Noise-Adaptive Diffusion Sampling for Inverse Problems Without Task-Specific Tuning
Authors: Yingzhi Xia
Setthakorn Tanomkiattikun
Liangli Zhen
ZAIWANG GU
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion models
Inverse problems
Generative model
Bayesian inference
Primary area: generative models

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.

Published as a conference paper at ICLR 2026
NOISE-ADAPTIVE DIFFUSION SAMPLING FOR IN- VERSE PROBLEMS WITHOUT PASK-SPECITE TUNING
Yingzhi Xia' Setthakorn Tanomkiattikun 1.3+ Liagli Zhen' Zaiwang Gu²
"Institute of High Performance Computing, Agency for Science, Technology and Research, Singapore Instiiute for Infocomm Research, Agency for Science, Technology and Research, Singapore
3 Johns Hopkins University {Xia_ Singging zhen.liangli, Gu.Zaiwang] sastt edu. sg
stanomk 10 juuu. g.u..ed. edu
ABSTRACT
Diffusion models (DMs) have recently shown remarkable performance on inverse problems (IPs). Aptimization-basee methods can fast solve IPs using DMs as powerful regularizers, but they are susceptible to local minima and noise overfit- ting. Although DMs can provide strong priors for Bayesian approaches, enforcing measurement consistency during the denoising process leads to manifold infea- sibility issues. We propose boise-space Hamiltonian Monte Carlo (N-HMC), posterior sampling method that treats reverse diffusion as a deterministic map- ping from initial noise clean images. N-HMC enables comprehensive explo- ration of the solution space, avoiding local optima. By moving inference entirely into the initial-noise space, N-HMC keeps proposals on the learned data mani- fold. We provide a comprehensive theoretical analysis of our approach and extend the framework to adaativ 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 supe- rior reconstruction quality with robust performance across different hyperparam- eters and initializations, significantly outperforming recent state-of ttae-ooftthe-at art meth- ods. The code is available at htttpps: /etttp::/qithuee com/ NA- HMC NA
NA-HMC HMC
INTRODUCTION
Inverse problems (IPs) have wide applications many domains, including computer vision (Janii et al. 2021:
Quan et al. 2024), protein science (Yi et al. [023] Ouuang--Zang et al.] 2023 Yang
ct al. 2019 medical imaging (Song et al.] 2022b] Chu et al. 2025 Dao et al. 2024). scientific
computing Zheng et al. 2025 Xia & Zabaras 2022 Xu et al. 2024 The goal is to reconstruct an
unknown x: E 12" from noisy measurements y R.
= A(x) +
where A is a known forward operator, and R is additive noise. Diffusion models (DMs) have recently shown powerful capabilities in modeling complex data distributions, which can provide powerful class of priors for high-dimensional data in solving IPs.
Existing diffusion- based methods have demonstrated remarkable success across diverse inverse problems (Chung et al., 2023 Daras et al.) 2024 Zheng et al.) [225 [Song et al. (022b). Although methods suf- fer from three complementary limitations and issues: (1) Iterative guidance methods such as DPS (Chung et al.) 2023). DDRM (Kawar et al., (022). MONMM(Wang et al., 2023), TIGDM (Song al.) 2023), and TMPD (Boys et | 2024) use the likelihood term to shift intermediate images directly, systematically pushing intermediate states off the learned data manifold and violating the training- time noise-conditioning of the denoiser, resulting in various failure reconstructions like accumulated artifacts as shown in Figure (a). (2) Stochastic MAP methods that optimize in image space. includ- ing ReSample (Song et 2024). DifIPIR (Zhu al. 2023), DAPS (Zhang al. 2024), SITCOM
remarkable progress has been made, as illustrated Figure I current diffusion
"qqual contribution.
Corresponding author.

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Published as a conference paper at ICLR 2026
match well with very sharp details but require carefully tuned hyperparameters to not overfit to noise. This limits their effective- ness in high unknown noise settings. (3) Deterministic MAP methods that optimize in the DM noise space (DMPlug, (Wang et al. 2024) remove randomness but often get stuck in a single mode, especially in severely ill-posed problems like phase retrieval, due lack of posterior exploration. In short, enforcing data consistency mid-diffusion can break prior adherence, while optimizing only for fidelity leads overfitting or mode collapse.
Allhouri al. 2025a), and DIIP (Chihaoui & Favaro 2025a)
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Boala (Cun)
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Figure 1: Comparison of existing methods and their limitations with the N-HMC method. (a) It- erative Guidance Methods (DPS) lead to manifold infeasibility. (b) Stochastic MAP methods (Re- Sample) (Song al., (024) are susceptible to overfitting noise. (c) Deterministic MAP methods (DMPlug) (Wang et (2224) become trapped in a local mode. in the noise space ær and maps samples images via a deterministic mapping X D(xT).
(d) Our method performs sampling
Sampling from the full posterior ensures that the learned prior acts automatically as regularizer, while an annealing schedule for noise standard deviation αy promotes efficient exploration and pre- vents the sampler from being trapped in early local modes. Importantly, the method relies only on a fixed set of hyperparameters that remain constant across tasks, datasets, levels of measurement noise, avoiding the repeated tuning required by many existing approaches.
To address the practical challenges that the measurement noise level is often unknown, we further introduce a Noise-Adaptive N-HMC (NA-NHMC). Instead of requiring fixed noise level, we take principled Bayesian approach, placing a non-informative prior on the noise variance and marginal- izing it out. This yields parameter-free free likelihood term that automatically adapts to the true un- derlying noise in the measurements. As shown in experiments, this allows NA-NHMC to achieve robust, high-quality reconstructions across varying and even unknown noise types and levels without any task task-specific hyperparameter tuning. In contrast, the performance of other methods depends on the hyperparameters listed in Section [A.5] which were specifically tuned for Gaussian noise. Our key contributions include: (1) In Section 3.1 we propose N-HMC, posterior sampling method that addresses the three key limitations of existing tfate-of-the-arr (SOTA) approaches. We further analyze its sampling behavior and provide theoretical guarantee of its robustness to measurement noise in Section 3.2 and Appendix A.2 (2) In Section 3.3) we extend our method to settings with unknown noise types and levels. We show that outperforms SOTA methods most metrics (Section 4.3) especially for non- non-linear and high noise problems. (3) In extensive experiments, NA- NHMC method solves diverse inverse problem tasks under unknown noise types and levels without any task- or noise-specific hyperparameter tuning, in contrast to many existing methods. (4) We demonstrate in Section 4.1 that the annealing schedule for dy helps promote early exploration and prevent local-mode collapse, especially in severely ill-posed tasks like phase retrieval.
PRELIMINARIES
2.1 DIFFUSION MODELS FOR INVERSE PROBLEMS
Daras et al. (204)) broadly classifies methods for solving inverse problems (IPs) into two categories. The first is maximum a posteriori (MAP) inference, which aims to find the single most probable x. An alternative is the Bayesian framework, where the goal becomes generating plausible reconstruc-
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tions by sampling from the posterior distribution p(zly). where p(zly) can be decomposed into the prior p(ær) and the likelihood p(y) MAP delivers fast optimization, but struggles with high noise and multimodal posteriors, easily converging to local minima. In contrast, the Bayesian approach samples from p(zly) to generate plausible reconstructions, quantify uncertainty, and handle mul- timodality. Both approaches critically depend on powerful prior models like DMs that encode the complex statistical structure of complex data and prior knowledge.
Most diffusion-based approaches to IPs are based on the denoising diffusion probabilistic models (DDPM) framework (Ho al. 2020 Song & Ermon 2020). The framework consists of forward and reverse diffusion processes. The forward process gradually corr the clean images zy towards standard Gaussian noise æy. This process can be described by a stochastic differential equation (SDE), dæ = 6; ædt + VBrdur, where w is the standard Wiener process. In practice, the process is discretized via variance schedule (8.]221 forming a Markov chain:
T
q( (00) = II (1,1 [=1
((11) = Na /1 1,11)1,
(2)
In order to generate clean images, the reverse process begins with noisy sample ør N 0, I),
and recursively refines it according to the reverse SDE, dæ - wd 3e Va log (r )dt + BBdw,, where w is the time-reversed standard Wiener process, and p. (*) is the marginal probability of the noisy manifold at time t. V log pı (z:) is called the score function and is usually approximated by neural network trained through score score-matching matching methods.
Using the same discretization, clean images can be generated from the prior using iterative de- noising process.
(o( = p(xT) II po po( == N (i t), Eg (Z f))
Building on the DDPM framework, accelerate the denoising process. process.[Song al. (202aa) proposes Denoising Diffusion Implicit Models (DDIM), which define a non-Markovian and fully determin- istic forward/reverse process (8, 0). Unlike DDPM, which injects stochasticity each step to improve robustness, DDIM iteratively maps the initial noise æy to a clean sample Do via determin- istic trajectory. For our method, this property is particularly beneficial, as it allows us to consider the entire reverse process as a deterministic mapping from Ty to Zp.
