Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data
Authors: Asad Aali
Giannis Daras
Brett Levac
Sidharth Kumar
Alex Dimakis
Jon Tamir
Venue: ICLR 2025 Poster (2025)
Keywords: generative models
inverse problems
ambient diffusion
mri
Primary area: generative models
TL;DR: We provide a framework for solving inverse problems with diffusion models learned from linearly corrupted data.

Abstract: We provide a framework for solving inverse problems with diffusion models learned from linearly corrupted data. Firstly, we extend the Ambient Diffusion framework to enable training directly from measurements corrupted in the Fourier domain. Subsequently, we train diffusion models for MRI with access only to Fourier subsampled multi-coil measurements at acceleration factors R$=2, 4, 6, 8$. Secondly, we propose $\textit{Ambient Diffusion Posterior Sampling}$ (A-DPS), a reconstruction algorithm that leverages generative models pre-trained on one type of corruption (e.g. image inpainting) to perform posterior sampling on measurements from a different forward process (e.g. image blurring). For MRI reconstruction in high acceleration regimes, we observe that A-DPS models trained on subsampled data are better suited to solving inverse problems than models trained on fully sampled data. We also test the efficacy of A-DPS on natural image datasets (CelebA, FFHQ, and AFHQ) and show that A-DPS can sometimes outperform models trained on clean data for several image restoration tasks in both speed and performance.

Published as a conference paper at ICLR 2025
AMBIENT DIFFUSION POSTERIOR SAMPLING: SOLVING INVERSE PROBLEMS WITH DIFFUSION MODELS TRAINED ON CORRUPTED DATA
Asad Aali' Stanford University UT Austin
Giannis Daras* MIT CSAIL IFML
Brett Levac UT Austin IFMI.
Sidharth Kumar
Alexandros G. Dimakis
Jonathan I. Tamir
UT Austin
UC Berkeley
UT Austin
IFML
Bespoke Labs
IFML
ABSTRACT
We provide framework for solving inverse problems with diffusion models learned from linearly corrupted data. Firstly, we extend the Ambient Diffusion framework enable training directly from measurements corrupted in the Fourier domain. Subsequently, we train diffusion models for MRI with access only to Fourier sub- sampled multi-coil measurements at acceleration factors R= 2, 4, 6, Secondly, we propose Ambient Diffussin Posterior Sampling (A-DPS), reconstruction al- gorithm that leverages generative models pre-trained on one type of corruption (e.g. image inpainting) to perform posterior sampling measurements from a different forward process (e-g. image blurring). For MRI reconstruction in high acceleration regimes, observe that A-DPS models trained on subsampled data are better suited solving inverse problems than models trained on fully sampled data. We also test the efficacy of A-DPS on natural image datasets (CelebA, FFHQ, and AFHQ) and show that A-DPS can sometimes outperform models trained on clean data for several image restoration tasks in both speed and performance|]
Training
Amtlerx Diffasio Thrining
Mup
Ampp?
Gantinn
Aduona
plat
Inference
Figure Ambient Diffusion Posterior Sampling (Ambient DPS). During training, we only have access to linearly corrupted data from a forward operator Arsin We use the Ambient Diffusion framework to learn a generative model, Gambiant. for the uncorrupted distribution, p(æo). At inference time, sample from the posterior distribution p(ZulyAA,.)) ), for measurements Yint coming from different forward operator, Aut-
INTRODUCTION
For some applications. it expensive or impossible acquire fully observed data (Collaboration
et al. 2019
Gao et al. 2023 Yaman et al. 2020) but possible to acquire partially observed samples.
Equual contribution.

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Published as a conference paper at ICLR 2025
Ambient Diffusion
EDM
NCSNV2
EDM Traiiin Rm1
Training Rm
Traning
Traning R-d
Training ft-2
Training find
Rhng
Traming
Traniing Rtd
X
K
X
K
Prior samples from diffusion models trained MRI scans. Columns 4: Diffusion models trained on subsampled MRI scans at acceleration factors 2, 4, 6, 8, using the Ambicnt Diffusion framework extended for Fourier subsampled training. Columns 5 - 8: EDM models trained with L.--waelett reconstructions of subsampled scans R = 2, 4, 6, 8. Columa 9: NCSNV2 trained with fully sampled scans. Columa 10: EDM trained with fully sampled scans. We observe that Ambient Diffusion models consistently produce high-quality and realistic MRI scans even high acceleration regimes.
