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

About this paper

Maximiliano Hertel, Ilja Klebanov, Manuel Schaller, Karl Worthmann

A verifiable criterion for the regularity properties of Conditional Expectation Operators (CEOs) and Conditional Mean Embeddings (CMEs) is established. This criterion links the functional-analytic properties of these operators directly to the analytical regularity of their underlying conditional probability densities, providing quantitative error bounds for data-driven approximations and validating key assumptions in various applications.

Key insights

• The functional-analytic properties of Conditional Expectation Operators, such as boundedness, compactness, and the Hilbert-Schmidt property, are directly characterized by the regularity of the underlying conditional probability density.
• The abstract 'representability assumption' in Conditional Mean Embedding theory can be translated into concrete, verifiable conditions based on the analytical regularity of conditional densities.
• The established regularity framework enables the derivation of explicit quantitative error bounds for various data-driven approximation schemes, linking performance directly to density smoothness and function space properties.

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