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:2610.09731v1 Announce Type: cross
Abstract: Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications. Their immense su...
By Jose Cribeiro-Ramallo, Florian Kalinke, Zolt\'an Szab\'o
arXiv:2607. 24235v1 Announce Type: cross Abstract: Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others.
By Jose Cribeiro-Ramallo, Florian Kalinke, Zolt\'an Szab\'o
arXiv:2608.29265v1 Announce Type: cross
Abstract: Kernel density estimation (KDE) is one of the most fundamental statistical estimators of density functions. Its direct implementation on a dataset of...
By Xie Wang, Nicolas Langren\'e, Wen Chen
arXiv:2606. 11255v1 Announce Type: new Abstract: Bernstein--Schur kernels are products of a finite-feature kernel (one with an explicit finite-dimensional feature map) and a completely monotone shift-invariant kernel: nonstationary kernels that fall between the shift-invariant and dot-product templates random features usually exploit, so in general neither Bochner sampling nor polynomial sketching applies to the full kernel directly.
By Taha Bouhsine
The paper introduces a unified framework for measuring divergences between conditional distributions using kernel embeddings, termed conditional maximum mean discrepancy (CMMD). It defines a family of CMMD metrics—CMMD_0, CMMD_1, and CMMD_2—each based on different RKHS embeddings, and generalizes them to a level‑s CMMD with clear assumptions and operator‑based smoothing connections. Additionally, the authors present a novel doubly robust estimator for CMMD and demonstrate through experiments that the metric captures complex conditional dependencies for statistical testing.
By Peter Moskvichev, Siu Lun Chau, Dino Sejdinovic
The paper explores the relationship between Gaussian processes and Gaussian random elements within reproducing kernel Banach spaces. It demonstrates that the covariance operator of a weak second‑order Radon probability measure on such a space is uniquely defined by a symmetric positive‑definite function, and in the Gaussian case it identifies which functions correspond to covariance operators via γ‑radonifying operators. Leveraging these findings, the authors extend Driscoll’s theorem to the setting of reproducing kernel Banach spaces.
By Toni Karvonen, Rasmus Kleist H{\o}rlyck S{\o}rensen
arXiv:2607. 21823v1 Announce Type: new Abstract: We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit.
By Sergio A. Alvarez
The paper proposes methods for estimating many high‑dimensional mean vectors from independent samples by forming convex combinations of empirical means. Two data‑dependent weighting strategies are introduced: one uses a testing procedure to pick low‑variance neighbouring means, yielding a closed‑form plug‑in formula; the other minimizes an upper confidence bound on quadratic risk. Theoretical results show these approaches asymptotically achieve oracle (minimax) risk improvements as the effective dimension grows, and experiments confirm their effectiveness on simulated and real kernel mean embedding tasks.
By Gilles Blanchard (LMO, DATASHAPE), Jean-Baptiste Fermanian (LMO), Hannah Marienwald (TUB)
arXiv:2602. 23006v2 Announce Type: replace-cross Abstract: Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations.
By Arsalan Jawaid, Abdullah Karatas, J\"org Seewig
arXiv:2610.34544v1 Announce Type: new
Abstract: Kernel techniques are among the most powerful methods in data science and statistics. One of the key properties of kernels contributing to their numero...
By Xiaoyi Wen, Zoltan Szabo
The paper introduces a geometric framework for measuring how far empirical datasets deviate from the Gaussian family using optimal transport theory. It defines two new quantities—the relative Wasserstein angle and the orthogonal projection distance—based on the cone structure of the relative translation invariant quadratic Wasserstein space, and shows that the usual moment‑matching Gaussian is not generally the $W_2$‑nearest Gaussian. Closed‑form expressions are derived for one‑dimensional and several location–scale families, while a numerical approximation is proposed for higher dimensions, with experiments demonstrating convergence, stability, and the angle’s robustness as a non‑Gaussianity indicator.
By Binshuai Wang, Peng Wei