Among successful DM- methods for inverse problems. DPS and its variants (Chung et al., 2023 Kawar et al. 2022 Wang et al., 2023 Song et al. 2023 Chung et al.) 2022) are best-known reconstruction algorithms. But they suffer from approximation errors from Tweedie's formula cor- rections. To mitigate noise sensitivity, TMPD incorporates second-order information to correct the guidance trajectory; however, like other iterative methods, it relies on modifying intermediate states, which risks drifting off the learned manifold. SITCOM (Alkhouri et al., (025b) operates the constraint: data fidelity, back-
noisy image at each diffusion step and enforces a triple-consistens ward consistency with the diffusion posterior mean, and forward consistency along the diffusion trajectory. DIIP (Chihaoui & Favaro 2025h) updates the initial noise dr with data fidelity gradients after the standard diffusion sampling process. DMPlug (Wang al., (2024) proposes noise-space formulation but treats inverse problems as optimization tasks, making it sensitive to noise. While early stopping can mitigate this, its criterion is task- and noise-dependent. At the high noise levels considered here, the optimizer often becomes trapped in local mode, rendering early stopping ineffective. Similar behavior is observed in other Maximum Posteriori (MAP) methods such as ReSample (Song et al. 2024), which optimizes directly in clean -image space (leading to noisy or blurry images under early stopping). DAPS (Zhang et al., 2024), despite being formulated as a pos- terior sampling method, uses a heuristic o, that is much smaller than its true value to strengthen the consistency of the measurement. This deviation from true posterior sampling makes DAPS effec- tively MAP-like, inheriting the same sensitivity to noise.
2.2 HAMILTONIAN MONTE CARLO (HMC)
Hamiltonian Monte Carlo (HMC) Duane et al.) 1987) is an MCMC Metropolis et al., 1953) sam- pling method that utilizes fictitious momentum variable and simulates Hamiltonian dynamics to
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efficiently explore distant regions. Due to its superior scaling properties in high dimensions com- pared to other simpler Metropolis methods Brooks et al. (2011). HMC is particularly well suited for
sampling in high-dimensional space, such the 3 x 256 x 256 pixel space of images.
The Hamiltonian is defined as И = + V. where U = p(z) and V = 1041 " Then, discretize the trajectory using the leapfrog integrator. For single leapfrog step with step size 6,
we have
OU vt + 5/2) = v(t) dx =nt))
(4)
x(t + 8) = æxt) + 8M
u(t + 8/2),
(5)
8 OU
v(t + = vtt + 6/2) 2 da (x+++))
(6)
where v00) ~ (w; 0, M). This process is repeated L times to form a full trajectory. Due to discretization error, the Hamiltonian no longer preserved, which introduces bias and violates the detailed balance. To correct for this, a Seetropolis-Hasting (MH) correction step is applied at the end of each trajectory with acceptance probability of a = min exp(-H, + Ho)), where Ho. H1 denotes the initial and proposed Hamiltonian, respectively.
METHODOLOGY
In this section, propose posterior sampling method, Noise-space Hamiltonian Monte Carlo (N-HMC), to solve IPs with pretrained DMs. We show its derivation in Section [3. and discuss its robustness to measurement noise in Section 3.2 In Section 3.3 our method is modified to allow for unknown types and levels of measurement noise.
3.1 NOISE-SPACE HAMILTONIAN MONTE CARLO (N-HMC)
The goal in solving inverse problems is sample from the posterior distribution p(æoly) o (( mo). Since direct sampling from p(æo) is intractable, pretrained diffusion models are employed provide a powerful prior. Standard diffusion-basee approaches draw ET p(ær) from Gaussian noise prior and iteratively denoise through intermediate timesteps, aiming to sam- ple from p(ær y), p(.r((T-11 y)...., p( y) in sequence. The key challenge is evaluating
intractable likelihood p(y xx.) at each intermediate timestep L. To address this, iterative guidance
methods (Kawar et al. 2002 Wang et al., 2023 Chung et al.) 2023; Song et al. 20233] Rozet et al.)
2024 Song et al.] (2024] Zhang et al. (024) introduce approximations and apply likelihood correc- tions of 24 <- Ze + V/ a, log (.)). However, these gradient-basee corrections systematically
push intermediate states æ, away from the distribution which the denoiser is trained, leading to
what we refer to as the manifold feasibility problem. Following SITCOM (Alkhouri et al. 2025b),
formalize this issue as:
Definition 3.1 (Manifold Feasibility). For pretrained diffusion model, let pe(az) denote the marginal distribution at noise level t, and let M, be its bigb-prrbuyillity generative manifold. An inverse-problem solver maintains manifold feasibility if the intermediate states {xx} fed into the denoiser remain close to Mi for all t, ensuring the final reconstruction Dy lies on the learned data manifold.
Geometrically, standard guidance methods update x, using the likelihood gradient Vx, logp(y (x).. In sigh--imensionnl spaces, this gradient vector often contains components orthogonal to the local tangent space of the data manifold M,. Consequently, adding this gradient systematically pushes the state x, into Low-probability regions (off-manifold), feeding out-of-distribution inputs to the denoiser and causing accumulated artifacts shown in Figure (a). To avoid such approximations, we propose posterior sampling by drawing from the initial noise space. The sampled noise is then unconditionally denoised to clean image. We adopt unconditional DDIM for the denoising process, which treats the entire denoising trajectory as a deterministic mapping X, = D(x;) ), so the problem becomes evaluating the posterior distribution of noise (Xia et al. 2023), i.c., p(ær (B). We refer to our approach as noise-space sampling because HMC updates are performed exclusively on the initial noise Øт AN T)- This differs from image-space and iterative guidance methods that
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directly modify intermediate states æ, using measuremeett-cnnssseee gradients. Sampling from the noise space offers two advantages: (i) the prior p(ær) is a simple Gaussian distribution, and (ii) the likelihood pyy (r)) ((y))((z)) is directly accessible without intermediate approximations.
We use HMC for efficient posterior sampling in the noise space. To strictly justify our sam- pling objective, we formulate the inference process noise Iy is the sole latent variable. We treat the unconditional DDIM process with steps as deterministic parameterized generator function, denoted as D I + R", which maps æt to a clean image Φp = D( Under this formulation, the measurement generation process is
latent variable model where the initial
defined by 2T 2 Øo An, y. Consequently, the conditional distribution of given øт de- pends entirely on the generated image øo- The likelihood term is thus mathematically exact: (r) p ( D(xr)) (g A(D(A()))))))))) o..). This allows us to perform posterior graden sampling directly the noise space using the exact gradient of the likelihood with respect TT. Then we can compute the conditional score using Bayes' rule:
øy logp(
V æe logp(ær)) = Var (ogp(y)))))
Since æT is Gaussian noise in the DDIM framework, the first term is simply
|
ay log = ar 2
TT.
(8)
For the case of Gaussian measurement noise, if the noise level of is known, the likelihood term becomes
V æer logp(y(<T)) = Ver logp(y(D(aT)) = -Va, y A(D(xr))
(9)
20
We define p(y( = p(y)D((r) by viewing the denoising trajectory as a deterministic mapping X0 D(xT). ). This term can be computed directly using automatic differentiation. Because D(ær) results from a multi-step denoising process, backpropagating through multiple score networks can be computationally expensive. Following WWan et al. (2024), we illustrate in Appendix A.9 that accurate samples can still be obtained with as few as two denoising steps.
Once the conditional score V log poggprrll y)) is computed, our method proceeds with standard Hamiltonian Monte Carlo (HMC) sampling. We use the identity matrix as the mass matrix for momentum sampling. During implementation, of very low posterior probability, which forces HMC to adopt tiny step size in order to maintain proper acceptance rate. To address this issue, we use an annealing schedule for oy, allowing øT to explore the noise space more freely with larger step size in the start-up stage. Once dy gradually declines to the target level, posterior samples are collected. The complete procedure is summarized in Algorithm] 1 along with the unconditional DDIM denoising process in Algorithm 2
observed that the initial noise may lie in regions
3.2 ROBUSTNESS TO MEASUREMENT NOISE
An additional benefit of N-HMC over MAP methods is that the Gaussian prior acts as a regulariza- tion term in the noise space, keeping the noise vector ær close the hypersphere of radius vn. Therefore, N-HMC produces samples that are robust to measurement noise, as justified by Proposi- tion 1. For simplicity, we assume Gaussian measurement noise and that the forward operator A is approximately linear along the clean image manifold.
