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Diffusion Schrödinger Bridge Matching

Yuyang Shi, Valentin De Bortoli, Andrew Campbell, Arnaud Doucet

arXiv:2303.16852v3stat.MLcs.LG

TL;DR

Existing Schrödinger Bridge methods are limited by poor scaling or error accumulation, while diffusion and flow methods are not guaranteed to approximate the optimal transport map. The paper introduces IMF and DSBM, showing convergence under mild assumptions and improved practical handling of these numerical issues.

  • Problem

    Existing Schrödinger Bridge approximations tend to be restricted to low-dimensional settings, while diffusion and flow methods need not produce transports close to the optimal transport map.

  • Method

    The paper introduces IMF for solving Schrödinger Bridges and DSBM algorithms that approximate IMF iterates through Bridge Matching.

  • Results

    Under mild assumptions, IMF converges in KL divergence to the unique Schrödinger Bridge, while DSBM mitigates time-discretization and bias-accumulation issues.

  • Takeaways & Limitations

    DSBM is most effective for general transport problems, whereas its improvements over Bridge and Flow Matching for CIFAR-10 generative modeling are minor.

  • Takeaways & Limitations

    The practical iterative algorithm can perform poorly when its approximate minimization step fails to provide a suitable next iterate.

Abstract

from arXiv · show

Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs) implement such a transport through a Stochastic Differential Equation (SDE) or an Ordinary Differential Equation (ODE). However, while it is desirable in many applications to approximate the deterministic dynamic Optimal Transport (OT) map which admits attractive properties, DDMs and FMMs are not guaranteed to provide transports close to the OT map. In contrast, Schrödinger bridges (SBs) compute stochastic dynamic mappings which recover entropy-regularized versions of OT. Unfortunately, existing numerical methods approximating SBs either scale poorly with dimension or accumulate errors across iterations. In this work, we introduce Iterative Markovian Fitting (IMF), a new methodology for solving SB problems, and Diffusion Schrödinger Bridge Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM significantly improves over previous SB numerics and recovers as special/limiting cases various recent transport methods. We demonstrate the performance of DSBM on a variety of problems.

1 Introduction

The introduction frames generative modeling as a mass-transport problem whose modern stochastic methods are not necessarily close to the appealing OT map. It motivates Schrödinger Bridges and introduces Iterative Markovian Fitting as a new procedure for computing them while preserving endpoint distributions.

  • Motivation: Generative modeling transports a reference noise distribution to data using deterministic or stochastic maps, including Denoising Diffusion Models built from learned-drift SDEs.Bridge Matching and Flow Matching models further develop dynamic transport maps using SDEs.
  • Motivation: Existing dynamic transport methods are not necessarily close to the Wasserstein-2-minimizing OT map, despite its appealing properties.This motivates methods designed to connect generative transport with OT-like objectives.
  • Schrödinger Bridges: A Schrödinger Bridge is a finite-time diffusion matching two endpoint distributions while minimizing Kullback–Leibler divergence to a reference diffusion.It is presented as the dynamic counterpart of entropy-regularized optimal transport.
  • Contributions: The paper introduces Iterative Markovian Fitting, which alternates projections onto Markov processes and the reciprocal class to compute Schrödinger Bridges.The reciprocal class contains measures sharing the reference measure’s bridges.
  • Contributions: Unlike Iterative Proportional Fitting, Iterative Markovian Fitting iterates always preserve the initial and terminal distributions.The introduction states that the paper also establishes various theoretical results for IMF.

2 Dynamic Mass Transport Techniques

Dynamic mass transport techniques construct stochastic or deterministic processes that connect two distributions, including Denoising Diffusion Models, Bridge Matching, Flow Matching, and Schrödinger Bridges. Schrödinger Bridges solve entropy-regularized optimal transport, while iterative projection methods such as IPF and DSB approximate them.

  • Denoising Diffusion Models: Denoising Diffusion Models define a forward noising SDE that approaches a standard Gaussian and generate samples using its learned time-reversal.The reverse drift uses the score of the noising marginal, approximated by a neural network trained with denoising score matching.
  • Bridge Matching: Bridge Matching builds bridges between two distributions and learns a mimicking diffusion process with matching intermediate marginals.The framework uses a mixture of pinned diffusion bridges and learns the required drift component through regression.
  • Bridge Matching: Letting σ →0 recovers Flow Matching models from the Bridge Matching framework.This limiting case connects stochastic bridge-based transport to deterministic flow-based transport.
  • Schrödinger Bridges: Schrödinger Bridges find the path measure closest to a reference diffusion in Kullback–Leibler divergence while satisfying initial and terminal distribution constraints.Their static formulation is an entropy-regularized optimal transport problem, and σ = 0 recovers classical optimal transport.
  • Schrödinger Bridge Numerics: IPF alternates KL projections enforcing the terminal and initial constraints, starting from the reference path measure Q.The resulting iterates correspond to diffusions related by time reversal, enabling iterative learning of Schrödinger Bridge dynamics.

3 Iterative Markovian Fitting

Section 3 develops Iterative Markovian Fitting (IMF) by alternating Markovian and reciprocal projections, preserving endpoint marginals while approaching the Schrödinger Bridge. Under mild assumptions, IMF has the Schrödinger Bridge as its unique fixed point and converges to it in reverse KL divergence.

  • Markovian Projection: Markovian projection maps mixtures of bridges to Markov path measures and preserves their time marginals, while Flow Matching appears as the deterministic σ→0 limit.The projection is formulated through an associated SDE and reverse Kullback–Leibler projection.
  • Reciprocal Projection: Reciprocal projection maps a path measure into Q’s reciprocal class, minimizing KL(P|Π) over measures sharing Q’s bridge.Unlike Markovian projection, reciprocal projection does not generally preserve the Markov property.
  • Iterative Markovian Fitting: IMF alternates Markovian and reciprocal projections, starting from a reciprocal measure with prescribed endpoint marginals.The alternating construction is motivated by the uniqueness of the Markov measure in Q’s reciprocal class with the required initial and terminal marginals.
  • Iterative Markovian Fitting: Every IMF iterate preserves both endpoint marginals, contrasting with IPF, whose initial and final marginals are not preserved across iterations.This establishes the stated duality between IMF and IPF.
  • Convergence: Under mild assumptions, KL(P_n+1|P_SB) ≤ KL(P_n|P_SB), KL(P_n|P_n+1)→0, and IMF converges to the unique fixed point P⋆=P_SB.Odd iterates approach the reciprocal class, even iterates approach Markov measures, and lim_n→+∞ KL(P_n|P⋆)=0.

4 Diffusion Schrödinger Bridge Matching

This section introduces Diffusion Schrödinger Bridge Matching (DSBM), which combines Iterative Markovian Fitting with Bridge Matching to solve Schrödinger bridge problems. DSBM alternates forward and backward Markovian projections to remove terminal-marginal bias and recovers classical IPF iterates under reference-process initialization.

  • Method: DSBM combines Iterative Markovian Fitting with Bridge Matching into a practical algorithm for solving the Schrödinger bridge problem.The method is summarized as an iterative procedure over reciprocal and Markov classes.
  • Method: Approximate forward projection can accumulate bias because the learned Markov process may not satisfy the terminal marginal exactly.This makes the straightforward IMF iteration perform poorly in practice.
  • Method: DSBM alternates forward and backward Markovian projections, with the backward projection removing bias on the terminal marginal accumulated by the forward projection.The forward process starts from π0, whereas the backward process starts from πT; alternating them yields Algorithm 1.
  • Initialization coupling: When initialized with the forward reference-process coupling Π0_0,T = Q0,T, DSBM recovers the IPF iterates used in DSB.Under sufficiently rich function families, the optimal DSBM and DSB sequences satisfy Mn = P̃n for every n ≥ 1.
  • Initialization coupling: DSBM uses only initial-terminal couplings to construct bridge measures, making trajectory caching more computationally and memory efficient and countering practical forgetting of the reference bridge.Explicit reciprocal-class projections preserve the bridge structure during iterations.

5 Related Work

Prior work connects Markovian projection to Bridge Matching, positions Flow Matching and DDIM as deterministic bridge methods, and develops Schrödinger Bridge algorithms for generative modeling. These lines of work establish the methodological context for diffusion Schrödinger bridges.

