arxiv: 2604.16919 · v1 · submitted 2026-04-18 · 💻 cs.LG · cs.AI· cs.CV

Recognition: unknown

Noise-Adaptive Diffusion Sampling for Inverse Problems Without Task-Specific Tuning

Liangli Zhen, Setthakorn Tanomkiattikun, Yingzhi Xia, Zaiwang Gu

Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:34 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CV

keywords diffusion modelsinverse problemsHamiltonian Monte Carloposterior samplingnoise adaptationimage reconstructionBayesian inference

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The pith

Moving inference to the initial noise space lets diffusion models solve inverse problems robustly without task-specific adjustments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors show that inverse problems can be solved by sampling initial noises whose corresponding denoised images match the observations, using Hamiltonian Monte Carlo in that space. This approach sidesteps problems like getting stuck in local minima or producing images off the model's learned manifold. They further introduce an adaptive version that works even when the noise level or type in the measurements is unknown. A reader would care because it promises reliable, tuning-free reconstruction for tasks like image deblurring or inpainting using powerful diffusion priors. If true, it broadens the practical use of diffusion models for real-world inverse problems where conditions vary.

Core claim

N-HMC samples in the noise space by treating the entire reverse diffusion as a deterministic mapping from initial noise to clean image, enabling Hamiltonian Monte Carlo to explore the posterior over solutions while staying on the data manifold. NA-NHMC extends this with noise adaptation to handle unknown noise without tuning. This yields superior reconstruction quality and robustness across hyperparameters and initializations on multiple inverse problems.

What carries the argument

Noise-space Hamiltonian Monte Carlo (N-HMC), which moves the sampling process into the space of initial noises by viewing reverse diffusion as a fixed mapping to images.

Load-bearing premise

The assumption that shifting all inference to the initial noise space keeps every proposal on the data manifold and allows full exploration of solutions without introducing new problems.

What would settle it

Observing that NA-NHMC produces reconstructions with higher error or less robustness than existing diffusion-based solvers on a standard inverse problem benchmark would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.16919 by Liangli Zhen, Setthakorn Tanomkiattikun, Yingzhi Xia, Zaiwang Gu.

**Figure 2.** Figure 2: Gaussian deblur task on FFHQ (256 × 256) with varying measurement noise levels σy. The estimated standard deviation of measurement noise ∥y − A(xˆ0)∥/ √ m demonstrates that our noise-adaptive method accurately recovers the true σy (indicated by dashed line) without overfitting across different noise levels. where m denotes the dimensionality of the measurement space. The derivation of this expression … view at source ↗

**Figure 3.** Figure 3: Comparative results are averaged over 100 independent runs. (Top) Mean absolute error (MAE) heapmaps. (Bottom) Standard deviation heatmaps across runs. Our method achieves the lowest standard deviation compared to DPS and DAPS, indicating reduced sensitivity to initialization. Main Results. Our method achieves comparable or superior performance across most tasks, as measured by PSNR and SSIM on both the … view at source ↗

**Figure 4.** Figure 4: Nonlinear deblurring results on FFHQ (256×256) dataset with σy = 0.2. Visual comparison across state-of-the-art methods shows our approach produces high-quality reconstructions with sharp details and minimal artifacts. 4.2 HIGHLY ILL-POSED IPS: PHASE RETRIEVAL Another challenge commonly faced by both MAP and sampling-based methods is becoming trapped in a local mode, particularly in highly multimodal IPs … view at source ↗

**Figure 5.** Figure 5: Phase retrieval task on FFHQ (256 × 256) with σy = 0.01. Each curve shows the median performance, with shaded areas denoting the 5th–95th percentile interval. Our method successfully solves the IP at a much higher rate than DPS and DMPlug. This is due to the annealing schedule of σy that allows for initial exploration of the noise space, resulting in a lower probability of being stuck on a local mode. We q… view at source ↗

**Figure 6.** Figure 6: Nonlinear deblurring results on FFHQ (256×256) dataset under different noise conditions. (Top) Impulse noise. (Bottom) Speckle noise [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Performance of NA-NHMC across four tasks for FFHQ [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: Performance of NA-NHMC on SR (×4) task for FFHQ (256 × 256) as a function of the number of diffusion steps. The initial step is fixed at T = 750 for all cases to avoid numerical instability, and the remaining steps are evenly spaced in [0, 750]. Ground Truth Measurement step size = 0.05 step size = 0.005 step size = 0.0005 1000 steps 1500 steps 8000 steps Discretization Error 0 % 100 % [PITH_FULL_IMAGE:fi… view at source ↗

**Figure 9.** Figure 9: Unadjusted Langevin Algorithm (ULA) with different step sizes. Larger step sizes accel [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

**Figure 10.** Figure 10: Comparison between GAN and Diffusion Model (DDIM) for SR [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗

**Figure 11.** Figure 11: SR(×4). (Top) FFHQ (256 × 256). (Bottom) ImageNet (256 × 256). σy = 0.05 23 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗

**Figure 12.** Figure 12: SR(×16). (Top) FFHQ (256 × 256). (Bottom) ImageNet (256 × 256). σy = 0.05 Ground Truth Measurement DiffPIR RED-diff DPS DAPS ReSample DMPlug Ours [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗

**Figure 13.** Figure 13: Random inpainting. (Top) FFHQ (256 × 256). (Bottom) ImageNet (256 × 256). σy = 0.05 24 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗

**Figure 14.** Figure 14: Gaussian deblurring. (Top) FFHQ (256 × 256). (Bottom) ImageNet (256 × 256). σy = 0.05 Ground Truth Measurement DiffPIR RED-diff DPS DAPS ReSample DMPlug Ours [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗

**Figure 15.** Figure 15: Nonlinear deblurring. (Top) FFHQ (256 × 256). (Bottom) ImageNet (256 × 256). σy = 0.05 Ground Truth Measurement DiffPIR RED-diff DPS DAPS ReSample DMPlug Ours [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗

**Figure 16.** Figure 16: Phase retrieval. FFHQ (256 × 256). 25 [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗

**Figure 17.** Figure 17: HDR reconstruction. (Top) FFHQ (256 × 256). (Bottom) ImageNet (256 × 256). σy = 0.05 26 [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗

read the original abstract

Diffusion models (DMs) have recently shown remarkable performance on inverse problems (IPs). Optimization-based methods can fast solve IPs using DMs as powerful regularizers, but they are susceptible to local minima and noise overfitting. Although DMs can provide strong priors for Bayesian approaches, enforcing measurement consistency during the denoising process leads to manifold infeasibility issues. We propose Noise-space Hamiltonian Monte Carlo (N-HMC), a posterior sampling method that treats reverse diffusion as a deterministic mapping from initial noise to clean images. N-HMC enables comprehensive exploration of the solution space, avoiding local optima. By moving inference entirely into the initial-noise space, N-HMC keeps proposals on the learned data manifold. We provide a comprehensive theoretical analysis of our approach and extend the framework to a noise-adaptive variant (NA-NHMC) that effectively handles IPs with unknown noise type and level. Extensive experiments across four linear and three nonlinear inverse problems demonstrate that NA-NHMC achieves superior reconstruction quality with robust performance across different hyperparameters and initializations, significantly outperforming recent state-of-the-art methods. The code is available at https://github.com/NA-HMC/NA-HMC.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's core move is running HMC in the initial noise space of a diffusion model so that proposals stay on the manifold while sampling the posterior for inverse problems.

read the letter

The main new piece is N-HMC, which treats the reverse diffusion process as a fixed deterministic map from an initial noise vector to a clean image. They then run Hamiltonian Monte Carlo entirely in that noise space, folding in the measurement likelihood at each step. This setup is meant to avoid the local-minima traps of optimization methods and the manifold-infeasibility problems that come from enforcing consistency during denoising. The noise-adaptive extension, NA-NHMC, adds a way to handle unknown noise levels and types without per-task retuning. They claim a theoretical analysis plus tests on four linear and three nonlinear inverse problems, with code released on GitHub. The reported gains in reconstruction quality and robustness to hyperparameters and random seeds are the practical payoff they emphasize. What the work does cleanly is shift the sampling domain so that every proposal is guaranteed to land on the learned data manifold. That is a direct response to the limitations they cite in earlier diffusion-based solvers. Releasing code also makes the claims easier to check. The soft spots are modest but real. The theoretical analysis is asserted in the abstract, yet its depth on mixing times or approximation error in high-dimensional noise space is not visible here and would need verification. For the nonlinear cases, the exact form of the likelihood term inside the HMC dynamics matters, and any hidden assumptions about the forward model could affect how general the gains really are. Minor implementation details around step-size adaptation or burn-in could also influence the robustness numbers. This is aimed at researchers who already use diffusion priors for imaging or signal recovery and want a Bayesian sampler that does not require heavy per-problem tuning. Readers who care about practical posterior sampling with pre-trained models will find the formulation and the multi-problem experiments useful. The paper deserves a serious referee because the noise-space formulation is distinct from the optimization and consistency baselines it compares against, and the experimental scope is wide enough to justify detailed review. I would send it out rather than desk-reject.

Referee Report

0 major / 3 minor

Summary. The paper proposes Noise-space Hamiltonian Monte Carlo (N-HMC), which treats the deterministic reverse diffusion process as a fixed mapping from initial noise to images on the learned data manifold, and performs posterior sampling via HMC entirely in the initial-noise space to solve inverse problems. This is extended to a noise-adaptive variant (NA-NHMC) that handles unknown noise type and level without task-specific tuning. The authors provide a theoretical analysis of the approach and report extensive experiments on four linear and three nonlinear inverse problems, claiming superior reconstruction quality, robustness to hyperparameters and initializations, and outperformance of recent state-of-the-art methods.

