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...
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.
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.
arXiv:2610.28561v1 Announce Type: new Abstract: Stochastic gradient descent (SGD) is the cornerstone of modern deep learning, yet its theoretical convergence properties remain poorly understood in no...
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 $l_{\sigma}$.
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.
The paper presents a fully data-driven method for optimal control of nonlinear control‑affine systems modeled as stochastic diffusions. It embeds state probability densities into a reproducing kernel Hilbert space to learn Markov transition operators, which are then used in convex operator‑theoretic Hamilton‑Jacobi‑Bellman recursions that scale linearly with state dimensionality. Numerical experiments show the approach can solve diverse nonlinear control tasks, such as depth regulation of an autonomous underwater vehicle.
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.
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: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.
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.