Related papers: Convergence Types and Rates in Generic Karhunen-Lo…

Gaussian processes (GPs) are the most common formalism for defining probability distributions over spaces of functions. While applications of GPs are myriad, a comprehensive understanding of GP sample paths, i.e. the function spaces over…

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…

This report examines numerical aspects of constructing Karhunen-Lo\`{e}ve expansions (KLEs) for second-order stochastic processes. The KLE relies on the spectral decomposition of the covariance operator via the Fredholm integral equation of…

The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed…

We study recursive regularized learning algorithms in the reproducing kernel Hilbert space (RKHS) with non-stationary online data streams. We introduce the concept of random Tikhonov regularization path and decompose the tracking error of…

We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including…

Kernel mean embeddings, a widely used technique in machine learning, map probability distributions to elements of a reproducing kernel Hilbert space (RKHS). For supervised learning problems, where input-output pairs are observed, the…

Regularized kernel methods such as support vector machines (SVM) and support vector regression (SVR) constitute a broad and flexible class of methods which are theoretically well investigated and commonly used in nonparametric…

The commonly accepted definition of paths starts from a random field but ignores the problem of setting joint distributions of infinitely many random variables for defining paths properly afterwards. This paper provides a turnaround that…

A generalized version of Groeneveld's convergence criterion for the virial expansion and generating functionals for weighted $2$-connected graphs is proven. The criterion works for inhomogeneous systems and yields bounds for the density…

This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that…

This article provides a primer on the spectral representation of random fields via the Karhunen-Lo\`eve Expansion (KLE). The goal is to bridge the gap between the theoretical foundations of the KLE and its application in computational…

This article studies global testing of the slope function in functional linear regression model in the framework of reproducing kernel Hilbert space. We propose a new testing statistic based on smoothness regularization estimators. The…

Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super-$\sqrt{n}$ convergence rates in numerical integration, and thus provide alternatives to Monte Carlo integration in challenging…

Statistical machine learning plays an important role in modern statistics and computer science. One main goal of statistical machine learning is to provide universally consistent algorithms, i.e., the estimator converges in probability or…

This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact…

Stochastic processes are often represented through orthonormal series expansions, a framework originating in the classical works of Lo\`eve and Karhunen and widely used for simulation and numerical approximation. While truncation error in…

Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is primarily useful for those processes for which we can identify the necessary…

We propose a novel Bayesian methodology for inference in functional linear and logistic regression models based on the theory of reproducing kernel Hilbert spaces (RKHS's). We introduce general models that build upon the RKHS generated by…

New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of nonstationary processes. An application of the…