Bridge-matching unification of Schrödinger bridge solvers and flow rectification

Connection

Iterative Proportional Fitting, Iterative Markovian Fitting, and reflow are three superficially different iterative algorithms that all converge to the same object — the Schrödinger bridge (or its zero-noise limit, the optimal transport map) — yet they do so through different intermediate iterates. The non-obvious unifying insight is that bridge matching is the shared computational primitive underlying all three. Each algorithm differs only in (a) how the initial and subsequent couplings are constructed, and (b) whether noise is present (sigma > 0) or absent (sigma = 0).

Concretely: IPF (as formalized in DSB, Propositions 2-3) alternates fitting forward and backward drifts against the reference measure coupling (Brownian motion between the marginals). IMF (as formalized in DSBM, Definition 6 and Theorem 8) alternates Markovian projection and reciprocal projection, where bridge matching implements the Markovian projection step. The critical distinction: DSBM-IPF initializes from the reference coupling, while DSBM-IMF initializes from the independent coupling (pi_0 tensor pi_1). Both converge to the Schrödinger bridge, but their intermediate iterates differ (DSBM Proposition 9 vs Proposition 10).

The connection to rectified flow and reflow emerges in the zero-noise limit. When sigma tends to 0, the stochastic bridges collapse to linear interpolants X_t = (1-t)X_0 + tX_1, and bridge matching reduces to velocity-field regression — precisely the rectified flow procedure of Flow Straight and Fast (Section 3). One pass of this is rectified flow; iterating it (using the previous flow’s induced coupling as the new input) is reflow. Thus the full equivalence chain is: one DSBM-IMF iteration at sigma > 0 equals the Goria et al. algorithm; taking sigma to 0 gives rectified flow; iterating gives reflow.

Bridged Concepts

From Diffusion Schrödinger Bridge Matching

bridge matching: The computational primitive — regress the drift of forward bridges conditioned on sampled endpoints. Implements Markovian projection onto the SDE class (Proposition 5).
Iterative Markovian Fitting: The outer loop — alternates Markovian and reciprocal projections. DSBM-IMF (Proposition 10) initializes from the independent coupling, making its first iteration equivalent to the Goria et al. single-pass algorithm.
reciprocal class: The constraint set for reciprocal projection — processes sharing the same bridge kernels. IMF stays within the reciprocal class of the Schrödinger bridge (Theorem 8).

From Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling

Iterative Proportional Fitting: The classical algorithm — alternates forward/backward score matching against the reference coupling. First forward half-step recovers standard score-based generative modeling (DSB Proposition 3). Convergence guaranteed but iterates are non-Markovian in general.

From Flow Straight and Fast

rectified flow: Bridge matching at sigma = 0. Given any coupling, simulate linear interpolants and regress the velocity field. One pass straightens transport paths.
reflow: Iterated rectified flow. Each iteration uses the previous flow’s induced coupling, progressively straightening paths toward the OT map.

From Connecting Brownian and Poisson Random Bridges with Rectified Flows and Random-Bridges as Stochastic Transports for Generative Models

The Goria et al. algorithm (single-pass stochastic bridge transport) is formally one DSBM-IMF iteration: sample (X_0, X_1) from the independent coupling, simulate the forward bridge X_0 → X_1, then fit a reverse drift. The Appendix D sketch in Shelley & Mengütürk labels this as “IPF” but it is more precisely bridge matching under the independent coupling, i.e., the first IMF step.

Why It Matters

This unification resolves a persistent ambiguity in the diffusion bridge literature where “IPF” is used loosely to refer to any alternating-projection scheme. The distinction is not merely terminological: IPF and IMF produce different intermediate iterates despite sharing the same fixed point. IPF iterates are non-Markovian in general (they solve Schrödinger’s static problem at each step), while IMF iterates remain Markovian by construction. This means that practical implementations using bridge matching — including the Goria et al. algorithm and its Poisson bridge extension — are performing IMF, not IPF, even when authors describe them as “IPF-like”.

For the Shelley & Mengütürk framework specifically, correctly identifying the algorithm as IMF (not IPF) has practical consequences: it means the single-pass convergence guarantees come from DSBM Proposition 10 (IMF convergence), not from DSB Proposition 2 (IPF convergence). It also clarifies that extending to multiple iterations would follow the IMF trajectory, and that the zero-noise limit rigorously recovers the rectified flow results already established in the paper’s main body.

Potential Directions

Formalize the sigma-to-0 limit for the Poisson bridge case: The Brownian bridge to linear interpolant collapse is standard, but the analogous statement for Poisson bridges (where the bridge is a pure-jump process) requires showing that the Poisson bridge conditioned on X_1 concentrates on the deterministic path as the jump rate tends to infinity. This would complete the “Poisson IMF to Poisson reflow” equivalence.
Multi-iteration analysis: Run multiple DSBM-IMF iterations on the random bridge framework and measure whether the coupling quality improves beyond single-pass, particularly for the Poisson case where overshoot correction is needed.
Hybrid IMF with mixed bridge types: Use Brownian bridges (which can correct overshoots via continuous paths) for the backward/reciprocal projection step, while retaining Poisson bridges for the forward step. This directly addresses the overshoot problem identified earlier by leveraging the IMF alternation structure.

Evidence

Bridge matching = Markovian projection: DSBM Proposition 5 proves that minimizing the bridge matching objective is equivalent to computing the Markovian projection of the bridge measure mixture onto the SDE class. This is the key technical result grounding the entire equivalence.
DSBM-IMF iteration 1 = Goria et al.: DSBM Proposition 10 and Appendix A.2 show that starting from the independent coupling and performing one bridge matching step recovers the single-pass stochastic transport of Random-Bridges as Stochastic Transports for Generative Models.
DSBM-IPF != DSBM-IMF at intermediate steps: DSBM Proposition 9 (IPF) vs Proposition 10 (IMF) give different update rules despite sharing the Schrödinger bridge as the common fixed point.
IPF iterates are non-Markovian: DSB Section 3.3 and Appendix show that IPF half-steps involve fitting against the full path measure of the reference process, producing non-Markovian intermediates.
Rectified flow = sigma-to-0 bridge matching: Flow Straight and Fast Section 3 defines rectified flow as ODE velocity regression on linear interpolants. When sigma = 0, the Brownian bridge X_t | (X_0, X_1) collapses to (1-t)X_0 + tX_1, so bridge matching reduces to this velocity regression.
Appendix D sketch is IMF, not IPF: The sketch in Connecting Brownian and Poisson Random Bridges with Rectified Flows Appendix D constructs bridges from X_0 to X_1 sampled from the independent coupling pi_0 tensor pi_1, then fits a reverse. This is the DSBM-IMF initialization (independent coupling), not the IPF initialization (reference coupling). The sketch is structurally correct as a bridge-matching procedure but mislabelled as IPF.
IPMF extends to adversarial training: Recent work (arXiv 2410.02601) introduces Iterative Proportional Markovian Fitting, combining IPF’s marginal constraints with IMF’s Markovian structure through adversarial discriminators, confirming these are genuinely different algorithmic families that can be hybridized.

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