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Reflected Schrödinger bridges can be trained like flow matching, keeping samples inside the data domain at almost no extra cost.

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2026-07-12 01:05 UTC pith:GPHL3XT6

load-bearing objection Clean, usable extension of α-IMF to reflected Brownian bridges on the cube; the new sampler and score make it practical with almost no overhead.

arxiv 2607.03626 v1 pith:GPHL3XT6 submitted 2026-07-03 cs.LG stat.ML

Reflected Schr\"odinger Bridge Matching

Marcus H\"aggbom , Viktor Nilsson , Pierre Nyquist , Joakim and\'en This is my paper
classification cs.LG stat.ML
keywords Schrödinger bridgereflected Brownian motioniterative Markovian fittingflow matchingconstrained generative modelingimage-to-image translationDoob h-transform
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Generative models that transport one high-dimensional distribution to another (for example, one image dataset to another) usually allow paths to leave the natural domain of the data, such as the unit hypercube of pixel values. Reflecting the dynamics at the boundary guarantees that every sample stays inside that domain, but earlier methods for learning such reflected Schrödinger bridges required expensive full-path simulation and higher-order derivatives. This paper shows that the same efficient, partially simulation-free training used for ordinary Schrödinger bridges and flow matching still works once the reference process is a reflected Brownian motion on the unit cube. The authors supply a closed-form regression target for the score of the reflected transition density and an exact midpoint sampler for the reflected bridge. On MNIST–EMNIST and AFHQ cat–wild pairs the reflected model matches or slightly improves FID and pixel-wise fidelity while adding negligible wall-clock time.

Core claim

The iterative Markovian fitting procedure that learns ordinary Schrödinger bridges extends verbatim to reflected Brownian motion on the unit hypercube: the Markovian projection preserves all intermediate marginals, the associated least-squares objective is simply regression against the score of the reflected transition density, and the resulting algorithm (α-RSBM) converges to the reflected Schrödinger bridge while remaining almost as cheap as its non-reflected counterpart.

What carries the argument

The reflected Markovian projection (Proposition 4.1) together with the image-sum formula for the reflected Brownian transition density and the exact midpoint bridge sampler (Algorithm 2). These three objects turn the abstract reflected Schrödinger bridge into an ordinary least-squares regression problem that can be trained online exactly as in α-DSBM.

Load-bearing premise

The reference process must be pure reflected Brownian motion (zero drift) confined to the unit hypercube, and the infinite image-sum that defines its transition density is truncated after a fixed number of reflections whose accuracy is not proved for high-dimensional image data.

What would settle it

Replace the truncated image-sum score by an exact eigenfunction expansion (or a higher truncation) on the same image pairs; if FID, MSD or sample quality change materially, the truncation assumption is false for the reported regime.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper extends iterative Markovian fitting (IMF / α-IMF) to Schrödinger bridges whose reference process is reflected Brownian motion on the unit hypercube. It derives the corresponding Markovian projection (Proposition 4.1) that preserves all intermediate marginals, supplies an exact bridge midpoint sampler (Algorithm 2 + Proposition 4.2), and obtains a component-wise closed-form score via the image-sum factorization of the reflected transition density. The resulting α-RSBM algorithm is trained by ordinary least-squares regression against this score, exactly as in flow matching. Experiments couple MNIST–EMNIST and AFHQ cat–wild and report that reflection adds negligible wall-clock cost while matching or slightly improving FID and MSD, with the built-in guarantee that samples remain inside the data domain.

Significance. If the claims hold, the work supplies a practical, partially simulation-free route to high-dimensional reflected Schrödinger bridges that previously required expensive forward–backward SDE training. The theoretical core (Girsanov treatment of the Doob h-transform under reflection, Fokker–Planck preservation for mixtures of reflected bridges) is carefully adapted from the unconstrained setting and is stated with explicit regularity assumptions. The released code, the exact bridge sampler, and the transparent limitation to zero-drift RBM on the cube are genuine strengths that make the contribution immediately usable and falsifiable. The method therefore fills a concrete gap between efficient SB matching and constrained generative modeling.

