Related papers: Operator Reproducing Kernel Hilbert Spaces
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…
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…
A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing…
We develop novel learning rates for conditional mean embeddings by applying the theory of interpolation for reproducing kernel Hilbert spaces (RKHS). We derive explicit, adaptive convergence rates for the sample estimator under the…
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…
This paper proposes a method for constructing one-step prediction tubes for nonlinear systems using reproducing kernel Hilbert spaces. We approximate a bounded reproducing kernel Hilbert space (RKHS) hypothesis set by a finite-dimensional…
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 introduce a vector differential operator $\mathbf{P}$ and a vector boundary operator $\mathbf{B}$ to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This…
We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) $\mathscr{H}$ over infinite, discrete and countable sets $V$. In this setting we analyze in detail the distributions of the corresponding Dirac…
In this paper, we present a unified approach to function approximation in reproducing kernel Hilbert spaces (RKHS) that establishes a previously unrecognized optimality property for several well-known function approximation techniques, such…
Consider the problem: given the data pair $(\mathbf{x}, \mathbf{y})$ drawn from a population with $f_*(x) = \mathbf{E}[\mathbf{y} | \mathbf{x} = x]$, specify a neural network model and run gradient flow on the weights over time until…
The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. However, existing data-driven approaches to learning the Koopman operator rely on batch data. In this work, we present…
We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by…
Spatial temporal reconstruction of dynamical system is indeed a crucial problem with diverse applications ranging from climate modeling to numerous chaotic and physical processes. These reconstructions are based on the harmonious…
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However,…
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…
Traditionally, kernel methods rely on the representer theorem which states that the solution to a learning problem is obtained as a linear combination of the data mapped into the reproducing kernel Hilbert space (RKHS). While elegant from…
Regularized empirical risk minimization using kernels and their corresponding reproducing kernel Hilbert spaces (RKHSs) plays an important role in machine learning. However, the actually used kernel often depends on one or on a few…
In this work, we develop and study an empirical projection operator scheme for solving nonparametric regression problems. This scheme is based on an approximate projection of the regression function over a suitable reproducing kernel…
A mathematical model for variable selection in functional regression models with scalar response is proposed. By "variable selection" we mean a procedure to replace the whole trajectories of the functional explanatory variables with their…