Related papers: On Kernels and Covariance Structures in Hilbert Sp…
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…
The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces i.e.; spaces generated by positive semidefinite sesquilinear forms. Let H be a Hilbert space and let A be a positive bounded operator on H…
Operator learning offers a powerful paradigm for solving parametric partial differential equations (PDEs), but scaling probabilistic neural operators such as the recently proposed Gaussian Processes Operators (GPOs) to high-dimensional,…
Category and multicategory theory provide abstract frameworks for describing structures and their compositions, with multicategories extending traditional categories to handle multi-input operations. These theories enable modular reasoning…
The purpose of this paper is to give an overview of the operator structure of frames, where the operator belongs to certain classes of linear operators and the element belongs to $H$. We discuss the size of the set of such elements. Also,…
The application of Gaussian processes (GPs) to large data sets is limited due to heavy memory and computational requirements. A variety of methods has been proposed to enable scalability, one of which is to exploit structure in the kernel…
Gaussian processes are powerful, yet analytically tractable models for supervised learning. A Gaussian process is characterized by a mean function and a covariance function (kernel), which are determined by a model selection criterion. The…
Since its introduction, the Discrete Variable Representation (DVR) basis set has become an invaluable representation of state vectors and Hermitian operators in non-relativistic quantum dynamics and spectroscopy calculations. On the other…
In this chapter, the Hilbert space framework in the mathematical theory of composite materials is introduced for studying the properties of effective operators. The goal is to introduce some of the key concepts and fundamental theorems in…
Gaussian processes are a powerful class of non-linear models, but have limited applicability for larger datasets due to their high computational complexity. In such cases, approximate methods are required, for example, the recently…
Covariance functions and variograms play a fundamental role in exploratory analysis and statistical modelling of spatial and spatio-temporal datasets. In this paper, we construct a new class of spatial covariance functions using the Fourier…
This paper presents new results on Functional Analysis of Variance for fixed effect models with correlated Hilbert-valued Gaussian error components. The geometry of the Reproducing Kernel Hilbert Space (RKHS) of the error term is considered…
We consider continuous structures which are obtained from finite dimensional Hilbert spaces over $\mathbb{C}$ by adding some unitary operators. Quantum automata and circuits are naturally interpretable in such structures. We consider…
We study classical structures in various categories of completely positive morphisms: on sets and relations, on cobordisms, on a free dagger compact category, and on Hilbert spaces. As an application, we prove that quantum maps with…
In this paper, we describe families of those bounded linear operators on a separable Hilbert space that are simultaneously unitarily equivalent to integral operators on $L_2(R)$ with bounded and arbitrarily smooth Carleman kernels. The main…
In this article we investigate the effects of conformal transformations on kernel functions used in Support Vector Machines. Our focus lies in the task of text document categorization, which involves assigning each document to a particular…
We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of…
Gaussian processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP $T u$ that is the image of another GP $u$ under a linear transformation $T$…
We analize the algebraic structure of consistent and covariant anomalies in gauge and gravitational theories: using a complex extension of the Lie algebra it is possible to describe them in a unified way. Then we study their representations…
We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in…