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 Matérn 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.

翻译：与正定核相关联的特征映射在核方法及学习理论中占据核心地位，其中利普希茨连续性等正则性质与鲁棒性和稳定性保障密切相关。尽管重要性显著，但核特征映射利普希茨常数的显式表征仅在有限情形下可得。本文研究可微性假设下积分核相关特征映射的利普希茨正则性。我们首先给出确保利普希茨连续的充分条件，并推导相应利普希茨常数的显式公式。随后识别特征映射非利普希茨连续的条件，并将这些结果应用于若干重要核类。对于具有各向同性高斯权重分布的无限宽双层神经网络，我们证明相关核的利普希茨常数可表示为二维积分的上确界，从而得到高斯核与ReLU随机神经网络核的显式刻画。我们还研究连续且平移不变的核，例如高斯核、拉普拉斯核及马特恩核，这些核可解释为具有余弦激活函数的神经网络。在此设定下，我们证明特征映射利普希茨连续当且仅当权重分布具有二阶有限矩，进而推导其利普希茨常数。最后，我们提出关于有限宽度神经网络中利普希茨常数收敛渐近行为的开放性问题，并通过数值实验支撑该行为。