Noise, Denoise, Correct: MCMC Posterior Sampling with Diffusion Priors in Three Steps

arXiv:2610.09407 · cs.LG · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Noise, Denoise, Correct".

Jane: The gist: Diffusion waltz, an MCMC method using SDEdit-style noising–denoising as a proposal, corrected via Metropolis–Hastings for exact posterior sampling without prior evaluation,

Tom: First, who's behind it and why it matters.

Paper summary: Jane: So, thinking about the title "Noise, Denoise, Correct: MCMC Posterior Sampling with Diffusion Priors in Three Steps," what’s the simplest way to put this work?

Tom: It's an MCMC method that uses diffusion priors to sample from a posterior exactly without needing a prior evaluation, and it handles forward models that aren't differentiable.

Lu: The diffusion waltz structure is simple: you noising, then denoising, then a Metropolis–Hastings step to correct the sample.

Meng: The conditional noising extension is the part that makes it practical for data-driven problems because it lets you inject observation information into that proposal using a gradient-free ensemble Kalman update.

Jane: So, if you only listen to this show, what does this mean for someone who just wants to know what's new? It means they're getting a framework where they can sample from complex posteriors exactly on problems that are notoriously hard because the forward operator is non-differentiable.

Tom: The title says "Noise, Denoise, Correct," which points to the three steps of the method: first adding noise, then denoising back down, and finally correcting it via Metropolis–Hastings.

Lu: It's about taking something inspired by SDEdit for proposal generation and making it exact through that MH correction step based only on the likelihood.

Meng: For deployment at scale, the key thing is that they show how to approximate the Kalman gain K(h) using an ensemble of particles to avoid needing gradients of A directly.

Jane: And what this changes for us is that we can now tackle tasks like recovering initial conditions for Navier–Stokes where standard methods struggle with high nonlinearity and non-differentiability by using this diffusion waltz approach.

Tom: We've seen it performs strongly on all scores across different noise and nonlinearity regimes on the Navier–Stokes initial condition recovery task, ranking first or second against other methods.

Lu: It’s a method that matches or outperforms strong derivative-free baselines in both accuracy and calibration for that specific problem.

Meng: The limitation they flag is that the computation of K(h) still requires running an ensemble of J MCMC chains in parallel, so it adds some overhead when you try to run it at production scale.

Jane: So, the big picture is a simple method for exact posterior sampling that works even when things are nonlinear and non-differentiable, with observation information injected into the proposal to make it more effective.

Conclusion: Tom: So, we’ve been looking at this paper from arXiv titled "Noise, Denoise, Correct: MCMC Posterior Sampling with Diffusion Priors in Three Steps."

Jane: The core idea is that they've put together a three-step cycle for sampling from a posterior exactly using diffusion ideas.

Lu: Exactly. It proposes constructing a proposal by first noising the data, then denoising it back down, and finally applying a Metropolis–Hastings step to correct the sample.

Meng: It’s about getting that exact sample without needing an upfront prior evaluation, which is what makes it interesting for real-world applications.

Tom: So, if I boil it down for someone listening on the bus: this method uses diffusion as a trick to generate proposals that stay close to the data manifold, and then a final check ensures you land in the right spot.

Jane: That sounds like it tackles a big problem when your forward model isn't smooth or differentiable.

Lu: Right. They do this specifically for tasks where the forward operator is non-differentiable, like in some physical systems.

Tom: And they tested it on Navier–Stokes initial condition recovery problems, which are known for being really nonlinear and messy.

Jane: The results show that this Diffusion Waltz method performs very well across different noise levels and nonlinearity settings.

Meng: They consistently ranked first or second against other methods on those tests, which is significant because we’re often seeing gradient-based guidance fail in high nonlinearity scenarios.

Lu: What really stands out to me is the conditional noising extension they added later on, which lets you put observation information directly into that proposal.

Tom: That's what makes it more practical for real data problems, because you’re not just sampling blindly; you’re using what you see.

Jane: So, in short, this paper gives us a robust framework for exact posterior sampling on hard problems by combining diffusion priors with observation-informed proposals.

Meng: It's a simple plug-and-play structure, but the conditional noising is what really boosts the mixing and convergence speed to the stationary distribution.

Lu: The authors show that this approach can match or even beat derivative-free baselines on complex initial condition recovery tasks.

Tom: We’ve seen it matches or outperforms those derivative-free methods in both accuracy and calibration for that specific problem, which is a strong signal.

Jane: It suggests that using these diffusion ideas systematically can be a much better way to handle complex inverse problems than relying solely on standard optimization techniques.

Meng: The authors did acknowledge that the conditional noising still needs running an ensemble of MCMC chains, so it adds some computational cost at scale.

Lu: That’s a fair point about the overhead; you can’t avoid the necessary sampling steps entirely if you want exactness.

Tom: So, it's a powerful tool, but we gotta keep an eye on how much that ensemble computation hurts when we try to deploy this on massive datasets.

