz function approximation. These
results identify that networks built upon norm-bounded affine layers and
Lipschitz activations intrinsically lose expressive power even in the
two-dimensional case, and shed light on how recently proposed Lipschitz
networks (e.g., GroupSort and $\ell_\infty$-distance nets) bypass these
impossibilities by leveraging order statistic functions. Finally, based on
these insights, we develop a unified Lipschitz network that generalizes prior
works, and design a practical version that can be efficiently trained (making
certified robust training free). Extensive experiments show that our approach
is scalable, efficient, and consistently yields better certified robustness
across multiple datasets and perturbation radii than prior Lipschitz networks.
Our code is available at https://github.com/zbh2047/SortNet.
