Measurement Score-Based Diffusion Model
Authors: Chicago Y. Park
Shirin Shoushtari
Hongyu An
Ulugbek S. Kamilov
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion models
generative models
inverse problems
learning from measurements
learning without ground-truth
Primary area: generative models
TL;DR: We propose the first measurement score-based diffusion model that directly learns partial measurement scores using only noisy and subsampled measurements, enabling the synthesis of fully sampled measurements and solving inverse problems.

Abstract: Diffusion models have achieved remarkable success in tasks ranging from image generation to inverse problems. However, training diffusion models typically requires clean ground-truth images, which are unavailable in many applications. We introduce the Measurement Score-based diffusion Model (MSM), a novel framework that learns partial measurement scores directly from noisy and subsampled measurements. 
By aggregating these scores in expectation, MSM synthesizes fully sampled measurements without requiring access to clean images.
To make this practical, we develop a stochastic sampling variant of MSM that approximates the expectation efficiently and analyze its asymptotic equivalence to the exact formulation. We further extend MSM to posterior sampling for linear inverse problems, enabling accurate image reconstruction directly from partial scores.
Experiments on natural images and multi-coil MRI demonstrate that MSM achieves state-of-the-art performance in unconditional generation and inverse problem solving---all while being trained exclusively on degraded measurements.

Published as a conference paper at ICLR 2026
MEASUREMENT SCORE-BASED DIFFUSION MODEL
Chicago Y. Park
Shirin Shoushtari
Hongyu An
Ulugbek S. Kamiloy
Uniiversity of Wisconsin Madison kami semilovewisc.cmm edu
?Washington University in St. Louis
ABSTRACT
Diffusion models have achieved remarkable success in tasks ranging from image generation to inverse problems. However, training diffusion models typically re- quires clean ground-truth images, which are unavailable in many applications. We introduce the Measurement Score-based diffiusion Model (MSM), a novel frame- work that learns partial measurement scores directly from noisy and subsampled measurements. By aggregating these scores in expectation, MSM synthesizes fully- sampled measurements without requiring access to clean images. To make this practical, we develop stochastic sampling variant of MSM that approximates the expectation efficiently and analyze its asymptotic equivalence to the exact formula- tion. We further extend MSM to posterior sampling for linear inverse problems, enabling accurate image reconstruction directly from partial scores. Experiments on natural images and multi-coil MRI demonstrate that MSM achieves state-of- the-art performance in unconditional generation and inverse problem solving-al while being trained exclusively on degraded measurements. Code is available at https: / gi thub. com/ cwmm/wustlll l-cig/ Baigg/MSS /MSM.
INTRODUCTION
Score-based diffusion models are powerful generative methods that sample from high-dimensions distributions by learning the score function the gradient of the log-density- from training data. They achieve state-of-the-arg art performance in generating natural images (Dhariwal & Nichol, 2021), medical images (Khader et al., 2023), and more (Chung al., 2024). Beyond generation, diffusion models can be adapted for conditional sampling to solve inverse problems. However, training requires large set of clean data, which is often costly or difficult to obtain, such as from hardware limits for high- high-resolution images or long MRI scan times that cause patient discomfort.
To overcome the need for clean training data, recent approaches train diffusion models subsam- pled (Daras et al., 2024c), noisy (Xiang et al., 2023; Daras et al., 2024b), or jointly subsampled and noisy observations (Kawar et al., 2024), aiming to approximate the clean image score from degraded measurements. However, this an unnecessarily difficult objective: since measurements typically lie in structured subspace, recovering the full image score from degraded data is inherently challenging.
more natural strategy is to learn partial measurement scores directly within these subspaces. This perspective is closely related to the success of patch-based methods in imaging, where training on local patches improves scalability (Alkinani & El-Sakka, 2017; Wang et al., 2023b; Hu et al., 2024a). Our work extends this principle from the image domain the measurement domain, enabling new framework for training diffusion models entirely from degraded measurements. Operating in the measurement domain brings a key advantage: each subsampled measurement is uniquely defined by the acquisition operator, whereas a corrupted image is not not-infinitely many images can map to the same measurement. Training the measurement domain removes this ambiguity and ensures that the model learns to denoise well-defined, physically meaningful inputs.
We introduce the Measurement Score-based diffusion Model (MSM), which extends the idea of patch-based learning from the image domain to the measurement domain. Instead of recovering full-image scores, MSM learns denoising score functions restricted to the observable regions of noisy, subsampled measurements- ent.eeemeeeeeeaaainnngngg self- supervieed training without clean data. By aggregating these partial scores through randomized subsampling, MSM defines an effective framework for both generating full measurements and solving inverse problems. To make MSM practical, propose

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efficient stochastic sampling algorithms for unconditional generation and posterior sampling in linear inverse problems. Extensive experiments on natural images and multi-coil MRI demonstrate that MSM matches surpasses oate--o-tthearr performance.
2
BACKGROUND
2.1 SCORE-BASED DIFFUSION MODELS
Score-based diffusion models (Song & Ermon, 2019; Ho al., 2020; Song et al., 2021c; Park et al., 2025) learn the score function using neural networks. Tweedie's is formula (Efron, 2011) relates the score function to the minimum mean square error (MMSE) denoiser, allowing the score to be estimated using only noisy inputs and their corresponding denoised outputs. Learning score function performed across varying noise levels, by considering noisy images 2; clean image, n ~ No, I), and o is the noise level at the timestep t.
J,n, where æ is
Given a denoiser Dg trained to minimize the mean squared eπror (MSE), Tweedie's formula approxi- mates the score function as V log Po, (zx) = (Dg( (xx) - xx)) /o. This relationship allows denoisers to serve as practical estimators of the score function at varying noise levels, providing gradients that guide reverse-time stochastic sampling (Robbins, 1956; Miyasawa, 1961; Vincent, 2011).
Sampling then proceeds through a sequence of random walks (Song et al., 2021c; Park et al., 2025) as Tz- 22 7: V log Po, (ax) + //2:TTn for t T,T 1,..., 1, where or, Tx, and T. denote the noise level, step-size, and temperature parameters. These parameters can be derived from theoretical frameworks (Ho et al., 2020; Song et al., 2021c) or tuned empirically (Park et al., 2025), and the initial sampling iterate Ør is drawn from a standard Gaussian to be consistent with the training input of the denoiser.
2.2 TRAINING DIFFUSION MODELS WITHOUT CLEAN DATA
Training diffusion models to learn the score of clean images typically requires access to high-quality. clean data. However, in many applications, data is often subsampled, noisy, or both.
Training with noiseless but subsampled measurements, Ambient diffusion (Daras et al., 2024c; Aali et al., 2025) is a recent method for training diffusion models from subsampled measurements by applying an additional subsampling operation during training. At each step, the model receives a further subsampled and noise-perturbed input and learns to reconstruct the original subsampled measurement. This procedure jointly encourages denoising and inpainting, guiding the model to approximate the conditional expectation of the clean image given a noisy, partially observed input.
Training with noisy but fully-sampled measurements. SURE-score (Aali et al., 2023) enables training score-based diffusion models without access to clean data by leveraging Stein's 's unbiased risk estimator (SURE) (Stein, 1981). SURE-score trains the diffusion model using only noisy mea- surements, combining two loss functions: a SURE-based loss for denoising the measurements and a denoising loss for the diffusion noise added top of the denoised estimate. Another approach (Daras et al., 2024b) considers two regimes based on the relationship between measurement noise and diffusion noise. When the measurement noise is smaller than the diffusion noise, the clean image is estimated using Tweedie's 's formula by predicting the measurement from the diffusion iterate. Conversely, if measurement noise exceeds diffusion noise, the model trained with a consistency loss (Daras et al., 2024a), which encourages stable denoising outputs across nearby timesteps by enforcing that predictions remain consistent along the model's 's reverse trajectory.
Training with noisy and subsampled measurements. GSURE diffusion (Kawar et al., 2024) trains diffusion models using only noisy, subsampled data by adapting the Generalized SURE loss (Eldar, 2008) to the diffusion setting. It reformulates the training objective as projected loss computable without clean images. While this objective function theoretically shown to be equivalent to the supervised diffusion loss under the assumption that the sampling mask and the denoising error are independent, GSURE diffusion has two limitations: it has so far only been demonstrated in the single-coil setting and extending it to multi-coil MRI remains computationally challenging due the need for singular value decomposition of the full measurement operator; it also requires the minimum diffusion noise level do to match the measurement noise level p. The latter can severely degrade
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Ve 8al1
Not available
Available
S S~p(S)
8
St.
Dø
80
Training scheme
P(S)
Sampling scheme
Stochastic denoising loop
Reinject Noise
S: p(S)
S
Weight
Zt
St
Dg
8g
20
Figure 1: Illustration of the Measurement Score-based diffusion Model (MSM) for training and sampling using subsampled data. Training: MSM is trained solely on degraded measurements. Diffusion noise is added to these measurements, and the model learns denoise them. Sampling: At each diffusion step, MSM randomly subsamples the current full-measurement iterate, denoises the resulting partial measurement, and aggregates multiple outputs. weighting vector compensates for overlapping contributions across partial measurements. See Figure 4 for the MRI-specific version.
sampling performance when P exceeds typical value of do (e.g., du 0.01 in (Ho et al., 2020; Dhariwal & Nichol, 2021)). Other recent methods (Bai et al., 2024; 2025) address the same setting by alternating between reconstructing clean images using diffusion priors pretrained limited clean data and refining the model to learn from noisy, subsampled measurements.
While these methods differ in how they handle subsampling and noise, they share a common challenge: they attempt to directly approximate the full image score from degraded data. In contrast, our approach leverages the partial measurement-baasee statisties to generate the full measurements by directly learning measurement scores restricted to observed measurements. A detailed comparison of our approach with related approaches provided in Appendix E.1.
2.3 IMAGING INVERSE PROBLEMS
Inverse problems aim to recover an unknown image a E RR from noisy, undersampled measurements y R" modeled as y = Ar + e, where A mRx is known forward operator and e ~ (1) is Gaussian noise with variance ?).
