Conjectured Laplace-transform approach for principled conditioning on partial observations

Investigate whether extending the Laplace transform of the Koopman operator—previously used for kernel-based forecasting of deterministic dynamics—to the conditional mean embedding operators within hidden Markov models enables principled conditioning of maximum mean discrepancy flows on partial sequence observations.

Background

A central limitation identified by the authors is conditioning the generative process on partial observations in a principled manner. They discuss kernel Bayesian inference as one path and propose, as an alternative, leveraging an operator Laplace transform that has been used in Koopman-based forecasting for deterministic systems.

Because conditional mean embedding operators and Koopman operators are closely related, extending the Laplace-transform technique to the HMM-based spectral mean flows could provide a principled route to conditional generation—if the conjecture is borne out.