Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators
Maximiliano Hertel, Ilja Klebanov, Manuel Schaller, Karl Worthmann
math.DS, cs.LG, math.NA, math.ST, stat.ML
Submitted: 2026-08-06
Comments: 37 pages, 3 figures
License: http://creativecommons.org/licenses/by-nc-nd/4.0/
The gist: Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric
Terminology
Abstract
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 Y into a prescribed function space on 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.
Sources
- Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition
- Nonparametric Sparse Online Learning of the Koopman Operator
- Nonparametric approximation of conditional expectation operators
- Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem
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