diffusion bridge

Diffusion bridges are a class of generative models that learn to transport between two arbitrary distributions, generalizing standard diffusion models which transport between data and noise. They are closely related to Schrödinger bridge problems in optimal transport and encompass both stochastic (Lévy random bridge) and deterministic (rectified flow) transport mechanisms.

Papers Analyzed

1. Connecting Brownian and Poisson Random Bridges with Rectified Flows (Shelley & Mengütürk, 2025) — Unifies Lévy random bridges with rectified flow via the non-anticipative semimartingale representation, showing rectified flow is the zero-volatility limit of the Brownian random bridge
2. Random-Bridges as Stochastic Transports for Generative Models (Goria et al., 2025) — Foundation paper defining (Φ,Ψ)-bridges; demonstrates dramatic low-step sampling efficiency
3. Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling (De Bortoli et al., 2021) — Reformulates SGM as first IPF iteration of a Schrödinger bridge problem; introduces DSB algorithm
4. Flow Straight and Fast (Liu et al., 2022) — Introduces rectified flow with provably non-increasing transport costs and reflow for one-step generation
5. Improved Denoising Diffusion Probabilistic Models (Nichol & Dhariwal, 2021) — learned reverse process variance and cosine noise schedule for competitive log-likelihoods and fast sampling
6. Score-Based Generative Modeling through Stochastic Differential Equations (Song et al., 2021) — Unified SDE framework with VE-SDE and VP-SDE, probability flow ODE, and predictor-corrector sampling
7. Diffusion Schrödinger Bridge Matching (Shi et al., 2023) — Iterative Markovian Fitting and DSBM algorithm unifying DDMs, Flow Matching, Rectified Flow, and DSB as special cases
8. Tweedie’s Formula and Selection Bias (Efron, 2011) — Foundational treatment of generalized Tweedie’s formula as exponential family posterior moments; connects score estimation to James-Stein estimation and empirical Bayes information
9. On the Relation Between Optimal Transport and Schrödinger Bridges (Chen, Georgiou & Pavon, 2014) — Unifies OT and Schrödinger bridges via stochastic control; the SB fluid dynamic formulation differs from Benamou-Brenier formula by a Fisher information term; introduces optimal transport with prior as the zero-noise limit of SB
10. A Survey of the Schrödinger Problem and Some of Its Connections with Optimal Transport (Léonard, 2014) — Definitive survey establishing the SB–OT connection: the (f,g)-transform characterizes SB solutions, Schrödinger potentials are second-order Kantorovich potentials, and the Γ-convergence of Schrödinger to Monge-Kantorovich proves OT is the zero-noise limit
11. Flow Matching for Generative Modeling (Lipman et al., 2022) — Introduces conditional flow matching (CFM): simulation-free CNF training by regressing conditional velocity fields. OT displacement interpolation paths yield straight trajectories; subsumes diffusion paths as special cases. 3676 citations.
12. Stochastic Interpolants A Unifying Framework for Flows and Diffusions (Albergo, Boffi, Vanden-Eijnden, 2023) — stochastic interpolant $x_t=I(t,x_0,x_1)+\gamma (t)z$ unifies flows and diffusions; tunable noise $\gamma (t)$ smooths intermediate densities; recovers Schrödinger bridge when optimizing over $I$ (Theorem 41); explicitly confirms rectified flow ≠ OT (Remark 48)
13. Generalized Schrödinger Bridge Matching (Liu et al., 2024) — Extends bridge matching / DSBM to the generalized Schrödinger bridge with task-specific state costs $V_t$ via conditional stochastic optimal control; unifies rectified flow, DSBM, and flow matching as special cases ($V_t=0$); Gaussian path approximation with splines + optional path integral control debiasing; achieves stable convergence and scales to d > 12K

Textbooks

• Optimal Transport Old and New — Brenier’s theorem, Kantorovich duality, Wasserstein distances, displacement interpolation (straight OT paths), Benamou-Brenier formula, non-crossing property (Ch. 1, 5, 6, 7)

Key Concepts and Connections

The field reveals a deep mathematical unity across seemingly different approaches:

Bridge-Flow Duality: The conditional drift of a Lévy random bridge, E[Y|ξ_t]-ξ_t)/(T-t), has the identical form to the optimal rectified flow velocity. Moreover, E[ξ_t^{x,y}] = R_t^{x,y} for any Lévy driver — the bridge expectation equals the rectified flow path. Setting σ=0 collapses the stochastic bridge to deterministic flow.

Hierarchy of Methods: Standard diffusion models (score matching + reverse-time SDE) are the first iteration of Iterative Proportional Fitting for the Schrödinger bridge problem. Additional iterations (DSB) improve transport quality. Iterative Markovian Fitting (DSBM) provides a principled alternative that preserves both marginals throughout. GSBM extends this hierarchy further: OT $\subset$ SB $\subset$ GSB, incorporating task-specific state costs $V_t$ while maintaining the same matching-based training.

Stochastic vs Deterministic: The probability flow ODE provides a deterministic counterpart to any SDE-based process. For Gaussian bridges, this ODE averages source-push and target-pull velocities. Rectified flow is the zero-noise limit. Reflow progressively straightens paths for few-step generation.

Score-Bridge Connection: The Brownian random bridge drift decomposes as (ξ_t-x)/t + σ²∇log p_t — source-push plus score — with Nelson’s osmotic velocity explaining the forward/backward drift discrepancy. The generalized Tweedie’s formula extends this to Poisson random bridges. Efron (2011) establishes the deeper principle: the score ∇log f(z) provides the optimal Bayesian denoising correction (E{μ|z} = z + σ²l’(z)), and selection bias correction in the tails is mathematically equivalent to the shrinkage performed by bridge drifts pulling noisy states toward the data manifold.

Stochastic Control Foundation: Chen, Georgiou & Pavon (2014) provides the control-theoretic underpinning: the optimal SB drift is b_+^{Q*} = b_+^P + ∇ln φ, which is exactly the score correction that the generalized Tweedie’s formula in Appendix C of Shelley & Mengütürk (2025) computes algebraically for exponential family bridges. The fluid dynamic formulation reveals that OT and SB differ by a Fisher information penalty — the kinetic energy of Nelson’s osmotic velocity — and optimal transport with prior is recovered as the zero-noise limit.

SB–OT Structural Parallel: Léonard (2014) makes precise the deep structural analogy between the Schrödinger bridge and optimal transport. The SB solution is an (f,g)-transform $\hat{P}=f_0(X_0)g_1(X_1)R$, whose Schrödinger potentials $\phi =\log f$, $\psi =\log g$ solve viscous (second-order) Hamilton-Jacobi equations — the regularized counterparts of the first-order HJ equations for Kantorovich potentials. The entropic interpolation (time-marginal flow of the SB) is positive and smooth, converging to the displacement interpolation (Wasserstein geodesic) as noise vanishes. The Γ-convergence of Schrödinger to Monge-Kantorovich proves this rigorously: slowing down the reference process recovers OT as a limit. This places all SB-based generative methods (DSB, DSBM, bridge matching) as entropic regularizations of OT, with the regularization providing smoothness, uniqueness, and tractability.

Open Questions

• Can the random bridge framework (with its low-step efficiency) scale to high-resolution generation while maintaining quality at 1000+ steps?
• What are the optimal driving processes beyond Brownian and Poisson for specific generative tasks?
• How does the bridge framework connect to consistency models and distillation approaches?
• Can jump-diffusion bridges (σW_t + P_t) improve generation of discrete or mixed-type data?

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