The paper presents time‑uniform self‑normalized concentration bounds for stochastic processes in Hilbert spaces with vector‑valued noise, enabling regression‑error guarantees for both linear and nonlinear parametric operators. These results apply to possibly infinite‑dimensional inputs and outputs without requiring independence or mixing assumptions, and are derived in the context of sequentially collected, dependent data such as adaptive experimental design and dynamical‑system modelling.
By Rafael Oliveira
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:2606. 06772v2 Announce Type: replace-cross Abstract: Characterizing the optimization dynamics and statistical performance of over-parameterized deep neural networks (DNNs) remains a central challenge in understanding the remarkable success of deep learning.
By Junyu Zhou, Puyu Wang, Dennis Wagner, Yunwen Lei, Marius Kloft, Yiming Ying
arXiv:2609. 11712v1 Announce Type: cross Abstract: In this paper, we investigate the generalization performance of distributed gradient descent algorithms in a reproducing kernel Hilbert space under a robust loss function $l_{\sigma}$.
By Jun-Yi Meng, Zheng-Chu Guo, Yuan Mao
arXiv:2603. 00819v2 Announce Type: replace-cross Abstract: This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory.
By Simone Brugiapaglia, Nicola Rares Franco, Nicholas H. Nelsen
arXiv:2603. 16481v3 Announce Type: replace Abstract: Non-conservative uncertainty bounds are essential for making reliable predictions about latent functions from noisy data, and thus, a key enabler for safe learning-based control.
By Amon Lahr, Anna Scampicchio, Johannes K\"ohler, Melanie N. Zeilinger
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...
By Tian-Yi Zhou, Namjoon Suh, Guang Cheng, Xiaoming Huo
arXiv:2312. 15341v1 Announce Type: cross Abstract: We provide an overview of recent progress in statistical inverse problems with random experimental design, covering both linear and nonlinear inverse problems.
By Abhishake Rastogi, Tapio Helin, Nicole M\"ucke
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
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.
By Lei Shi, Jia-Qi Yang, Ding-Xuan Zhou
arXiv:2503. 18219v2 Announce Type: replace Abstract: This work studies the sampling complexity of learning with ReLU neural networks and neural operators.
By Philipp Grohs, Samuel Lanthaler, Margaret Trautner
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.
By Hugo Latourelle-Vigeant, Sinho Chewi, Aram-Alexandre Pooladian, John Sous, Theodor Misiakiewicz