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We propose a novel adaptive learning algorithm based on iterative orthogonal projections in the Cartesian product of multiple reproducing kernel Hilbert spaces (RKHSs). The task is estimating/tracking nonlinear functions which are supposed…
Reproducing kernel Hilbert spaces (RKHSs) are key spaces for machine learning that are becoming popular also for linear system identification. In particular, the so-called stable RKHSs can be used to model absolutely summable impulse…
We consider the problem of supervised learning with convex loss functions and propose a new form of iterative regularization based on the subgradient method. Unlike other regularization approaches, in iterative regularization no constraint…
When concept shifts and sample scarcity are present in the target domain of interest, nonparametric regression learners often struggle to generalize effectively. The technique of transfer learning remedies these issues by leveraging data or…
We study kernel least-squares estimation under a norm constraint. This form of regularisation is known as Ivanov regularisation and it provides better control of the norm of the estimator than the well-established Tikhonov regularisation.…
We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a…
Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates…
An extension of reproducing kernel Hilbert space (RKHS) theory provides a new framework for modeling functional regression models with functional responses. The approach only presumes a general nonlinear regression structure as opposed to…
We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2 . The class includes, as a special case, the usual empirical norm…
Based on the variable Hilbert scale interpolation inequality bounds for the error of regularisation methods are derived under range inclusions. In this context, new formulae for the modulus of continuity of the inverse of bounded operators…
Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward…
Depth measures are powerful tools for defining level sets in emerging, non--standard, and complex random objects such as high-dimensional multivariate data, functional data, and random graphs. Despite their favorable theoretical properties,…
We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic…
The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $n^{-1/2}$. However, re-weighting of random points can sometimes be used to improve the convergence order.…
For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the…
This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents…
Learning with Reproducing Kernel Hilbert Spaces (RKHS) has been widely used in many scientific disciplines. Because a RKHS can be very flexible, it is common to impose a regularization term in the optimization to prevent overfitting.…
We introduce and study the theory of training neural networks using interpolation techniques from reproducing kernel Hilbert space theory. We generalize the method to Krein spaces, and show that widely-used neural network architectures are…
We propose a nonparametric method to learn the L\'evy density from probability density data governed by a nonlocal Fokker-Planck equation. We recast the problem as identifying the kernel in a nonlocal integral operator from discrete data,…
In this paper we investigate the connection between supervised learning and linear inverse problems. We first show that a linear inverse problem can be view as a function approximation problem in a reproducing kernel Hilbert space (RKHS)…