Figure 2:
Furthermore, it may be desirable to train generative models with corrupted data since that reduces
memorization of the training set (Daras et al.) 2023b) Carlini et al. 20033 Somepalli et al. 2022).
Prior works have shown how to train Generative Adversarial Networks (GANs) (Bora al.
2018
Cole et al. (221), flow models (Kllarr et al. 2023) and restoration models (Lehtinen et al. 2018
Krull et al. 2019 Yaman al.) 2020 Millard & Chiew (023) with corrupted data. More recently,
there has been a shift towards traming diffusion generative models given corrupted data (Daras et al.,
2024 22023a by Aali al. 2023 Kawar et al. 2023 Cui et al. [2022 Kim & Ye (022). What remains unexplored is how generative models trained on a certain type of corruption (e.g. inpainted data) can be used to solve inverse problems with a different forward process (e.g. blurring).
We propose the first framework to solve inverse problems with Ambient Diffusion models (Daras al. 20236). These models are trained using only access linear measurements and they estimate the ambient score, i.e. how to best reconstruct given an input with a corrupted linear forward operator corrupted noisy imput. We show how use these models for solving linear inverse problems outside their training distribution. Our experiments on datasets of natural images and multi-coil MRI show something surprising: Ambient Models can outperform (in the high corruption regime) models trained on clean data while being substantially faster. Our algorithm extends Diffusion Posterior Sampling (Chung et al. 2023 to Ambient Diffusion models. Our contributions
We propose Ambient Diffusion Posterior Sampling (Ambient DPS), algorithm that uses diffusion models trained on linearly corrupted data as priors for solving inverse problems with arbitrary linear measurement models.
We extend the Ambient Diffusion framework to train models using Fourier subsampled measurements. Specifically, we train on subsampled multi-coil MRI scans various retro- spective acceleration factors (R=2, 4, 6, 8); we observe that models trained on subsampled data are better priors for solving inverse problems in the high acceleration regime.
We use pre-trained Ambient Diffusion models to solve inverse problems (compressed sensing, super-resolution) on natural image datasets (CelebA, FFHQ, AFHQ) and show that
they can even outperform models trained on clean data in the high corruption regime.
'We open-source code and Ambient Diffusion models; gibbub.com/uucsilbbammmiee http://sswla.aabbbiiiiiittsoooon
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Fully-Sampled (FS) Ground Truth
Ambient DPS
DPS
Training R-2
Training Rod
Training R-6
Trainin RuB
Training R-1
Inference
Sampling Mask
PSe 12.0
10W 1234
(10x)
Inference R=8
Sampling
196: 320
PAR 2.21
HoW 2015
Error (10x)
Figure 3: Posterior sampling reconstructions for MRI scans using models trained on Fourier subsampled data various acceleration factors (Ambient DPS, columns 2 and model trained clean data (FS-DPS (Chung al. 2023), column 6). Rows
show reconstructions and 4 display the difference to the ground truth on a 10x scale. At high inference acceleration Ambient DPS, outperforms FS-DPS, despite that the underlying models were trained solely on corrupted data.
and 8, respectively, while rows
Ambient
METHOD
2.1 BACKGROUND AND NOTATION
Diffusion Posterior Sampling (DPS). Diffusion models are typically trained to reconstruct
clean image Zg (=() from noisy observation z, 2o o,m. 17 ~ (O, Despite the simplicity of the training objective, diffusion models can approximately sample from p(x) by running a discretized version of the Stochastic Differential Equation (Song et al. 2020 Anderson 1982):
dæ 22, (IE DD [x,] xt)dt + g(t)dw,
(2.1)
where w is the standard Wiener process and E[ xy] is estimated by the network. Given measure- ment Yint Ai, sample from the posterior p(zolymnt) by running the process:
E(zp xx]
dx -2
log p(y xx))
dt g())dw.
(2.2)
T
likelihood term
For most forward operators it is intractable to write the likelihood in closed form. Hence, several approximations have been proposed to diffusion models for inverse problems (Chung et al.) 2023
Kawar et al. Jalal et al. 2021] Song et al. 2021 Chung et al., 2022 Feng et 20.3 Graikos al. 2022). One of the simplest and effective approximations is Diffusion Posterior Sampling (DPS) (Chung et al. 2023). DPS estimates 2p using æ, and uses the conditional likelihood p p(yin (p)
instead the intractable term, i.e. DPS approximates p(yut xx)) with p (aytt (20 E[zo xx)),, where is tunable guidance parameter. The update rule becomes:
dæ -20,0: E[ [2)
+ Tie Va, poyut x [[zo|/z])) dt + g())dw, (2.3)
Ambient Diffusion. In some settings do not have large training of clean data but have access to large set of lossy measurements that we would like to leverage to train diffusion model for the clean distribution.

