Chiara Segala

Christian Fiedler, Michael Herty, Michael Rom et al.

Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of the limiting reproducing kernel Hilbert space. Furthermore, several examples of kernels, that allow a rigorous mean field limit, are presented.

Sara Avesani, Gaia Fumagalli, Michael Multerer et al.

We develop a novel framework for sparse multiscale kernel approximation of large scattered data problems based on a samplet representation. Samplets form a multiresolution analysis of localized discrete signed measures and enable quasi-sparse representations of kernel matrices associated to asymptotically smooth kernels as well as smoothness detection of scattered data. Building on the latter, we introduce an adaptive data site selection strategy based on the localization of the native reproducing kernel Hilbert space norm in the samplet expansion coefficients. The selection results in a small set of representative data sites, significantly reducing the effective problem size. On the corresponding reduced kernel subspace, we solve an $\ell^1$-regularized least-squares problem using a trust-region semismooth Newton method in a normal-map formulation, stabilized by an online low-rank SVD on the active set to handle the notorious ill-conditioning of kernel matrices. Numerical experiments in two and three dimensions, including multi-kernel models with varying lengthscales, demonstrate that the proposed approach achieves accurate reconstructions with considerably sparser representations and good computational efficiency.

Michael Multerer, Paul Schneider, Chiara Segala

We propose a scalable and theoretically grounded low-rank conditional expectation model for recursive Monte Carlo optimal stopping problems, in particular American option pricing. Our method reformulates the estimation of continuation values as a learning problem in a reproducing kernel Hilbert space, in which the conditional expectation is represented as a linear operator acting on future payoffs. This perspective yields an offline-online decomposition: the operator is learned once from simulated data and subsequently reused across all exercise dates, eliminating the need to recompute regression models at each step of the backward recursion. We establish convergence guarantees and derive bounds quantifying the approximation errors across exercise dates. Numerical experiments demonstrate the speed and accuracy of the proposed approach relative to extant methods.

Giacomo Albi, Sara Bicego, Michael Herty et al.

Feedback control synthesis for large-scale particle systems is reviewed in the framework of model predictive control (MPC). The high-dimensional character of collective dynamics hampers the performance of traditional MPC algorithms based on fast online dynamic optimization at every time step. Two alternatives to MPC are proposed. First, the use of supervised learning techniques for the offline approximation of optimal feedback laws is discussed. Then, a procedure based on sequential linearization of the dynamics based on macroscopic quantities of the particle ensemble is reviewed. Both approaches circumvent the online solution of optimal control problems enabling fast, real-time, feedback synthesis for large-scale particle systems. Numerical experiments assess the performance of the proposed algorithms.