Related papers: A universal reproducing kernel Hilbert space for l…
We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which…
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert…
Motivated by the surge of interest in Koopman operator theory, we propose a machine-learning alternative based on a functional Bayesian perspective for operator-theoretic modeling of unknown, data-driven, nonlinear dynamical systems. This…
In this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions: a data is represented by a set of functions, and the label is also a function. We focus on the use of…
Motivated by the need of processing functional-valued data, or more general, operatorvalued data, we introduce the notion of the operator reproducing kernel Hilbert space (ORKHS). This space admits a unique operator reproducing kernel which…
This paper considers the design of nonlinear data-enabled predictive control (DeePC) using kernel functions. Compared with existing methods that use kernels to parameterize multi-step predictors for nonlinear DeePC, we adopt a novel,…
This manuscript presents an algorithm for obtaining an approximation of a nonlinear high order control affine dynamical system. Controlled trajectories of the system are leveraged as the central unit of information via embedding them in…
The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in…
Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we…
This work presents a nonparametric framework for dissipativity learning in reproducing kernel Hilbert spaces, which enables data-driven certification of stability and performance properties for unknown nonlinear systems without requiring an…
Traditional machine learning models, particularly neural networks, are rooted in finite-dimensional parameter spaces and nonlinear function approximations. This report explores an alternative formulation where learning tasks are expressed…
This paper reviews the functional aspects of statistical learning theory. The main point under consideration is the nature of the hypothesis set when no prior information is available but data. Within this framework we first discuss about…
The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and…
Traditional linear methods for forecasting multivariate time series are not able to satisfactorily model the non-linear dependencies that may exist in non-Gaussian series. We build on the theory of learning vector-valued functions in the…
We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and…
We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This…
This paper introduces a computational framework to identify nonlinear input-output operators that fit a set of system trajectories while satisfying incremental integral quadratic constraints. The data fitting algorithm is thus regularized…
We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding…