Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators

2026-08-06

AI Summary

This work investigates when the conditional expectation operator (CEO) defines a bounded and Hilbert–Schmidt map from a function space into a reproducing kernel Hilbert space (RKHS). By analyzing the regularity of the Radon–Nikodym density of the conditional distribution, the authors establish a unified and verifiable sufficient condition that integrates probabilistic regularity, operator theory, and kernel methods. The criterion is validated across three distinct settings—nonparametric regression, Bayesian inverse problems, and Koopman operator theory—providing a direct pathway to verify the well-definedness and error bounds of conditional mean embeddings. This result establishes a cohesive theoretical framework applicable across diverse domains.

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 $\mathcal{Y}$ 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.

Problem

• Conditional Expectation Operators
• Conditional Mean Embeddings
• Reproducing Kernel Hilbert Space
• Regularity Criterion
• Sobolev Regularity

Innovation

• Conditional Expectation Operators
• Conditional Mean Embeddings
• Reproducing Kernel Hilbert Space
• Sobolev Regularity
• Radon–Nikodym Density