Research map / Image generation / Fast diffusion sampling
Stochastic Optimal Control for Diffusion Bridges in Function Spaces
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Derives Doob's h-transform and stochastic optimal control for diffusion bridges in infinite-dimensional Hilbert spaces, using an explicit Gaussian Radon-Nikodym density, and applies it to function-space bridge matching and Bayesian posterior sampling (DBFS).
Problem
Diffusion bridges and sampling in infinite-dimensional function spaces lack a theory: there is no Lebesgue density, so Doob's h-transform and SOC-based diffusion models cannot be directly applied.
Approach
Extends SOC and Doob's h-transform to Hilbert spaces by defining h via an explicit Radon-Nikodym derivative with respect to a Gaussian reference measure (N(0,Q_inf)), enabling bridge matching and relative-entropy-based Bayesian posterior learning on function-valued data.
Key results
- On unpaired EMNIST to MNIST transfer, DBFS reaches test FID 9.1 versus 8.2 for IDBM and 6.0 for DSBM.
- On Physionet imputation at 10% missingness, DBFS achieves RMSE 0.50 versus 0.52 for DSDP-GP and 0.60 for CSDI.
- On 1D function generation, DBFS achieves kernel two-sample test power of 5.1% on Quadratic, comparable to SP-SGM at 5.4%, while NDP is at least 99.0%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DBFS instead of SP-SGM | 1D function generation · Quadratic | kernel two-sample test power (%): 5.4 ± 0.7 -> 5.1 ± 0.4 | Helped |
| Use DBFS instead of SP-SGM | 1D function generation · Melbourne | kernel two-sample test power (%): 5.3 ± 0.7 -> 9.67 ± 0.45 | Hurt |
| Use DBFS instead of SP-SGM | 1D function generation · Gridwatch | kernel two-sample test power (%): 4.7 ± 0.5 -> 3.9 ± 0.4 | Helped |
| Use DBFS instead of DSBM | Unpaired image transfer · EMNIST → MNIST | FID: 6.0 -> 9.1 | Hurt |
| Use DBFS instead of DSBM | Unpaired image transfer · AFHQ-64 Wild → Cat | FID: 25.4 -> 44.4 | Hurt |
| Use DBFS instead of DSDP-GP | Physionet imputation · 10% observed | imputation RMSE: 0.52 ± 0.04 -> 0.50 ± 0.04 | Helped |
| Use DBFS instead of DSDP-GP | Physionet imputation · 50% observed | imputation RMSE: 0.64 ± 0.05 -> 0.61 ± 0.04 | Helped |
| Use DBFS instead of DSDP-GP | Physionet imputation · 90% observed | imputation RMSE: 0.81 ± 0.03 -> 0.77 ± 0.03 | Helped |
| Use DBFS instead of CNP | Functional regression · RBF context | log-likelihood: 0.97 ± 0.01 -> 1.02 ± 0.01 | Helped |
| Use DBFS instead of CNP | Functional regression · RBF target | log-likelihood: 0.45 ± 0.01 -> 0.47 ± 0.01 | Helped |
| Use DBFS instead of CNP | Functional regression · Matérn 5/2 context and target | log-likelihood: 0.85± 0.01 -> 0.93 ± 0.01; 0.21 ± 0.02 -> 0.25 ± 0.01 | Helped |
| Use DBFS instead of CNP | Functional regression · Periodic context | log-likelihood: -0.16± 0.01 -> -0.15 ± 0.01 | Helped |
Surprising findings
- DBFS has slightly worse FID than finite-dimensional DSBM on unpaired image transfer (44.4 vs 25.4 on AFHQ-64), which the authors attribute to resolution-agnostic design.
Limitations
- Time-independent coefficients prevent defining a noise schedule, which may limit performance.
- Gradient of the Bayesian learning loss is computationally demanding.
- Experiments limited to regular 1D and 2D domains.
Experimental setup
Datasets: MNIST EMNIST AFHQ Physionet Quadratic Melbourne Gridwatch Gaussian process synthetic data
Benchmarks: MNIST EMNIST AFHQ-64 Wild to Cat Physionet Quadratic Melbourne Gridwatch Gaussian process regression
Metrics: kernel two-sample test power FID imputation RMSE log-likelihood
Baselines: IDBM DSBM NDP SP-SGM CSDI DSDP-GP CNP NP
Builds on: Doob's h-transform Schrödinger bridge bridge matching iterative density fitting (IDBM) Girsanov theorem
Closest papers by meaning
- Generative Modeling with Phase Stochastic Bridge (ICLR 2024 · similarity 0.90)
- UniDB: A Unified Diffusion Bridge Framework via Stochastic Optimal Control (ICML 2025 · similarity 0.90)
- Stochastic Control for Fine-tuning Diffusion Models: Optimality, Regularity, and Convergence (ICML 2025 · similarity 0.89)
- Underdamped Diffusion Bridges with Applications to Sampling (ICLR 2025 · similarity 0.87)
- Neural Guided Diffusion Bridges (ICML 2025 · similarity 0.87)
- Conditioning Diffusions Using Malliavin Calculus (ICML 2025 · similarity 0.87)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.