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Scale-Consistent Posterior Dynamics for Diffusion Inverse Problems

Zhaoqiang Liu, Tongyao Pang, Ruibing Wang, Yang Zheng

arXiv:2608.15144v1stat.MLcs.AIcs.LG

TL;DR

Posterior sampling with pretrained diffusion priors requires tractable likelihood handling without losing posterior behavior. This paper develops scale-consistent, corrected SDEs and variance-matched discretization, achieving strong reconstruction fidelity on FFHQ and ImageNet under 100 score evaluations.

  • Problem

    Posterior inverse-problem sampling lacks an available exact conditional score, motivating tractable reusable-prior methods that preserve posterior behavior.

  • Method

    The paper combines scale-consistent posterior proxies, frozen-target Langevin correction, and variance-matched split-step IMEX discretization with post-solve innovation.

  • Results

    Under a fixed 100-score-evaluation budget, the method achieves strong reconstruction fidelity on FFHQ and ImageNet, particularly for motion deblurring.

  • Takeaways & Limitations

    The analysis separates target consistency, continuous sampler tracking, and finite-step approximation as distinct requirements for posterior sampling.

  • Takeaways & Limitations

    The convergence and weak-error guarantees assume an exact score, while learned-score error generally does not vanish under grid refinement.

Abstract

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Posterior sampling with a pretrained diffusion prior is governed by a conditional score whose intermediate likelihood component is generally intractable. We begin from an ideal one-parameter posterior SDE family in which a stochasticity parameter controls probability-flow transport and stochastic exploration without changing the posterior marginals. To obtain a tractable model, we express the likelihood in a rescaled clean-image coordinate and use log-SNR to organize the resulting posterior proxies. Projecting the diffusion uncertainty through the forward operator then yields a noise-conditioned covariance path whose targets approach the clean posterior. Because endpoint consistency of these targets does not ensure that a surrogate transport follows them, we interleave the transport with a frozen-target Langevin corrector, producing a continuous surrogate SDE. We discretize this model with an outer Lie--Trotter splitting and a variance-matched split-step IMEX predictor that treats the learned prior explicitly, the linear likelihood implicitly, and the stochastic innovation after the implicit solve. We prove marginal invariance of the ideal family, posterior convergence of the continuous surrogate under mixing and transport-defect conditions, and a first-order weak error bound for the discrete algorithm. Experiments on FFHQ and ImageNet with 100 score evaluations demonstrate competitive reconstruction fidelity for super-resolution and deblurring. A controlled 100-image ablation separates scale consistency from the finite-step effects of stochastic-increment placement, continuation, and corrector allocation. A separate noiseless box-inpainting study shows that large exploration reaches a performance plateau only when the matched innovation is injected after the stiff likelihood solve.

1. Introduction.

The paper addresses the intractable intermediate likelihood score in diffusion inverse problems by constructing a scale-consistent, continuously corrected posterior surrogate. It combines this model with a variance-matched split-step IMEX discretization and establishes theoretical guarantees alongside empirical evaluations and ablations.

  • Motivation: The intermediate likelihood score is generally unavailable, making accurate and stable posterior guidance the central difficulty in diffusion inverse problems.Existing approaches trade generality and computational cost against likelihood-guidance accuracy and numerical stability.
  • Exact posterior family: An exact one-parameter posterior SDE family preserves posterior marginals while its stochasticity parameter controls the balance between probability-flow transport and stochastic exploration.ζ = 0 gives probability-flow transport, ζ = 1 the usual reverse SDE, and larger ζ increases exploration without changing ideal posterior marginals.
  • Scale-consistent surrogate: The tractable surrogate uses a rescaled clean-image coordinate, log-SNR continuation, noise-conditioned covariance targets, and a frozen-target Langevin corrector to achieve continuous posterior convergence.The corrector addresses the gap between consistency of continuation targets and convergence of transport marginals.
  • Numerical method: The discrete method applies Lie–Trotter predictor-corrector splitting with a variance-matched split-step IMEX predictor and centered Crank–Nicolson corrector.The likelihood is solved implicitly, the learned prior remains explicit, and calibrated stochastic innovation is added after the implicit solve without filtering by the likelihood resolvent.
  • Theory and experiments: The framework provides posterior convergence and first-order weak accuracy, while FFHQ and ImageNet experiments use 100 score evaluations to assess reconstruction fidelity and finite-budget trade-offs.Controlled ablations separate scale consistency, stochastic-increment placement, continuation, and predictor-corrector score allocation, including noiseless box inpainting.

