Related papers: An Operator Learning Approach via Function-valued …
In this article, a novel approach to learning a complex function which can be written as the system of linear equations is introduced. This learning is grounded upon the observation that solving the system of linear equations by a…
We present Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions based on the foundational ideas of function encoders. We decompose the task of learning operators into two parts:…
A linear operator in a Hilbert space defined through the form of Riesz representation naturally introduces a reproducing kernel Hilbert space structure over the range space. Such formulation, called H-HK formulation in this paper, possesses…
In this paper, we give some results concerning atomic decompositions for operators on reproducing kernel Hilbert spaces, using frame theory techniques. We provide also generalizations for atomic decompositions of some theorems for…
Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable…
Proper splittings of operators are commonly used to study the convergence of iterative processes. In order to approximate solutions of operator equations, in this article we deal with proper splittings of closed range bounded linear…
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…
Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation…
We propose a kernel-spectral embedding algorithm for learning low-dimensional nonlinear structures from high-dimensional and noisy observations, where the datasets are assumed to be sampled from an intrinsically low-dimensional manifold and…
The basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely to efficiently perform computations in an intractably large Hilbert space. In this paper we explore some theoretical…
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…
Incorporating spatial information into hyperspectral unmixing procedures has been shown to have positive effects, due to the inherent spatial-spectral duality in hyperspectral scenes. Current research works that consider spatial information…
In this paper, we present a fast and accurate numerical scheme for the solution of fifth-order boundary-value problems. We apply the reproducing kernel Hilbert space method (RKHSM) for solving this problem. The analytic results of the…
The performance of reproducing kernel Hilbert space-based methods is known to be sensitive to the choice of the reproducing kernel. Choosing an adequate reproducing kernel can be challenging and computationally demanding, especially in…
This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of the…
This paper investigates a general regularization framework for unsupervised domain adaptation in vector-valued regression under the covariate shift assumption, utilizing vector-valued reproducing kernel Hilbert spaces (vRKHS). Covariate…
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…
Kernel-based learning algorithms are widely used in machine learning for problems that make use of the similarity between object pairs. Such algorithms first embed all data points into an alternative space, where the inner product between…
In the present paper, we consider the integral operator, which acts in Hilbert space and has sine kernel. This operator generates two operator identities and two corresponding canonical differential systems. We find the asymptotics of the…
Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation…