  This paper proposes that Lipschitz continuity is a natural outcome of
regularized least squares in kernel-based learning. Lipschitz continuity is an
important proxy for robustness of input-output operators. It is also
instrumental for guaranteeing closed-loop stability of kernel-based
controlllers through small incremental gain arguments. We introduce a new class
of nonexpansive kernels that are shown to induce Hilbert spaces consisting of
only Lipschitz continuous operators. The Lipschitz constant of estimated
operators within such Hilbert spaces can be tuned by suitable selection of a
regularization parameter. As is typical for kernel-based models, input-output
operators are estimated from data by solving tractable systems of linear
equations. The approach thus constitutes a promising alternative to
Lipschitz-bounded neural networks, that have recently been investigated but are
computationally expensive to train.