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Super Resolution (Downscaling Factor = 4)
+
Gaussian Noise (a 0.05)
Corrupted Input
FS-DPS
A-DPS
Ground Truth
Figure Posterior sampling reconstructions for FFHQ. showing: (I) Corrupted Input: Examples from FFHQ dataset corrupted using Resolution Downscaling (4x) and Gaussian Noise (a = 0.05). (2) FS-DPS: Reconstruction using Diffusion Posterior Sampling (DPS), with diffusion model trained with clean fully- sampled data (p 0.0). (3) A-DPS: Reconstruction using Ambient-DPS, with diffusion model trained on randomly inpainted data with crasure probability (p- 0.6), (4) Ground Truth: Original uncorrupted examples from FFH dataset. observe that Ambient DPS, provides better reconstructions (qualitatively) even though the underlying models were trained on corrupted data (p 0.6).
The authors of Daras et al. (2023b) consider the setting of having access to linearly corrupted data {yo = Atrain Atrain J. where the distribution of Atrain, denoted as p( p(Aeraiu). is assumed to be known. The ultimate goal is to learn the best restoration model given linearly corrupted noisy input Vt,train Araai (2o + o.n), at all noise levels l. The authors of Daras et al. (2023b) form further corrupted iterates Vr,train Atrain (æg + ann) using matrix Atrain (that is perturbation of the given matrix A ) and train with the following objective:
||| Acrain ha (uo,train: Atrain) - Ye,train
J(0)= E mg. Ausin Awain
(2.4)
that provably learns EE [ Atrain Vtatrainl as long as the matrix E[AT A Atrain A is full-rank. In certain cases, it is possible to introduce minimal additional corruption and satisfy this condition. For example, if Atrain a random inpainting matrix (i.e. Ai ~ Be(1 - p)), then Arrain can be formed by taking Atrain and erasing additional pixels with any non-zero probability 5 > 0.
Multi-coil Magnetic Resonance Imaging. MRI is prototypical use case for framework that can learn generative models from linearly corrupted data, as in many cases it is not feasible to collect large training set of fully sampled data (Tibrewala et al., 2023 Desai et al., 2022, Tariq et al.). In settings such 3D+time dynamic contrast-enhanced MRI (Zhang et al. 2015) it impossible to collect fully sampled data due to the time-varying dynamics of the contrast agent (Zhang et al.).
In the multi-coil MRI setting, the acquisition involves collecting measurements of an image directly in the spatial frequency, known as k-space, from a set of spatially localized coils. Mathematically, there are Ne coils, each of which gives measurements:
(2.5) where x is the (complex-valued) image of interest, S; represents the coil-sensitivity profile of the ith coil, F is the Fourier transform, P represents the Fourier subsampling operator and w, is complex- valued Gaussian i.i.d noise. We assume the noiseless case and we point the reader to Daras et al. (202a): Aali et al. (2023): Kawar et al. (2023) for approaches that handle the noisy case.
; PFS + wi, E [Nc].
C".. the composition of For the discrete approximation of the continuous signal as an image P. P,F.S. can be written as a matrix A, E Cm where the number of measurements, m, depends on the subsampling operator P. It is common to denote with R the ratio , which is known as the acceleration factor. At inference time (i.e., for a new patient), we typically want to acquire data with high acceleration factor because this reduces scan time and patient discomfort.
Due to the time required to collect k-space measurements, it is often not possible to acquire fully sampled k-space. Hence, training cannot rely fully sampled image to guide reconstruction
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quality as is done in the fully supervised setting (Aggarwal al. 2019 Hammernik al. 2018
two independent measurements are available for the same underlying signal, restoration models can be trained without having access to uncorrupted data (Lehtinen et al. 2018 Gan al. 2023).