minor comments (5)
  1. Table 1 reports single-run FID/MSD/LPIPS without error bars or multiple random seeds; a short note on variability (or a second independent run) would strengthen the claim of “maintaining or slightly improving” performance.
  2. Section 4.2 and Appendix B fix the image-sum truncation at 11 reflections per coordinate without a quantitative error bound; a one-sentence remark on the observed magnitude of the neglected terms (or a small ablation) would make the heuristic fully transparent.
  3. In Algorithm 1 the pre-train / fine-tune split and the precise role of the EMA are described only in the experimental appendix; a brief forward reference in the main text would improve readability.
  4. Figure 2 caption and the surrounding text use “clipped α-DSBM” without stating the clipping rule; a parenthetical “(project onto [0,1]^d)” would remove any ambiguity.
  5. Typographical: “IMF2” footnote on page 3 and the occasional missing space before citations (e.g., “Shi et al., 2023, Proposition 2”) should be cleaned for the camera-ready version.

Circularity Check

0 steps flagged

No significant circularity: Markovian projection, bridge sampler and score are derived from Girsanov/reflection maps without self-definitional or fitted-as-prediction steps.

full rationale

The paper's central theoretical claims (Proposition 4.1 on marginal preservation and KL minimality of the reflected Markovian projection; Proposition 4.2 on exactness of the reflected Brownian bridge sampler; the component-wise image-sum score (15)–(16)) are obtained by standard Girsanov change-of-measure arguments adapted to reflection (Appendix A) and by the elementary reflection map R of the unit cube. No quantity is fitted to data and then re-presented as a first-principles prediction; the regression target is the closed-form (truncated) score of the reference RBM, not a quantity derived from the learned network. Self-citations to Shi et al. (2023), Peluchetti (2023) and De Bortoli et al. (2024) supply the non-reflected IMF/α-IMF scaffolding that is then extended; they are not uniqueness theorems that force the present construction, nor are they used to smuggle an ansatz that already encodes the claimed result. Empirical FID/MSD/LPIPS numbers are independent external metrics. The acknowledged limitations (zero-drift reference, unit-cube domain, fixed truncation of 11 reflections) are stated as such and do not create circularity. Consequently the derivation chain is self-contained against its own inputs and scores 0.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central algorithmic claim rests on classical existence/uniqueness of reflected SDEs, Girsanov under finite entropy, and the explicit reflection map on the cube; the only free numerical choices that affect reported metrics are the diffusion scale σ, the truncation depth of the image sum, and standard optimizer hyper-parameters. No new physical entities are postulated.

free parameters (3)
  • diffusion scale σ = 1.0 / 0.75
    Chosen by hand (1.0 for MNIST, 0.75 for AFHQ); controls the strength of the entropic regularisation and therefore the FID/MSD trade-off.
  • image-sum truncation depth = 11
    Fixed at 11 reflections per coordinate; the infinite sum for the reflected transition density is replaced by a finite approximation whose error is not quantified for d≈10^4.
  • N_pretrain / N_finetune = 50k-100k / 75k-20k
    Training schedule lengths (50k/100k for MNIST, 75k/20k for AFHQ) chosen to match the baseline of De Bortoli et al.; affect final FID.
axioms (3)
  • standard math Weak existence and uniqueness of solutions to the reflected SDE (8) under the stated coefficients.
    Invoked throughout Section 4 and Appendix A; standard for Lipschitz coefficients on a convex domain.
  • domain assumption dΠ/dQ^r is sufficiently regular for the Doob h-transform and Girsanov change of measure to be well-defined (C^{1,2} density).
    Explicitly listed as a hypothesis of Proposition 4.1 and Proposition A.1.
  • ad hoc to paper The unit hypercube reflection map R and the factorisation of the transition density into independent coordinates remain valid after truncation.
    Used to obtain the practical score formula (16) and the bridge sampler; truncation error is not controlled.

pith-pipeline@v1.1.0-grok45 · 22097 in / 2340 out tokens · 25002 ms · 2026-07-12T01:05:07.357615+00:00 · methodology

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read the original abstract

Recent advances in generative modeling have enabled the efficient computation of Schr\"odinger bridges (SB) in high-dimensional settings by leveraging partially simulation-free training methods inspired by flow matching. However, these have not covered SBs with reflecting dynamics, a useful model choice with built-in guarantees that generated samples stay in the data domain. Existing alternatives for reflected SBs instead rely on more complex training based on forward--backward SDE theory, requiring expensive higher-order derivatives and sampling entire paths during training. In this article, we introduce a partially simulation-free framework that allows reflected SBs to be trained similarly to flow matching, using a new sampling method and regression target. We demonstrate our results by coupling pairs of well-known high-dimensional image datasets. Using reflected dynamics incurs negligible additional wall-clock time during both training and inference while maintaining or slightly improving generative performance.