When clean training data unavailable, self-supervised learning is often used for training neural networks directly degraded measurements, without the need for clean ground-truth data (Chen et al., 2021; 2022: Hu et al., 2024b; Yaman et al., 2020; Millard Chiew, 2023). For example, Self- Suff-uuperviioo via Data Undersampling (SSDU) (Yaman et al., 2020; Millard & Chiew, 2023) is a widely-used approach for training mod-to-end restoration networks using distinct subsets of the measurements. In the context of score-based diffusion models, Ambient Diffusion Posterior Sampling (A-DPS) (Aali et al., 2025) replaces the dlan-imagge-rrineel model in the DPS method (Chung et al., 2023) with Ambient diffusion model trained subsampled measurements.
3 MEASUREMENT
DIFFUSION MODEL
We present the training and sampling procedures for MSM. Our approach generalizes the idea of patch-base basch learning-wiin widely used to enable computationally efficient supervised training- to self- supervised learning in the measurement domain. A key feature of MSM is that operates solely

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subsampled measurements during training. This enables learning partial measurement scores without access to clean ground-trume truth data and naturally extends self-superviee denoising to the challenging setting of noisy, subsampled measurements. We also introduce conditional sampling algorithm that uses the pretrained MSM to solve linear inverse problems.
3.1 LEARNING PARTIAL MEASUREMENT SCORES
We first consider the setting where MSM has access only to partial, noiseless observations of an unknown sammled measurement R". Let R denote subsampled measurement
= Sz E R
where S E {0, 1 with m < n. is subsampling mask drawn from distribution p(S). We assume that in the absence of noise, the fully-sampled measurement uniquely determines the underlying image x. The definition of 2 depends on the application: for natural images, we set = æ, where FCx, where the Fourier transform and C' is coil-sensitivity operator (Fessler, 2020). More broadly, we suppose that sallles measurements are form that MSM is designed to operate solely on partial measurements s Sz, with S ~ p(S), without access to any fully-sampled z or clean image æ.
IR is the clean image; for MRI, we set
Tx, where E R=* (p Z n) an invertible transformation. It is important to note
We define forward diffusion process that adds Gaussian noise to the subsampled measurement
= S oEn,
72 NNO,
where E (1...., 7), 80 8, and 8T approaches a known distribution such as standard Gaussian. At each step f,, the diffusion model Dg takes as input the noisy subsampled measurement 8, and is conditioned on the noise level os, and outputs a denoised estimate of the subsampled measurement 89 R
$ o,) = Da(s oz).
(1)
The model is trained by minimizing the mean squared error (MSE) loss between the predicted and true subsampled measurements:
(()) F... [||s (s 0.)
Once trained, the partial measurement score function can be approximated using Tweedie's for- mula (Efron, 2011), which estimates the gradient of the log-probability of the measurement iterate:
(s oz, S) of (So(s:: σ1) -s)),
where S9 E nd denotes the learned partial measurement score, explicitly conditioned on the known subsampling mask S that generated 8t. Illustrations for training are provided in Figure and Figure 4.
3.2 UNCONDITIONAL SAMPLING WITH MSM
Implementing diffusion model on the fully-sampled measurement requires access to the score function V log Po, (z)), where denotes the noisy version of the mallyssmppled measurement Instead, we train our model to approximate the partial measurement score Pog (s:).
The goal of MSM sampling is to generate mully-sampler measurement z given the partial measure- ment scores in (2) for ~ p(S). To that end, we define the MSM score as the expectation over all possible mask-conditioned partial scores
V log Lo, (zx) W Es<p(S)
S V log Por (8: SS)
la, S..
where each subsampling operator S Rmx drawn from distribution p(S), and W R" is a weighting vector that compensates for overlapping contributions across sampling masks. W is defined as the reciprocal of the expected total coverage:
W
[max (Es-p(S) [diag(S S)] 1)] 1)]-
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Algorithm I Measurement Score-Based Sampling
Require: T. ((S) {ot){=1
1: Initialize 2T ~ (0, I), 2g + 0
2: for = T to do
3: for to w do
4:
S ~ P(S),
t 5:
Partial score-based denoising
5:
89
(1) 8t + to (p (1)
; S(1)
(4) p(s
(0)
6:
Si
Reinject noise & update iterate
2t - SOOT 8
+ (I S s()) ((()))(())))))))))))))))))))))))
7:
8:
end for
9:
C' D diag( (S(O) S(O)
Compute weight for aggregation
10:
W [ 1)]
11:
20 -
W Σ 5 g
+ Ic=0 20
Aggregate partial denoised results
12:
2t-1 ( 21, 20)
end for
14:
return z0
where the maximum is applied elementwise to avoid division by zero in regions not covered by any subsampled measurement. Note that the MSM score in (3) can also be interpreted as the score of Product-of-experte (compositelikellhoon)
model, in which each expert corresponds to mask- conditioned partial score V log Po, (8x S), and MSM score is obtained by aggregating these partial scores across random masks.
To efficiently approximate the expectation in (3), we propose a stochastic sampling algorithm that uses randomly selected subset of partial scores. Specifically, masks ( P(S) for i 1,..., 22, where ())) E IR denotes subsampling operator i. The corresponding subsampled measurement obtained as s (i) ( zt. An unbiased estimator of the MSM score is then defined as
stochastically sample sampling
v log To. (zx) = W
(()
Pa. (s s(o)
(5)
W
S000
where each transpose operator ((i) E |nxx; maps the partial score from the subsampled measure-
ment space back the fully-sampled measurement space. Note that the reweighting vector W in (4)
can be empirically estimated as: W + mmx (Σm diag (S()) g(O). 1)]
Concretely, MSM performs sampling by iteratively reconstructing a fully-sampled measurement iterate z, through a stochastic loop of w partial denoising operations. At each diffusion time t, the algorithm draws w random subsampling masks S(1), each revealing a different subset of coordinates. (i) For mask (1)), the corresponding partial measurement s 5 2, is denoised using the pretrained MSM model to obtain 8(
Within this stochastic loop, MSM updates z, after each partial denoising step by (1) drawing noisy partial estimate (1) p(s
( s( and (2) reinserting the noisy partial estimate into the current
iterate: 2; ( s
(6) (I- ( ((() zi. This update ensures that every subsequent partial
denoiser operates on an iterate that already incorporates the information extracted in the previous loops, enabling the stochastic loops refine complementary regions of 2;-

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After completing the stochastic loops, MSM aggregates all partial denoised estimates to form an MMSE estimate of the salpy-sampled measurement:
En W
((i)
(0) 80 + 1c-0 Ep,
(6)
where C= = Di=1 dSag(S() 5()) records which coordinates covered, and the indicator term preserves previous values at coordinates that receive no coverage by the w stochastic loops.
The estimate En treated the clean prediction for the reverse update 2t-1 p(zz-1 2;, E0). following the standard reverse-diffusion step (Ho et al., 2020, Equation 6). Repeating this process from T down to 1 produces fully-sampled measurement 20, which is mapped to the final output image when relevant. The full sampling procedure is outlined in Algorithm 1, with illustrative examples shown Figure and Figure 4.
In Appendix A, we provide a theoretical guarantee showing that the distribution ((z) obtained using the stochastic MSM score converges to the ideal MSM distribution ((z) as the number of stochastic iterations increases. Specifically, under bounded-variance assumption the score estimator, the KL divergence satisfies
DRL (q(2) [()))) C,
where is the boundedvvrrianee constant defined in Assumption 1 and C' is finite constant indepen- dent of w.
3.3 POSTERIOR SAMPLING WITH MSM
We extend MSM to sample from the posterior distribution for solving linear inverse problems described in Section 2.3. We consider measurement operators A of the form A = HT, where H E m is the linear measurement operator-such as downsampling, blurring, box inpainting, or random projection-and T E Raxp is an invertible transformation introduced in Section 3.1. This allows us to express the measurement model as y = Hz + e, where IR" the unknown fully-sampled measurement and (0 m/) is Gaussian noise with variance
To estimate the posterior score, we combine the stochastic score estimate from (6) with corre- sponding fully-sampled prediction éq. The posterior score is approximated
V log Po. (z: y) = log (z)) + V log Po. (y x1) 2 V log To, (=e) V log (y
20),
where the log- log-likeld gradient is given by
V llg.Po. (y) 2n) V y H H 11
(7)
with yt denoting the step-size parameter. Note that H may differ from the randomized subsampling operators ~ PSS) used during MSM training.
We incorporate the likelihood term by updating the denoised estimate via
2S 7t V ly H
This update is inserted between lines 11 and 12 in Algorithm to convert unconditional MSM sampling into posterior sampling.
A related posterior sampling strategy, using diffusion models trained on clean data, was proposed in (Wang al., 2023a). Further simplification of our method for cumpressed-sensingg MRI presented in Appendix D.3.
3.4 LEARNING PARTIAL MEASUREMENT SCORE FROM NOISY AND SUBSAMPLED MEASUREMENTS
We now show how MSM framework can be extended to train directly noisy and subsampled measurements by integrating with self-supervised denoising methods to address the noise on the subsampled measurements.

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We formulate our observed measurement
+M -~ O~,(
where p the measurement noise level. and the remaining notations follow Section 3.1. We first define the sequence diffusion noise level {or) At each time step t, we compare the diffusion noise level or with the measurement noise level p. and apply one of the following strategies accordingly:
Case 1: oz > p. We add residual noise to match the diffusion level:
8t 8 v ot p2n.
The training objective is: ((0) Esg,
|| Eis sc]ll]] + (SURE((;
o - p² E (o of (80(81501)) -81) + +88 11:
+ LSURE (0; p).
where the first term is inspired by (Daras al., 2024b), which shows that noisier image can be denoised using a less noisy reference; we apply this idea to subsampled measurements. The second term is the SURE loss, following (Chen al., 2022, Equation 9), which enables the model to learn to denoise measurement noise and plays key role in the next case.
Case 2: noise as
p. We first denoise using the MSM with the noise conditioned of p, then add diffusion
8t + (s :) + o,n.
Training minimizes the discrepancy within the non-subsampled region:
( = E.,, (s p) $s(s; o ) + (s,
Here, (s;s;p) p) serves as a pseudo-clean reference. Its quality is crucial but improves naturally during training, since the same prediction is refined in Case 1. In practice, most o, larger than p. making Case more frequently sampled. As a result, the pseudo-clean reference used in Case 2 is continuously improved, ensuring stable training across both cases.