If only one measurement is available (as in our setting), some end-to-end techniques use loss on the measurement domain by partitioning the training measurements for each sample into (1) measurements for reconstruction, and (2) measurements for applying the loss function (Yaman et al. 2020). Other approaches leverage structure in the MRI acquisition learn from limited-resolution data (Wang et al., 2023). More recently, works have begun leveraging signal set properties such as group invariance assist in learning from subsampled measurements for variety of inverse problems (Tachella et al., 2022b; Scanvic et al. 2023 Chen et al.) 2022a Tachella et al. 2022a). The idea of additional corruption (as we do our work), more related to the Noisier2Noise framework (Moran et al., 2020) which has been initially developed for denoising. MMlllrdd & Chiew (2023) extended this idea to the MRI setting to learn models for MRI restoration without access to clean data. All these approaches learn a restoration model without access to reference data. However, as they are inherently end-to-end methods, their performance on out-of-distribution tasks (e.g., due to different acquisition trajectories) is known to degrade (Jalal et al. 2021 Zach et al. 2022).
2.2 AMBIENT DIFFUSION FOR MRI
The MRI acquisition process results in linearly corrupted measurements and thus should be possible use the Ambient Diffusion framework to learn from corrupted observations. Yet, we identified three important changes that differ from the setting studied by Daras et al. (2023b): i) the inpainting happens in the Fourier Domain, ii) the inpainting has structure, i.e. whole vertical lines in the spatial Fourier are either observed or not observed (instead of masking random pixels) and, iii) the image is measured from an array of spatially varying receive coils. In what follows, we show how account for these factors and apply the Ambient Diffusion framework to MRI data.
First, as in Ambient Diffusion, we will corrupt further the given measurements, but in our case, the additional corruption will happen in the Fourier space. We create further corrupted measurements for each coil by decreasing the Fourier subsampling ratio, i.e. we create iterates:
-x, PFS,æ.
(2.6)
This is possible do because we can just subsample the available data z To further corrupt the given measurements, we keep the inpainting structure that there is in the measurements, i.e. we delete (at random) 2-D lines in Fourier space (instead of pixel masking as in Daras al. (2023b). Due to the multiplicity of coils, we combine their measurements to form a crude estimator of ar by taking the adjoint of Atraini Ptrain = 2, S F Notice that for discrete signal æ E C all these operations are linear and hence ÿ can be written as:
Uurain
Σ F
S x.
(2.7)
Aurain As in Ambient Diffusion, we will train the network to predict the corresponding signal before the additional corruption, i.e. our target will be:
Btrain
Σ SM PFS: x.
(2.8)
A
We are now ready to state our main Theorem.
Theorem 2.1 ((Informal)). Let Yi,trains Itatrain represent the noisy versions of Utrain Ptrain respec- tively, iee.
Yt,train Atrain (x o.9)
(2.9)
Ut.train Aurain (x + o.n)
Then, the minimizer of the objective:
J(Ø)= EE_
yo, crai Wr. train Areain. Aursiutoo (Wt, p)) / yo,traia
(2.10)
is: hoo train, P) = 12(cco Ur, trains P].
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Proof overview. The formal proof of this Theorem is given in the Appendix. We provide sketch of the proof here. The techniques are based on Theorem 4.2 of Ambient Diffusion (Daras et al. 2023b). In Theorem 4.2, the condition that needs to satisfied is that tran Atrain Aurin full-rank. Our objective is slightly different because the network doesn't see the whole forward operator Aerain. but only part of it, i.e. the matrix P. By following the steps of the Ambient Diffusion proof, we arrive at the necessary condition for our case, which is to prove that [[Arrain P] is full-rank.
We start the argument by noting that E[P[P] is full-rank. Intuitively, this is because there is a non-zero probability of observing any Fourier coefficient and hence any deleted Fourier co-efficient could be due the extra corruption introduced by P. Since FF an invertible matrix, then also B[F PJ is full-rank. Finally, use the properties of the sensitivity masks to show that is also full-rank and this completes the proof.
E
S F P]
2.3
AMBIENT DIFFUSION POSTERIOR SAMPLING
DPS requires access to E[zg æ to approximately sample from p(zolyimr). Since Ambient Diffusion models can only work with corrupted inputs, we propose the following update rule instead:
Efarolrr. A +7v Vx, log mo If Arraia))
dæ: -2
dt g())dm
Ambient Score
(2.11)
for fixed Atrain p(Agrain) E Comparing this to the DPS update rule (E.q. 2.3 all the B[ [ar]] terms have been replaced with their ambient counterparts, i.e. with E[ (t, train. Ataai). The latter is approximated by neural network that is trained with Ambient Diffusion.