Figures

Figures reproduced from arXiv: 2607.03626 by Joakim And\'en, Marcus H\"aggbom, Pierre Nyquist, Viktor Nilsson.

Figure 1
Figure 1. Figure 1: A reflected Schrödinger bridge between two 2D distributions resembling Erwin Schrödinger [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Transports between classes in AFHQ (64 × 64) and MNIST/EMNIST. For each direction, the first column contains the initial (true) sample, followed by generated samples from α-DSBM, clipped α-DSBM and α-RSBM. The out-of-bounds pixels in the α-DSBM are colored magenta. The two rows correspond to two different samples picked at random from the test data. h-transform term of the process, and marginally sampling … view at source ↗
Figure 3
Figure 3. Figure 3: MNIST-to-EMNIST transfer, where (c) and (d) contains the 16 samples for which [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: AFHQ (64 × 64) transfer, where (c) and (d) contains the four samples for which α-DSBM produced highest out-of-bounds pixel count. Out-of-bounds pixels are colored in magenta. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

18 extracted references · 1 canonical work pages

  1. [1]

    Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797,

    Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797,

    Pith/arXiv arXiv

  2. [2]

    Score matching for bridges without learning time-reversals.arXiv preprint arXiv:2407.15455,

    Elizabeth L Baker, Moritz Schauer, and Stefan Sommer. Score matching for bridges without learning time-reversals.arXiv preprint arXiv:2407.15455,

    Pith/arXiv arXiv

  3. [3]

    The pedestrian’s guide to local time.arXiv preprint arXiv:1512.08912,

    Tomas Björk. The pedestrian’s guide to local time.arXiv preprint arXiv:1512.08912,

    Pith/arXiv arXiv

  4. [4]

    Jonathan Ho and Tim Salimans

    doi: 10.1093/biomet/asaf048. Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance.arXiv preprint arXiv:2207.12598,

    Pith/arXiv arXiv doi:10.1093/biomet/asaf048

  5. [5]

    Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

    Pith/arXiv arXiv

  6. [6]

    Girsanov theory under a finite entropy condition

    11 Christian Léonard. Girsanov theory under a finite entropy condition. InSéminaire de Probabilités XLIV, pages 429–465. Springer, 2012a. Christian Léonard. From the Schrödinger problem to the Monge–Kantorovich problem.Journal of Functional Analysis, 262(4):1879–1920, 2012b. Christian Léonard. A survey of the Schrödinger problem and some of its connection...

    1920

  7. [7]

    Mirror diffusion models for constrained and watermarked generation.Advances in Neural Information Processing Systems, 36: 42898–42917, 2023a

    Guan-Horng Liu, Tianrong Chen, Evangelos Theodorou, and Molei Tao. Mirror diffusion models for constrained and watermarked generation.Advances in Neural Information Processing Systems, 36: 42898–42917, 2023a. Qiang Liu. Rectified flow: A marginal preserving approach to optimal transport.arXiv preprint arXiv:2209.14577,

    Pith/arXiv arXiv

  8. [8]

    Large deviations for scaled families of Schrödinger bridges with reflection.arXiv preprint arXiv:2506.03999,

    Viktor Nilsson and Pierre Nyquist. Large deviations for scaled families of Schrödinger bridges with reflection.arXiv preprint arXiv:2506.03999,

    Pith/arXiv arXiv

  9. [9]

    doi: 10.1007/978-1-4614-7687-0

    ISBN 9781461476870. doi: 10.1007/978-1-4614-7687-0. Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion Schrödinger bridge matching.Advances in Neural Information Processing Systems, 36:62183–62223,

    work page doi:10.1007/978-1-4614-7687-0

  10. [10]

    The proof is essentially a straightforward extension of that of Shi et al

    13 A Theory and proofs In this appendix, we provide Proposition 4.1. The proof is essentially a straightforward extension of that of Shi et al. (2023), but with a difference in treatment of the Doob h-transform. We drop the superscript of the reflected path measures Qr and the Markov class Mr from the main text. Recall the reflected SDE (8): dXt =µ(t, X t...

