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Denoising Diffusion Bridge Models

Linqi Zhou, Aaron Lou, Samar Khanna, Stefano Ermon

arXiv:2309.16948v3cs.CVcs.AI

TL;DR

Standard diffusion models are poorly suited to translating between paired, non-noise distributions. DDBMs learn conditional scores for diffusion bridges and reverse them between endpoints, improving image translation while matching EDM performance in unconditional generation.

  • Problem

    Standard diffusion models assume random-noise priors, making translation between paired image distributions difficult and requiring cumbersome conditioning or sampling modifications.

  • Method

    DDBMs learn the score of a diffusion bridge over paired endpoints and transport between their distributions using a reverse SDE or probability-flow ODE.

  • Results

    DDBMs achieve better image quality and significantly better translation faithfulness on standard image translation tasks, while matching EDM performance in unconditional generation.

  • Takeaways & Limitations

    DDBMs provide a general framework that unifies score-based diffusion models with transport-based distribution translation and reuses diffusion-model design choices.

  • Takeaways & Limitations

    The method assumes paired endpoints drawn from an arbitrary but unknown joint distribution qdata(x, y).

Abstract

from arXiv · show

Diffusion models are powerful generative models that map noise to data using stochastic processes. However, for many applications such as image editing, the model input comes from a distribution that is not random noise. As such, diffusion models must rely on cumbersome methods like guidance or projected sampling to incorporate this information in the generative process. In our work, we propose Denoising Diffusion Bridge Models (DDBMs), a natural alternative to this paradigm based on diffusion bridges, a family of processes that interpolate between two paired distributions given as endpoints. Our method learns the score of the diffusion bridge from data and maps from one endpoint distribution to the other by solving a (stochastic) differential equation based on the learned score. Our method naturally unifies several classes of generative models, such as score-based diffusion models and OT-Flow-Matching, allowing us to adapt existing design and architectural choices to our more general problem. Empirically, we apply DDBMs to challenging image datasets in both pixel and latent space. On standard image translation problems, DDBMs achieve significant improvement over baseline methods, and, when we reduce the problem to image generation by setting the source distribution to random noise, DDBMs achieve comparable FID scores to state-of-the-art methods despite being built for a more general task.

1 INTRODUCTION

Standard diffusion models assume random-noise priors, making paired image translation difficult and motivating DDBMs as a unified alternative for distribution transport.

  • Standard diffusion models map data distributions to random noise, a setup that does not naturally support translation between paired images.
  • Conditioning and manually altered sampling incorporate non-noise inputs but are theoretically unprincipled, directional, and lose cycle consistency.
  • ODE flow-matching methods model transport between arbitrary distributions but have mainly been applied to image generation rather than image translation.
  • DDBMs use diffusion bridges to unify unconditional diffusion generation with transport-based distribution translation.
  • DDBMs generalize diffusion-model design choices and achieve better image quality and significantly better translation faithfulness on standard image translation tasks.

2 PRELIMINARIES

Diffusion models define forward stochastic processes from data to a prior and reverse them using learned scores; fixed-endpoint conditioning extends this framework to diffusion bridges.

  • Diffusion models gradually transport data qdata(x) to a prior pprior(x), then reverse the process to generate realistic samples.
  • The forward process is an SDE with drift, diffusion coefficient, and Wiener noise, while its reverse uses the marginal score ∇xt log p(xt).
  • A deterministic probability-flow ODE has the same marginal distributions as the reverse diffusion process.
  • Denoising score-matching: The score ∇xt log p(xt) is learned through a tractable denoising score-matching objective enabled by Gaussian transition kernels.
  • Diffusion process with fixed endpoints: Doob’s h-transform conditions a diffusion process to reach a fixed endpoint y almost surely by adding an endpoint-dependent drift adjustment.
  • Diffusion process with fixed endpoints: When the initial point is fixed as well, the resulting diffusion bridge connects a given x0 to a specified xT, supporting image-to-image translation.

3 DENOISING DIFFUSION BRIDGE MODELS

DDBMs learn to reverse diffusion bridges between paired endpoint distributions by matching a tractable conditional score, using stochastic or deterministic reverse dynamics.

