Bridge Graphical Models: Coupling, Projection, and Current-Preserving Dynamics for Generative Modeling

Continuous-time generative models are often built from endpoint-conditioned bridges, but generation requires a different object: a non-anticipative Markov decoder that only observes the current state and time. We identify this bridge-to-decoder compression as a structural bottleneck shared by diffusion models, flow matching, rectified flow, Schrödinger bridges, and field-based generative models. We introduce the \emph{Markovization gap}, the time-integrated conditional variance of the bridge velocity given the Markov state. It is the MMSE of predicting endpoint-conditioned motion from the information available to a sampler, and it measures an irreducible loss incurred before any neural ne...

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