  • Markovian projection and Bridge Matching: Markovian projection was proposed in machine learning to define Bridge Matching models and later analyzed theoretically and applied to discrete and constrained domains.The concept originated earlier in probability and stochastic-process literature, with Peluchetti (2021) introducing its machine-learning use.
  • Bridge and Flow Matching: Flow Matching uses deterministic bridges with deterministic ODE samplers, while DDIM can be formulated as a discrete-time version of Flow Matching.The passage notes active study of Flow Matching and extensions to Riemannian settings.
  • Diffusion Schrödinger Bridge: Schrödinger Bridges have longstanding roles in probability theory and stochastic control and were later adapted for generative modeling through several related algorithms.The passage also notes work on the case of a Dirac delta terminal distribution and subsequent extensions.

6 Experiments

Experiments show that DSBM produces accurate, scalable Schrödinger-bridge transports across low-dimensional, high-dimensional, and image or fluid-transfer tasks. It outperforms existing SB baselines in several settings while maintaining realistic samples and source-consistent reconstructions.

  • 2D Experiments: In 2D experiments, DSBM outperforms FM and CFM without OT solvers and outperforms DSB across all datasets, although OT-CFM performs best overall.Evaluation uses 2-Wasserstein distance and integrated path energy, with lower path energy indicating shorter trajectories.
  • High-Dimensional Gaussian Experiment: In the d = 50 Gaussian experiment, DSBM avoids the inaccurate variance and covariance estimates observed for RF, IMF-b, and DSB as iterations increase.The experiment compares convergence of learned means, variances, and endpoint covariances against the analytically computable true SB.
  • High-Dimensional Gaussian Experiment: DSBM achieves KL divergence similar to SB-CFM at d = 5 and substantially higher accuracy in higher dimensions.Accuracy is measured by the KL divergence between learned marginal distributions and the true SB.
  • MNIST, EMNIST transfer: For MNIST-to-EMNIST transfer, OT-CFM becomes less applicable in higher dimensions and produces lower-quality samples than the evaluated alternatives.The comparison includes DSBM, DSB, Bridge Matching, CFM, OT-CFM, and RF, using generated samples and FID convergence.
  • CelebA transfer: On CelebA, increasing σ improves FID until it becomes too high, while reducing LPIPS-based alignment; on AFHQ, DSBM generates realistic samples similar to inputs.The CelebA ablation varies σ, whereas the AFHQ experiment uses 512 × 512 cat-to-wild and wild-to-cat transfer.
  • Unpaired Fluid Flows Downscaling: For unpaired fluid-flow downscaling, DSBM reconstructs high-resolution fields from slightly noised low-resolution inputs and achieves smaller distances across frequency classes than Diffusion-fb.The task uses unpaired 64 × 64 and 512 × 512 fields, and the reconstructions remain consistent with the low-resolution source.

7 Discussion … A.1 Bridge Matching and Flow Matching Models

The paper introduces IMF and its DSBM algorithms for learning Schrödinger Bridges while mitigating time-discretization and bias-accumulation issues. The appendix relates DSBM to Flow Matching, Conditional Flow Matching, Bridge Matching, stochastic interpolants, and related reconstruction methods.

  • 7 Discussion: IMF provides an alternative, dual interpretation of classical IPF for learning Schrödinger Bridges.The paper introduces DSBM-IPF and DSBM-IMF as practical algorithms built on IMF.
  • 7 Discussion: DSBM-IPF and DSBM-IMF mitigate the time-discretization and bias-accumulation issues of existing Schrödinger Bridge methods.The discussion presents these as practical algorithms for learning Schrödinger Bridges.
  • 7 Discussion: The authors identify further comparison of DSBM-IMF and DSBM-IPF as future work and relate IMF to Rectified Flow through a shared limiting-case perspective.They also propose deriving a Schrödinger Bridge analogue for Rectified Flow extensions to general convex costs.
  • Outline of the Appendix: The appendices clarify relationships among existing methods, analyze linear SDEs, connect bridge parameterizations with stochastic interpolants, and provide proofs and Gaussian-case IMF results.The supplied outline states these topics but is truncated before describing the remaining appendices.
  • A.1 Bridge Matching and Flow Matching Models: Flow Matching and Conditional Flow Matching objectives are exactly the same under the stated Gaussian setting, while CFM also applies when the initial distribution is non-Gaussian.Thus, CFM is presented as a generalized version of FM; when σmin > 0, their smoothing procedures differ.
  • A.1 Bridge Matching and Flow Matching Models: Bridge Matching recovers the Flow Matching objective in the Brownian-motion, σ → 0 limit and generalizes Flow Matching to arbitrary initial and terminal distributions.The passage specifically notes that the initial distribution need not be Gaussian.
  • A.1 Bridge Matching and Flow Matching Models: Aligned-data inverse-problem and interpolation objectives are equivalent to the Bridge Matching objective, differing mainly in their choices of the endpoint coupling Π0,T.The cited examples use aligned data or pair clean images with corrupted images through a degradation kernel.
  • A.1 Bridge Matching and Flow Matching Models: A reconstruction objective yields the optimal Flow Matching vector field after exchanging endpoint roles, while its stochastic variant remains an open question for interpretation within Bridge Matching.The equivalence uses a conditional-expectation vector field and an L2 or Bregman-loss justification.

A.2 On DSBM and Existing Works · A.3 DSBM and Rectified Flow · B The Design Space of Brownian Bridges

DSBM recovers several existing transport algorithms through specific choices of bridge and coupling, including Brownian Bridge Matching, Flow Matching, Rectified Flow, OT-CFM, and a one-iteration Schrödinger Bridge procedure. Compared with Rectified Flow, DSBM uses forward and backward bridge matching to improve marginal accuracy across iterations, whereas Rectified Flow can accumulate errors when its initial transport is inaccurate.

  • A.2 On DSBM and Existing Works: DSBM recovers existing algorithms by varying the bridge Q|0,T and coupling Π0,T.The framework is presented as a unifying procedure for these methods.
  • A.2 On DSBM and Existing Works: With independent coupling Π0,T = π0 ⊗ πT and a constant-diffusion Brownian bridge, DSBM recovers the Brownian Bridge Matching loss.The corresponding loss is equation (10), matching the Brownian Bridge Matching loss in equation (5).
  • A.2 On DSBM and Existing Works: As σ → 0, DSBM recovers Flow Matching, and repeated forward projections recover Rectified Flow.The Rectified Flow result follows by repeating lines 7–9 of Algorithm 1.
  • A.2 On DSBM and Existing Works: Given an estimated OT-map coupling between π0 and πT, the first DSBM iteration recovers OT-CFM.This connects the initial DSBM iteration to OT-CFM under an estimated optimal-transport coupling.
  • A.2 On DSBM and Existing Works: For general bridges and the optimal Schrödinger Bridge static coupling Π0,T = ΠSB 0,T, DSBM converges in one iteration and recovers Somnath et al. (2023).The one-iteration result requires the optimal Schrödinger Bridge static coupling.
  • A.3 DSBM and Rectified Flow: DSBM and Rectified Flow both build transport maps between general distributions, but DSBM uses an SDE while Rectified Flow uses an ODE.The SDE formulation is theoretically important because the uniqueness characterization of SB requires σt > 0.
  • A.3 DSBM and Rectified Flow: Rectified Flow cannot correct accumulated marginal errors after its first iteration, whereas DSBM’s forward and backward bridge matching can improve marginal accuracy with iteration.Rectified Flow may improve flow straightness while sacrificing sampling accuracy, especially when its initial transport is poor; this occurs in moons-8gaussians, Gaussian transport, and MNIST–EMNIST transfer.

B.1 Relationship to Stochastic Interpolants … C.2 Proof of Proposition 4

The appendices relate the bridge parameterization to stochastic interpolants, derive explicit samplers for linear diffusion bridges, and establish the assumptions and KL projection results underlying the method. They also identify how Markov and non-Markov settings differ and show the Ornstein–Uhlenbeck formulas reduce to Brownian motion.