  • DDBMs learn from paired samples drawn from an arbitrary joint distribution qdata(x, y) and approximate sampling from qdata(x | y).
  • The constructed process replaces the Gaussian endpoint relation of ordinary diffusion with the observed paired endpoint distribution qdata(x0, xT).
  • Time-reversed SDE and probability flow ODE: The learned reverse process can use a time-reversed SDE or probability-flow ODE conditioned on xT.
  • Marginal distributions: VE and VP bridge constructions provide Gaussian conditional sampling distributions whose means interpolate between scaled endpoints and approach point masses near either endpoint.
  • Denoising bridge score matching: Matching the closed-form conditional score ∇xt log q(xt | x0, xT) trains a neural network to approximate the target score ∇xt log q(xt | xT).
  • The framework supports both stochastic bridge sampling and deterministic probability-flow transport between the endpoint distributions.

4 GENERALIZED PARAMETERIZATION FOR DISTRIBUTION TRANSLATION

DDBM extends diffusion-model parameterizations to distribution translation by conditioning bridge scores on endpoint information. Its scaling and drift controls generalize existing diffusion and transport formulations.

  • DDBM adapts score-network parameterizations, noise schedules, and efficient ODE sampling from diffusion models to distribution translation.
  • The pred-x parameterization predicts x0 and uses newly derived scaling functions designed for distribution translation.
  • Bridge-score reparameterization provides the model output used to represent the score of the diffusion bridge.
  • The scaling and loss-weighting functions depend on the bridge signal-to-noise structure and endpoint statistics.The additional hyperparameters σT and σ0T characterize the endpoint distribution and its correlation with x0.
  • In the EDM case, the generalized scaling functions reduce to EDM’s scaling functions.
  • The generalized time-reversal parameter w controls the strength of drift adjustment and permits a wider class of resulting marginal densities.

5 STOCHASTIC SAMPLING FOR DENOISING DIFFUSION BRIDGES

DDBM sampling combines higher-order ODE integration with scheduled stochastic updates. The stochastic component addresses blurry outputs caused by deterministic backward paths from fixed bridge endpoints.

  • Pure probability-flow ODE sampling can produce averaged or blurry outputs because bridge endpoints are fixed and backward paths are deterministic.
  • The Denoising Diffusion Bridge Hybrid Sampler starts from a sample of qdata(y) and returns x0 using guidance strength and step-ratio controls.
  • The scheduled stochastic steps approximately preserve the marginal distribution at each sampling step while adding noise for quality and diversity.
  • The sampler interleaves higher-order ODE steps with scheduled Euler–Maruyama steps following the backward SDE.

6 RELATED WORKS AND SPECIAL CASES

DDBM connects diffusion bridges with diffusion generation, flow matching, and related bridge methods. Specific parameter choices recover unconditional diffusion and OT-Flow-Matching or Rectified Flow as special cases.

  • Diffusion models: Diffusion models transport data toward a prior and learn the reverse process for realistic sample generation.
  • Diffusion bridges, Schrödinger bridges, and Doob’s h-transform: Diffusion bridges and Schrödinger bridges model processes conditioned on endpoint distributions, with prior work using path simulation, simulation-free generation, or IPF.
  • Flow and Optimal Transport: Flow-matching methods learn ODE-based transport maps between distributions, including deterministic straight-line paths in OT-Flow-Matching and Rectified Flow.
  • Special cases: With a Gaussian endpoint, the DDBM marginal matches a regular unconditional diffusion process, and its samplers reduce to the reverse SDE and ODE.
  • Special cases: A variance-scaled bridge with a suitable VE schedule connects DDBM’s probability-flow drift to the straight-line path term used by OT-Flow-Matching and Rectified Flow.
  • Experiments: Figure 3 provides a qualitative comparison between DDBM outputs and relevant baselines.

7 EXPERIMENTS

Experiments evaluate DDBM for pixel-space translation, sampler and preconditioning choices, and unconditional generation when one bridge endpoint is Gaussian. DDBM provides strong translation quality and faithfulness, while matching EDM performance in unconditional generation.

  • Image-to-image translation: DDBM is evaluated on pixel-space image translation across 64 × 64 and 256 × 256 image resolutions.The experiments use Edges→Handbags and DIODE-Outdoor datasets.
  • Image-to-image translation: At N = 40 function evaluations, DDBM is compared with Pix2Pix, SDEdit, DDIB, Rectified Flow, and I2SB.The shared low-step regime is intended to demonstrate sampler effectiveness when sampling steps are limited.
  • Image-to-image translation: DDBM achieves high generation quality and faithfulness; VP bridges outperform VE bridges in some cases, while Rectified Flow and DDIB struggle on substantially different domains.I2SB is the closest competing method but falls short when NFE is low.
  • Ablation studies: Increasing Euler step ratio s introduces stochasticity and improves FID up to a dataset-dependent value, while VP bridges rely heavily on guidance scale w = 1.For Edges→Handbags, the best-performing step ratio is reported as s = 0.3; the ODE sampler produces blurry images compared with the hybrid sampler.
  • Ablation studies: Each introduced sampler and preconditioning component further boosts generation performance in the ablation study.The baseline removes output reparameterization and reuses the EDM ODE sampler.
  • Unconditional generation: When one bridge endpoint is Gaussian, DDBM reduces to diffusion models and matches EDM on CIFAR-10 with negligible FID degradation while marginally improving FFHQ-64 × 64.The unconditional-generation evaluation uses FID on 50K generated images and compares against diffusion- and optimal-transport-based models.

