发表机构
Humboldt University of Berlin; École Polytechnique Fédérale de Lausanne; Heidelberg University（柏林洪堡大学; 洛桑联邦理工学院; 海德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究希尔伯特空间间全纯算子学习的极小极大收敛速率，给出度量熵与统计下界，并应用于Darcy流方程的参数映射。
AI 中文摘要
我们考虑从有限个含噪观测中学习非线性算子 $G_0: \mathcal X\to \mathcal Y$ 的问题。虽然有限正则性类别（如Lipschitz算子或$C^k$算子）受制于样本复杂度的维数灾难（即代数样本复杂度不可能实现），但对于全纯算子可以获得代数收敛速率。本文研究估计全纯算子的信息论极小极大速率。我们提供了全纯类度量熵的代数上界和下界（在指数上匹配至任意小的损失），以及在随机设计和高斯白噪声模型下的统计极小极大下界。对于均方误差，我们新的下界与上界匹配（在指数上至任意小的损失），并与先前在相关全纯类上获得的收敛速率一致（arXiv:2412.17582）。最后，我们将该理论应用于椭圆Darcy流方程的参数到解映射和参数到感兴趣量映射。
英文摘要
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.
Comments36 pages, no figures