  • B.1 Relationship to Stochastic Interpolants: The bridge parameterization connects to Albergo et al.’s stochastic interpolants through explicit relationships between their coefficient functions and Markovian projections.The paper recovers the probability-flow result of Albergo et al. (2023).
  • B.1 Relationship to Stochastic Interpolants: In the Ornstein–Uhlenbeck setting, Markovian bridges are fully parameterized by two functions, whereas non-Markov bridges require three.The framework focuses on Markov reference measures but can extend to non-Markovian settings.
  • B.2 Linear SDE and Bridge Matching: For linear SDE bridges, the appendix targets integral, forward, and backward samplers with explicit coefficients and endpoint-dependent conditioning.The considered processes have drift −αβ_tX_t and diffusion coefficient σβ_t^1/2, with β continuous and positive.
  • B.2.1 Brownian motion: For Brownian-motion bridges, the appendix derives integral, forward, and backward sampling representations using explicit conditional diffusion formulas.The backward sampler is obtained by time-reversing the forward unconditional process initialized at x0.
  • B.2.2 Ornstein-Uhlenbeck: For Ornstein–Uhlenbeck bridges, the appendix likewise derives integral, forward, and backward samplers, including formulas involving hyperbolic cotangent and cosecant functions.The backward sampler is again derived from time reversal of the forward unconditional process.
  • B.2.2 Ornstein-Uhlenbeck: As α →0, the Ornstein–Uhlenbeck bridge formulas recover the Brownian-motion setting.This follows from tanh(x) ∼x and sinh(x) ∼x as x →0.
  • C.1 Proof of Proposition 2: Under assumptions A1, A2, and A3, the Doob h-transform is well-defined, the conditioned process is Markov, and its Markovian projection is characterized by the stated drift.The assumptions impose regularity, absolute continuity with bounded density ratio, and a growth bound on ∇log φ_t|0.
  • C.2 Proof of Proposition 4: The proof of Proposition 4 uses KL additivity to show that, when the conditional bridge is fixed, the KL minimizer matches the prescribed endpoint coupling while retaining that bridge.The minimizer is constructed by integrating Q|0,T(·|x0, xT) against P0,T(dx0, dxT).

C.3 Proof of Proposition 5 … C.10 Impact of the resolution on the entropic regularization

The appendix establishes existence, uniqueness, and convergence properties for Schrödinger Bridges and IMF, clarifies the nonconvexity of Markov measures, and shows that upsampling changes effective entropic regularization. In particular, resolution scaling transforms ε to ε/f^2, so Schrödinger Bridges are not invariant under upsampling.

  • C.3 Proof of Proposition 5: Under the stated assumptions, a unique Schrödinger Bridge exists, and any Markov measure in Q’s reciprocal class with finite KL(P|Q) is that bridge.The result follows from the cited Schrödinger Bridge existence and uniqueness theorems.
  • C.4 Proof of Lemma 6: Lemma 6 establishes the reverse-KL decomposition used in the proofs, contrasting with Csiszár’s forward-KL projection identity on convex sets.The comparison highlights that the set of Markov measures is not treated through the usual convex projection framework.
  • C.5 Proof of Proposition 7: The IMF sequence decreases KL(Pn|PSB), remains finite, and satisfies limn→+∞ KL(Pn|Pn+1) = 0.This is Proposition 7 under the conditions of Proposition 5 and Lemma 6.
  • C.6 Proof of Theorem 8: IMF has the unique fixed point P⋆ = PSB, and its iterates converge with limn→+∞ KL(Pn|P⋆) = 0.The proof uses KL coercivity and compactness of the relevant sublevel set.
  • C.7 Proof of Proposition 9: The forward and backward process equations can be viewed as SDEs whose drifts are conditional expectations of Q’s drift under Π0,T|t.The backward direction uses the time-reversal of Q conditioned on its endpoints.
  • C.8 Proof of Proposition 10: The optimal DSB sequence is Markov and belongs to M ∩ R(Q), while its forward and reverse projections reproduce the corresponding DSB iterates.Initialization satisfies Π0,T = Q0,T and Π0 = P̃0 = Q.
  • C.10 Impact of the resolution on the entropic regularization: For upsampled marginals, P⋆ solves the static Schrödinger Bridge with regularization ε exactly when down#P⋆ solves the original problem with regularization ε/f^2.Thus, for the same σ, DSBM at 64 × 64 and 128 × 128 produces different results even after downsampling.

D Convergence of IMF in the Gaussian setting · E Discrete-Time Markovian Projection · F Comparing DSBM-IPF and DSB

In the Gaussian setting, IMF has the true Schrödinger bridge coupling as a fixed point and converges rapidly in the analytic example. The discrete-time Markovian projection recovers the continuous-time projection in a limiting case, while DSBM-IPF reduces bridge bias accumulation relative to DSB through reciprocal and Markovian projections.

