arXiv:1812.09632v3 Announce Type: replace
Abstract: We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributi...
By Bal\'azs Csan\'ad Cs\'aji, Kriszti\'an Bal\'azs Kis
arXiv:2510. 25599v2 Announce Type: replace Abstract: Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classification-focused.
By Christopher B\"ulte, Yusuf Sale, Gitta Kutyniok, Eyke H\"ullermeier
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:2504.18184v5 Announce Type: replace
Abstract: We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert sp...
By Jia-Qi Yang, Lei Shi
arXiv:2402.04691v5 Announce Type: replace-cross
Abstract: This study investigates the use of stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We study weak and str...
By Lei Shi, Jia-Qi Yang
The paper tackles the challenge of predicting multiple high‑dimensional physical fields that must satisfy linear equality constraints, a common scenario in physics‑informed machine learning. It critiques the conventional approach of deducing one field from others, showing its sensitivity to arbitrary choices and its impact on accuracy and uncertainty. To address this, the authors introduce a symmetric framework that first applies a row‑wise PCA to preserve constraints in a latent space, then trains a linearly‑constrained multi‑output Gaussian process using a specially parametrized kernel, and validate the method on population dynamics and CFD problems involving Reynolds stress tensors.
By Mahamat Hamdan Nassouradine, Cl\'ement Gauchy, Pierre-Emmanuel Angeli, S\'ebastien da Veiga