Proposition 1. Assume that the distribution of the decoded sample wo around the ground truth zt is well-aaprox by Gaussian distribution Po(āo) z N2oi कद o. Then, the residual Ado satisfies
A = (BB((BBT) + tr( (A Prese A
where
A
A
I
AΣ
post
Σ Epost
B
Im
02
6
o3
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Algorithm 1: N-HMC
Require: HMC iterations K, leapfrog steps initial integration step size 6, measurement noise schedule
{0 2T. y. A.
1: for 0 to K do
2: repeat
3:
N/
// Initial momentum
3o (ZIM
5:
He =|rrr''' 1
2. 11 A(xo)] + 2/p"
// Current // Initialize proposal ær
6:
ET - 2T
7:
for = to L do
8:
p - 1/4
TT. +
20 Vazi lly - A
// Update momentum
9:
21 37 + 6p
// Update x
3 DDIM(
11:
p g+ト
2
x7 +
2n y.N
Vay 1y - A(15))
// Updato momentum
12:
end for
13:
H1
2 112 +
1
A(E)) 2T
// Proposal Hamiltonian
20
14:
u ~~ Unif(O:) (. 1)
15:
if < exp( Ho Hi) then
16:
Accept proposal
else 5 + 75 end if
18:
// Anneal step size
19:
20:
until Proposal accepted
21:
2T 27
// Accepp the proposal
22:
end
23:
return FT
and In is the m identity matrix. m denotes the dimension of y.
The expected residual decomposes into two contributions: a noise-dependent term that appears only when measurement noise present, and a second term that persists in all settings due intrinsic uncertainty of prior diffusion models. In Corollary 1.1 below, we show that both terms behave in way that yields residual whose magnitude matches the true measurement noise.
Corollary 1.1 Under the assumptions of Proposition 1, if 0o/0, < 1, the residual A(#o) satisfies
E (f y A(xa) mo?
In other words, the magnitude of residual aligns with the true known level of measurement noise, indicating that N-HMC remains robust and does not overfit to noise.
3.3 NOISE- ADAPTIVE NHMC
In practice, the type and level of measurement noise are often unknown, making the likelihood term p(y ØT) intractable. To address this, other methods usually have tunable hyperparameters that control the strength of the likelihood term use task-specific early stopping criteria. Instead of this heuristic approach, we introduce a noise-adaptive sampling method, NA-NHMC, which extends N-HMC the unknown noise setting without any additional hyperparameter tuning.
We treat the noise variance as a latent variable and adopt the Jeffreys prior, principled noninfor- mative choice due its parameterization invariance. It scale-invariant and represents maximal uncertainty about the noise level, making it appropriate when no prior information about ag is avail- able. It can also be viewed as the limiting case of an Inverse-Gamma prior og Inv-I 3) as o, 9 - 0. The Inverse-Gamma distribution is the conjugate prior for the variance of Gaussian likelihood. Proposition 2 characterizes the resulting behavior under additional assumptions.
1
p(

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Under the assumptions of Proposition and that the pretrained diffusion model unconditionally generates images that lie on the high-quality manifold (0o/oy y < 1), then the update rule of NA-NHMC follows:
Proposition 2
V æv log p(yler NHMC =
V ly A(D(xr))1
20
By marginalizing oy.. the likelihood term becomes
p xT) = p(y æT, oy) p do
(10)
-m/2
y A(D-A(DD()))))
(11)
where m denotes the dimensionality of the measurement space. The derivation of this expression provided Appendix A.I Substituting this marginalized likelihood into the N-HMC framework yields our proposed noise- adaptive Algorithm 3] Proposition 2 below shows that,
0.20
0 =0.01
Std. Measurement
0.16
9y 0.02
0.05
with an appropriate measurement noise prior, the likeli- hood term V log hoggp(ylann (()))r)) of NA-NHMC is identical to that of N-HMC (with known noise level).
d,
0.08
Figure demonstrates Proposition 2 in practice. All ex- periments use an identical setup across different noise
0.04
levels, highlighting that our method does require any
0.00
hyperparameter tuning for a specific noise level. Despite the absence of such tunable parameters, the estimated standard deviation of the measurement noise, computed as |/y A(=oo)ll///mm closely matches the true, unknown noise level dy- This confirms that NA-NHMC effectively adapts to varying noise without specific tuning. The com- plete NA-NHMC is summarized in Algorithm
0%
20%
40%
6.0%
80%
100%
Progress (%)
Figure 2: Gaussian deblur task on FFHQ (256 256) with varying mea- surement noise levels dy. The esti- mated standard deviation of measure- ment noise |y - A( lloo)ll//m demon- strates that our noise-adaptive method accurately recovers the true oy (indi- cated by dashed line) without overfitting across different noise levels.
The flexibility of NA-NHMC goes beyond noise-level ro- bustness. Although NA-NHMC is formulated assuming Gaussian measurement noise, experiments with alterna- tive noise types (Section 4.3) show that the method re- mains effective across other common noise distributions, demonstrating its broader robustness.
EXPERIMENTS
Following previous approaches (Wang et al. 2024) (Zhang et al. 2024). we evaluate our method
on two datasets: FFHQ 256 256 (Karras al. 2019) and ImageNet 256 256 (Deng et al. 2009). using 100 images from the validation set of each dataset. We utilize the same pretrained DM trained by CChun al. (2023) for FFHQ and by Dhariwal & Nichol (202T) 1) for ImageNet except for ReSample. For ReSample, we use pretrained LDD by Rombach et al. (2022). All measurements are corrupted by additive Gaussian noise with standard deviation dyy.
We compare our method against several representative baselines, including DifPPIR (Zhu et al., 2023), RED-diff (Mardani et al.) 2023), DPS (Chung et al. 2023), DAPS (Zhang et al. 2024). ReSample (Song et al. 2024). SITCOM (Alkhouri et al. 2025b), and DMPlug (Wang et al. 2024 The implementation details for all the baseline methods are provided in Appendix A.55 We evaluate reconstruction quality using three standard metrics: peak signal-to- -onal-to--oise ratio (PSNR), structural similarity index measure (SSIM) (Wang et al. 2004)), and Learned Perceptual Image Patch Similarity (LPIIS) (Zhang et al. 2018)

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4.1 EXPERIMENT RESULTS
Linear IPs. We evaluate our approach on four linear inverse problems. For super-resolution tasks, we consider both 4x and 16x downsampling using 4 x 4 and 16 16 average pooling opera- tions, respectively. We also examine random inpainting with 92% of pixels randomly masked, and anisotropic Gaussian deblurring using blur kernels with standard deviations of 20 and in orthogo-
nal directions. The results for linear IPs are presented in Tables 10 13
Nonlinear IPs. We further assess performance on three challenging nonlinear inverse problems. The first is nonlinear deblurring using encoded blur kernels from Tran et al. (2021). The second is phase retrieval, where only the Fourier magnitude is observed as measurements. Finally, we consider HDR reconstruction, which aims to recover images with higher dynamic range by factor of from tone-mapped observations. The results for nonlinear IPs are presented in Tables
14 15
Main Results. Our method achieves comparable
superior performance across most tasks, as measured by PSNR and SSIM on both the FFHQ and ImageNet datasets. Notably, the improvement over SOTA meth- ods is more pronounced for nonlinear tasks, which are substantially more challenging than linear IPs. Many ex- isting SOTA approaches are MAP-based by design (e.g.. ReSample, SITCOM, and DMPlug) or become MAP-like heuristically (e.g.. DAPS). While these methods perform well in loise regimes, they often overfit when the noise level is higher. Since the noise levels used in our ex- periments (a,, 0.05, 0.20) exceed those commonly re- ported in prior work, our results further demonstrate that the proposed noise-adaptive method is more robust and consistently outperforms alternatives across most tasks and metrics without any hyperparameter tuning. Fig- ure | contains visual examples for the nonlinear deblur- ring problem. See Appendix 12 for more examples. A fundamental distinction lies in the generalization ca- pability across diverse degradation conditions. Standard guidance-based methods (e.g.. DPS) inherently rely manual hyperparameter calibration to bal- ance measurement fidelity against the diffusion prior. As evidenced in Table 4 the optimal step size is highly deskddepnndent (ranging from Ç 0.4 to Ç 10.0), meaning static configuration fails to generalize. In contrast, NA-NHMC derives its dynamics from the marginalized posterior, which effectively acts an automatic gradient normalization mechanism. This structural advantage allows a single configuration to robustly generalize across varying tasks and noise levels without task-specific recalibration.
DPS
DAIS
NA-NTIMC (Qurs)
MAE
Stll Dev.
Figure 3: Comparative results are aver- aged over 100 independent runs. (Top) Mean absolute еπтоr (MAE) heapmaps. (Bottom) Standard deviation heatmaps across runs. Our method achieves the lowest standard deviation compared to DPS and DAPS, indicating reduced sen- sitivity to initialization.
