Lipschitz bounds for integral kernels
New paper provides explicit formulas for the Lipschitz constants of key kernels like Gaussian and ReLU.
A team of researchers including Justin Reverdi and Sixin Zhang has published a significant paper, 'Lipschitz bounds for integral kernels,' on arXiv. The work tackles a core problem in machine learning theory: providing explicit, calculable bounds for the Lipschitz constants of kernel feature maps. These maps are fundamental to kernel methods, and their Lipschitz continuity is directly tied to the robustness and stability guarantees of AI models. Until now, such explicit characterizations were only available in limited cases.
The paper establishes sufficient conditions for Lipschitz continuity and delivers concrete formulas for the constants. A key finding is that for infinite-width neural networks with isotropic Gaussian weights, the Lipschitz constant of the associated kernel can be expressed as the supremum of a two-dimensional integral. This leads to explicit characterizations for the Gaussian kernel and the ReLU random neural network kernel. For continuous, shift-invariant kernels (like Gaussian, Laplace, and Matérn), the researchers proved the feature map is Lipschitz continuous if and only if the underlying weight distribution has a finite second-order moment, and they derived the corresponding constant. The work concludes by raising an open question about the asymptotic behavior of these constants in finite-width networks, supported by numerical experiments.
- Provides explicit formulas for Lipschitz constants of Gaussian, Laplace, and Matérn kernels.
- Proves infinite-width neural network kernel Lipschitz constant is a 2D integral supremum.
- Links kernel Lipschitz continuity directly to the finite second-order moment of weight distributions.
Why It Matters
Enables stronger mathematical guarantees for AI model robustness, improving reliability in critical applications.