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Underdamped Diffusion Bridges with Applications to Sampling
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Proposes a general diffusion-bridge framework with degenerate (underdamped) noise, derives an ELBO and score-matching equivalence, and introduces the underdamped diffusion bridge sampler (DBS) with splitting integrators that outperforms overdamped baselines on sampling benchmarks.
Problem
Learning diffusion bridges for sampling unnormalized densities and generative modeling when the noise is degenerate (noise only in some dimensions, as in underdamped Langevin dynamics), where existing overdamped methods converge slowly and discretization of degenerate SDEs is poorly handled.
Approach
Generalizes controlled diffusion path-measure frameworks to degenerate noise coefficients, derives discrete Radon-Nikodym derivatives for splitting integrators (OBABO, BAOAB, OBAB), gives an ELBO for underdamped generative bridges, and proposes underdamped DBS with learned hyperparameters (mass matrix, diffusion coefficient, terminal time, prior).
Key results
- On Funnel (d=10), underdamped DBS achieves Delta log Z 0.010 vs 0.021 for overdamped DBS, with ESS 0.779 vs 0.603.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use underdamped rather than overdamped dynamics in DBS | DBS · Funnel (d=10) | ESS: 0.603 \scriptstyle \pm 0.014 -> 0.779 \scriptstyle \pm 0.009 | Helped |
| Use underdamped rather than overdamped dynamics in DBS | DBS · ManyWell (d=50) | ESS: 0.887 \scriptstyle \pm 0.004 -> 0.898 \scriptstyle \pm 0.002 | Helped |
| Use underdamped rather than overdamped dynamics in DBS | DBS · LGCP (d=1600) | log Z (LB): 486.376 \scriptstyle \pm 1.020 -> 497.545 \scriptstyle \pm 0.183 | Helped |
| Use underdamped rather than overdamped dynamics in DBS | DBS · LGCP (d=1600) | ESS × 10: 0.032 \scriptstyle \pm 0.002 -> 0.174 \scriptstyle \pm 0.017 | Helped |
| Use OBAB or BAOAB integration instead of classical Euler integration (ablation) | DBS · Sampling benchmarks | ESS | Helped |
| Use OBABO integration instead of other considered schemes (ablation) | DBS · Sampling benchmarks | ESS | Helped |
| Learn hyperparameters end-to-end, including mass matrix, diffusion matrix, terminal time, and prior (ablation) | Underdamped DBS · Sampling benchmarks | ESS | Helped |
| Learn terminal time and prior distribution end-to-end (ablation) | Underdamped DBS · Sampling benchmarks | ESS | Helped |
Surprising findings
- Underdamped versions of CMCD and DIS did not uniformly improve over overdamped versions (e.g., underdamped DIS diverged on LGCP).
Limitations
- Theory suggests small KL between path measures requires step size scaling linearly in d, in tension with the empirical results.
- Non-uniqueness of optimal controls; generative-modeling extension is only derived, not benchmarked.
Experimental setup
Datasets: Funnel ManyWell LGCP Credit Cancer Ionosphere Sonar Seeds Brownian
Benchmarks: Funnel ManyWell LGCP Credit Cancer Ionosphere Sonar Seeds Brownian
Metrics: log-normalizing constant error effective sample size Sinkhorn distance log Z lower bound
Baselines: ULA MCD CMCD DIS DBS
Builds on: Diffusion Bridge Sampler Time-Reversed Diffusion Sampler Controlled Monte Carlo Diffusion Monte Carlo Diffusion Strang splitting
Closest papers by meaning
- Diffusion Bridge Implicit Models (ICLR 2025 · similarity 0.90)
- Denoising Diffusion Bridge Models (ICLR 2024 · similarity 0.90)
- Improved sampling via learned diffusions (ICLR 2024 · similarity 0.89)
- Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models (NeurIPS 2024 · similarity 0.89)
- Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics (ICLR 2025 · similarity 0.89)
- Improved off-policy training of diffusion samplers (NeurIPS 2024 · similarity 0.89)
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