Ground Truth Measurement
DIfRPIR
RED-dft
DPS
DAPS
ReSample
DMPlug
Ours
Nonlinear deblurring results on FFHQ (256 256) dataset with og = 0.2. Visual compar- ison across state-of-the-art methods shows our approach produces high-quality reconstructions with sharp details and minimal artifacts.
Figure 4:
4.2 HIGHLY ILL-POSED IPS: PHASE RETRIEVAL
Another challenge commonly faced by both MAP and sampling- based methods becoming trapped in local mode, particularly in highly multimodal IPs such as phase retrieval. Figure 5 this
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issue. While DPS and DMPlug occasionally recover the correct solution, most initializations verge to spurious local modes and are thus counted as failures. In contrast, our method incorporates early exploration through By scheduling, making it more robust to initialization and substantially more likely to recover the global solution.
0.8
24
0.6
PSNR
0.75
16
0.4
0.0
0%
25%
50%
75%
100%
0%
50%
75%
100%
0%
25%
50%
75%
100%
Progress (%)
DMPlug
NA-NHMC
Figure 5: Phase retrieval task FFHQ (256 x 256) with By 0.01. Each curve shows the median performance, with shaded areas denoting the 5th-95th percentile interval. Our method successfully solves the IP at much higher rate than DPS and DMPlug. This is due to the annealing schedule of ay that allows for initial exploration of the noise space, resulting in a lower probability of being stuck local mode.
We quantify robustness to initialization using the standard deviation map in Figure 3 Our method achieves mean absolute error (MAE) comparable to that of DPS, but with substantially lower pixel-wise standard deviation. While DPS can, on average, produce accurate reconstructions, its performance sensitive initialization and may introduce artifacts in both the face and background (red circle). In contrast, such artifacts never appear in any of the 100 runs with our method. Notably, despite exhibiting lower overall uncertainty, our method still assigns uncertainty in complex regions, which aligns with areas of high MAE (orange).
Table Non-linear IPs Results on FFHQ (256 x 256) with Gaussian Noise og 0.05, (Bold: best, underline: second best)
Nonlinear Deblurring
Phase Retrieval
IIDR Reconstruction
PSNR
SSIM
LPIPS
PSNR t
SSIM
I.PIS 4
PSNR
SSIM
LPIPS
DinPIR
26.12
0.743
0.289
16.77
0.482
0.543
25.20
0.814
0.223
RED-diff
18.12
0.217
0.680
11.83
0.213
0.769
21.44
0.525
0.458
DPS
23.26
0.672
0.300
10.87
0.296
0.714
27.46
0.849
0.168
DAPS
27.00
0.736
0.283
18.52
0.414
0.528
26.03
0.758
0.259
ReSamplc
24.57
0.637
0.432
13.95
0.377
0.677
23.65
0.722
0.386
SITCOM
24.97
0.569
0.328
11.89
0.216
0.723
26.97
0.753
0.256
DMPlug
27.15
0.784
0.266
25.17
0.783
0,260
NA-NHMC (ours)
27.66
0.792
0.249
19.30
0.554
0.482
28.45
0.849
0.217
Table 2: Non-linear IPs on ImageNet (256 x 256) with Gaussian Noise o, = 0.05. (Bold: best, underline: second best)
Nonlinear Deblurring
HDR Reconstruction
PSNR
SSIM f
LPIPS
PSNR t
SSIM 1
LPIPS
DifIPIR
24.24
0.638
0.381
23.29
0.730
0.273
RED-diff
17.94
0.244
0.623
20.98
0.524
0.415
DPS
17.60
0.427
0.482
25.31
0.763
0.248
DAPS
24.28
0.632
0.404
23.57
0.709
0.283
SITCOM
24.00
0,556
0.355
24.76
0.708
0.276
DMPlug
22.30
0.576
0.421
20.61
0.562
0.431
NA-NHMC (ours)
24.98
0.694
0.308
25.86
0.779
0.253
4.3 ROBUSTNESS TO UNKNOWN MEASUREMENT NOISE
In practice, the measurement noise may be unknown and its type may not be Gaussian. This can pose problem as many methods require multiple hyperparameters that tuned for specific level

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Published as a conference paper at ICLR 2026
and type of measurement noise. In this section, we evaluate method's robustness to unknown measurement noise on two tasks and two noise types: impulse and speckle. For impulse noise, each pixel in each channel is randomly replaced by 0 or 1 with probability p/2 each, where p Un 0.2). For speckle noise, the noise takes the form y(1 + e), where y is the measurement tensor and (0, 0.4). Table shows that method achieves superior performance metrics while using the exact same hyperparameters' as the Gaussian noise experiment. As illustrated in Figure 6 methods such as DifiPIR, which do not suffer from noise overfitting in the Gaussian setting, now struggle with impulse noise. In contrast, even under high noise levels, NA-NHMC remains robust, producing hugh-quality reconstructions.
most
Truth
Measurement
RED-dIf
DAPS
ReSemple
DMPlug
Ours
Figure 6: Nonlinear deblurring results on FFHQ (256x 256) dataset under different noise conditions. (Top) Impulse noise. (Bottom) Speckle noise.
Table 3: The Performance Comparison with Different Types and Levels of Measurement Noise on FFHQ (256 256). (Bold: best, underline: second best)
Super Resolution (x4)
Nonlinear Debluming
Impake
Speckle
Impuise
Speckle
PSNR
SSIM
1.PIPS
PSNR
SSIM
LPIPS
PSNR 1
SSIM
LPIPS
PSNR 1
SSIM
LPPIS
DifPPIR
0.492
0.549
25.91
0.733
0.324
21.00
0.402
0.526
25.96
0.731
0.299
RED-diff
21.81
0.516
0.109
0.781
DPS
21.99
0.581
0.395
27.00
0.761
0.246
21.64
0.595
0.322
23.42
0.678
0.302
DAPS
15.00
0.361
0.702
24.48
0.597
0.442
17.94
0.259
0.657
26.46
0.698
0.308
ReSample
22.98
0.639
0.483
26.17
0.733
0.387
22.74
0.616
0.471
24.64
0.692
0.409
SITCOM
0.425
0.259
0.295
DMPlug
19.52
0.358
0.562
26.79
0.689
0.336
23.79
0.662
0.335
25.82
0:740
0.308
NA-NHIM (ours)
23.42
0.631
0.382
27.36
0.768
0.290
24.16
0.677
0.319
27.97
0.796
0.253
5 CONCLUSION
In this work, we introduce N-HMC, posterior sampler that operates in the noise space using reverse diffusion as a deterministic mapping from initial noise to a clean image, enabling posterior explo- ration while keeping proposals the learned data manifold. The developed adastivee variant, NA-NHMC, eliminates task-specific hyperparameter tuning by automatically adapting to unknown noise types and levels, which is a significant practical advantage over existing approaches. Theory establishes the correctness and efficiency of noise-space sampling, and experiments across diverse linear and nonlinear inverse problems on FFHQ and ImageNet show state-of-the-ar reconstruc- tions, robustness to initialization and noise, competitive runtimes with few denoising steps, and incertainty-awart estimates. The provided analysis and experiments also show that our method can mitigate peassrrement-consiseeeee drift, noise overfitting, and local-mode collapse without relying on any task-fpe hyperparameter tuning.
While NA-NHMC shows promising results, it incurs higher computational cost compared to other methods such as DPS due to HMC sampling. Additionally, its reliance on small number of diffu- sion steps may limit its immediate applicability to more complex applications. Moreover, the high dimensionality of the posterior leads to long warmup phases before reaching stationarity. Future work could address these challenges by developing more efficient gradient estimation techniques and incorporating faster warmup strategies that relax the requirement exact detailed balance.
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ACKNOWLEDGMENT
This research was supported by the Agency for Science, Technology and Research (A*STAR) un- der its RIE2025 Human Health and Potential (HHP) Industry Alignment Fund Pre-Positioning Pro- gramme (IAF-PP) (Grant No. H24J4a0145). We gratefully acknowledge the insightful comments and suggestions provided by the anonymous reviewers.
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ETHICS STATEMENT
We affirm adherence to the ICLR Code of Ethics. This work uses publicly available datasets and standard benchmarks; no human subjects, personal data, sensitive attributes are collected or pro- cessed. We assessed potential risks (misuse, bias, fairness, privacy, and security) and found none beyond those commonly associated with generic image/model evaluation. All institutional and le- gal requirements were respected, and we disclose that we have no conflicts of interest or external sponsorship that could unduly influence the results.
