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

What the math gives to ML

The paper provides a verifiable regularity mechanism for conditional expectation operators: regularity of the conditional density implies that the operator maps between prescribed function spaces and can be bounded or Hilbert–Schmidt. The transferable asset is a computable certificate for learned stochastic transition or regression operators, rather than an informal smoothness assumption. A practical neural implementation is to train a conditional model or neural Koopman operator together with a Sobolev/Jacobian penalty on its conditional density or conditional feature map, and to monitor the resulting Hilbert–Schmidt norm and singular-value tail. This gives a falsifiable prediction about finite-rank approximation error and detects when a learned operator leaves the regularity regime required for stable long-horizon use.