NUMERICAL EVALUATIONS
We evaluated MSM on unconditional generation and conditional sampling for natural images and multi-coil MRL All models used the same diffusion architecture (Dhariwal & Nichol, 2021), trained from scratch on a single NVIDIA A100 GPU for 1M iterations (see Appendix D.4 for architectural details). Experiments used 69k FFHQ faces (128 128 RGB) and 2k fastMRI T2-weighted slices (256 x 256, complex-valued) (Zbontar et al., 2018; Knoll et al., 2020), We performed inverse problem evaluations on 100 test images per domain.
4.1 RGB FACE IMAGE EXPERIMENT
Training data. We considered training settings: (1) subsampling only and (2) subsampling with added Gaussian noise 0.1. In both cases, the masked-pine -iieel ratio p is set to 40% using 32 x 32 patches, applied identically across RGB channels (see Figure 2 for example). Full training details of our method and baselines are provided in Appendix D.4 and Appendix D.5.
Unconditional sampling. MSM used stochastic loop parameter = with 200 sampling steps. We compared MSM to three baselines: an oracle diffusion model trained on clean images, Ambient diffusion (Daras et al., 2024c) trained on noiseless masked inputs, and GSURE diffusion (Kawar et al., 2024) trained on noisy masked inputs. All baselines used 200-step accelerated sampling (Song et al., 2021a). As shown in Table 1, MSM achieves better FID scores than all baselines trained without clean data, evaluated clean images despite being trained without clean data. Additional results showing how w influences sampling quality and time efficiency are provided in Appendix E.6.
10,000 generated samples. Figure 2 further shows that MSM generates
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Table FID scores for unconditional image samples under different training scenarios on human face im- ages. Best values are highlighted for each training scenario, with comparisons shown when corresponding haseline methods available. Note how MSM con- sistently achieves substantially lower FID scores than alternative methods across the evaluated settings.
FID scores unconditional image sam- ples under different training scenarios on multi-coil brain MR images. Best values are highlighted for each training scenario, with comparisons shown when responding haseline methods available. Note how MSM achieves a lower FID compared the alternative methods.
Table 2:
Training data
Metbods
FID.
Training data
Methods
FID.
No degradation
Oracle dillusion
10.21
degradation
Oracle diffusion
28.41
= 0.4, =
Ambient diffusion
MSM
29.14
55.90
R 4,
MSM
64.37
Ambient diffusion
70.07
MSM
37.14
0.4, p
GSURE diffusion
It 4, 0.1
MSM
82.17
Training set: cleen data
0.4,1
p 0.A.7
clean data
Acc. 4, n
Aoc. 4, = 0.1
Generated samples
Figure 2: Generated samples from MSM trained under three degradation settings (first row: training data; second row: samples generated by models trained the corresponding data). Note how despite never seeing ground-trui -tuthh data, MSM can generate high-quality images.
Conditional sampling. We compared MSM with A-DPS (Aali al., 2025), which the most closely related inverse-probles problem solver that also uses pretrained diffusion model trained on incomplete data. We evaluated two inverse problems: box inpainting with a 64 x 64 missing region and 4x bicubic super-resolution. Both methods used models trained on the same noiseless 40% masked data without retraining. A-DPS took 1000 steps, while MSM took 200 steps with w = 3, and step sizes in (7) set to Y 1.75 for both inpainting and super-resolution. As shown in Table 3 and Figure 3, MSM outperforms A-DPS in PSNR, SSIM, and LPIPS. We also observe that A-DPS performs worse than the input image in PSNR and SSIM, likely due limitations of the Ambient diffusion prior in generating fine details when trained on non-sparse subsampling patterns-suce as box masks -rater than more localized patterns (e.g., dust masks) mainly used in the original training setup of Am- bient diffusion (Daras et al., 2024c). Detailed hyperparameter setups for A-DPS are provided in Appendix D.6. Additional experiments showing that MSM achieves comparable performance to dlean-data-oe dara---att-- diffusion based inverse problem solvers are provided in Appendix E.5. Results using an MSM prior trained on noisy and subsampled data are provided in Appendix E.3.
4.2 MULTI-COIL MRI EXPERIMENT
Training data. We considered two training settings: (1) subsampling only and (2) subsampling with added Gaussian noise p = 0.1. We applied a 1-D Cartesian subsampling operation in k-space using random masks with acceleration rate of R 4, while fully sampling all vertical lines and the central 20 lines for autocalibration. Training details for our method and baselines are provided in Appendix D.4 and D.5.
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Table 3: Quantitative results on two natural im- age inverse problems comparing methods using diffusion priors trained without clean images. Best values are highlighted per metric. MSM achieves the best performance across both distortion -aseed and perception oriented metrics.
Table 4:
Quantitative results on multi-coil com- pressed sensing MRI comparing diffusion based and self- supervised methods, all trained without clean data. Best values are highlighted per metric. MSM outperforms the baselines in both PSNR and LPIPS, including the restoration speciifc baseline SSDU.
Setup
Toput
A-DPS
MSM
Sctup
Input
A-DPS
SSDU
MSM
PSNR1
18.26
20.14
24.71
PSNR
22.75
27.28
29.65
30.71
Inpainting
SSIMT
0.749
0.621
0.867
CS-MRI(X4)
SSIMT
0.648
0.801
0.847
0.839
LPIPS
0.304
0.305
LPIPS| 0.306
0.173
PSNR1
23:21
22.61
28.11
PSNR
21.94
26.29
28.02
28.86
SR (x
SSIMT
0.728
0.702
0.868
CS-MRI (x6)
SSIMT
0.617
0.763
0.820
0.805
LPIPS
0.459
0.277
0.117
LPIPS
0.342
0.201
0.186
0.168
Input
A-DPS
MSM
Groundtruth
Input
A-DPS
SSDU
MSM
Groundtruth
0227
18/37 7/33
29 0.07
PSNRY LPIPS
22.607033
25
28 7037
29 79.55
PSNR PLPPS
CS-MRI (c4)
Inpainting
24.13 0.51
23.85 0.27
27.93 0.12
PSNR PSN//PIPP LPIPS
23.29
28.62/0.18
28.29 /.16
29.62/0.12
PSNR LPIPS
CS-MRI (266)
SR (De4)
Figure 3: Visual comparison of methods trained on subsampled data for inverse problems. Note how MSM leads to the best results in both applications.
Unconditional sampling. MSM used stochastic loop parameter of w 1 with 200 sampling steps (see Appendix E.6 for how w affects sampling quality). We compared against an oracle diffusion model trained clean images and Ambient diffusion (Daras et al., 2024c) trained on noiseless subsampled inputs: GSURE diffusion (Kawar et al., 2024) is omitted since its extension from single- coil MRI to multi-coil MRI remains computationally infeasible and no practical approach has been proposed, All baselines used accelerated sampling with 200 steps (Song et al., 2021a). As shown in Table 2, MSM achieves better FID scores than Ambient diffusion, based on 10,000 generated samples. Figure 2 further demonstrates that MSM generates realistic images, even when trained on noisy and subsampled measurements, FID computation details are provided Appendix D.7, and generation results under extreme subsampling are shown in Appendix E.2.
Conditional sampling. We evaluated accelerated MRI reconstruction using random masks with acceleration rates R 4 and R 6, and measurement noise = 0.01. We used stochastic posterior sampling algorithm introduced in Appendix D.3 with MSM pretrained on noiseless measurements without retraining, with step size 7t 2 in (19). We compared against two baselines trained on the same subsampled data: diffusion based method (A-DPS (Aali et al., 2025)) and self-supervised end-to-end method (Robust SSDU (Millard & Chiew, 2024)). A-DPS used 1000 steps, SSDU performed single forward pass, and MSM used 200 steps with w 3. As shown in Table and Figure 3, MSM outperforms both baselines in PSNR and LPIPS. Detailed hyperparameter setups for A-DPS and Robust SSDU are provided in Appendix D.6. Comparisons with dlean-dat--triing diffusion- based inverse problem solvers are in Appendix E.5, and additional results using MSM prior trained on noisy and subsampled MRI data are in Appendix E.3.
5 CONCLUSION
We introduced the Measurement Score-based diffusion Model (MSM) framework for generating the full measurements using score functions learned solely from noisy, subsampled measurements. Our

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key idea is the MSM score, defined as expectation over partial scores induced by randomized subsampling. We develop a stochastic sampling algorithm for both prior and posterior inference that efficiently approximates this expectation, enabling clean image generation and inverse problem solving. We demonstrate that MSM achieves state-of-the-art performance among Liffusion-hased methods trained without clean data, for both unconditional image generation and conditional sampling in linear inverse problems. The framework applies broadly to settings where only subsampled measurements are available but collectively cover the full data space, making it valuable for generative modeling in limited-data regimes and high- dimensiinnii sampling from oow-dimesioonl observations.
6 ACKNOWLEDGMENTS
This work was supported in part by the National Science Foundation under Grants No. 2504613 and No. 2043134 (CAREER), and by the National Institutes of Health (NIH) under Grants RO1EB032713 and RFINSI16565.
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A THEORETICAL ANALYSIS
We have introduced the MSM framework, which can generate fully-samples measurements using partial measurement scores. To make the expectation of the MSM score (3) efficient, our algorithm approximates using a minibatch of w sampled operators in (5), where S(1), s()),..., S)() are sampled independently and identically from the distribution p(S). This implies that for fixed W, we have an unbiased estimator of the MSM score: E !V log 9o, (z+) = V log go, (ze), where the expectation over the randomness in the sampled minibatch.
Assumption 1. There exists such that for all R"
E V log fo, (z) V log Co. (z)
where the expectation taken over S ~ P(S).
This assumption implies that the gradient estimate of MSM score has bounded variance, assumption commonly adopted in stochastic and online algorithms (Welling & Teh, 201 Ghadimi & Lan, 2016; Liu et al., 2022).
Theorem 1. Let q(=) and q(z) denote the distributions of samples generated by using the MSM score log go (zt) and its stochastic appreximation V log 9o. (zt). respectively. Under Assumption the KL divergence between the two distributions is bounded
DKL(g(2) (()) S 2
C,
where C is a finite constant independent of
Proof. Our proof invokes Girsanov's theorem, which characterizes how the distribution of Brownian-driven stochastic process transforms when we transition from one probability measure to another (Chen et al., 2024: Baker et al., 2024; Huang et al., 2021; Song et al., 2021b).
