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.
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.
The paper studies random projections (RP) for spectral algorithms, providing a comprehensive analysis of their generalization performance under broad conditions. By exploiting the RKHS embedding property and operator similarity, it establishes optimal learning rates in Sobolev norms that match minimax lower bounds up to logarithmic factors. For common RP implementations—randomized sketches and Nyström sub-sampling—the required projection dimension scales with the effective dimension, reducing computational cost while preserving statistical efficiency.
arXiv:2410. 11116v4 Announce Type: replace-cross Abstract: In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces.
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...
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.
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.
The paper studies operator learning on function spaces using encoder–decoder architectures. It shows that as input and output resolutions grow, the induced kernels converge to a limiting kernel, enabling regularity assumptions independent of resolution. The authors derive upper and lower bounds for regularized stochastic gradient descent, extend the analysis to neural networks via the limiting neural tangent kernel, and provide error bounds and complexity guarantees for various kernel and encoding constructions.
arXiv:2609. 18901v1 Announce Type: cross Abstract: In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+\xi_j$ or by a known physical constraint $Du^*=v$.
arXiv:2403.12187v2 Announce Type: replace-cross Abstract: Motivated by the abundance of functional data, such as time series and images, we study the approximation and statistical learning of nonline...
arXiv:2607. 20119v1 Announce Type: cross Abstract: We introduce the Directional Kernel Mean Difference (DKMD), a signed statistic for univariate distribution comparison that preserves the direction of distributional shifts.
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...