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REPRODUCIBILITY STATEMENT
We provide implementation details, hyperparameters, and training/cvaluation protocols in the main text and appendix; key equations and assumptions stated with complete proofs in the appendix. All datasets and preprocessing steps are described and referenced; code and configuration files have been released in repository to enable exact reproduction of results.
A
APPENDIX
A.I ROBUST-N-HME DERIVATION
Assumptions
1. Measurement noise 1) R. follows gaussian distribution with unknown oz:
1
| - A(D(xT)) 21
ply ØT, og))
(2To3)m//2 exp
(12)
20
By follows a Jeffreys prior distribution:
p(a & oy
(13)
Marginalizing of yields
x
ply oyy p(oz) d
(14)
y A(D(ET)))I
(2ma m/2 mm/2 exp
do
(15)
20
y
(1/2)lly A(D(
(o +
do
(16)
ob
1
-m/2
2 lly A(D(xr)))12
(17)
And then, have
log 1ogp(y)or)= log
2 y A(D(ar)))1))
(18)
A.2 PROOFS
Measurement model.
Let zq E R" denote the ground truth signal. The measurement is given by
y A(xi) 17,
r) ~ (0,N0.,
(19)
where A: R" + Rm is the measurement operator and 17 represents Gaussian measurement noise. In the following proofs, A is assumed to be approximately linear around æt. Thus, A(xo) Awo.
Generative model.
Consider the DDIM sampler defined by
c = D(rr), 2rr N In):
(20)
where D denotes the deterministic decoder via the diffusion model.
Lemma 1. Product of two Gaussian probability density functions (PDFs).
@ (w) N (x; μ1, Σ,), (2(x) = (a;;; μ12 Σz). Then, the product qı (x) and q2 (xx) is proportional to a Gaussian PDF A(x, μ. E), where
# Σ (M 15'12)
Σ (Σ + 1ー
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Lemma 2. trix Σ, i.e.,
decomposition of random variable with mean and covariance ma-
E
|
| μl -=l*+tt(22)
Proposition 1. Assume that the distribution of the decoded sample zo around the ground truth To is well-aaproximmate by Gaussian distribution po(tt) = Nato Tp. InI. Then, the residual y A satisfies
(b Sppo.y)-polae (co (2))
A = ozjr((B") + tr( Aoat )
where
A T A ozy
I
AE pos
E post
B
Im
Proof First, consider the distribution of øo given fixed zi and y
(a) po (o (o
(o
(21)
A(yo) Im) (áto zū. In
(22)
N (zo: (A (ATA) A y. (A (ATA) !)- (d 08
(23)
N (0o; poot (z); Epos)
(24)
pa (A y) N (do: Appost (y), AΣ Ao ) where (a) follows from Bayes' theorem. with
(25)
(b) is from local linearity of A. (c) is the result of Lemma
ATg y A
A
ATA
A
I
Ppos (B)) = Epost
Epost
o
post
0
00
7
The expected squared residual conditioned y is (x w) 11y/
A lyy
Ap (1) +
AAE )
(26)
which is the result of Lemma 2. Then, integrate y conditioned zū
pr (1 by Acoll2
(27)
Eysg(yia m
B (# 1 A
(28)
Eyrg(y mi)
||y A Mposu(
(y) (1))(
+ (r(AE post A T
(29)
A A
A p
Ey mz))
y
>tr(AA post AT)
(30)
o
y
AΣ poat A
A 20
I
Axo
tr(
B (o + tr( AE (rE A
(31)
og
- tB
+ tr
AA post
A
(32)
AE A
where (a) is the result of Lemma 2, and B I,
Corollary 1.1 Under the assumptions of Proposition 1, if 00/0, < 1, the residual A(zo)
satisfies
E (2oo w)- (dto.u))))) vvz;)) y A( may
Proof If 1o/oo & 1.
AT
A
I
GAATA
0g
-1
Σ post
=
o/
+ In
+
I
(33)
0
of
A
A
A
GAAT
Im
Im
Im
(34)
Elyy A(xr) 2= oi BB + tr
(A
AA
AT)
ma²
(35)
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Proposition 2 Under the assumptions of Proposition 1 and that the pre-trained diffusion model unconditionally generates images that lie on the high quality manifold (0v/oy < 1), then the update rule of NA-NHMC follows:
pr logplyyar NHMM
V y A(D(xT))] 2
Proof We can consider the likelihood term of NA-NHMC
log p(ylar)
m
log
A(D((T)))1))
(36)
2
V Φp (ogp(y(<TT)
Værl -
(37)
2 A(D(A (T |>
V ær y A(D(xT)))
(38)
20
which follows from Corollary 1.1. This likelihood term is exactly the same that of N-HMC, where the true noise level σy is known.
A.3 PSEUDOCODE OF UNCONDITIONAL DDIM
The DDIM method (Sogg et al. 2022a) we used our experiment follows the Algorithm 2
below:
Algorithm 2: DDIM
Require: diffusion steps T, diffusion model 86, initial seed ær
I: for
to
2: Êlς1 = (0(0111, ++1)
// Compute the score
@o( Va++i (x1+1 VI (upiém)) // Predict &o with Tweedie's formula
4: # = Vorco (Z141) 1++ + v1 T.e1+1
// Unconditional DDIM step
5: end for
6: return
A.4 PSEUDOCODE OF THE NOISE-ADAPTIVE NHMC
Following the reasoning in Proposition we assume an uninformative prior on oy. Under this assumption, the likelihood term can be written as
log pyyla) = (-7) log 2 A(D(=r)))?)
where dim(y). The factor 1/2 inside the logarithm be omitted for the Hamiltonian and gradient computations. The corresponding gradient is
V कe log p(g)(()))) (-
V Var A(D(xT)))
2 A(D(
A.5 IMPLEMENTATION DETAILS FOR BASELINE METHODS
DiffPIR
Number of diffusion steps: 100
Number of optimization steps: 50
We follow the recommended hyperparameter 77 = 1.0 and À 7.0 from Zhang et al. 20025). The learning rate of the schedule-free AdamW optimizer is set to 0.1.
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Algorithm 3: NA-NHMC
Require: HMC iterations K, leapfrog steps initial integration step size ó, ær. V. A. 7
I: for K
2: repeat
3:
((0,))
// Initial um
4:
DDIM(zr)
5:
Ho 2|
+ log (1w - A( + 3/p²
// Current Hamiltonian 11 Initialize proposal ær
6:
IT - IT
7:
for to --15 do
p p
A( v -
Vaylly A(
// Update nomentum
9:
fi < ei + 6p
// Update wi
10:
a DDIM(x77)
p -
ar + 2 A(=))
m
Va 12 A(
// Update momentum
12:
end for
13:
Hi | 1 + 2 log (lly> A(#5))) + 12p²
// Proposal Hami Itonian
14:
Unif(O, 1)
15:
if < exp( Hu- H1) then
16:
Accept proposal
17:
else
18:
5 <- 78 end if
// Anneal step size 8
19:
20:
until Proposal accepted
21: Ør <- x7
// Accept the proposal
22:
end for
23:
return
RED-diff
Number of optimization steps: 1000
We follow the recommended the hyperparameter X 0.25 and Adam optimizer with Ir 0.5 as in Zhang et al. (2025)
DPS
Number of diffusion steps: 1000
We follow the learning rate form in Chung al. (2023) with ç; adjusted for different tasks, as shown in Table
Table 4: Tuned learning rate C for DPS
SR 4x:4
SR (x16)
Inpainting (92%)
Gaussian Deblurring
Nonlinear Deblurring
Phose Retrieval
HDR
FFHQ
1.0
0.6
1.0
1.0
1.0
0.4
1.0
ImageNet
1.0
0.6
1.0
0.4
0.5
1.0
DAPS
Number of diffusion steps: 250
Number of ODE solver steps:
We follow the hyperparameter settings of Zhang et al. (2024). as listed in Table 5 and adopt their heuristic o 0.01 in place of the actual value. ò 0.01 for all tasks. The number of MCMC sampling steps = 100 for FFHQ (256 x 256) and = 40 for ImageNet (256 x 256). Otherwise, the hyperparameter for each task the same for both datasets.
ReSample
Number of diffusion steps: 500
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Table 7/o for DAPS
SR (x4)
SR (x16)
Inpainting (92%)
Gaussian Deblurring
Nonlinear Deblurring
Phase Retrieval
HDR
le-4
le-4
le-4
le-4
Se-5
5e-5
2e-5
For both noise levels (ry = 0.05, 0.20) tested this paper, the recommended optimization steps lead to overfitting to noise and poor performance. Instead, we used 50 steps for pixel optimization and 25 steps for latent optimization.
SITCOM
Number of diffusion steps:
We follow the hyperparameter settings of Alkhouri et al. (2025a), as listed in Table The stopping criterion 8 for σy {0.05, 0.2) chosen 0.051 Vm 0.201 ym respectively, with m denoting the dimension of y.
