# Lipschitz bounds for integral kernels

arXiv: https://arxiv.org/abs/2604.02887 · HTML: https://mlsanity.com/paper/2604.02887/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 39 references that are in the map.
- First posted on arXiv 3 Apr 2026.

## Builds on (most cited first, 19 of 19)

- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators](https://mlsanity.com/paper/1903.01014/) (2019)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Approximation and Learning with Deep Convolutional Models: a Kernel Perspective](https://mlsanity.com/paper/2102.10032/) (2021)
- [Improving the Accuracy-Robustness Trade-Off of Classifiers via Adaptive Smoothing](https://mlsanity.com/paper/2301.12554/) (2023)
- [Adversarial robustness of sparse local Lipschitz predictors](https://mlsanity.com/paper/2202.13216/) (2022)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Scalable Global Alignment Graph Kernel Using Random Features: From Node Embedding to Graph Embedding](https://mlsanity.com/paper/1911.11119/) (2019)
- [D2KE: From Distance to Kernel and Embedding](https://mlsanity.com/paper/1802.04956/) (2018)
- [Kernel-based models for system analysis](https://mlsanity.com/paper/2110.11735/) (2021)
- [The Effect of Prior Lipschitz Continuity on the Adversarial Robustness of Bayesian Neural Networks](https://mlsanity.com/paper/2101.02689/) (2021)
- [Training Lipschitz continuous operators using reproducing kernels](https://mlsanity.com/paper/2112.03344/) (2021)
- [Upper and lower bounds for the Lipschitz constant of random neural networks](https://mlsanity.com/paper/2311.01356/) (2023)
- [AVATAR: Adversarial Autoencoders with Autoregressive Refinement for Time Series Generation](https://mlsanity.com/paper/2501.01649/) (2025)
- [Feature maps for the Laplacian kernel and its generalizations](https://mlsanity.com/paper/2502.15575/) (2025)

## Abstract (arXiv)

Feature maps associated with positive definite kernels play a central role in kernel methods and learning theory, where regularity properties such as Lipschitz continuity are closely related to robustness and stability guarantees. Despite their importance, explicit characterizations of the Lipschitz constant of kernel feature maps are available only in a limited number of cases. In this paper, we study the Lipschitz regularity of feature maps associated with integral kernels under differentiability assumptions. We first provide sufficient conditions ensuring Lipschitz continuity and derive explicit formulas for the corresponding Lipschitz constants. We then identify a condition under which the feature map fails to be Lipschitz continuous and apply these results to several important classes of kernels. For infinite width two-layer neural network with isotropic Gaussian weight distributions, we show that the Lipschitz constant of the associated kernel can be expressed as the supremum of a two-dimensional integral, leading to an explicit characterization for the Gaussian kernel and the ReLU random neural network kernel. We also study continuous and shift-invariant kernels such as Gaussian, Laplace, and Mat'ern kernels, which admit an interpretation as neural network with cosine activation function. In this setting, we prove that the feature map is Lipschitz continuous if and only if the weight distribution has a finite second-order moment, and we then derive its Lipschitz constant. Finally, we raise an open question concerning the asymptotic behavior of the convergence of the Lipschitz constant in finite width neural networks. Numerical experiments are provided to support this behavior.