Jane: Exactly. The next thing to watch is whether this method scales up well enough for the huge problems we’re actually trying to solve in science and engineering.

Takao Gregory David Bellchambers, Sanmitra Ghosh, Michalis Michaelides

PhysicsX

cs.LG

Submitted: 2026-10-07

Updated: 2026-10-07

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 89/100

The gist: The gist: Diffusion waltz, an MCMC method using SDEdit-style noising–denoising as a proposal, corrected via Metropolis–Hastings for exact posterior sampling without prior evaluation, outperforms

Key concepts

Diffusion Waltz Method: A three-step MCMC cycle: (i) noise the current state, (ii) denoise it back to a new sample, and (iii) use Metropolis–Hastings to accept or reject the new sample. This is inspired by SDEdit and aims for exact posterior sampling without prior evaluation.
Conditional Noising Extension: An enhancement that incorporates observation data ($y$) into the noising step. Instead of simple noise, it uses a kernel like Ensemble Kalman Conditional Noising (EKCN) to propose a new state that nudges the current estimate towards the observed data, improving proposal quality.
Statistical Linearisation and EKCN Kernel: This technique approximates complex nonlinear forward operators by linearizing them. This allows for gradient-free computation of necessary components like Jacobians and Kalman gains, making the proposal kernel tractable even when explicit derivatives are unavailable.
Navier–Stokes Initial Condition Recovery: The specific problem tested where the forward operator is non-differentiable. Diffusion waltz and its extension are validated here, showing they perform strongly across various noise levels and nonlinearity regimes.

Summary

The gist: Diffusion waltz, an MCMC method using SDEdit-style noising–denoising as a proposal, corrected via Metropolis–Hastings for exact posterior sampling without prior evaluation, outperforms existing baselines across different noise and nonlinearity regimes on a non-differentiable Navier–Stokes initial condition recovery task.

Diffusion waltz Method

Diffusion waltz is introduced as a three-step cycle to sample from the posterior exactly, assuming a perfect denoiser. The proposal kernel q(x′0 x0) is constructed by (i) noising x0 to an appropriate level t, then (ii) denoising back to level 0 to yield a new sample x′0, and finally (iii) a Metropolis–Hastings (MH) accept/reject step. The first two steps—perturbing x0 via noising/denoising to produce a new sample x′0 that remains close to it on the data manifold—are inspired by SDEdit [Meng et al., 2022]. The third step is an MH correction, which guarantees that the resulting chain converges to the posterior as the number of iterations grows, and crucially, the acceptance probability in this step depends only on the likelihood. This follows from the reversibility of the proposal kernel with respect to the generative prior pθ(x0), i.e., q(x′0 x0)pθ(x0) = q(x0 x′0)pθ(x′0), which cancels the prior terms in the acceptance ratio.

Conditional Noising Extension

An extension is proposed to take into account observation information into the proposal, where the noising step is replaced by xt ∼ q(xt x0, y) for an arbitrary noising kernel q that depends on y. The Ensemble Kalman conditional noising (EKCN) kernel is used as an example of such a proposal, defined by xt = αt x0 + g(x0, y; h) + βtx1, where g(x0, y; h) nudges A(x0) towards y with learning rate h > 0. The ensemble Kalman Levenberg-Marquardt (EK-LM) update rule is employed to define g, which minimizes the data misfit loss J (x) = 1/2 y − A(x)2Γ. This results in a proposal kernel q(xt x0, y) that is reversible with respect to the posterior pθ(x0 y), which ensures tractability provided q can be evaluated.

Statistical Linearisation and EKCN Kernel

The ensemble Kalman conditional noising (EKCN) kernel is derived by applying statistical linearisation to approximate the nonlinear forward operator A(x) ≈ Ax + b. This allows for the computation of an approximate Jacobian DA(x) ≈ (Cxy)⊤(Cxx−1), which can be computed without explicitly taking derivatives. The resulting Kalman gain K(h) is approximated in a gradient-free manner using ensembles, leading to the expression K(h) ≈ hCxy (hCyy + Γ−1). This approximation is made tractable by using a pushthrough identity to reduce the matrix inverse complexity. The full conditional noising waltz kernel Kθ is shown to be reversible with respect to pθ(x0 y), meaning Kθ(x0 → x′0)pθ(x0 y) = Kθ(x′0 → x0)pθ(x′0 y).

Experimental Validation

The method is validated on the Navier–Stokes initial condition recovery problem, where the forward operator A is non-differentiable. The experiments explore different regimes in terms of the level of nonlinearity (controlled by T) and noise intensity (controlled by σy). Diffusion waltz with ensemble Kalman conditional noising (DW-EKCN) performs strongly on all scores across all regimes, consistently ranking first or second. Specifically, DW-EKCN performs strongly on all scores across all regimes, consistently ranking first or second. The conditional noising is instrumental in accelerating convergence to the stationary distribution, leading to rapid convergence within the first ∼ 50 MCMC steps.