  • D Convergence of IMF in the Gaussian setting: The Gaussian Markovian projection is represented by a linear SDE with an explicitly derived time-dependent drift and diffusion coefficient σ.The drift denominator is positive, so the drift is well-defined, smooth, and sublinear, yielding a unique strong solution.
  • D Convergence of IMF in the Gaussian setting: IMF covariance updates are analytically characterized through the scalar map f(c²) = exp[G(1, c²)].The update follows from the Gaussian representation of X1 and the resulting covariance calculation.
  • D Convergence of IMF in the Gaussian setting: IMF recovers the true static Schrödinger bridge solution as a fixed point of its Gaussian covariance iteration.The result is established by showing f(c²_SB) = c²_SB and identifying Π^SB_0,1 as the fixed point.
  • D Convergence of IMF in the Gaussian setting: The analytic Gaussian fixed-point procedure appears to converge very rapidly, typically requiring only two or three iterations.This behavior is visualized across a variety of parameter settings in Figure 12.
  • E Discrete-Time Markovian Projection: The discrete-time Markovian projection provides a minimizer for each transition loss and recovers the target discrete marginals when the initial marginal is correctly specified.Under the stated Gaussian transition assumptions and the bound ∥x_k+1 − x_k∥ ≤ γ^1/2, the proposition establishes the marginal-identification result.
  • E Discrete-Time Markovian Projection: When the conditional drift uses f(x_k, x_N) = ∇log q(x_N|x_k), the discrete-time construction corresponds to a discretization of the bridge.The associated Markovian drift is obtained from the bridge conditional score.
  • F Comparing DSBM-IPF and DSB: DSBM-IPF alternates reciprocal and Markovian projections, whereas DSB-type algorithms compute IPF iterates using time reversals.DSBM explicitly projects onto the reciprocal class of Q, incorporating the reference measure as an inductive bias.
  • F Comparing DSBM-IPF and DSB: DSBM can reduce bridge bias accumulation because an IMF-refined iterate preserves the correct initial marginal and bridge, providing a better IPF initialization when it is closer to M⋆ than Q in KL divergence.Algorithm 1 performs the Markovian and reciprocal projection once before switching between forward and backward directions.

G Joint Training of Forward and Backward Processes … I.2 Gaussian Experiment

The paper extends DSBM with joint forward–backward training, variance-scaled losses, and experiments spanning two-dimensional transport, reference-noise effects, and Gaussian bridges. The experiments use specified Brownian reference dynamics and evaluate transport quality and Gaussian marginal convergence under controlled settings.

  • G Joint Training of Forward and Backward Processes: Joint DSBM trains forward and backward drift models together, then alternates Markovian updates using their learned processes.The procedure initializes a bridge coupling, jointly learns vϕ and vθ, and constructs the next Markovian update.
  • G Joint Training of Forward and Backward Processes: The joint-training method combines forward and backward updates symmetrically, while leaving the optimal mixture weighting for future work.The proposed mixture averages the available forward and backward updates; the paper explicitly identifies optimal combination as unresolved.
  • G Joint Training of Forward and Backward Processes: A denoising score matching consistency loss links the forward and backward models without requiring divergence computation.The consistency relation also recovers the deterministic time-reversal sign flip as σt approaches zero.
  • H Loss Scaling: Loss scaling downweights the forward objective by 1 + σ^2t/(1 −t) near t = 1, where noise makes the unscaled loss large.The same reasoning applies to the backward loss.
  • I Experiments: All experiments use Brownian motion as the reference measure, its Brownian bridge, T = 1, Adam with learning rate 10−4, and SiLU activations unless specified otherwise.The runs use mixed CPU and GPU computing clusters.
  • I.1 2D Experiments: The 2D experiments use synthetic datasets, 10000 test samples, probability-flow ODE samples, and 2-Wasserstein distance for evaluation.The larger test set addresses substantial variation in the distance estimate observed with only 1000 samples.
  • I.1.1 Variance of the reference measure Q: Increasing σ produces increasingly curved transport paths in the 3gaussians problem, with Table 4 reporting 2-Wasserstein distances across σ values for two transport tasks.The second task is moons-8gaussians.
  • I.2 Gaussian Experiment: The Gaussian experiment uses two-layer, 256-unit MLP drift networks, batch size 128, 20 diffusion sampling steps, 10000 training steps per outer iteration, and 20 outer iterations.For d = 50, the experiment finishes in approximately 200 minutes on one CPU; Table 3 estimates Gaussian marginal KL divergence analytically from sample means and variances.

I.3 MNIST Transfer Experiment · I.4 CelebA Transfer Experiment · I.5 AFHQ Transfer Experiment

The paper evaluates DSBM transfer on EMNIST/MNIST, CelebA, and AFHQ across increasing image resolutions and reports faithful MNIST transfers with better input-output similarity than Bridge Matching. The CelebA and AFHQ experiments additionally test scalability to high-resolution image transfer.

