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

arXiv: https://arxiv.org/abs/2608.19144 · HTML: https://mlsanity.com/paper/2608.19144/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 25 of its 40 references that are in the map.
- First posted on arXiv 16 Jun 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 25 of 25)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Poisson Flow Generative Models](https://mlsanity.com/paper/2209.11178/) (2022)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [PFGM++: Unlocking the Potential of Physics-Inspired Generative Models](https://mlsanity.com/paper/2302.04265/) (2023)
- [Theoretical guarantees in KL for Diffusion Flow Matching](https://mlsanity.com/paper/2409.08311/) (2024)
- [Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fitting](https://mlsanity.com/paper/2410.02601/) (2024)
- [Field Matching: an Electrostatic Paradigm to Generate and Transfer Data](https://mlsanity.com/paper/2502.02367/) (2025)
- [Interaction Field Matching: Overcoming Limitations of Electrostatic Models](https://mlsanity.com/paper/2506.02950/) (2025)
- [A Unified Framework for Diffusion Bridge Problems: Flow Matching and Schr"odinger Matching into One](https://mlsanity.com/paper/2503.21756/) (2025)
- [Latent Stochastic Interpolants](https://mlsanity.com/paper/2506.02276/) (2025)
- [Unlocking the Duality between Flow and Field Matching](https://mlsanity.com/paper/2602.02261/) (2026)
- [Foundations of Schr"odinger Bridges for Generative Modeling](https://mlsanity.com/paper/2603.18992/) (2026)

## Abstract (arXiv)

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, Schr"odinger bridges, and field-based generative models. We introduce the 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 network is trained. To make this bottleneck comparable across model families, we define Bridge Graphical Models (BGMs), which separate endpoint coupling, bridge law, Markovian projection, and current-preserving dynamics representation as independent design choices. The same formalism also represents Poisson and electrostatic models as field-line bridge kernels with a corresponding field-line Markovization gap. Across synthetic, latent, and pixel-space pilots on CIFAR-10 and Fashion-MNIST, a feature-space proxy gap estimated in minutes before training ranks design choices in the same direction as downstream training loss and FID under fixed architecture, bridge, sampler, and compute. These results support the Markovization gap as a pre-training diagnostic for bridge and coupling design.