Consider the two stochastic processes {=(t))eeju.n and {²(1))ee(o.1) corresponding to the Eu- ler Maruyama discretizations of the following reverse-time SDEs
dz log 9o. (z) + /2T,w z(T) zT N(O,I),
d = V log 7n (2)dt + / 2Tdd. 2(T) = 2r ~ N)O,T),
Let [[zor] and [[ denote the path measures induced by the respective processes. The process (e(t))eejo11 corresponds to the reverse diffusion trajectory driven by the true measurement score log yo (2), while ($(1)) is generated by the reverse process using an approximate (stochastic) measurement score log To (z).
By using the chain rule of KI. divergence from Lemma 3, we have
DRL (Q || Q = BEy~N(o.) |DK (Q(1 27) || @112== = [(1))-
(8)
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Using the definition of KL divergence and the fact that Mr dQ//Q from Lemma 1, we have
dQ
DKL (Q(.Iz zT) 11 Q(.12 21)) = EQ
log
Eg log My]
dQ
V log You, (zt)
V log aore (Ze)
T
V log 9o. (z)) V log To. (z1)
E
dwo
+
J
V2T2
2
2Tc
log yo (zt) - V log 20, (zc)
1
T
V log Jo. (z+) - V log Cor C, (2t)
Ee
dwe.
B2
db
V2T
2
2
2T.
I
T
log 9o (2e)
1
v log qo (s
dw,
2T
W
( 21
+ Ba
T
1
E
log yo (2t)
(8) log Yo
27
dl
4T.
W
=S()) x1
!
T
2²
x²
Ee
E
V log 9o, (2z)
v log Co (8
dt
dt
C,
4T,
a. 55())
w
4Tc
w
where C' = Sor 1/(4T)) is a finite constant. the first line, we use the definition of KL divergence between Q and @ and the result from Lemma 1. In the third line, expectations over w sampling masks S ~ p(S) for = ...... w. Note that since E V log To. (z)) an unbiased estimator of MSM score log qo, (z)).
the law of iterated
have E|V log 9o, (z,)] v log qo, (z²).
which yields the expectation in the fourth line to be 0. In the last line, we use the bounded variance in Assumption 1.
Following this result with (8) and Lemma 2, we have
82
DKL
To) s DRL (Q
(I
C.
w
Lemma 1. (The Girsanov Theorem III.) Let (z(1))=z and (E(()}=_r be Itå process of the forms
dz V log yo (z)dl+ + 2T.d. =(T) = 2T ~ NO, dē V log ao, (Z)dt + v 2T,W. 2(T) ΣΛ ~ ~(0,11
where 0 < T S oo a given constant, and w E I2' is a nn-dimessonnal Brownian motion. Suppose that there exist a process a(z,t) such that
v Vog (zt) - V log ão, (z+)
alz, t)
(10)
V2T.
which satisfies Novikov's condition
E erp (1 1 T
2 (-"' a a² t)dtt t)de
1 T
V log fo (zt) V log To. (zi)l2
] oo.
dt dt)
erp
2
2T.
where E Eg is the expectation with respect to Q. probability measure induced by the process (z: 1- Then, we can define Mr and probability measure Q induced by process {2,00- as
Mr
V log yo (Ze)
V log Co,, (zx)
1 [ IV log qo. (21) V log qo. (zi) dt
/'
(-Jo
exp
dw; 2 2
2T
2T.
where
1 T and d MrdQ.
(11)
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Proof of the Girsanov Theorems can be found in (Oksendal, 2013, Theorems 8.6.3, 8.6.4, and 8.6.5). Remark. Note that it can be shown that Novikov's condition is satisfied for function a(2, t) in (10)
| log 9o (2,) Vlog To. (2x) dt
E
exp
2 2
2T.
1
IE
V log ar, (Zt)
V log To, (ze) dt
eap
2
2T.
22 erp
1
2Tc dl
<0,
w
2
where in the second line, we use the total law of expectation (i.e., E[a] = [|E[[]b]] we use the law of iterated expectations over sampling masks S p(S) for 1...., w. Here, we use Assumption land the fact the fo" (1/(2T)) dt is a finite constant.
Lemma 2. Let Q and Q be the path measure of two stochastic processes {=(()) and (3(1)) We denote qo and To as the marginal distribution of z (0) and (0)) Then, we have
DK go) S DKL QQ Q)
Proof. From the chain rule of KL divergence, we have
DKL (Q || ② = DKL (Qz(0)==2) @=)())=0))
+ +ʃ. (QL x00) = z0) || @( (()) = =x)) (dzz
/ (Q =0) zo) || @ 3()) (0)
DKL(go (go || To)
From the non-negativ of KL divergence, obtain the desired results.
Lemma 3. (Chain Rule of KL Divergence.)
Let Q and Q be the path measure induced by the two following reverse-time SDEs
dz V log Yo (z) + 22Tdw =(T) = 2T 10,
d log qo. (2) /TTdw 2(T) 2g ~ (0,
From the chain rule of KL divergence, we have
DKL (Q || 11()=
DKL (Qx(T)=* @=(T)==))
DKL (Q(((())(T)) 2T) Q((((T)) = -2r) (=(T)=or(dz)
DKL (Qx()))) ( + Ber~A((0.)) [ = 2T) @(112 2T))
= =Ezr~Noon
[KL = 2T) || Q612 = 2T))
where in the last two equalities, we use the fact that Qzr = Q., = IO,
B PROOF OF SYMMETRY PROPERTY OF THE MSM SCORE JACOBIAN
We now show that the MSM score has a symmetric Jacobian, as in the true score. Assuming the underlying distribution is twice differentiable and well-behaved (Song & Ermon, 2019; Song et al., 1b) differentiating the MSM score expression in (3) with respect to 2, gives
2 log Yo (z,) = W (ES-p((
S ST Po. (8, S) S5
(12)
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Since V log (s. [) is symmetric by assumption, and both the transformation (-)S and expectation preserve symmetry, the result remains symmetric, multiplying by the diagonal matrix W also maintains symmetry, the MSM score admits symmetric Jacobian.
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C PRACTICAL RELEVANCE OF ASSUMPTIONS AND THEORY
C.I PRACTICAL VALIDITY OF OUR ASSUMPTIONS ON FORWARD-OPERATOR
Our theory assumes two conditions: (i) the training measurements collectively cover the full mea- surement domain, and (ii) the subsampling operators S used during image sampling are drawn independently from a distribution p(S). Both conditions are realistic in many imaging modali- ties. For (i), different acquisitions naturally provide complementary subsets of measurements. For example, in magnetic resonance imaging (MRD, different k-space sampling masks are routinely used across acquisitions; in computed tomography (CT), projection angles can be varied; in optical tomography, source-detector positions or wavelengths can change; in positron emission tomography (PET), detector configurations or energy windows may differ; in ultrasound imaging, one can vary transducer firing patterns; and in light-field photography or electron microscopy, system parameters such as aperture, focal length, sample orientation, or defocus levels can be adjusted. Beyond scientific imaging, maage-restoratin settings also satisfy this condition when different samples reveal different observable regions- for instance, box inpainting, random (dust-like) inpainting, or partially blurred regions that vary across images. MSM accommodate such scenarios because the observable regions collectively cover the full domain. However, degradations that apply uniformly across the entire image-Such global Gaussian blur, motion blur, nonlinear measurement effects-do expose complementary regions and therefore are not handled by the current formulation. Extending MSM to handle such globally applied degradations remains an interesting direction for future work.
For (ii), once the set of forward operators observed in the training data defines the empirical dis- tribution p(S). MSM simply samples new operators independently from this distribution during generation. Note that this assumption is only required during sampling and does not require the physical hardware to acquire measurements independently; simply reflects how MSM simulates operator variability during generation. Importantly, sampling new operators in this way also ensures that all measurement coordinates are repeatedly visited throughout the diffusion process. each stochastic loop drops proportion p of coordinates, then the probability that given coordinate is never selected across all diffusion steps is pl (#)ttpp which becomes negligibly small in practice. For instance, with p = 0.4, 100 diffusion steps, and w = 1, this probability is 0.4 x 1.6 x 10 10-40, Thus, MSM effectively achieves full coverage of the measurement domain during sampling, avoiding the undesirable case where any region is left untouched.
C.2 EMPIRICAL ACCESSIBILITY OF BOUND PARAMETERS IN THEBRY
Theorem 1 establishes that the KL divergence between the true distribution g(2) and its stochastic approximation @(z) is bounded as
2
DKL (g()) (2))
<
C,
(13)
where e quantifies the variance of the stochastic score approximation and is a finite constant determined by the diffusion process.
To estimate the range of v in practice, we computed the squared error between the MSM score, and its stochastic approximation:
approximated using w = 64
v log go. (201) V log To, (2(0) l|
(14)
averaged over 3,000 training images perturbed at timesteps t = 200, 400, 600. By varying the number of batiib E {1, 4, 16). directly observed that the approximation error with MSM score (which is approximated with w = 64) decreases with w, consistent with the theoretical scaling v² //w as in Table X for both FFHQ and fastMRI datasets.
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Table Empirical squared error between MSM score (reference 64) stochastic approximation for different timesteps and minibatch sizes w. Results are averaged over 3,000 training images for FFHQ and fastMRI.
Time step
Minibatches uo
E V log lo
V log Co, (1
FFHQ
fastMRI
16
3.57 10-3
4.15 x 10-5
200
10
1.15 10
6.17 x 10
4.05 x
8.20 x 10
4.65 x 10
400
5.70 x 10"3
1.12 x 10"5
3.20 x 10
2.79 x 10
5.60 10
1.42 x 10106
2.70 10"3
1006
3.70 x 10-3
6.21 x 10-6
For C, we recall from the proof of Theorem that
C [
T
dl,
(15)
4T.
where Te is the temperature associated with the pariance-preservve (1)) art/ar, in terms of the signal scaling factor a that defines the noisy image at diffusion step as specified in (Park et al., 2025, Table 1). Hence,
diffusion in our experiment setup.
Under this setting, T, = } -
T 1 at dt.
C
(16)
4a
Using the schedule parameters from prior work, obtain C' = 101.01 for unconditional MSM setup.