8 CONCLUSION

Denoising Diffusion Bridge Models build stochastic bridges between paired distributions and transport between them using learned scores and differential equations.

  • Denoising Diffusion Bridge Models construct stochastic bridges between paired samples with tractable intermediate marginal distributions.
  • The model learns the conditional score of a tractable bridge distribution.
  • A learned score enables transport between endpoint distributions through a reverse SDE or probability flow ODE.

A PROOFS

The proofs derive bridge marginals and establish score-based reverse-time dynamics. They connect conditional bridge distributions to product-rule identities, Fokker–Planck equations, and reverse SDE or probability flow formulations.

  • Bayes’ rule derives the marginal distribution of x_t for tractable bridge transition kernels.
  • Denoising Bridge Score Matching uses samples from q(x_t | x_0, x_T) and paired data to approximate the conditional score.
  • The time evolution of q(x_t | x_T) has a time-reversed SDE involving the bridge score s(x,t,y,T) = ∇_x_t log q(x_t | x_T).
  • The conditional bridge score decomposes into ∇_x_t log p(x_T | x_t) and ∇_x_t log p(x_t | x_0).
  • Marginalizing over x_0 yields the conditional Fokker–Planck evolution for q(x_t | x_T), under the stated regularity conditions.

A.4 SPECIAL CASES OF DENOISING DIFFUSION BRIDGES

The framework recovers unconditional diffusion and connects to OT-Flow Matching and Rectified Flow as special cases. In the noiseless limit, the learned drift becomes the endpoint displacement, while conditioning on x_T remains a caveat.

  • Unconditional diffusion models: The resulting probability flow ODE is that of a regular diffusion in this unconditional special case.
  • Unconditional diffusion models: Setting q_data(x_0,x_T) to p(x_T | x_0)q_data(x_0) recovers unconditional diffusion models.
  • OT-Flow Matching and Rectified Flow: lim c→0 D = x_1 − x_0 for T = 1, so the network learns the endpoint displacement in the noiseless OT-Flow Matching and Rectified Flow limit.
  • OT-Flow Matching and Rectified Flow: As bridge noise approaches zero, the bridge score can explode, motivating direct matching to the limiting drift D.
  • The presented framework requires x_T as an additional condition, which is handled through a generalized score parameterization.
  • EDM as a special case: EDM scaling functions and Heun discretization are reused for unconditional diffusion experiments.

B EXPERIMENT DETAILS

Experiments use established architectures, bridge schedules, optimization settings, matched baselines, and low-step sampling across unconditional generation and image translation tasks.

  • Architecture: Karras et al. architectures are reused for unconditional CIFAR-10 and FFHQ-64×64 generation, while ADM architectures support pixel- and latent-space translation.
  • VE and VP bridge parameterization: VE bridges use σ_t = t and α_t = 1, whereas VP bridges use a time-invariant drift with β_1 = β_0.
  • Training: Training uses AdamW with learning rate 0.0001, no weight decay, and batch size 256 below 256×256 resolution.
  • Training: At 256×256 resolution, the effective batch size is 64 and training terminates at 500K iterations.
  • Training: Image-to-image translation uses σ_0 = σ_T = 0.5 and random-flip augmentation.
  • Sampling: All experiments use low-step sampling, with N = 18 for 32×32 translation and N = 40 for other resolutions.

B.1 ADDITIONAL RESULTS.

The paper investigates whether DDBMs can adapt to latent-space image translation, using Day→Night translation as an experiment. The model shows high-quality qualitative results despite trailing Rectified Flow and I2SB on the reported metrics.

  • Latent space translation: DDBMs are evaluated for latent-space image-to-image translation on the Day→Night task.The experiment examines adaptation to latent representations used to reduce computational burden in diffusion-model applications.
  • Latent space translation: DDBMs perform less well than Rectified Flow and I2SB but rank second in IS and MSE.The passage reports the comparative metric outcome without specifying the numerical values.
  • Latent space translation: Figure 9 visualizes Day→Night translation with day images on top and night translations below.
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