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:2603. 07221v2 Announce Type: replace Abstract: Margin-based learning, exemplified by linear and kernel methods, is one of the few classical settings where generalization guarantees are independent of the number of parameters.
arXiv:2606. 15812v1 Announce Type: new Abstract: Constructing mathematically tractable function spaces that capture hierarchical compositional representations remains a central challenge in statistical learning theory.
arXiv:2606. 14954v1 Announce Type: cross Abstract: We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers.
arXiv:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.
arXiv:2509. 26371v3 Announce Type: replace-cross Abstract: Recently, there has been growing interest in characterizing the function spaces underlying neural networks.
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:2503. 24092v2 Announce Type: replace-cross Abstract: Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces.
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.
arXiv:2503. 18219v2 Announce Type: replace Abstract: This work studies the sampling complexity of learning with ReLU neural networks and neural operators.
arXiv:2608. 01357v1 Announce Type: new Abstract: Traditional approximation theory measures convergence rates in terms of the number of parameters or degrees of freedom.
arXiv:2509. 23544v2 Announce Type: replace-cross Abstract: Many modern applications involve predicting structured, non-Euclidean outputs such as probability distributions, networks, and symmetric positive-definite matrices.
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.