2. Related Work.

Prior diffusion inverse-problem methods approximate likelihood guidance, exploit sensing-operator structure, or avoid pointwise likelihood scores through variational, intermediate-state, Monte Carlo, and annealing strategies. This work positions itself around an exact posterior SDE reference and three explicit approximation layers leading to a split-step sampler.

  • Related work: DPS evaluates measurement-error gradients at Tweedie denoised estimates, while related methods incorporate local data-manifold geometry or conditional clean-estimate covariance.These methods replace the unknown clean image with a denoised estimate before enforcing measurement consistency.
  • Related work: SNIPS, DDRM, and DDNM exploit sensing-operator structure by separating observed and unobserved or range-space components.SNIPS and DDRM use singular-vector decompositions, while DDNM preserves the measurement-determined range-space component and uses the diffusion prior for complementary components.
  • Related work: Alternative approaches use variational score-matching surrogates, intermediate-state guidance, auxiliary-variable or particle methods, annealing, and consistency-based solvers.RED-Diff, MGPS, Monte Carlo methods, DAPS, and consistency-based inverse solvers each avoid or modify direct pointwise likelihood-score approximation.
  • Related work: Continuation methods embed difficult problems in parameterized families, while Bayesian annealing bridges a tractable reference and posterior through intermediate distributions.Numerical continuation commonly tracks solutions with predictor and corrector steps, and score-based models provide progressively less smoothed distributions.
  • Related work: The paper uses an exact one-parameter posterior SDE reference and separates scale-consistent likelihood, covariance-path, and frozen-target Langevin-corrector approximations.A split-step IMEX discretization then converts the continuous surrogate into a practical predictor-corrector sampler.

3. Method.

The method constructs a scale-consistent continuation of tractable posterior targets, then combines transport with a frozen-target Langevin corrector. An outer Lie–Trotter and variance-matched split-step IMEX discretization yields a weakly convergent posterior sampler under stated transport, mixing, and score assumptions.

  • Ideal posterior SDE family: A stochasticity parameter ζ interpolates between posterior probability-flow transport and stochastic exploration while preserving the exact posterior marginals.ζ = 1 gives the stochastic posterior reverse SDE, while ζ = 0 gives the posterior probability-flow ODE; both converge weakly to the clean posterior at the endpoint.
  • Scale-consistent posterior continuation: A rescaled clean-coordinate likelihood proxy becomes a log-SNR continuation whose projected covariance softens uncertain measurement directions and vanishes toward the clean posterior.The continuation targets converge in total variation to p(· | y), with local-uniform score convergence under positive C1 prior and Gaussian-mollification assumptions.
  • Transport and correction: Because changing continuation targets need not be preserved by transport, the surrogate interleaves continuation transport with a frozen-target Langevin corrector of intensity γλ.The continuous surrogate retains the ideal exploration parameter while allocating auxiliary Langevin time γλdλ for target tracking.
  • Split-step IMEX discretization: Variance matching places stochastic innovation after the implicit likelihood solve, avoiding the covariance damping caused by injecting new noise inside a stiff backward-Euler resolvent.Backward Euler retains only (1 + ωkν)^−2 of injected variance in a likelihood eigenmode of curvature ν, whereas variance matching preserves the predictor’s first-order weak limit.
  • Convergence guarantees: The continuous surrogate approaches the posterior when the centered score interaction is controlled, and the discrete algorithm is first-order weakly accurate when η∗ = O(h) and δbud = O(h).The weak-error terms arise from discretization, numerical correction, and corrector-time discrepancy; the output law converges weakly to p(· | y).
  • Scope of guarantees: The convergence theorems assume an exact score, so learned-score error remains a separate modeling error that generally does not vanish under grid refinement.This limitation separates numerical convergence from approximation error introduced by the pretrained score network.