These results demonstrate that both and C are not abstract constants but measurable quantities. In particular, v can be empirically estimated from score approximation error, and C' admits a closed-form expression based the diffusion temperature schedule.
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D IMPLEMENTATION DETAILS
D.1 MSM FRAMEWORK MR IMAGES
The main manuscript illustrates the measurement score-based diffusion model (MSM) framework's training and sampling schemes, but omits domain-specifie transformations for clarity. These trans- formations are essential in the MRI setting, which requires conversions between measurement and image spaces before and after denoising.
Specifically, we apply the inverse Fourier transform FT followed by the adjoint coil-sensitivity operator C to project the measurements into image space before denoising. After denoising, we map the denoised image back to measurement space by applying the forward coil-sensitiiity operator C' and the Fourier transform F.
This results in modified version of (1), given by:
(8 σ1) F 8:; 0t);
(17)
where S is the subsampling operator. A visual illustration is provided in Figure 4.
V 8
Not available
Available
S S~p(S)
C F
FC
St
D
$0
Training scheme
P(S)
Sampling scheme
Stochastic denoising loop
Reinject Noise
S, pSS)
F
FC
S
Weight
Zt
8t
D
$0
28
Figure 4: Illustration of the Measurement Score-based diffusion Model (MSM) for training and sampling using subsampled MRI measurements. MSM operates directly on k-space measurements, with minor domain transformations between k-space and image space only the input and output of the diffusion model Dø.
D.2 DETAILED MSM POSTERIOR SAMPLING ALGORITHM
This appendix provides the dedicated algorithm for the MSM- MSM-based posterior sampling procedure introduced in Section 3.3. Algorithm 2 extends the unconditional MSM sampler by inserting, at each diffusion step, a cata-consistency update derived from the posterior gradient. This makes explicit how MSM transitions from unconditional sampling to solving inverse problems.
D.3 STOCHASTIC POSTERIOR SAMPLING FOR COMPRESSED SENSING MRI
Our posterior sampling algorithm is described in Section 3.3. For compressed-sensing MRI. we apply an additional simplification based on directly approximating the partial posterior score for each partially subsampled measurement within the stochastic algorithm.
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Algorithm 2 Measurement Score-Based Posterior Sampling
Require: T. ((S) (o y, H
1: Initialize 2T ~ (0, 1), 2g + 0
2: for T to do
3: for
w do
4:
S ~ P(S),
t 5:
Partial score-based denoising
5:
89
(() it + to (n (1)
; (()
(4) p(s
(0)
6:
Si
Reinject noise & update iterate
2t - SOO
+ (I S s()) (
7:
8
8:
end for
9:
C' Ei diag( (S(O) S())
Compute weight for aggregation
10:
W | 1)]
11:
20 t
W Σm= ( / + LC=0 20
Aggregate partial denoised results
12:
2g 20 7/t V ly H 12
2t-1 z
21; 20)
14: end for
15:
return 2u
To perform posterior sampling for compressed-sensine MRI, we estimate the posterior score using stochastic ensemble, similar the MSM score ensemble Section 3.2:
V log (zt y) V log Po. (zt) + V log Po. (y
2x)
W
S
log Po.
(0) [
(1)
oi = S(()
W
T
V log Po, (5
V log (1)
2/0))
(18)
where the second line replaces the partial prior score in the stochastic MSM approximation (5) with the corresponding partial posterior score, and the third line expands this term using Bayes' rule into prior and likelihood components. All partial prior and likelihood distributions are conditioned on the corresponding subsampling operator ( for notational simplicity, we omit this conditioning
in Po, (s{ (5)) and Po, (x(() s ) For each subsampling operator S E " x we define y
( H project the observed measurement y into the same partial measurement space s, The log-likelihood gradient approximated by
logPr, Pr, (g()) 18)
V log Po. (3
(()))
(19)
y (1)
(1) (1)
80
where "t is tunable step size, and we define H (()) H H the degradation operator
H H restricted to the coordinates selected by (
D.4 MODEL ARCHITECTURE AND TRAINING CONFIGURATION
We adopted the U-Net architecture (Ronneberger et al., 2015), following the design used in (Ho et al., 2020; Dhariwal & Nichol, 2021), as our diffusion model backbone. Models were trained with the A optimizer (Loshchilov & Hutter, 2019) and used exponential moving average
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(EMA) to stabilize training by averaging model weights over time, using decay rate of 0.9999 for gradual updates. The diffusion process consisted of 1000 timesteps, with linearly increasing noise variance schedule starting from 0.0001 and reaching 0.2 the final step. All diffusion models, including diffusion-based baselines, were trained with the same architecture for each application. The hyperparameter setup and architectural details summarized in Table 6.
Table 6: Diffusion model architecture and training hyperparameters for each dataset.
RGB Images
Multi-Coll MRI
Base channel width
128
Attention resolutions
[32, 86
Áttention heads
4
Residual blocks
2
Batch size
128
Learning rate
5c 5
1e 5
Channel multipliers
[1, 1, 3,
[1, 2, 2,4,4
# Input/Output channels
D.5 COMPARISON METHODS FOR UNCONDITIONAL SAMPLING
We now provide detailed setups for used baselines for unconditional sampling experiments.
Oracle diffusion. For both natural images and multi-coil MRI, we train the oracle diffusion model using clean images without any degradation. The model is trained predict the noise component of noisy images for both data types.
Ambient diffusion (Daras et al., 2024c). For natural images, under the training setup as MSM in noiseless and subsampled data scenario, following the recommendation of (Daras et al., 2024c) to apply minimal additional corruption, we define the further degradation operator Š by dropping one additional 32 32 pixel box. The model is trained directly predict the clean image, which found to perform better than predicting the noise component.
For MRI data, under the same training setup as the noiseless and subsampled setting of MSM, to define the further degradation operator S, we drop an additional 10% of the sampling pattern while preserving the autocalibration signal region. Unlike the natural image case, the model is trained to predict the noise component, which we found to perform better than direct clean image prediction.
During sampling, we use 200 steps of denoising diffusion implicit models (DDIM) (Song et al., 2021a) and apply the same further degradation configuration used during training to subsample diffusion iterate.
GSURE diffusion (Kawar al., 2024). We only apply this method to the RGB face data, not the multi-coil MRI data, because GSURE diffusion's extension from single-coil MRI to multi-coil MRI remains computationally infeasible, and no practical approach has been proposed. Under the same training setup of MSM's noisy and subsampled training setup, train the GSURE diffusion model to predict the clean images and follow exactly the same training configuration as described in (Kawar et al., 2024).
Note that we exclude recent expectation-maximiiee maximization based methods (Bai et al., 2024; 2025), as their reliance clean-image initialization is incompatible with setting, where no clean images are available.
D.6 COMPARISON METHODS FOR IMAGING INVERSE PROBLEMS
We now describe the baseline methods used for solving inverse problems.
Diffusion posterior sampling (DPS) (Chung et al., 2023). DPS estimates the gradient of the likeliioddooo using the MMSE estimate 2e (ax) from a pretrained diffusion model
V loggppy
ax) z V log (o)())))
(20)
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where the gradient is taken with respect Zt.
Following the original implementation. we set the step size for the likelihood gradient as "
Ty A where i selected via grid search within the recommended range [ We used
= 2 for the super-resolution experiment, 0.7 for box inpainting, and 10 for compressed sensing MRI.
Ambient diffusion posterior sampling (A-DPS) (Aali et al., 2025). A-DPS follows the same posterior sampling strategy as DPS but uses diffusion model trained on noiseless subsampled data. For natural-image experiments, we use the pretrained Ambient diffusion model trained with dropping ratio p 0.4. For MRI experiments, we use the pretrained Ambient diffusion model trained acceleration rate R 4, consistent with the standard setup diffusion- based MRI reconstruction.
The step size is set in the same form DPS: "/ A- we l with the constant chosen according each inverse problem: c 2 for super-resolution, 0.7 for box inpainting, and 10 for compressed sensing MRI.
Robust self-supervision via data undersampling (Robust SSDU) (Millard & Chiew, 2024). Robust SSDU is designed to handle noisy, subsampled measurements by introducing additional sub- sampling and noise the observed measurements during training. In our implementation, the primary sampling mask S1 exactly matches the test acquisition patterns (acceleration factors {2, 6, 8)), and the training noise level set the same value used test time (a,, E {0.005, 0.01, 0.02, 0.03)). Given the noisy subsampled input s = S1 + nu, where n is additive Gaussian noise with standard deviation on Robust SSDU forms further corrupted input
g S28 n
(21)
where S2 has acceleration factor of R = 2, and ñ is independent Gaussian noise with standard deviation on, matching the original measurement noise level, following the original implementation.
Denoising diffusion null-space model (DDNM) (Wang et al., 2023a). DDNM also uses diffusion model trained on clean data and introduces projection-based update that blends the prior estimate and the measurement:
E B
xt, y) R (I Σ,A4A)BB
xx] + E,A"y y,
(22)
where A is the pseudoinverse and Σ, is a weighting matrix, such as Σ; = A or a spectrally tuned version.
We followed the enhanced version of DDNM described in (Wang et al., 2023a, Section 3.3 and Equation (19)) specify weight matrix E, in (22).
D.7 MEASURING FID SCORE
To compute the Fréchet Inception Distance (FID), we used the implementation provided in the following r For each method, we generated 10,000 images, then computed FID using features extracted from a pretrained inception network. For MRI images, which have complex-valued channels, we converted them to magnitude images and replicated the single-channel magnitude three times to form 3-channel input compatible with the pretrained Inception network. Note that although the pretrained FID model was trained on natural images, it still reflects perceptual quality on MRI data (Bendel et al., 2023).
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D.8 ILLUSTRATION OF SUBSAMPLED MEASUREMENTS AND THEIR SCORES
The main manuscript establishes that MSM learns the score of each subsampled measurement- implicitly conditioned on the corresponding subsampling mask-and shows how these measurement scores are leveraged to synthesize fully-sampled measurements. The purpose of this section is to visually illustrate how these measurement scores behave under different masks and clarify the dimensional structure underlying MSM sampling.
Recall that MSM operates on subsampled measurement 8, E R formed 8; S Sz where 2, R is not accessible fully-sampled measurement and S {0, 10 with < n is binary subsampling mask. Because each 8; is produced by specific mask S, the measurement score V log Pr, (8x) is implicitly mask-conditioned: it describes the distribution of 8, restricted to the coordinates selected by that mask, rather than a score marginalized over masks.