Table Optimization Steps K for SITCOM
SR (x4)
SR (x 16)
Inpainting (92%)
Gaussian Deblurring
Nonlinear Deblurring
Phase Retrieval
HDR
20
20
30
30
30
30
40
DMPlug
Number of diffusion steps:
t= [250, 500, 750]
We set the Adam optimizer learning rate 0.01. We follow the recommended stopping criteria in Wang et al. 2024).. For linear tasks, we use window size = 10, patience = 100, and maximum iterations 5000. For nonlinear tasks, we the window size 50 and patience 300 with
maximum iterations = 10000.
A.6 IMPLEMENTATION DETAILS FOR OUR METHOD
Number of diffusion steps:
t = [375, 750]
We implement NA-NHMC with the same hyperparameter configuration 20, dn 0.05, 0.95 for both datasets. For all tasks except phase retrieval, an initial step size óg = 0.05 and an annealing schedule gy 0.5 + 2(1 kt/ 10) applied during the first 10 HMC iterations, after which, the noise-adaptive scheme (Algorithm 3 is used.
For phase retrieval, we use initial step size & 0.2 and keep other hyperparameters unchanged.
annealing schedule g. 1.0 + 20 / k/50 applied during the first 50 HMC iterations, after which, the noise-adaptive scheme (Algorithm is used.
These annealing schedules are chosen encourage sufficient exploration the posterior the early stage. Our results are not sensitive to the exact schedule: using slower schedule does not degrade performance.
A.7 HYPERPARAMETES SENSITIVITY ANALYSIS
We evaluate several hyperparameter choices for the Hamiltonian Monte Carlo sampler the super- resolution (x4) inverse problem using the FFHQ 256 x 256 dataset. When varying a particular hyperparameter, all remaining hyperparameters are kept their default values used in all other experiments. For different choices of the number of leapfrog steps L, we also adjust the number of HMC iterations ensure that each setting uses the same amount of computational resources. The
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resulting performance is summarized in the tables below. The results indicate that the performance of NA-NHMC exhibits little sensitivity to the choice of hyperparameters.
Table 7: Different step sizes for Super Resolution (x4) on FFHQ (256 x 256) with Gaussian Noise By 0.05. (Bold: best)
0.02
0.05
0.10
0.15
0.20
PSNR
27.12
27.29
27.31
27.31
27.31
SSIM
0.745
0.770
0.771
0.771
0.772
LPIPS
0.299
0.291
0.288
0.288
0.286
Table 8: Different number of leapfrog steps L for Super Resolution (x4) on FFHQ (256 x 256) with Gaussian Noise Bg 0.05. (Bold: best)
10
15
20
25
30
PSNR
26.86
27.18
27.29
27.36
27.34
SSIM
0.749
0.765
0.770
0.771
0.777
LPIPS
0.318
0.299
0.291
0.286
0.281
Table 9: Step size decay factor y for Super Resolution (x4) on FFHQ (256 x 256) with Gaussian
Noise By 0.05. (Bold: best)
0.91
0.93
0.95
0.97
0.99
PSNR
SSIM
0.770
0.769
0.770
0.771
0.770
A.8 ADDITIONAL EXPERIMENT RESULTS
Linear IP' results
All experiments Section 3.3 are repeated with a higher level of Gaussian measurement noise (a, 0.20). The results are shown below.
A.9 ABLATION STUDIES
Number of HMC iterations
Sampling with HMC requires a warmup phase, since the initial noise æt may be far from the solu- tion. As shown in Figure 7 the quality of sampled images improves monotonically with the number of iterations, as expected. Performance begins to plateau after roughly 120 iterations. Unlike MAP- based methods such as ReSample (Song et al., (024) and DMPlug (Wang al. (024).. does not deteriorate beyond this point. This stability provides evidence that the prior term acts an effective regularizer, preventing overfitting.
Number of diffusion steps and memory usage
In this section, we evaluate NA-NHME with varying numbers of diffusion steps, using fixed pa- rameters K 80 and 7, = 20. Both runtime (in seconds) and memory usage (in GB) increase linearly with the number steps. We ran all experiments on NVIDIA H200 GPU. The baseline cost is 90 seconds and 3.63 GB for two steps, with each additional step adding roughly 45 sec- onds and 1.84 GB. The quantitative evaluations are shown in Figure g While three diffusion steps achieve the highest PSNR and lowest LPIPS, the improvement over steps is marginal. To avoid incurring roughly 50% additional runtime and memory overhead, we use two diffusion steps in all experiments.
Note that performance appears to decline when using more than three diffusion steps. This effect arises because the sampler converges more slowly its stationary distribution as the number of
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Table 10: Linear IPs on FFHQ (256 256) with Gaussian Noise By 0.05. (Bold: best, underline: second best)
Super Resolution (x-))
Super Resolution (x16)
Randomn Inpuining (92%)
Gaussian Debluring
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNK 1 SSIM
LPIPS
PSNR
SSIM
LPIPS
DiAPIR
96
0.735
0.323
19.84
0.541
0.444
20.93
0.595
0.405
27.48
0.778
0.287
RED-dRf
21.55 .5%%
0.3390
086002
17.60
0.391
0.567
23.70)
0.651
01.344
17.077
0.211
D.692
DPS
26. 8484
0.762
0.239
20,06
0.522
0.380
25.74
0.745
0.245
26.88
0.761
0.234
0.685
ReSample
0.237
B.3822
20.01
0.532
0.576
24-12
0.509
0.728
SITCOM
27.35
0.787
0.268
20.82
0,574
0.400
26.56
0,785
0.265
27.94
0.79%
0.266
DMPlug
26.73
0.697
0.321
17.42
0.280
0.607
26.15
0.769
0.270
27.81
0.769
0.289
NA-NHMC
26.72
0.785
0.798
Table 11: Linear IPs ImageNet (256 256) with Gaussian Noise By 0.05. (Bold: best, underline:
second best)
Super Resolution (x-)
Super Resolution (x16)
Random Inpainiing (92%)
Gaussian Debluring
PSNR
SSIM
LPIPS
PSNR
LPIPS
DiAPIR
23.99
0.626
0.426
18.18
0.387
0.626
19.30
0.443
0.583
25.24
0.678
0.378
RED-dif
17.67
0.266
0.613
12.45
0.151
0.726
17.25
0. 0.360
0.541
13.99
0.170
0.688
DPS
23.36
0.623
0,345
17.15
0.339
0.524
22.31
0.593
0.347
22.55
0.555
0.401
DAPS
14.29
0.139
0.753
0.585
0.132
0.558
SITCOM
24.93
0.684
0.318
18.58
0.404
0.525
23.89
0.684
0.320
25.63
0.712
0.311
DMPlug
24.52
0.667
0.378
16.74
0,311
0.590
23,49
0.668
0.358
23.55
0.605
0.423
NA-NHMC (ours)
24.99
0.665
0,355
19.09
0.396
0.580
24.10
0.676
0.324
25.76
0.699
0.327
Table 12: Linear IPs on FFHQ (256 256) with Gaussian Noise By 0.20. (Bold: best, underline:
second best)
Super Resolution (x-)
Super Resolution (x (x1)6)
Random Inpaintngg (92%)
Gaussian Debluriig
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR 1 SSIM
LPIPS
PSNR
SSIM
LPIPS
RED-dif
12.64
0.101
0.824
11.19
0.152
0.799
15.33
0.168
0.731
8.47
0.037
0.868
DPS
21 80
0.556
0.385
16.13
0.377
0.502
21.60
0.531
0.404
24.45
0.678
0.290
DAPS
13.48
0.121
0.792
17.08
0.420
0.549
20.64
0.353
0.587
8.12
0.031
0,862
ReSample
17.93
0.468
SITCOM
23.04
0.647
0.362
17.49
0.469
0.495
23.23
0.683
0.389
24.97
0.709
0.323
DMPlug
15.95
0.140
0,706
11.69
0.093
0.773
19.55
0.347
0.564
17.33
0.181
0.660
NA-NHMC (ours)
23.29
0.636
0,391
17.36
0.393
0.552
23,69
0.670
0.365
25.57
0,710
0.327
Table 13: Linear IPs on ImageNet (256 256) with Gaussian Noise By 0.20. (Bold: best,
underline: second best)
Super Resolution (x-1)
Super Resolution (x16)
Ramdom Inpuininng (92%)
Gaussian Debluring
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
18.48
RED-JM
11.35
0.197
0.776
9.29
0.09%
0.811
11.37
0.079
01782
8.08
0.043
0.817
DPS
19.29
0.406
0.481
11.25
0.145
0.728
18.64
0.361
(1.005
20.33
0.455
0.468
DAPS
13.71
0.151
0.760
14.27
0.139
0.755
18.89
0.248
0.598
9.06
0.060
0.789
SITCOM
(.440)
0.45%
16.16
0.325
0.628
21.00
0.494
01457
22.73
0.584
0.492
DMPlig
18.58
0.412
0,500
9.53
0.125
0.782
19.15
0.447
0.474
23.05
0.591
0.423
NA-HHMC (ours)
21.53
0.510
0.492
16.43
0.276
0.675
21.80
0.536
0.456
23.45
0.597
0.405
diffusion steps increases. While increasing the number of HMC iterations could offset this effect, would further amplify runtime costs to impractical level.