4. Experiments.

Experiments on FFHQ and ImageNet evaluate reconstruction across super-resolution and deblurring under a 100-score-evaluation budget. The sampler leads distortion metrics on most tasks, while ablations show task-dependent continuation, corrector, exploration, and innovation-placement effects.

  • Experimental setup: Experiments use 100 images per dataset for bicubic ×4 super-resolution and Gaussian or motion deblurring with σd = 0.05, reporting PSNR, SSIM, and LPIPS.FFHQ and ImageNet images are 256 × 256, with dataset-specific pretrained diffusion priors.
  • Experimental setup: The comparison includes DPS, DiffPIR, DDRM, RED-Diff, MGPS, and DAPS under matched outer levels and a 100-NFE budget for the proposed sampler.Corrector calls count toward the proposed budget, while repeated internal score calls for MGPS and DAPS are reported separately.
  • Main results: The sampler achieves the best PSNR and SSIM on all FFHQ tasks, plus the best Gaussian-deblurring PSNR and SSIM and motion-deblurring performance under all three metrics on ImageNet.DDRM remains strongest on ImageNet super-resolution, while DDRM and MGPS retain the best LPIPS on several FFHQ tasks.
  • Component ablation: SC increases PSNR by 10.001 dB on FFHQ and 13.776 dB on ImageNet, whereas NC changes FFHQ PSNR by 0.021 dB but improves ImageNet motion deblurring by 2.856 dB.These results separate scale consistency from continuation and show that continuation benefits depend on the task.
  • Component ablation: At the same 100-NFE budget, PC improves FFHQ PSNR slightly and reduces ImageNet LPIPS from .1968 to .1802, but lowers ImageNet PSNR by .179 dB.The corrector supplies the convergence-theorem mechanism while introducing a finite-budget fidelity–perception trade-off.
  • Split-step stress test: At ζ = 256, VM-IMEX reaches 19.410 dB missing-square PSNR and .2195 LPIPS, versus 6.527 dB and .8435 for direct BE in noiseless box inpainting.VM-IMEX injects matched innovation after the stiff likelihood solve, avoiding the degradation caused when direct BE filters it through the resolvent.

5. Conclusion.

The paper separates posterior-target accuracy, continuous-sampler tracking, and finite-step discretization, using variance matching and Langevin corrections to stabilize likelihood dynamics while preserving stochastic innovation.

  • Conclusion: The framework distinguishes whether the target approaches the posterior, whether the continuous sampler tracks that target, and whether discretization approximates continuous dynamics.These three questions are explicitly separated rather than conflated.
  • Conclusion: A variance-matched split-step predictor stabilizes the likelihood drift without filtering the stochastic innovation.The predictor addresses likelihood-drift stability while retaining stochastic innovation.
  • Conclusion: Langevin corrections track the evolving target, while projected diffusion uncertainty supplies counts for the likelihood covariance.The passage links target tracking to Langevin corrections and covariance construction to projected diffusion uncertainty.

Appendix A. Proof of the Posterior SDE Family.

The appendix proves that conditioning the initial distribution preserves the forward diffusion structure, while Gaussian smoothing yields smooth positive-time densities satisfying the forward Fokker–Planck equation. Reverse-time Fokker–Planck uniqueness then establishes equality of reverse-process marginals with the conditioned forward marginals, including convergence to the posterior at time zero.

  • Conditioning the initial distribution on the observation y does not alter the forward diffusion’s governing structure.
  • For every t > 0, Gaussian smoothing makes the conditioned transition density strictly positive and smooth, even without an initial density.
  • The smoothed conditioned density satisfies the forward Fokker–Planck equation.
  • Rewriting the reverse process in the original time variable produces a negative diffusion term, justified by the increasing reverse clock s = T − t.
  • Fokker–Planck uniqueness gives q_t = p_t(· | y) for all t ∈ (0, T], and forward-semigroup continuity yields p_t(· | y) ⇒ p_0(· | y) as t ↓ 0.