  • I.3 MNIST Transfer Experiment: The MNIST experiment uses a shared U-Net with approximately 6.6 million parameters, batch size 128, 30 diffusion sampling steps, and 5000 training steps per outer iteration.The setup uses the first five EMNIST letters, yielding 10 classes across the two domains.
  • I.3 MNIST Transfer Experiment: DSBM transfers EMNIST letters toward MNIST digits, while CFM and Rectified Flow samples with two rectified steps are less clear and identifiable.OT-CFM improves slightly over CFM, but many samples remain unclear; DSB has not converged after 10 iterations and still produces many letter-like samples.
  • I.3 MNIST Transfer Experiment: DSBM transfers MNIST samples faithfully while preserving more similarity to inputs than Bridge Matching.The Mean Squared Distance between initial and final samples confirms that DSBM methods transfer more closely to the original inputs.
  • I.4 CelebA Transfer Experiment: CelebA transfer is evaluated at 64×64 and 128×128 resolutions between male/old and female/young image classes.The study evaluates dependence on σ > 0 at 64×64 and showcases scalability at 128×128, using 20000 samples per class with horizontal-flip augmentation.
  • I.4 CelebA Transfer Experiment: The CelebA σ > 0 ablation runs DSBM for 20 iterations while comparing multiple σ values before full convergence.Loss and quality continued improving after 20 iterations, but early stopping enabled comparisons across more σ values.
  • I.4 CelebA Transfer Experiment: CelebA inference and cache refresh use 100 sampling steps, while each outer DSBM iteration trains for 20000 iterations.The experiment uses modified Euler-Maruyama sampling with constant stepsizes, batch size 64, and EMA rate 0.999.
  • I.5 AFHQ Transfer Experiment: AFHQ transfer tests cat-to-wild translation at 512×512 resolution using approximately 5000 samples per class.Networks are pretrained with Bridge Matching for 100000 steps, followed by 20 DSBM iterations with 25000 steps per outer iteration; σ2 = 5 and 100 sampling steps are used.

I.6 CIFAR-10 Generative Modeling Experiment · I.7 Fluid Flows Experiment · J Broader Impact

The paper evaluates DSBM on CIFAR-10 generation and fluid-flow reconstruction, where it improves or matches competing methods under reported conditions. It closes by describing potential applications and warning that malicious use of generative models could cause societal harm.

  • I.6 CIFAR-10 Generative Modeling Experiment: DSBM-IMF improves CIFAR-10 sample quality over Bridge Matching at the same σ, indicating benefits from additional outer iterations.FID is computed between 50000 training-set samples and 50000 generated samples.
  • I.6 CIFAR-10 Generative Modeling Experiment: The best DSBM-IMF result uses σ2 = 0.2 and is slightly better than FM with σ2 = 0 using 100 Euler steps.Using dopri5, FM achieves a FID of 4.055 with on average 148 integration steps.
  • I.6 CIFAR-10 Generative Modeling Experiment: FID worsens for Bridge Matching and DSBM as sampler stochasticity increases, while RF and DSBM-IMF improve sampling quality at low diffusion-step counts.The CIFAR-10 setup uses a 39.6-million-parameter U-Net with batch size 128 and EMA rate 0.9999.
  • I.7 Fluid Flows Experiment: The fluid-flow dataset contains unpaired 64 × 64 and 512 × 512 fields, plus context fields and two channels representing supersaturation and vorticity.The model follows prior processing and architecture choices, using an approximately 11.3-million-parameter network.
  • I.7 Fluid Flows Experiment: DSBM-IPF and DSBM-IMF produce reconstructions consistent with low-resolution sources, whereas Diffusion-fb and Bridge Matching produce dissimilar samples.DSBM-IMF achieves comparable statistical-profile performance to Diffusion-fb and can be comparatively more accurate.
  • J Broader Impact: The work targets transport-map learning for general distributions with high accuracy and scalability, with potential applications in machine learning, physics, biology, and geosciences.The fluid-flow experiment is presented as evidence of these potential applications.
  • J Broader Impact: As with generative models generally, intentional malicious use could cause detrimental societal impacts.This is identified as a broader-impact risk alongside the stated scientific and practical potential.
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