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Conditioning Diffusions Using Malliavin Calculus
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Proposes BEL algorithms that learn diffusion-bridge/conditioning drifts via a generalised Tweedie score formula derived from Malliavin integration by parts, robust to singular (Dirac) rewards.
Problem
Conditioning a reference diffusion on a terminal observation when the reward/likelihood is singular (e.g. diffusion bridges with Dirac delta rewards) or lacks gradients, where existing stochastic optimal control methods that need differentiable rewards are unstable.
Approach
Replaces the gradient of the terminal reward with a Malliavin-calculus integration-by-parts representation: the conditional score equals the conditional expectation of a stochastic-integral score process weighted by a free matrix function alpha, yielding a gradient-free, amortised least-squares loss (BEL algorithms) that generalises Tweedie's formula to nonlinear SDEs.
Key results
- On the shape process bridge, BEL average reaches mean pointwise distance 0.085 vs 0.090 for time reversal, 0.498 for adjoint paths, and 1.396 untrained.
- On the 10-dimensional Brownian motion bridge, BEL average achieves the best Dist (2.3e-2) versus 9.4e-2 for the reparametrisation trick.
- On the double-well 1D problem, BEL first achieves the best MV (2.9e-1) and Dist (8.4e-2) among the tested variants, while BEL last performs worst (Dist 1.1).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BEL average rather than the reparameterization trick | Conditioned Brownian motion, dimension 10 · Brownian motion, dimension 10 | Dist: 9.4 \times 10^{-2} -> 2.3 \times 10^{-2} | Helped |
| Use BEL average rather than BEL last weighting (ablation) | Conditioned Brownian motion, dimension 1 · Brownian motion, dimension 1 | Dist: 5.6 \times 10^{-2} -> 3.2 \times 10^{-3} | Helped |
| Use BEL first weighting rather than BEL last weighting (ablation) | Conditioned double-well SDE, dimension 1 · Double-well, dimension 1 | Dist: 1.1 \times 10^{0} -> 8.4 \times 10^{-2} | Helped |
| Use BEL optimal rather than the reparameterization trick | Conditioned double-well SDE, dimension 10 · Double-well, dimension 10 | Dist: 5.2 \times 10^{-1} -> 3.0 \times 10^{-1} | Helped |
| Use BEL average rather than time reversal | Shape-process bridge · Shape processes | Dist: 0.090 -> 0.085 | Helped |
| Use BEL average rather than adjoint paths | Shape-process bridge · Shape processes | Dist: 0.498 -> 0.085 | Helped |
| Use BEL first weighting rather than BEL average weighting (ablation) | Conditioned Brownian motion, dimension 1 · Brownian motion, dimension 1 | Dist: 3.2 \times 10^{-3} -> 2.1 \times 10^{-2} | Hurt |
| Use BEL last weighting rather than BEL average weighting (ablation) | Conditioned double-well SDE, dimension 10 · Double-well, dimension 10 | Dist: 3.0 \times 10^{-1} -> 1.1 \times 10^{0} | Hurt |
Surprising findings
- BEL last and the reparametrisation trick perform worst on the double-well problem, which the authors note aligns with their empirical experience.
- The reparametrisation trick performs best on 1D Brownian motion but BEL average is best in 10 dimensions.
Limitations
- Variance-optimal alpha is derived only for the simplified Brownian-motion setting.
- Time-reversal and Gaussian approximations are compared empirically only on small experiments; the forward-KL objective has different finite-sample properties.
Experimental setup
Datasets: Fashion-MNIST simulated Brownian motion, double-well and shape SDE trajectories
Benchmarks: Brownian motion bridge (1D, 10D) double-well SDE (1D, 10D) shape process bridge Fashion-MNIST upper-left corner conditioning
Metrics: Dist MV mean pointwise distance
Baselines: Reparametrization Trick Time reversal Adjoint paths BEL last BEL first BEL average
Builds on: Bismut-Elworthy-Li formula Tweedie formula denoising score matching reparametrisation trick for stochastic optimal control Doob h-transform
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