Year: 2026
Type: preprint
Abstract: We consider the problem of learning nonlinear operators $G_0: \mathcal X\to \mathcal Y$ from finitely many noisy observations. While finite-regularity classes such as Lipschitz- or $C^k$-operators suffer from the curse of sample complexity (i.e. algebraic sample complexity is impossible), algebraic convergence rates can be obtained for holomorphic operators. In this paper we study information-theoretic minimax rates for estimating holomorphic operators. We provide algebraic upper and lower bounds of the metric entropy of holomorphy classes (matching up to an arbitrarily small loss in the exponent) as well as statistical minimax lower bounds in both the random-design and a Gaussian white noise model. For the mean squared error, our novel lower bounds match our upper bounds (up to an arbitrarily small loss in the exponent) and coincide with the convergence rates obtained previously for related holomorphy classes (arXiv:2412.17582). Finally, we apply the theory to the parameter-to-solution map and the parameter-to-quantity-of-interest map arising from the elliptic Darcy flow equation.
Source: arXiv (Cornell University)
Authors: Hoang, Ly Duc, Wang, Sven, Zech, Jakob
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Awards: G442688124, G987523819
OpenAlex ID: W7221027574