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Bridge Graphical Models: Coupling, Projection, and Current-Preserving Dynamics for Generative Modeling
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Introduces the Markovization gap, a pre-training MMSE diagnostic of how much endpoint-conditioned bridge velocity is lost when compressed into a Markov decoder, within a Bridge Graphical Model framework, and shows a minutes-scale feature-space proxy ranks coupling choices consistently with downstream loss and FID on toy, latent, and CIFAR-10/Fashion-MNIST pilots.
Problem
Continuous-time generative models compress endpoint-conditioned bridges into non-anticipative Markov velocity fields, but there is no diagnostic for how much information this compression discards before training.
Approach
Define the Markovization gap as the time-integrated conditional variance of the bridge velocity given the Markov state (the irreducible MMSE floor of flow matching), and unify diffusion, flow matching, Schrödinger bridges and field methods as Bridge Graphical Models whose coupling, bridge, projection and dynamics choices are separable and comparable via this gap.
Key results
- On the 2x2 CIFAR-10 proxy table, minibatch OT lowers the proxy gap from 3.25 to 2.34 (straight bridge) versus independent coupling, and the same ordering holds for Brownian bridges.
- In the CIFAR-10 pixel pilot, minibatch OT reduces FID from 50.48 to 49.17 and training loss from 0.182 to 0.168; on Fashion-MNIST FID drops from 30.14 to 21.96.
- In the latent experiments, minibatch OT roughly halves the estimated gap and reduces neural eval MSE by about 67%-69% and terminal MMD^2 by about 45%-51%.
Surprising findings
- A perfectly Gaussian independent straight bridge still has a strictly positive irreducible gap (pi*d/2 for unit variances), so the floor is not an architecture effect.
- On CIFAR-10 the KID and Inception Score differences between couplings are inconclusive at pilot scale.
Limitations
- The feature-space kNN proxy can oversmooth the conditional mean and partly measure marginal velocity variance or transport cost; it is not a certified lower bound.
- Image experiments use 20k-step pilots, limited seeds and modest architectures; rank agreement only, not a statistically meaningful correlation across scales.
- Brownian bridge results are diagnostic-only with no downstream image training.
Experimental setup
Datasets: CIFAR-10 Fashion-MNIST scikit-learn handwritten digits LFW face subset (scikit-image) synthetic eight-Gaussian ring
Benchmarks: CIFAR-10 Fashion-MNIST scikit-learn digits latent LFW latent
Metrics: Markovization gap proxy training loss FID KID Inception Score neural eval MSE terminal MMD^2 1NN accuracy
Baselines: independent coupling minibatch OT
Builds on: flow matching rectified flow diffusion flow matching generator matching stochastic interpolants
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