Diffusion schedule
The pre-trained DMs used in this paper have 1000 diffusion steps. While other methods usually use evenly-spaced schedule with the first step bing pure Gaussian noise (ā, = 0), we found this choice using two steps for unconditional DDIM, the natural choice to use timesteps in the middle. Thus, we choose t = [375, 750], which is spread evenly and avoids numerical stability. Table [6] confirms that this schedule yields superior performance in PSNR and SSIM while being close to optimal for LPIPS. We adopt this diffusion schedule for all main experiments.
to be numerically unstable for our few-step setting. Since
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Table 14: Non-linear IPs on FFHQ (256 256) with Gaussian Noise By 0.20. (Bold: best,
underline: second best)
Nonlincar Deblurring
Phase Retrieval
HDR Reconstruction
PSNR
SSIM
LPIPS
PSNR
SSTM
LPIPS
PSNR
SSIM
LPIPS
DiAPIR
23.34
0.641
0.374
16.76
0.482
0.543
21.85
0.694
0.344
RED-diff
12.85
0.063
0.816
0.061
0.855
16.73
0.222
0.649
DPS
22.83
0.643
0.307
10.60
0.267
0.701
24.92
0.703
0.321
DAPS
17.38
0.154
0.728
12.93
0.103
0.797
18.04
0.299
0,607
ReSample
23.30
0.635
0.477
12.51
0.335
0.712
22.51
0.677
0.428
SITCOM
16.26
0.173
0.656
10.19
0.082
0.810
20.11
0.346
0.534
DMPlug
22.08
0.544
0.437
16.17
0,473
NA-NHMC (ours)
24.89
0.705
0.317
0.434
0.570
26.61
0.793
0.271
Table 15: Non-linear IPs on ImageNet (256 x 256) with Gaussian Noise By 0.20. (Bold: best,
underline: second best)
Nonlinear Deblurring
HDR Reconstruction
PSNR
SSIM 1
LPIPS
PSNR
SSIM 1
LPIPS
DiIPIR
21.52
19.94
0.418
RED-diff
12.47
0.071
0.759
16.46
0.267
0.593
DPS
16.11
0.340
0.551
21.92
0.519
0..48
DAPS
17.84
0,201
0.617
18.08
0.351
0.536
SITCOM
14.49
0.156
0.668
19.82
0.441
0,500
DMPlug
21.80
0.561
0.420
20.54
0.562
0.430
NA-NIIMC (ours)
22.61
0,585
0,382
24.12
0.701
0,320
A.10 ALTERNATIVE SAMPLING SCHEMES
Sampling in a space as high-dimensional as (3 x 256 x 256) is challenging task. Many standard sampling algorithms are not suitable in this setting. A key requirement for efficiency is the use of gradient information to accelerate convergence. The simplest such method the Unadjusted Langevin Algorithm (ULA).
However, because ULA lacks a Metropolis-Hasting (MH) correction, its step size must tuned carefully. Figure illustrates this trade-off: large step sizes enable rapid exploration but cause sig- nificant discretization error as By approaches the target value, resulting in poor samples; conversely, small step sizes reduce error but lead to very slow exploration and long runtime.
Since different stages the sampling chain require different effective step sizes, algorithms with a Metropolis- Hastings (MH) correction more attractive, the acceptance test provides natural criterion for adapting step size. We therefore consider the detropolis-Adjuster Langevin Algorithm (MALA). the No-U-Turn Sampler (NUTS), and Hamiltonian Monte Carlo (HMC). In practice, how- ever. both MALA and NUTS tend to settle on excessively small step sizes in this high- sigh-dimensiooall setting, resulting in impractically long runtimes. By contrast, HMC accommodates larger step sizes and achieves a more favorable trade-off between accuracy and efficiency, making it the most suitable choice for our framework.
A.11 ALTERRATTVV PRIOR MODEL: GAN-BASED INFERENCE
In place of the diffusion models, experimented with StyleGAN2 Karras et al. 2020 the prior model for FFHQ 256 x 256 dataset. The quantitative results are presented in Table The quality of image samples significantly inferior to diffusion models across all tasks.
A.12 ADDITIONAL QUALITATIVE RESULTS
In this section, we present additional qualitative results. Since we don't 't have access to an LDM for ImageNet (256 x 256), ReSample cannot applied this dataset.
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28.0
0.32
0.21
0.76
0.72
0.20
500
100
50
100
50
100
Number HMC iterations
Super Resolution O41
Gausstan Deblurang
Nenimear Debluriin
HDR Rectestruction
Figure 7: Performance of NA-NHMC across four tasks for FFHQ (256 x 256) as function of the number HMC iterations K. For all tasks, performance increases monotonically with more steps, but with diminishing improvements.
27.5
0.425
27.0
0.400
0.375
PSNR
26.0
0.350
25.5
0.325
0.300
25.0
0.275
3 Number of Diffusion Steps
6
Figure 8: Performance of NA-NHMC on SR (x4) task for FFHQ (256 x 256) as a function of the number of diffusion steps. The initial step is fixed T 750 for all cases avoid numerical instability, and the remaining steps are evenly spaced in [0, 750).
Discretisation Error
step sise stepes
step 11,00 1500 stops
Ground Truth
step size 0.0005
8000 stops
Measurement
0%%
100 %
Unadjusted Langevin Algorithm (ULA) with different step sizes. Larger step sizes accel- crate convergence but introduce greater discretization error, substantially degrading sample quality.
Figure 9:
Table Performance of NA-NHMC on SR (x4) for FFHQ (256 x 256). Each schedule is defined by two parameters: (i) the first timestep (rows: 600, 750, 900) and (ii) the final timestep (columns: 250, 375, 500).
Metries
PSNR
SSIM
LPIPS
Schedule
750
900
600
600
750
250
27.12
27.24
26.82
0.744
0.767
0.718
0.290
0.287
0.335
375
27.03
27.29
27.07
0.736
0.770
0.738
0.304
0.291
0.319
500
26.87
27.17
27.01
0.723
0.763
0.741
0.330
0.305
0.317
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Table 17: GAN-Based Inference for Linear IPs FFHQ (256 256) with Gaussian Noise By 0.05.
Super Resolution (x4)
Random Inpainting (92%)
PSNR
SSIM
LPIPS
PSNR
SSIM
LPTPS
NA-NHMC (GAN)
18.27
0.454
0.513
17.80
0.440
0.528
NA-NHMC (DDIM)
27.29
0.770
0.291
26.72
0.785
0.268
Ground Truth
Measurement
NA-NHMC (GAN)
NA-NHMC (DDIM)
Figure 10: Comparison between GAN and Diffusion Model (DDIM) for SR(x4). FFHQ (256 x 256). By 0.05.
Measurement
RED-dff
ReSample
DMPlug
Spe Booa
Speni
Sher
Spac
Bodid
Booo
Boadd
Boda
Bodad
Buase
Boda
Figure 11: SR(x4). (Top) FFHQ (256 256). (Bottom) ImageNet (256 256). ay = 0.05
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Truth
Measuremant
RED-dff
DAPS
ReSample
Curs
Figure 12: SR(x16). (Top) FFHQ (256 256). (Bottom) ImageNet (256 256). oy 0.05
Measurement
ReSample
UMPlug
Figure 13: Random inpainting. (Top) FFHQ (256 x 256). (Bottom) ImageNet (256 256). 0g 0.05
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Ground Truhh
Messurement
DIIPIR
RED-dif
DPS
DAPS
ReSample
DMPlug
Ours
Figure 14: Gaussian deblurring. (Top) FFHQ (256 x 256). (Bottom) ImageNet (256 x 256). Ty = 0.05
Figure 15: Nonlinear deblurring. (Top) FFHQ (256 256). (Bottom) ImageNet (256 256).
By = 0.05
Ground Truth
Moasurament
DIfIPRR
RED-dff
DPS
DAPS
ReSample
DMPlug
Curs
Figure 16: Phase retrieval. FFHQ (256 x 256).
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RED-dif
DAPS
ReSemple
Figure 17: HDR reconstruction. (Top) FFHQ (256 256). (Bottom) ImageNet (256 256).
