发表机构
Technische Universität Ilmenau; Freie Universität Berlin; Chemnitz University of Technology（伊尔梅瑙工业大学; 柏林自由大学; 开姆尼茨工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出了条件期望算子与条件均值嵌入的可验证正则性准则，验证其在三类场景下的适用性，为相关领域提供了统一分析框架。
AI 中文摘要
条件期望算子（CEOs）及其相关的条件均值嵌入（CMEs）在应用数学和机器学习中处于核心地位，出现在非参数回归、贝叶斯逆问题和Koopman算子理论中。一个基本问题是，CEOs何时将$\boldsymbol{\textit{Y}}$上的函数空间映射到$\boldsymbol{\textit{X}}$上的指定函数空间，尤其是再生核希尔伯特空间（RKHS）。我们证明，此类映射特性由条件律的Radon–Nikodym密度的正则性刻画，并建立了一个简单、可验证的充分条件，在此条件下CEOs是有界的且为希尔伯特-施密特算子。对于与Sobolev空间范数等价的RKHS，该条件简化为条件密度的Sobolev正则性。该结果为验证CME表示以及基于Galerkin型和CME的估计量的误差界提供了直接途径。我们在三个场景中验证了该正则性条件：非参数回归、贝叶斯逆问题和随机动力系统的Koopman算子理论。我们证明，在每种场景下，潜在概率模型上的经典正则性结果都蕴含所需的映射特性。所得框架为概率、算子理论、核方法和随机动力学中的条件期望算子提供了统一视角。
英文摘要
Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon--Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert--Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.
Comments37 pages, 3 figures