Figure 5 provides a concrete visual example using 64 x 64 image. The figure illustrates: (i) how different masks produce different subsampled measurements 8t, (ii) how the dimensionality changes from the fully-sampled measurement to each subsampled measurement; and (ii) how MSM performs score- score-based denoising under several predefined masks. This visualization makes explicit how the measurement scores inherently conditoond and how MSM integrates them during sampling to reconstruct the sully-sampled measurement.
Partial Score Denolsing
S
Vlagp(s)" ((
gaaes
IR 64x33
3, E
R 644x64
En E R44
Partial Score Denoising
S
()
(3)
G4X Vingp(s) get) gaixa
R
64x32
Figure 5: Visual illustration of subsampled measurements, their associated masks, and the corre- sponding measurement scores used in MSM sampling.
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E ABLATION STUDIES AND DISCUSSIONS
E PRIOR WORK AND DISTINCTION OF OUR APPROACH
Among recent approaches that train diffusion models without clean images, ambient diffusion (Daras et al., 2024c) and its pnsteior-sampling extension A-DPS (Aali et al., 2025) are the most directly related, as they also learn from degraded natural images and multi-coil MRI. However, both operate in the image domain, approximating clean-image scores by further subsampling images and predicting original subsampled images. contrast, MSM generalizes the idea of patch-based learning- widely applied to enable computationally efficient supervised training to self-supervised learning in the measurement domain. learns partial measurement scores restricted to observable regions from only noisy, subsampled measurements, and combines them in expectation form a full measurement score. With our efficient stochastic algorithm, MSM enables both unconditional sampling and posterior sampling for solving linear inverse problems. Table highlights the key differences.
Table Comparison of MSM with ambient diffusion (Daras al., 2024c) and its direct extension Ambient Diffusion Posterior Sampling (A-DPS) (Aali et al., 2025). MSM is the first to directly learn measurement scores from noisy, subsampled measurements and use them for both generating full measurements and solving inverse problems.
Ambient Diffusion
Ambient-DPS
MSM (Ours)
Training data
Subsampled data, with further random
Only noisy, subsampled
subsampling in training
mcasurements
Learned object
Learns the approximation of the true
measurement scores, restricted to observable regions
Partial
score by predicting the original subsampled image from a further subsampled input
Sampling domain
Image domain
Measurement domain: stochastic generation of the full measurements
Posterior sampling
Limited: can only solve inverse prob- lems when the test-time degradation matches the training degradation
Direct extension of diffusion posterior sampling (Chung et al., 2023) applied to pretrained ambient diffusion
New posterior score formulation using partial measurement posterior scores; stochastic condi- tional sampling of full measurements
Key novelty
First diffusion frame- work trained purely on subsampled data
Extension of ambient diffusion to posterior sampling
First to directly learn measurement scores and use them in stochastic measurement-space dif- fusion process for both full measurement and posterior sampling
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E.2 IMAGE SAMPLING WITH MSM USING EXTREMELY SUBSAMPLED DATA
We evaluated the unconditional sampling capability of our framework in a more challenging training scenario with extremely subsampled data. Specifically, we used MRT data with k-space subsampling via random masks at an acceleration factor of R 8, including fully-sampled vertical lines and the central 20 lines for autocalibration.
MSM was configured with a stochastic loop parameter = and took 200 sampling steps. As baseline, ambient diffusion took 200 sampling steps. As shown in Table and Figure MSM achieves a lower FID score than the Ambient diffusion on the same data setup, demonstrating the robustness of our approach under high subsampling rates.
Table FID scores under different training settings on multi-coil brain MR images. Best values are highlighted for each training scenario, with comparisons shown when corresponding baseline methods are available. Note how MSM consistently achieves lower FID scores than the Ambient diffusion, even in extremely subsampled data scenarios.
Training
Methods
FID.
No degradation
Oracle diffusion
29.25
MSM
13.60
Ambient diffusion
47.80
MSM
74.92
8,
Ambient diffusion
84.77
Ambient
MSM
Ambient
MSM
Training data: ==4
Training data: R
Generated samples
Figure 6: Visual comparison of MSM trained under extreme subsampling (R = 8) with MSM and baseline methods trained under less degraded conditions.
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E.3 SOLVING INVERSE PROBLEMS WITH MSM TRAINED ON NOISY AND SUBSAMPLED DATA
We showed that MSM trained on subsampled data can solve inverse problems for both natural images and multi-coil MRI. We further verified that MSM, when trained noisy and subsampled data, can achieve comparable performance using the same step size for the log-likelihood gradient as in the subsampled-oni scenario, as summarized in Table and Figure 7.
Table 9: Results natural image inverse problems and two compressed sensing MRT tasks. MSM trained on subsampled data is compared with MSM trained on noisy and subsampled data (training type shown in parentheses). Both training setups yield comparable performance across all metrics.
Testing data
Input
MSM (Noisy & Suhsumpled)
MSM (Noiseless & Subsampled)
PSNR1
18.26
24.16
24.71
Box Inpainting
SSIMF
0.749
0.864
LPIPSL
0.304
0.081
0.076 ................................
.ww.............................
PSNR1
23.21
27.99
28.
SR (X4)
SSIM
0.728
0.868
0.868
LPIPS
0.459
0,127
0.117
PSNRT
22.75
29.74
30.71
CS-MRI (x4)
SSIMT
0.648
LPIPS
0.306
PSNRT
21.91
28.11
28,
CS-MRI (x6)
SSIMT
0.617
0.795
0. 805
LPIPS
0.342
0.192
0. 0.168
Input
MSM
MSM Metiee
Groundtruth
Input
MSM
MSM
Groundtruth
18.04 0033
22.33 00.08
23.20 0.05
PSNR AARZLLFT /LIIS
1..3/
30:81 70.6
3 00 1
PSNR /LPISS PSS
CS-MRI (D4)
Inpainting
20:62 70.48
25:59/ / D.13
26 / 70.11
PSNA//LIPR
23.47 0.28
29.47/0.14
29.56 /0.12
PSNR / LPIPS
CS-MRI (X6)
SR
Figure 7: Visual comparison between MSM trained on noisy and subsampled data and MSM trained on only subsampled data. Both models produce high-quality results on natural image and MRI tasks.
E.4 COMPARISON WITH CLASSICAL CS-MRI RECONSTRUCTION METHON
We have shown that MSM outperforms diffusion- and self-supervised methods in multi-coil compressed sensing MRI, all trained without clean data. We now compare MSM with the classical CS-MRI method: total variation (TV) regularization.
Table 10: Quantitative results on two compressed sensing MRI tasks. MSM. trained using only subsampled data, is compared with the classical CS-MRI method: total variation (TV) regularization.
Setup
Input
TV
MSM
PSNR1
22.75
25.71
30.71
(SMRI(X4)
SSIMI
0.750
0.839
LPIPS
0.238
PSNRT
21.94
21.18
28.86
CS-MRI (X6)
SSIMI
0.617
0.702
0.805
I.PIPS
0.342
0.282
0.168
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E.5
COMPARISON WITH DIFFUSION-BASE INVERSE PROBLEM SOLVERS TRAINED ON CLEAN DATA
We compared MSM against diffusion-based inverse problem solvers- (Chung et al., 2023) and DDNM (Wang et al., 2023a)-that use diffusion priors trained on clean images, on both RGB face images and multi-coil compressed sensing MRI. Detailed configurations of these methods are provided in Section D.6.
The measurement noise level was set to 17 0.01, and all experimental setups for both MSM and the baselines followed those described in Section 4.1 and Section 4.2. The results are summarized in Table 11 and Figure 8. Notably, although MSM was trained only on subsampled measurements, it performs comparably to the baselines that leverage clean data-based pretrained diffusion models and even surpasses DPS on several inverse problems across multiple metrics. This result highlights that reconstruction quality depends not only the diffusion prior but also the data consistency strategy. Because DDNM employs a data consistency strategy similar to ours, it achieves slightly better results than ours; in contrast, DPS relies different data consistency strategy, which leads to its distinct and generally lower performance even when using the clean diffusion prior.
Table 11: Quantitative results on two natural image inverse problems and two compressed sensing MRI tasks. MSM, trained using only subsampled data, is compared with methods that use pretrained diffusion models trained on clean data. The number of iterations used for each method shown in parentheses. Best and sccond-best values are highlighted per metric (PSNR, SSIM, LPIPS). Note that despite not having access to the clean data, MSM approaches the performance of the clean data-based methods.
Testing data
Input
DPS (1000)
DDNM (200)
MSM (200)
PSNRT
18.26
23.61
25.16
21.71
Inpainting
SSIM
0.867
LPIPS.L
0.304
0.077
0.071
0.076
.w..............................
PSNRT
23.21
27.20
28.82
18
(x-1)
SSIM
LPIPS
0.459
0.128
0.126
0.117
PSNRT
22.75
31:31
32.84
30.71
CS-MRI (X4)
SSIM
0.648
0.845
0.895
0.839
LPIPS4
0.112
PSNRT
21.94
29.19
29.95
28.86
CS-MRI (X6)
SSIM
0.728
0.795
0.851
0.805
LPIPS
0.459
0.149
Input
DPS
DONM
MSM
Groundtruth
Input
DPS
DONM
MSM
Groundtruth
/9.3
0.13
2P:56 0.15
2P.54 0.16
FANR LFIPS
23.01.10:32
CS- MRI (xc))
Impaining
24.95/ 0.
29.89774 0.
30,34/00 0.
PSNR / LONRR
23.53/0.25
29307
30.4/00.5
29 2982
PSNR/
LPIPS
CS OS-MRI (26)
SR (x44
Figure 8: Visual comparison of reconstructed images using diffusion based inverse problem solvers.
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E.6 EFFECT OF STOCHASTIC LOOP ITERATIONS ON SAMPLE QUALITY
Our MSM framework includes a configurable parameter w, which controls the number of stochastic iterations used to denoise the full measurement iterates. As theoretically justified in Appendix A, using a larger w yields a closer convergence to the ideal MSM sampling algorithm that averages infinitely many partial scores at each iteration.