0y = 0.05
26


Reviews:
[{"review_id": "F9fcTOmUTz", "reviewer": "Reviewer_iaDC", "summary": "This paper presents Noise-space Hamiltonian Monte Carlo (N-HMC) sampler, using a pretrained diffusion prior to solve general inverse problems. N-HMC directly samples from $p(x_T|y)$ with a few-step diffusion rollout estimating $\\hat x_0^*$. While reminiscent of DMPlug, it formulates recovery as sampling rather than optimization, naturally accommodating measurement noise. The authors establish theoretical guarantees for N-HMC, showing its robustness to measurement noise under mild assumptions. Based upon N-HMC, NA-NHMC is proposed to adapt to the possibly unknown measurement noise level without hyperparameter tuning. Experiments on natural images show consistent gains over prior methods, with especially strong performance in noisy inverse-problem settings.", "questions": "- The proposed N-HMC performs $p(x_T|y)$ sampling in the noisy space with the help of a few-step sampler that estimates $\\hat x_0$. Similar sampling strategy is discussed in [1], which also samples in the noisy space but follows an noise annealing scheme as ReSample and DAPS. Can the authors comment on the differences between these methods? In particular, what are the pros and cons of sampling $p(x_T|y)$ vs. sampling $p(x_t|y)$ with an annealing noise schedule? Also, it seems an empirical comparison against SITCOM is necessary as it reported better results than DMPlug and DAPS in the considered experimental setups.\n- What is the reason of choosing inverse-gamma prior for $\\sigma_y$? Is it a specific trick to derive proposition 2?\n\n[1] Alkhouri et al. \"SITCOM: Step-wise Triple-Consistent Diffusion Sampling for Inverse Problems\", ICML 2025.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This paper is overall well-written. It categorizes and clearly explains the strengths and weaknesses of existing methods, especially highlighting how and why existing methods are sensitive to measurement noise and rely on extensive hyperparameter tuning.\n- The proposed N-HMC sampler is well justified by Proposition 1 that indicates its robustness to measurement noises.\n- NA-NHMC extends N-HMC to a blind inverse problem setting where the noise level is unknown. NA-NHMC coincides with N-HMC with known noise level under inverse-gamma prior assumption of the noise level, which is demonstrated both in theory and in practice.\n- Experimental results show clear advantage of NA-NHMC over existing diffusion posterior samplers on image restoration tasks, especially with varying measurement noise levels.", "weaknesses": "- I found no major weaknesses in this paper, but I believe some justifications are needed. See questions below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CPyBQd3Qzs", "reviewer": "Reviewer_4fZg", "summary": "This paper proposes a noise-adaptive method for solving inverse problems using diffusion models as priors. The main goal of the paper is to develop methods that adapt to the manifold structure of data, hence obtaining better performance for inverse problems under noisy scenarios.", "questions": "- Why is the method called noise-space sampling? This is a bit confusing.\n\n- In general, there is a lot of mentions of \"manifold\" but I found this non-rigorous. There's really no geometric insight in any of these comments. For example, the authors mention \"manifold feasibility problem\" as the main motivation of their work, but this is not defined or explained. Is there any theoretical result regarding this?\n\n- Please clarify the equation in line 211. How does first equality work? Since $\\mathcal{D}(x_T) \\approx x_0$, I don't understand why the first equality in line 211 would work.\n\n-  Please provide a remark after Proposition 1 that explains and clarifies the result.\n\n- The paper misses a reference for comparison, which also addresses the noise-robustness issue of standard solvers by adopting a second-order view:\n\n> *Boys, B., Girolami, M., Pidstrigach, J., Reich, S., Mosca, A., & Akyildiz, O. D. Tweedie Moment Projected Diffusions for Inverse Problems. Transactions on Machine Learning Research, 2024.*\n\nPlease add this benchmark to your comparisons in your experiments. \n\n- To see the introduced bias of the method in a simple setting, the paper would benefit from a simple experiment, see Figure 1 of the paper cited above. Please consider adding this.\n\nstyle comment: I do not think using bold text in such frequency is appropriate -- in fact, I think standard academic writing only allows italics for emphasizing - no bolds please.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "This paper addresses an important problem, that is the noise sensitivity and intractability of general methods used for solving inverse problems via diffusion priors.", "weaknesses": "The paper has a few issues that needs to be addressed comprehensively (see questions for details):\n\n- the mathematical justification is thin and derivations are unclear\n- experimental results are missing some baselines proposed to address the same issues, notably noise sensitivity of inverse problem solvers.\n- the method is inherently expensive as DDIM mapping should be autodiffed - the paper avoids this by using a few steps, which is not very principled.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "x7zGYuKHkB", "reviewer": "Reviewer_kd9g", "summary": "To avoid local minima or noise overfitting problem in inverse problem solving with diffusion model, the paper proposed to search a good initial noise by Hamiltonian Monte Carlo (HMC), which leads to the following sampling through an ODE staying on the data manifold. For this, the proposed method repeat updating initial noise with sampling only with 2 denoising steps. The paper also introduces noise adaptive sampling which provides robustness on various measurement noises.", "questions": "- Could authors explain the reason of large gap of performance between the original baseline paper and this paper?\n- Could authors provide runtime comparison by setting the same number of function evaluation? Or Could authors provide performance comparison by setting the same computational budget? \n- Is there any reason for empty boxes for ReSample in Figure 10 - 16?\n- What if we use the GAN or consistency model instead of diffusion model? The reviewer cannot find a strong reason to use the diffusion model from the algorithm 1.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper presents motivation and methods clearly.\n- The paper considers various measurement noise including impulse and speckle noises, which increases the effectiveness of the proposed method in real world.\n- Extensive experiments support the effectiveness of the proposed method and gives sufficient analysis on its behavior.", "weaknesses": "- The major difference from DAPS is twofolds: the paper uses HMC instead of Langevin dynamics, and the search space is changed from image to noise space. However, both changes seems to introduce additional computational cost, which results in slower sampling.\n- Missing related work [1] that update the initial noise with data fidelity gradient after sampling.\n- The performance reported in Table 1 has a huge gap from the original paper. For example, DAPS for Phase Retrieval originally achieves 30.63dB of PSNR with the same setting, but it is 18.52dB in this paper.\n\n\nReferences\n\n[1] Diffusion Image Prior, ICCV 2025", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EXhNg6Nv9J", "reviewer": "Reviewer_eggp", "summary": "The paper proposes Noise-space Hamiltonian Monte Carlo (N-HMC), which uses HMC to search for a good initial noise for solving inverse problems. The authors additionally propose a noise adaptive version of N-HMC, which adjusts the algorithm to work with unknown noise levels.", "questions": "1. The authors assume that $\\sigma_y$ follows an inverse-gamma prior, but later admits that they use an uniformative (i.e. uniform) prior. Any clarification on this?\n\n2. Why is phase retrieval branded as a *multimodal IP*? All inverse problems are inherently multimodal, and this may confuse the readers.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The performance seems to be good, outperforming several widely established baselines.\n\n2. The method of using HMC for searching better noise initialization is new.", "weaknesses": "1. *What* N-HMC is solving is unclear. Is this doing posterior sampling? The mathematical statement should be precisely provided. Currently, the derivation starts with (7), which is ad-hoc. *Where* is the posterior scores used?\n\n2. One of the motivations for this method is that the performance is inherently free form hyperparameter tuning, which does not seem to be the case. As the method is based on HMC, there are actually *more* hyperparameters that one can adjust, including how you would define the burn-in period. Reading the appendix, I am not convinced that the method requires less efforts for hparam tuning. It actually seems to require more effort, as opposed to methods such as DPS where one can just choose a step size.\n\n3. In the experiments, two more metrics should be reported. PSNR/SSIM/LPIPS are all distortion metrics, and reporting them all does not give a more informed picture. 1) Report the FID values (perception metric) with more than 1k, 2) Report the computational cost. The computational cost is reported in the appendix, but it should be more accessible. 90 seconds is relatively slow, which is another drawback of the method.\n\n4. The equality for $\\nabla_{x_T} \\log p(y|x_T)$ is, at best, an approximation. This holds across the entirety of the derivations.\n\n5. The noise-adaptive part is confusing. $\\sigma_y$ is undefined in the main text. It starts by stating that they model the noise variance with an inverse-gamma prior, which is arbitrary. How this leads to Alg. 3, is again, ambiguous. How is $m$ set?\n\n6. Following 5, even for unknown noise levels, methods such as DPS are fine off with choosing a static step size (e.g. 1.0), which works well across all noise levels. If the authors were to truly argue that the noise adaptive part is important, then the experiments should be conducted on real-world degradations that are off the inverse crime setting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)