Comparative Performance

In comparison against standard gradient-based guidance methods (e.g., DPS) using a differentiable neural surrogate for the NS system, DW-EKCN and other MCMC based approaches mostly outperform these methods. While DPS and MMPS achieve competitive performance and perform the best in the low nonlinearity / low noise setting, they are mostly outperformed by MCMC based methods in other settings, especially high nonlinearity. The conditional noising is instrumental in accelerating convergence to the stationary distribution.

Conclusion

Diffusion waltz provides a simple plug-and-play framework for posterior sampling via MCMC that is asymptotically exact (assuming a perfect denoiser) and handles non-differentiable forward models. Its conditional noising extension injects observation information into the proposal while preserving this exactness, substantially improving mixing. On Navier–Stokes initial condition recovery, diffusion waltz matches or outperforms strong derivative-free baselines in both accuracy and calibration.

Acknowledgments and Disclosure of Funding

We would like to thank Martin Accou, Andrew Boardman, Thomas Gessey-Jones, Ayush Jain, James-Michael Leahy and Daniel Owen-Lloyd for the insightful discussions and careful review of the manuscript.

References

Bilal Ahmed and Joseph G Makin. Solving diffusion inverse problems with restart posterior sampling.

Michael Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.

Christophe Andrieu and Johannes Thoms. A tutorial on adaptive MCMC.

Gregory Bellchambers. Exploiting the exact denoising posterior score in training-free guidance of diffusion models.

Gabriel Cardoso, Sylvain Le Corff, Eric Moulines, et al. Monte carlo guided denoising diffusion models for Bayesian linear inverse problems.

Hyungjin Chung, Jeongsol Kim, Michael T Mccann, Marc L Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems.

Florentin Coeurdoux, Nicolas Dobigeon, and Pierre Chainais. Plug-and-play split Gibbs sampler: embedding deep generative priors in Bayesian inference.

Simon L Cotter, Gareth O Roberts, Andrew M Stuart, and David White. MCMC methods for functions: modifying old algorithms to make them faster.

Giannis Daras, Hyungjin Chung, Chieh-Hsin Lai, Yuki Mitsufuji, Jong Chul Ye, Peyman Milanfar, Alexandros G Dimakis, and Mauricio Delbracio. A survey on diffusion models for inverse problems.

Prafulla Dhariwal and Alexander Nichol. Diffusion models beat GANs on image synthesis.

Simon Duane, Anthony D Kennedy, Brian J Pendleton, and Duncan Roweth. Hybrid Monte Carlo.

Geir Evensen.

Improvements for AI systems

textbfDiffusion Waltz for Exact Posterior Sampling on Non-Differentiable Models (DW-EKCN): Enables Accurate Inference Under Complex Conditions. This method targets the posterior exactly, assuming a perfect denoiser and handles non-differentiable forward models, allowing inference in scientific applications where standard gradient methods fail. Furthermore, its conditional noising extension inject[s] observation information into the proposal while preserving this exactness.
textbfAdaptive MCMC Tuning for Robust Sampling Convergence. The paper proposes using the Robbins-Monro method to adaptively tune hyperparameters, stating that adaptively tuning the hyperparameters starting from high t, h improves the performance and drives parameters to smaller values as chains converge to higher likelihood regions. This prevents issues where decreasing t creates another problem, where the acceptance rate tends to become vanishingly small.
textbf Ensemble Kalman Conditional Noising (EKCN) for Enhanced Mixing. EKCN waltz is shown to accelerate convergence by guiding particles: EKCN guides each ensemble member towards high-likelihood regions, leading to rapid convergence within the first ∼ 50 MCMC steps. This results in higher acceptance rates initially and better mixing compared to RWMH waltz, as evidenced by trace plots showing slow convergence of RWMH waltz is due to many chains getting stuck early on.
textbf Tractable Computation of Proposals via Ensemble Kalman Updates. The EKCN kernel allows for the computation of the proposal using ensembles: we use all 100 members for the computation of the Kalman gain (Appendix C.1) and show that with a size J ensemble, the computation of (72) can be made tractable. This makes complex, data-informed proposals feasible even in high-dimensional observation spaces.
textbf Derivative-Free Inference for Non-Differentiable Forward Operators. By employing techniques like the Ensemble Kalman Levenberg-Marquardt (EK-LM) update rule, the system can minimize data misfit in a derivative free manner. This allows for inference on problems where the forward operator A is non-differentiable, such as in Navier–Stokes simulations.

Abstract

Pretrained diffusion models are powerful priors for inverse problems, but posterior sampling under nonlinear, non-differentiable forward models remain hard. We introduce diffusion waltz, an MCMC method using SDEdit-style noising-denoising as a proposal, corrected via Metropolis-Hastings for exact posterior sampling without prior evaluation. We further propose injecting observations into the proposal while preserving exactness, using a gradient-free ensemble Kalman update. On a non-differentiable Navier-Stokes initial condition recovery task, diffusion waltz outperforms existing baselines across different noise and nonlinearity regimes.