To empirically validate this, we fixed the number of diffusion sampling steps to 10 and explored three different values of in {1, 2, 4]. We intentionally fixed the low number of sampling steps to isolate the effect of tt, as increasing the number of diffusion steps improves sample quality. Figure 9 illustrates that larger w leads to visually more plausible generations: while w can still produce reasonable samples, it often results in artifacts, such as visible boundaries in some regions. As w increases, the results become more stable and visually coherent. Table 12 and Table 13 further support these observations quantitatively, showing that larger achieves better FID scores across both datasets.
Table 12:
FID and average time per sampling for MSM with varying stochastic loop iterations w on face images. Note that the trade-off exists where larger w reduces FID but increases sampling time.
Table 13:
FID and average time per sampling for MSM with varying stochastic loop iterations on MR images. Similar to face images, increasing reduces FID at the cost of longer sampling time.
Training data
Methods
FID
Time (s)
Training data
Methods
FID
Time (s)
MSM (w
4)
85.02
1.02
MSM
75.08
2.18
0.4,
2)
90.65
0.51
R 4.
MSM 1)
125.59
0.27
MSM (w
102.89
0.57
Stochastic loop=
Generated samples
Figure 9: Samples generated using different values of the stochastic loop parameter
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E.7 SOLVING INVERSE PROBLEMS UNDER DIFFERENT TEST NOISE LEVELS USING MSM
We additionally evaluate how MSM's posterior solver behaves under varying measurement noise at inference time by measuring reconstruction quality across multiple test noise levels while keeping the pretrained measurement-domaii diffusion model fixed. As summarized in Table 14, MSM maintains strong performance across all noise levels and generally outperforms competing methods trained without clean images.
Table 14: Reconstruction results across different test noise levels for inpainting, super-resolution, and CS-MRI.
Best values per metric highlighted. MSM demonstrates robust performance across noise levels and generally outperforms alternative methods trained without clean images.
Setup
Noise
Input
A-DPS
SSDU
MSM
level
PSNR1
18.29
19.26
N/A
24.75
Inpaintinng
SSIMT
0.757
0.654
0.871
LPIPSL
0.299
0.288
PSNR1
18.26
20.14
24.71
Inpainting
0.01
SSIM
0.749
0.621
N/A
0.867
LPIPS
0.304
0.305
N/A
0.076
PSNR1
18.19
19.16
N/A
24.68
Inpainting
= 0.02
SSIMT 0.726
0.057
0.854
1.PISS.
0.322
0.288
N/A
0.089
PSNRT
23.27
22.13
N/A
28.29
(x4)
0.005
SSIMT
0.734
0.696
N/A
0.880
LPIPSL
0.456
0.287
0.107
PSNR1
23.21
22.61
N/A
28.11
SR (x
= 0.01
SSIMF
0.728
0.702
N/A
0.868
LPIPS
0.459
0.277
N/A
0.117
PSNRT
23.09
22.41
N/A
27.59
4x
17 0.02
SSIMT
0.711
0.702
N/A
0.836
LPIPSL
0.470
0.278
N/A
0.147
CS-MRI (x-4) 0.005
SSIMT
0.652
0.777
0.855
0.858
LPIPSL
0.305
0.197
0.165
0.127
PSNRT
22.62
27.28
29.65
30.71
CS-MRI (X4) 7 0.01
SSIMT
LPIPSL
0.306
0.173
0.160
0.145
PSNR1
22.52
27.12
28.99
29.07
CS-MRT 0.02
SSIMT
0.623
0.792
0.833
0.769
LPIPSL
0.178
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E.8 EFFECT OF NOISE REINJECTION IN MSM SAMPLING
As introduced in Section 3.2. MSM includes a noise-reinjection step inside each stochastic loop. At every loop iteration, the diffusion noise is added back the denoised subsampled estimate before proceeding to the next update. This step helps ensure that coordinates updated in earlier iterations remain compatible with the current iterate. In this section, we compare reconstruction performance with and without this noise-reinjection step.
Table 15: Ablation study on the effect of noise reinjection in MSM sampling. We report quantitative results natural image inpainting, 4x super-resolution, and CS-MRI acceleration rates and 6. Best values highlighted per metric. MSM achieves the best performance across both distortion based perception-rriented metrics.
Setup
Input
A-DPS
MSM
without noise reinject)
MSM (wr with noise reinject)
PSNR
18.26
20.14
24.63
24.71
Inpainting
SSIMT
0.749
0.621
0.867
0.867
LPIPS
0.304
0.305
0.076
0.076
PSNRT
23.21
22.61
27.99
28.11
(x:4)
SSIM
0.728
0.702
0.806
0.868
LPIPS
0.159
0.277
0.117
0.117
PSNR
22.75
27.28
29.71
30.71
CS-MRI (x4)
SSIMT
0.648
0.804
0.814
0.839
LPIPS.
0.306
0.173
0.164
0.145
PSNRT
21.94
26.29
27.67
28.86
(x6)
SSIMT
0.617
0.763
0.769
0.805
LPIPS
0.342
0.201
0.193
0.168
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E.9 LEARNING PARTIAL MEASUREMENT SCORES FROM NOISY MEASUREMENTS WITH INCORRECT NOISE ASSUMPTIONS
In practice, the noise level of the measurements is rarely known exactly, even though prior work on training diffusion models from noisy or incomplete data typically assumes access to a noise estimate (Aali et al., 2023: Chen al., 2022: Daras al., 2024b; Kawar al., 2024). To assess MSM's sensitivity to this assumption, we evaluate the moisy-measuremen training procedure from Section 3.4 under both a matched and mismatched setting. The model is trained assuming a measurement noise level of Passumed = 0.1 1 and we compare its performance when the true test-time noise matches this value (Pree 0.1) versus when it is lower (frme 0.05). As shown in Table 16, MSM exhibits only minor change in FID, indicating that it is robust to moderate mis-specification of the measurement noise level during training.
Table 16: FID scores for unconditional generation under noise-level miss-pecifiicatinn The model trained true noise level Pire may differ. MSM shows only minor degradation when the assumed noise level does not match the actual one, demonstrating robustness noise level mis-spccification.
assuming Passumed 0.1, while the
Training data
Methods
FTD.
0.4, Passune
0.1, Puuee 0.1
MSM (Matched)
P 0.4, Passinat
0.1, Pree
MSM (Mismatched)
38.28
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E.10 COMPARISON OF INFERENCE EFFICIENCY FOR CONDITIONAL SAMPLING: MSM VS. AMBIENT DPS
We compare the inference efficiency of MSM and Ambient DPS for conditional sampling in inverse problems. Since MSM requires w stochastic loops per diffusion step while Ambient DPS does not, it is important to assess this cost-performance trade-off. All methods are evaluated under the 200 diffusion sampling steps. We test MSM with E 2,3 across inpainting. super-resolution, and peapressed-sensingg MRI. Even with the minimal configuration w 1, MSM achieves higher reconstruction quality than Ambient DPS while requiring only single loop per diffusion step. Increasing w provides the expected accuracy-efficiene trade-off characteristic of our stochastic score aggregation. Results are summarized in Table 17.
Table 17: Inference efficiency comparison between Ambient DPS and MSM under different stochastic loop w.
Setup
Input
A-DPS
MSM (w = 1)
MSM (w
2)
MSM (wr
3)
PSNR
18.26
20.14
24.48
24.64
24.70
Iopainting
SSIMT
0.749
0.621
0.868
0.867
0.868
LPIPS
0.076
0.075
PSNR
SR (x:4)
SSIM
0.728
0.702
0.872
0.870
0.869
LPIPS
0.459
0.277
0.120
0.117
0.116
PSNRT
22.75
27.28
29.12
30.50
30.71
CS-MRI
SSIM
0.648
0.804
0.806
0.828
0.839
LPIPS
www.............................
.ww.............................
PSNR
CS-MRI (X6)
SSIM
0.617
0.763
0.759
0.789
0,805
LPIPS
0.342
0.201
0.202
0.178
0.168
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Published as a conference paper at ICLR 2026
F USE OF LARGE LANGUAGE MODELS
In preparing this manuscript, we used large language models (LLMs) solely for minor editorial assistance, such correcting grammar and fixing typographical errors. LLMs were used for research ideation, methodological development, data analysis, or substantive writing, All scientific contributions and writing decisions are entirely those of the authors.
G REPRODUCIBILITY STATEMENT
We have made significant efforts ensure reproducibility of our work. An anonymous supplementary code package is provided, which fully reproduces our proposed method for both training and unconditional sampling, as well as for solving inverse problems. Detailed descriptions of the models. training configurations, and evaluation protocols are included in the main paper and Appendix D..4 Additional implementation details and data preprocessing steps are provided in the supplementary materials.
