arXiv Machine Learning
Aug 28

Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences

The monograph explores the relationships between Gaussian processes and reproducing kernel Hilbert spaces (RKHS), two widely used approaches that rely on positive definite kernels. It examines how these frameworks connect and are equivalent across key topics such as regression, interpolation, numerical integration, distributional discrepancies, statistical dependence, and Gaussian process sample path properties. By establishing a unifying perspective based on the equivalence between the Gaussian Hilbert space and the RKHS, the work aims to bridge methods developed independently by the machine learning, statistics, and numerical analysis communities.

By Motonobu Kanagawa, Philipp Hennig, Dino Sejdinovic, Bharath K. Sriperumbudur
arXiv Statistics ML
Sep 4

Online Learning of Functional Principal Component Analysis for Multidimensional Functional Data

The paper introduces an online framework for functional principal component analysis (FPCA) tailored to multidimensional functional data streams. It models functional principal components with tensor product splines, enforcing smoothness and orthonormality via a penalized approach on a Stiefel manifold. The authors present efficient Riemannian stochastic gradient descent and AdaGrad algorithms, along with a dynamic smoothing parameter tuning strategy based on rolling block validation, and provide asymptotic normality results and confidence intervals for the estimators.

By Muye Nanshan, Nan Zhang, Jiguo Cao
arXiv Machine Learning
Aug 31

Generalized Splines and Gaussian Processes

arXiv:2608.28446v1 Announce Type: cross Abstract: For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes...

By Michael Unser