    2023

  11. [11]

    A typical example is that of the Brownian bridge

    in a certain point. A typical example is that of the Brownian bridge. More generally, our aim is to, conditioned on F0, reweight the terminal distribution of Q so that it becomes Π1|0 while preserving the bridge distribution Q|0,1. Shi et al. (2023) achieve this via the infinitesimal generator, identifying the change in drift from the generator of the pro...

    2023

  12. [12]

    = 1 + Z t 0 Aφs|0(Xs |x 0)ds+ Z t 0 ∇φs|0(Xs |x 0)Tσ(s, Xs)dWs + Z t 0 ∇φs|0(Xs |x 0)Tn(Xs)dLs. Rearranging the terms as φt|0(Xt |x 0)−1− Z t 0 ∇φs|0(Xs |x 0)Tσ(s, Xs)dWs = Z t 0 Aφs|0(Xs |x 0)ds+ Z t 0 ∇φs|0(Xs |x 0)Tn(Xs)dLs, since the terms on the left-hand side are continuous martingales, the right-hand side is also a continuous martingale. But this h...

    2016

  13. [13]

    Finally, the computation in Shi et al

    dt+σ(t, X t)dfWt +n(X t)dLt. Finally, the computation in Shi et al. (2023, Proof of Prop

    2023

  14. [14]

    A.2 Mixture of diffusion bridges The last ingredient needed to prove Proposition 4.1 is a means of getting the dynamics of the Π0- mixture of diffusion bridges Π|0 given by (18)

    =E Π1|0,t[∇logQ 1|t(X1 |X t)|X 0 =x 0, Xt =x t] follows analogously in the reflected setting, resulting in (18). A.2 Mixture of diffusion bridges The last ingredient needed to prove Proposition 4.1 is a means of getting the dynamics of the Π0- mixture of diffusion bridges Π|0 given by (18). This is given by an extension of Peluchetti (2023, Theorem

    2023

  15. [15]

    For completeness, the theorem is restated here in our notation before we provide the extension to our setting

    to reflected SDEs. For completeness, the theorem is restated here in our notation before we provide the extension to our setting. Theorem A.2(Peluchetti (2023, Theorem 1)).Consider the set of SDEs indexed byλ∈Λ dX λ t =µ λ(t, Xλ t )dt+σ λ(t, Xλ t )dW λ t , t∈[0,1], X λ 0 ∼P λ 0 , (19) corresponding to the path measures {Pλ}λ∈Λ and with marginal densities ...

    2023

  16. [16]

    Then the quadratic variation must be the same for both processes, σM(σM)T =σσ T, almost surely

    Assume DKL(Π||M)<∞ . Then the quadratic variation must be the same for both processes, σM(σM)T =σσ T, almost surely. Since the local time term has bounded variation, we may use Girsanov’s theorem of Léonard (2012a, Theorem 2.1) to identify the change in drift going fromM to Πby Z t 0 σσT(s, Xs)βsds= Z t 0 σσT(s, Xs)E Π1|0,s[∇logQ r 1|s(X1 |X s)|X 0, Xs]−v...

    2023

  17. [17]

    = 1 2 EΠ Z 1 0 (ut(X0, Xt)−v t(Xt))T(σσT(t, Xt))−1(ut(X0, Xt)−v t(Xt))dt = 1 2 Z 1 0 EΠ0,t (ut(X0, Xt)−v t(Xt))T(σσT(t, Xt))−1(ut(X0, Xt)−v t(Xt)) dt, where the change in order of integration is valid by Fubini’s theorem. The KL divergence is minimized whenM 0 = Π0 and, sinceσσ T is positive definite, for eacht, v∗ t (x) =E Π0|t [ut(X0, Xt)|X t =x] =E Π0|...

    2024

  18. [18]

    In both experiments, we sample using an EMA model with decay rate 0.999, both in training in the finetuning step and for evaluation

    with β= (0.9,0.999)is used as optimizer and gradients are clipped if exceeding unit norm. In both experiments, we sample using an EMA model with decay rate 0.999, both in training in the finetuning step and for evaluation. We used 11 reflections per dimension in the truncation of (14). B.1 Compute resources The experiments were run on a HPC cluster with 4...

    2024