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Reviews:
[{"review_id": "l9BtM6ALvw", "reviewer": "Reviewer_8nET", "summary": "The proposed Measurement Score-based Model (MSM) extends patch-based diffusion to the measurement domain by decomposing the forward operator in inverse problems (assume $y=Ax + e$) as a composition of a non-invertible operator H and an invertible operator T. The score functions are learned on (1) subsampled clean measurements $s=Sz$ and (2) subsampled noisy measurements, where $s=Sz + \\eta$, in both cases measurement scores are learned.\n\nAn algorithm for unconditional generation and posterior sampling is presented, using the expectation over partial (subsampled) measurement scores to approximate the full measurement score. Experiments on inpainting and MRI reconstruction demonstrate competitive performance compared baselines in particular Ambient-DPS.\n\nWhile the core idea of learning measurement-space scores without clean data is interesting and well motivated, the paper lacks self-consistency and clear organization, making it difficult to follow when reading it linearly.", "questions": "1) What is the advantage of directly training on the space of subsampled measurements in comparison to training on the corrupted space (i.e., in image domain)? Since this is part of the core contribution of this paper, it should be clearly discussed.\n\n2) Measurement noise $\\rho$ is usually not known in practice. Since the paper emphasizes the absence of clean samples as a key motivation, a discussion on the practical validity of assuming a known noise level seems important. How sensitive is the setup with respect to the measurement noise level?\n\n3) The theoretical analysis assumes i.i.d. sampling of the measurement operators $\\mathcal{S}$. In practice, acquisition patterns in MRI, CT, or PET are typically structured and hardware-constrained rather than i.i.d., so the practical validity of this assumption should be discussed.\n\n4) In Eq. (4), the maximum operation in computing the weighting function is reasoned by avoiding division by 0. However, in my understanding this would anyways be detrimental to the sample quality, if there are regions not covered at all by the generation process. Are there any insights on how many sampling steps and stochastic loop iterations are needed to ensure this? \n\n5) Evaluating the FID based on 3k samples (L1126) is a rather low sample number, at least 10k should be used. This is particularly important for unconditional generation, where diversity matters and FID variance can be high. In any case, if the choice is substantially lower than common practice, it should be discussed and justified.\n\n6) Given that MSM requires stochastic loops w and aggregation of partial measurement scores, which is not the case for Ambient DPS, a comparison on inference time would be interesting. \n\n\nMinor comments\nL262: box inpainting is not a degradation operator\nL166: $s$ is not defined\nL868: $\\mathcal{S}$ is not a forward operator, it should say subsampling operator", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "Addresses a clinically important setting in MRI, where fully sampled data are rarely available, and demonstrates strong performance for both natural image inpainting and MRI reconstruction. \n\nProvides a theoretical formulation which links subsampled measurement scores to the full measurement score, thereby the authors extend prior score-decomposition ideas (patch-based) to the measurement setting.", "weaknesses": "The method is developed and demonstrated primarily for subsampled measurements (e.g., incomplete k-space or inpainting), where spatial subsampling applies naturally. It is less clear how the formulation extends to other partial-measurement settings such as limited-angle tomography in CT, where missing data correspond to absent projection views rather than spatially masked measurements. Thus, the main practical advantage currently appears in MRI-like scenarios.\n\nThe paper lacks self-consistency in the description of its sampling procedures. Algorithm 1 details how the full measurement score is obtained, yet line 12, the actual reverse diffusion update, is not mentioned in Section 3.2. Since this step constitutes the core of the sampling process, it should be explicitly described or at least referenced later. Moreover, the posterior sampling introduced in Section 3.3 receives very limited explanation and no dedicated algorithm, despite being highlighted as one of the paper’s main contributions. From the written explanation I would assume that line 12 the posterior score is used, where the full measurement variable is already denoised and substituted for $z_t$ as a posterior approximation.", "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": "vFPPIHIums", "reviewer": "Reviewer_hKzq", "summary": "This paper introduces a new method for learning diffusion-based priors on under sampled + corrupted data. Unlike previous approaches which learn image based denoisers/samplers the authors propose a method that learns partial score in the measurement domain of their sampling operators. Their results show improved unconditional and conditional sampling performance compared to previous SOTA self-supervised methods.", "questions": "1. what acceleration level was SSDU trained at?\n2. for the mri experiments were the sampling masks always 1-D?\n3. how robust is the method to different training/inference noise levels?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper attempts to solve an important problem in generative inverse problem solvers that has been previously explored by earlier work. The paper provides an original method for learning measurement scores. The method is communicated clearly and is backed by solid experiments in which they compare their technique to existing SOTA self-supervised generative and end-to-end techniques. Their experiments are convincing that their technique is better than existing self-supervised approaches.", "weaknesses": "There are some experiments that I think would help strengthen the paper. An ablation over measurement noise would be good to show how performance varies over more than just a single noise level. This goes for both training and inference time. At a bare minimum we should see performance of the posterior solver at the same noise level as the training measurement noise (apologies if I misread and this is the case).  Along this same idea, It would also be good to make sure that when you are running inference for conditional sampling the results that are shown use test samples at the same under sampling level of the training data. For example, in Table 4  is it that case that CS-MRI (x6) inference is attempted using the model trained on x4 data? This is fine, but I would also like to see how inference on CS-MRI(x6) preforms using a model trained with x6 data. This is especially important when comparing to end-to-end methods like SSDU. Perhaps I have misread the results section, if so please just clarify.", "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": "eCw4VAUpFX", "reviewer": "Reviewer_a8c4", "summary": "The paper proposes a new framework called the Measurement score-based Diffusion model (MSM), which aims to learn the score function of partial measurements directly from noisy and subsampled measurements, i.e., without requiring clean data. Later, by aggregating the partial measurement scores in expectation, MSM can generate samples of full clean data. The paper also proposes a method for conditional generation using MSM, which can be used for solving inverse problems. Experiments on unconditional and conditional image generation show significant improvements over other related methods, such as ambient diffusion.", "questions": "Q1. From the explanation, it looks like Eqn 2 is the marginal score, i.e., $\\nabla \\log p(s_t)$  and not the mask conditional score, i.e.,  $\\nabla \\log p(s_t| S)$.  However, because of Eqn3. I’m confused about this. Can the authors confirm/clarify that the goal of Eqn. 2 is to learn the marginal score function and not the mask conditional score? \n\nQ2. My major concern is why MSM is defined as in Eqn 3, and how it can be a good approximation of the true score $\\nabla \\log p(z_t)$ ? The definition seems to be based more on intuition rather than a principled approximation. The MSM score is the key component for everything else later, i.e., for both conditional and unconditional generation. So, I believe it is quite important to show (1) Eqn3 is an approximation of  $\\nabla \\log p(z_t)$  and (2) show that it is a good approximation (may be empirically on a toy use case if not in theory). Also, it is quite unclear how the approximation in Eqn 19 holds. Also, in Section 4 again, the theoretical analysis essentially talks about how efficient the stochastic approximation of MSM is compared to the true MSM, and I don’t see why this is more important than checking how efficiently the MSM score and its stochastic approximation can model the true score instead?\n\nQ3. In Algorithm 1, line 11, what is the mentioned Identity condition, i.e., the 2nd term? Is this a typo? Also, the need for the reinject and update procedure in lines 6 and 7 is not clearly explained in the paper.\n\nQ4. The whole section 3.3 is very unclear regarding the specific practical implementation of the posterior sampling with MSM. Specifically, the following text needs more explanation: “We solve the inverse problem by replacing the score function in the random walk sampling process with our posterior approximation. This approach avoids the need for automatic differentiation when computing the likelihood gradient, thereby reducing computational overhead.”? Is MSM posterior sampling not using DPS-style approximation? In that case, comparison with A-DPS for inverse problem solving seems unfair, because DPS is not exactly perfect either.\n\nQ5: What is P=0.4 in Table 1? What does this hyperparameter denote?\n\nOverall, I believe the paper has fundamental concerns, with a major concern being that the MSM score seems to be hand-designed. I believe that it is crucial to explain why/how this is a valid approximation in a principled manner (not just the fact that its Jacobian is symmetric).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 1, "presentation": 1, "contribution": 2, "strengths": "The proposed MSM framework addresses the same problem as other works, such as ambient diffusion, G-SURE, etc, but is quite novel in terms of the methodology and with broader applicability.\n\nMSM can also be trained with noiseless subsampled measurements, offering a useful solution in practice, where the measurements are often noisy.\n\nTheorem 1 adds more validity to the proposed stochastic approximation of MSM, which shows that the approximation becomes more accurate with more Monte-Carlo samples, i.e., more computation.\n\nIn both unconditional and conditional generation tasks, MSM significantly outperforms methods such as Ambient diffusion, A-DPS, respectively, revealing its effectiveness in practice.\n\nThe paper is generally organized well. The way the authors positioned their method and contrasted it with related works made it easier to distinguish the proposed methodology.", "weaknesses": "The paper presents a very practical and impactful methodology overall, but the method needs many clarifications regarding the MSM score and why it is defined as such, which I believe lies at the core of the proposed methodology, its novelty, and empirical effectiveness. Also, additional experiments are required to verify the arguments, including a fairer comparison in the case of inverse problem solving. Also, there are a lot of instances in the main text where the explanations are very poor and sometimes even incomplete. Please see the questions below for more specific comments and references.", "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": "fijqJeEmA5", "reviewer": "Reviewer_5Qe2", "summary": "This paper proposes the Measurement Score-based diffusion Model (MSM), a way to train diffusion models without any clean ground-truth images and the corresponding sampling method. Instead of learning full image scores from degraded data, MSM learns partial measurement scores directly on the observed parts of noisy, subsampled measurements. MSM also introduces its two stage of training strategy, adjusting for different levels of diffusion noise regarding to the measurement noise. At sampling time, it randomly subsamples, denoises those partials, maps back to original shape and pooling them with a overlap-aware weighting to reconstruct a full measurement. Experiments are conducted on posterior sampling for noisy inverse problems, such as inverse problems on face and multi-coil MRI images, by adding a log-likelihood gradient term for data consistency.", "questions": "See Weaknesses 1, 2, 3.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "## 1. Design novelty\n\nThis paper introduces novel MSM framework that learns partial measurement scores directly on observed measurements, then stitches them into a full score during sampling.\n\n## 2. Well-crafted sampling design\n\nThe proposed sampling pipeline 1. a stochastic mask-subsample; 2. partial measurement denoise; 3. coverage-aware weighting aggregation of multiple partial scores, is well-intuitive and reasonable, preserving plug-and-play modularity with standard diffusion poterior sampling.\n\n## 3. Strong empirical results\n\nUnconditional generation: Consistently outperforms Ambient Diffusion when trained only on degraded data (better FID/visual fidelity across masking and noise settings).\n\nConditional/inverse problems: Beats Ambient-DPS (A-DPS) on tasks like inpainting, super-resolution, and accelerated MRI, improving PSNR/SSIM/LPIPS while using competitive sampling budgets.\n\n## 4. Solid theoretical analysis\n\nProvides clear bounds for partial measurement score estimation and a KL divergence guarantee showing the stochastic MSM sampler approaches the optimal as the number of masks increases with error shrinking with more aggregates.", "weaknesses": "1. The measurement corruption setting of MSM is only regular masking and the sampling method is keen to this design, which constraint the gneralization of MSM for different kinds of inverse problems, such as motion deblur, Gaussion deblur, or nonlinear inverse problems. Could MSM handle these inverse problems?\n\n2. The 200 NFEs of MSM is ambigous, for default setting with $w=3$, the model pass is 600. Could authors provide efficiency analysis and comparasions?\n\n3. In table 11 of E.5, it shows that MSM is no better than training-free DDNM. It seems because MSM trained without clean images is sensitive with test data and inverse problem settings. Could MSM be finetuned based on a diffusion model already trained on clean images?", "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)