Appendix B. Proof of Terminal Target and Score Consistency.

The appendix proves that the rescaled terminal target converges to the clean posterior in total variation and that its surrogate score converges locally uniformly to the exact posterior score.

  • Terminal target consistency: The Gaussian-convolution rescaled prior converges to p0 in L1 because the kernels form an approximate identity.This establishes the prior component of endpoint consistency.
  • Terminal target consistency: Pointwise likelihood convergence and dominated convergence imply convergence of the normalizing constants, Zλ → Z∞.The likelihood factors remain in (0, 1], while the limiting normalizer is positive.
  • Terminal target consistency: Combining prior and normalizer convergence proves total-variation convergence of the normalized terminal target.The normalized unnormalized densities therefore converge to the stated limiting posterior.
  • Score consistency: Bayes’ rule and the Gaussian measurement model reduce score consistency to prior-score and likelihood-gradient differences.The score difference is expressed as ∇z log πλ(z) − ∇z log p0(z) − A⊤(Γ−1.
  • Score consistency: The prior-score difference converges locally uniformly by assumption, while boundedness on compact sets yields the same for the likelihood term.Together these limits prove the stated score convergence.

Appendix C. Proof of Continuous Posterior Convergence.

The proof tracks the surrogate through relative-entropy dissipation, showing that the continuous tracking error vanishes under stated mixing and defect conditions. Endpoint consistency then converts this entropy convergence into posterior convergence.

  • Fokker–Planck decomposition: The Fokker–Planck equation decomposes into the predictor operator and a frozen-target corrector operator.The decomposition uses (1 + ζ)qλ = qλ + ζqλ.
  • Entropy tracking: Cancellation in the entropy calculation leaves rλ as the relevant continuous tracking defect rather than a numerical local error.The cancellation follows from integrating by parts and using the evolution equation for ¯π.
  • Convergence bound: Under conditions (3.38), the integrating-factor Grönwall bound implies H(λ) → 0.A simpler sufficient condition is obtained by splitting the convolution at a fixed large value and using kλ ≥ k∗ > 0 on the remaining tail.
  • Posterior convergence: Pinsker’s inequality, the triangle inequality, and endpoint consistency yield convergence to the posterior.The final step combines entropy control with the endpoint consistency established in Subsection 3.2.

Appendix D. Proof of the Weak Error Bound for the Complete Scheme.

Appendix D proves the complete scheme’s weak-error bound by comparing exact and numerical kernels through local Lie–Trotter, predictor, and corrector estimates. The proof telescopes these discrepancies across levels and corrector steps to establish Theorem 3.4.

  • Kernel decomposition: The proof defines exact continuous, split, predictor, corrector, and numerical kernels, then compares them level by level.These kernels provide the decomposition used to analyze one transition of Algorithm 3.1.
  • Local estimates: Local weak estimates control Lie–Trotter splitting, the semi-implicit predictor, and the centered-implicit corrector under smoothness, moment, and fixed-interval regularity assumptions.The predictor estimate uses a stochastic Taylor expansion, while the corrector treats its likelihood implicitly and prior score explicitly.
  • Variance-matched predictor: The variance-matched split step preserves the required local order because moving the innovation covariance after the likelihood solve changes covariance only by O(hk^2).This follows from τk+1 = τk e−hk and Rk = I + O(hk), under bounded coefficients.
  • Corrector discretization: Replacing an exact corrector of duration Mk+1ηk+1 with Mk+1 numerical steps costs at most CMk+1ηk+1^2.The argument combines telescoping over corrector steps with smooth dependence of the corrector semigroup on auxiliary time.
  • Global weak-error bound: Telescoping exact and numerical kernels over the full interval, using backward Kolmogorov regularity and moment stability, proves Theorem 3.4.The accumulated estimates are substituted into the global comparison to obtain equation (3.41).
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