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Provable Posterior Sampling with Denoising Oracles via Tilted Transport
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Introduces tilted transport, which uses the quadratic log-likelihood of linear inverse problems plus a prior denoising oracle to map the posterior to a boosted posterior that is strongly log-concave under conditions on conditioning, SNR, and prior susceptibility.
Problem
Provable posterior sampling for linear inverse problems y = Ax + noise when the prior is only available via a denoising oracle (score) from diffusion models; existing heuristic posterior samplers lack quantitative guarantees.
Approach
Exploit the quadratic structure of the Gaussian log-likelihood: a time-varying quadratic tilt of the prior evolves under the OU/heat semigroup via a Riccati-type ODE, so the posterior is exactly equivalent to sampling a boosted posterior at an intermediate time T, which is strongly log-concave when the prior's susceptibility is controlled and the measurement matrix is well conditioned; then run th
Key results
- Tilted transport sufficient condition gives strong log-concavity of the boosted posterior when chi_{||Q||}(pi) < ||Q||^{-1} kappa/(kappa-1); for Ising models the guarantee matches the threshold lambda_max(Q)-lambda_min(Q)<1.
- On high-dimensional Gaussian mixtures, boosted Langevin (LMC plus tilted transport) yields lower sliced Wasserstein error than plain LMC across SNR from 1e-5 to 1e-1, especially at low and high SNR.
- On the 128x128 phi^4 lattice, tilted transport accelerates Langevin autocorrelation decay for beta = 0.2, 0.4, 0.6.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply tilted transport to boost Langevin dynamics | Langevin dynamics · Gaussian mixture, d=20, 40, and 80 | Sliced Wasserstein distance | Helped |
| Apply tilted transport to boost Langevin dynamics | Langevin dynamics · Gaussian mixture, mid-SNR regime | Sliced Wasserstein distance | Helped |
| Use the tilted posterior near blowup time instead of the original posterior | Langevin dynamics · Scalar field φ⁴, 128 × 128 lattice, β=0.2, 0.4, and 0.6 | Autocorrelation | Helped |
Surprising findings
- Even with a perfect prior denoising oracle, posterior sampling for Ising-type tilts is computationally hard once the conditioning is poor (gap >= 1 + epsilon), so generic guarantees require well-conditioned measurements.
Limitations
- Guarantees are in continuous time; discrete-time complexity and score singularity near t=0 are not analyzed.
- Theory covers well-conditioned linear measurements; poorly conditioned problems such as inpainting are not handled.
- The susceptibility chi_t(pi) is generally not explicit and may be infinite for heavy-tailed priors.
Experimental setup
Datasets: synthetic Gaussian mixture priors (25 components) 2D lattice scalar phi^4 field
Benchmarks: high-dimensional Gaussian mixture posteriors (d=20,40,80) 2D lattice scalar phi^4 model (128x128)
Metrics: sliced Wasserstein distance autocorrelation relaxation time
Baselines: Langevin Monte Carlo importance sampling Glauber dynamics
Builds on: score-based diffusion models Bakry-Emery criterion Polchinski renormalization group flow stochastic localization
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