Approximation theory for 1-Lipschitz ResNets

TL;DR

This paper establishes the first universal approximation guarantees for 1-Lipschitz ResNets, demonstrating their ability to densely approximate scalar 1-Lipschitz functions and represent piecewise affine functions, with practical training considerations.

Contribution

It provides the first rigorous universal approximation results for 1-Lipschitz ResNets, including density and exact representation capabilities under norm constraints.

Findings

ResNets are dense in scalar 1-Lipschitz functions on compact domains.

They can exactly represent scalar piecewise affine 1-Lipschitz functions.

Density holds with fixed hidden width when inserting norm-constrained linear maps.

Abstract

1-Lipschitz neural networks are fundamental for generative modelling, inverse problems, and robust classifiers. In this paper, we focus on 1-Lipschitz residual networks (ResNets) based on explicit Euler steps of negative gradient flows and study their approximation capabilities. Leveraging the Restricted Stone-Weierstrass Theorem, we first show that these 1-Lipschitz ResNets are dense in the set of scalar 1-Lipschitz functions on any compact domain when width and depth are allowed to grow. We also show that these networks can exactly represent scalar piecewise affine 1-Lipschitz functions. We then prove a stronger statement: by inserting norm-constrained linear maps between the residual blocks, the same density holds when the hidden width is fixed. Because every layer obeys simple norm constraints, the resulting models can be trained with off-the-shelf optimisers. This paper provides the first universal approximation guarantees for 1-Lipschitz ResNets, laying a rigorous foundation for their practical use.