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:2607. 07468v1 Announce Type: cross Abstract: We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning.
By Abhishake Rastogi, Tatiana A. Bubba, Tapio Helin, Luca Ratti
arXiv:2602. 13906v2 Announce Type: replace-cross Abstract: Stochastic approximation (SA) is a method for finding the root of an operator perturbed by noise.
By Shaan Ul Haque, Zedong Wang, Zixuan Zhang, Siva Theja Maguluri
The paper proves global universal approximation theorems for non‑anticipative and general path‑dependent functionals on spaces of piecewise linear paths, showing that linear functionals of the corresponding signatures are dense in $L^p$ and weighted norms. It demonstrates that these results apply to piecewise linear interpolations of Gaussian processes, including Brownian motion and fractional Brownian motion with Hurst parameter $H>1/4$, and provides quantitative convergence rates for signatures of such approximations toward their continuous‑time counterparts. Consequently, the authors obtain $L^p$‑approximation results for path‑dependent functionals of Gaussian processes and for random ordinary and stochastic differential equations driven by Brownian motion.
By Mihriban Ceylan, David J. Pr\"omel
arXiv:2608. 06155v1 Announce Type: cross Abstract: Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory.
By Maximiliano Hertel, Ilja Klebanov, Manuel Schaller, Karl Worthmann
We develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces. Under mild conditions on the loss we establish existence and measurability of the estimator, covering a wide range of convex and non-convex losses, including bounded robust losses.
arXiv:2512. 16396v2 Announce Type: replace-cross Abstract: We establish $L^p$-universal approximation theorems for general path-dependent and non-anticipative functionals on suitable rough path spaces, showing that linear functionals acting on signatures of time-extended rough paths are dense with respect to the $L^p$-distance.
By Mihriban Ceylan, David J. Pr\"omel
arXiv:2507. 07008v2 Announce Type: replace Abstract: Used as priors for Bayesian inverse problems, diffusion models have recently attracted considerable attention in the literature.
By Emile Pierret, Bruno Galerne
The paper investigates how many linear samples are needed to learn Lipschitz operators under Gaussian measures. It establishes both lower and upper bounds on the Hermite polynomial approximation error and shows that the minimal worst‑case error cannot converge algebraically with the number of samples. However, if the covariance operator of the Gaussian measure decays rapidly, convergence rates arbitrarily close to any algebraic rate can be achieved.
By Ben Adcock, Michael Griebel, Gregor Maier
arXiv:2606. 07931v1 Announce Type: cross Abstract: We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes.
By Yunbei Xu
arXiv:2609.17802v1 Announce Type: cross
Abstract: Measures on function spaces arise throughout Bayesian inverse problems and generative modeling, often with low-dimensional structure relative to a tr...
By Ricardo Baptista, Bamdad Hosseini, Alexander W. Hsu
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