Minimax Operator Learning for Holomorphic Maps between Hilbert Spaces
Abstract
We consider the problem of learning nonlinear operators G0: 𝒳→ 𝒴 from finitely many noisy observations. While finite-regularity classes such as Lipschitz- or Ck-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.
Related
- Norm of the generalized Hilbert operator on weighted Bergman spaces
- Holomorphic sectional curvature and the Calabi operator on Kähler C-spaces
- Norms of the Hilbert Matrix Operator on Some Analytic Function Spaces
- The Differential Hilbert Operator Between Weighted Bergman Spaces
- On the Spectra of Multiplicative Hilbert Operator on Bergman Spaces of Dirichlet Series
BibTeX
@misc{hoang2026minimax,
title = {{Minimax Operator Learning for Holomorphic Maps between Hilbert Spaces}},
author = {Ly Duc Hoang and Sven Wang and Jakob Zech},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07295},
eprint = {2610.07295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07295},
}
RIS
TY - UNPB
TI - Minimax Operator Learning for Holomorphic Maps between Hilbert Spaces
AU - Ly Duc Hoang
AU - Sven Wang
AU - Jakob Zech
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07295
UR - https://arxiv.org/abs/2610.07295
AB - We consider the problem of learning nonlinear operators G_0: 𝒳→ 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.
ER -