Approximation theory for 1-Lipschitz ResNets

Davide Murari, Takashi Furuya, Carola-Bibiane Schönlieb

Keywords: Lipschitz neural networks, Universal approximation, Residual neural networks, Stone–Weierstrass theorem

TL;DR: We study the approximation properties of 1-Lipschitz Resnet, and prove their density in the set of 1-Lipschitz functions.

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.

Primary Area: Deep learning (e.g., architectures, generative models, optimization for deep networks, foundation models, LLMs)

Submission Number: 8115