arXiv:2502. 11665v3 Announce Type: replace-cross Abstract: The classical kernel ridge regression problem aims to find the best fit for the output $Y$ as a function of the input data $X\in \mathbb{R}^d$, with a fixed choice of regularization term imposed by a given choice of a reproducing kernel Hilbert space, such as a Sobolev space.
arXiv:2510.02532v2 Announce Type: replace-cross Abstract: Deep neural networks excel in high-dimensional problems, outperforming models such as kernel methods, which suffer from the curse of dimensio...
The paper introduces a compositional variant of kernel ridge regression where the predictor reweights input coordinates, framing the approach as a variational problem to study feature learning in compositional architectures. It demonstrates that both global minimizers and stationary points can discard Gaussian noise variables while retaining relevant ones, and shows that α1-type kernels (e.g., Laplace) recover features contributing to nonlinear effects at stationary points, whereas Gaussian kernels recover only linear ones.
arXiv:2606. 08799v1 Announce Type: cross Abstract: We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual--curvature term.
The paper investigates diffusion models trained in a lazy high‑dimensional regime, extending benign overfitting theory to generative settings. By analyzing denoising score matching in a vector‑valued RKHS with an inner‑product kernel, the authors derive exact risk trajectories under gradient flow when the number of samples scales proportionally with dimensionality. These trajectories reveal three distinct phases—spectral generalization, noise‑dominated interpolation, and empirical Bayes memorization—whose interplay shapes the distribution of generated samples.
arXiv:2501. 10870v2 Announce Type: replace-cross Abstract: The principal objective of this work is twofold within nonparametric regression settings: (1) to establish the minimax optimal convergence rates for fixed-bandwidth Gaussian kernel spectral algorithms when the true regression function resides in a Sobolev space, and (2) to apply Gaussian spectral algorithms for achieving robust and adaptive transfer learning under concept shift.
arXiv:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.
arXiv:2406. 04425v2 Announce Type: replace Abstract: A fundamental problem in machine learning is understanding the effect of early stopping on the parameters obtained and the generalization capabilities of the model.
The paper introduces Total Sensitivity Kernels (TSKs), a weighted ANOVA kernel framework that learns the importance of individual inputs and their interactions for approximating a multivariable black-box function from limited data. By selecting an RKHS where the target function has minimum norm, the authors derive a unique solution and prove consistency for finite-data interpolation. Numerical experiments show that adapting the kernel to the learned multivariable structure can significantly improve approximation accuracy compared to a standard product kernel.
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:2607. 02003v1 Announce Type: cross Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics.
arXiv:2510. 12249v2 Announce Type: replace Abstract: In performative learning, the data distribution reacts to the deployed model - for example, because strategic users adapt their features to game it - which creates a more complex dynamic than in classical supervised learning.
arXiv:2607. 00257v1 Announce Type: new Abstract: Accurate prediction of complex dynamical systems from noisy measurements remains a significant challenge in scientific computing.