证明1-Lipschitz残差网络可逼近任意1-Lipschitz函数，为生成模型和鲁棒分类提供理论支持。
Approximation theory for 1-Lipschitz ResNets
- 基于负梯度流的显式欧拉步，构建1-Lipschitz ResNet结构。
- 宽度深度无限时，可逼近任意标量1-Lipschitz函数；固定宽度时通过范数约束仍可逼近。
- 结果首次给出1-Lipschitz ResNet的通用逼近保证，适合生成建模与鲁棒学习场景。
1-Lipschitz神经网络在生成建模、反问题求解和鲁棒分类中具有基础性作用。本文研究基于负梯度流显式欧拉步的1-Lipschitz残差网络（ResNets），分析其逼近能力。利用受限Stone-Weierstrass定理，我们首先证明：当宽度和深度趋于无穷时，这类1-Lipschitz ResNet在任意紧集上稠密于标量1-Lipschitz函数集合。同时，它们能精确表示标量分段仿射1-Lipschitz函数。进一步证明：通过在残差块间插入范数约束线性层，即使隐藏层宽度固定，相同稠密性仍成立。由于每层均满足简单范数约束，模型可用标准优化器训练。本文首次为1-Lipschitz ResNet提供通用逼近保证，为其实际应用奠定严格理论基础。
原文摘要 · Abstract (English)
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.
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