Significance. If the theoretical analysis is sound and the experimental protocols are reproducible, the method offers a principled way to avoid local minima and manifold-infeasibility issues common in optimization-based and consistency-enforcing diffusion approaches for inverse problems. The noise-adaptive extension without per-task tuning could broaden applicability, and the open-source code supports verification.

minor comments (3)

§3.2: clarify the precise form of the Hamiltonian and the discretization scheme used for N-HMC, including any Metropolis-Hastings correction steps, to ensure the sampler is exactly targeting the posterior.
§4.3 and Table 2: the reported PSNR/SSIM gains for NA-NHMC versus baselines should include standard deviations over multiple random seeds and initializations to substantiate the robustness claim.
[§5] §5: the theoretical analysis would benefit from an explicit statement of the assumptions under which the noise-space proposals remain on the manifold (e.g., regarding the diffusion model's training distribution and the forward operator).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. We appreciate the recognition of N-HMC's ability to perform posterior sampling in noise space to avoid local minima and manifold infeasibility, as well as the potential of the noise-adaptive NA-NHMC variant.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces N-HMC by reinterpreting the deterministic reverse diffusion process as a fixed mapping from initial noise to manifold images, then performing HMC sampling directly in that noise space to target the posterior. This construction is presented as an independent methodological shift that avoids local minima and manifold infeasibility without relying on fitted parameters renamed as predictions or self-referential definitions. The noise-adaptive extension NA-NHMC is described as a practical generalization with its own analysis. No load-bearing step reduces by construction to prior inputs or self-citations; the central claims rest on the proposed sampling strategy, theoretical analysis, and cross-problem experiments, which remain externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on domain assumptions about diffusion model manifolds and the benefits of noise-space sampling; no free parameters or invented physical entities are described in the abstract.

axioms (2)

domain assumption Diffusion models provide strong priors for Bayesian approaches to inverse problems
Stated as background motivation in the abstract.
domain assumption Enforcing measurement consistency during the denoising process leads to manifold infeasibility issues
Presented as a known limitation of prior Bayesian methods.

invented entities (2)

Noise-space Hamiltonian Monte Carlo (N-HMC) no independent evidence
purpose: Posterior sampling method that treats reverse diffusion as a deterministic mapping from initial noise to clean images
Newly proposed sampling framework.
Noise-adaptive variant (NA-NHMC) no independent evidence
purpose: Extension that handles inverse problems with unknown noise type and level
Adaptive extension of the core method.

pith-pipeline@v0.9.0 · 5517 in / 1450 out tokens · 50765 ms · 2026-05-10T07:34:24.508641+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages

[1]

2021 , url =

URLhttps://arxiv.org/abs/1907.05600. Yang Song, Liyue Shen, Lei Xing, and Stefano Ermon. Solving inverse problems in medical imag- ing with score-based generative models, 2022b. URLhttps://arxiv.org/abs/2111. 08005. 12 Published as a conference paper at ICLR 2026 Phong Tran, Anh Tuan Tran, Quynh Phung, and Minh Hoai. Explore image deblurring via encoded b...

work page doi:10.1109/cvpr46437.2021.01178 1907
[2]

The measurement is given by y=A(x ∗

Measurement noiseη∈R m follows gaussian distribution with unknownσ 2 y: p(y|xT , σ2 y) = 1 (2πσ 2y)m/2 exp − ∥y− A(D(x T ))∥2 2σ2y .(12) 2.σ y follows a Jeffreys prior distribution: p(σ2 y)∝ 1 σ2y .(13) Marginalizingσ 2 y yields p(y|xT ) = Z ∞ 0 p(y|xT , σ2 y)p(σ 2 y)dσ 2 y (14) ∝ Z ∞ 0 1 (2πσ 2y)m/2 exp − ∥y− A(D(x T ))∥2 2σ2y 1 σ2y dσ2 y (15) ∝ Z ∞ 0 (σ...
[3]

In the following proofs,Ais assumed to be approximately linear aroundx ∗

+η, η∼ N(0, σ 2 yIm),(19) whereA:R n →R m is the measurement operator andηrepresents Gaussian measurement noise. In the following proofs,Ais assumed to be approximately linear aroundx ∗
[4]

Generative model.Consider the DDIM sampler defined by ˆx0 =D(x T ),x T ∼ N(0,I n),(20) whereDdenotes the deterministic decoder via the diffusion model

Thus,A(x 0) =Ax 0. Generative model.Consider the DDIM sampler defined by ˆx0 =D(x T ),x T ∼ N(0,I n),(20) whereDdenotes the deterministic decoder via the diffusion model. Lemma 1.Product of two Gaussian probability density functions (PDFs). q1(x) =N(x;µ 1,Σ 1), q 2(x) =N(x;µ 2,Σ 2). Then, the product ofq 1(x)andq 2(x)is proportional to a Gaussian PDFN(x, ...

work page arXiv 2026