We term our approximate sampling algorithm for solving inverse problems with diffusion models learned from corrupted data Ambient DPS (A-DPS). We remark that similar to DPS, the proposed algorithm an approximation to sampling from the true posterior distribution. Similarly to Theorem 1 of the DPS paper, for given measurement yinr at noise level By and given matrix Ainfs one can upper-bound the Jensen gap A-DPS as follows:
exp( exp(-1/20)) [1 / ||ر| 2 z (lp(aaolx))de (
JS 2
(2.12)
where wo = E E[zoo/A Asrain Lt, A for fixed Arran sampled from p( Aevain). We omit the proof of this proposition since it follows the exact steps as the proof of Theorem in the DPS paper.
2.4 AMBIENT DIFFUSION FOR IN-DOMAN RECONSTRUCTIONS.
Ambient DPS can be used to solve any inverse problem for which the forward operator is known, using a diffusion model trained on linearly corrupted data of some form. It is important to underline though that if the inverse problem that want solve inference time has the same forward operator as the one that was used for the training measurements, then we can use the Ambient Diffusion as supervised restoration model. This because Ambient Diffusion models are trained Utrain Atrain], and hence if Ainf comes from the distribution Atrain then, the one-step prediction of the model, Ambient One Step (A-OS), is the Mean Squared Error (MSE) minimizer. Similarly, if we are interested in unconditional generation, we can simply run Ambient DPS without the likelihood term.
estimate EE[zo|yrrain
"The Ambient MRI models take input P instead of A, in Equation 2.10

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Table 1: Unconditional and Conditional Sampling FID for models trained MRI data different acceleration factors.
Training
Training Method
Unconditional
EDM NCSNV2
10.41
R
13.20
LI-EDM
18.55
= 2
Ambient Dillusion
30.34
LI-EDM
27.64
Ambient Diflusion
32.31
LI-EDM
51.43 13
Ambient Diffusion
LI-EDM
102.
R = 8
Ambient Diffusion
48.15
Training Data
Inference Data
Reconstruction Method
Conditional FID
R =
R = 8
FS-DPS
4.16
R
0.71
0.87
R =
A-DFS (our method)
1:29
2.00
EXPERIMENTS
3.1 AMBIENT MRI DIFFUSION MODELS
In this section, we extend Ambient Diffusion to the multi-coil Fourier subsampled MRI setting (as detailed in Section 2.2 and we train our own MRI models from scratch. We give details about our dataset preparation in Section B in the Appendix.
Sampling masks. We retrospectively subsample the k-space training data by applying randomly subsampled masks that correspond to acceleration factors R {2, 4, 6, 8). The sampling masks include fully sampled vertical (readout) lines corresponding to the observed Fourier coefficients (phase encodes). We always sample the central 20 lines for autocalibration. To form further corrupted measurements, we randomly remove additional lines such that we create measurements at acceleration factor 1. For example, we take data R 2, we create further corrupted measurements R = 3 and we train the model to predict the clean image by measuring its eттоr with the available data at R given its prediction for the R corrupted input.
Comparison Models. We train model for each acceleration factor R {1,2,4,6,3 The R = model is trained on clean data (no extra corruption) based on the EDM approach (Karras et al.) 2022). The R > models trained with modified Ambient Diffusion framework. While focus on EDM for fully sampled training, we also train an NCSNV2 model for
following the approach in Jalal al. (2021). As baselines, we also train EDM models after LI-Wavelet compressed sensing reconstruction (LI-EDM) of the training set at each acceleration factor (Lustig et al. (007).
Unconditional generation evaluation. We evaluate the unconditional generation performance of each model and show prior samples for Ambient Diffusion and EDM models at R - in Figure Similar to the results in Daras al.) (2023b), we observe that the Ambient Diffusion samples qualitatively similar to EDM models low acceleration, and become slightly blurry at higher accelerations. Notably, the samples generated by Ambient Diffusion at R 8 have no residual aliasing artifacts, in contrast to the samples generated from LI-Wavelet reconstructions.
To quantify unconditional sample quality, we follow the approach in Bendel et al. (2022) and calculate FID scores from 100 samples using a pre-trained VGG network. Table shows FID scores for the diffusion models. While there is an increase in FID for the Ambient models, we see that those trained higher acceleration factors outperform L1-EDMM models trained L.1-Wavelet reconstructions.
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Training Method
Training Data
Reconstraction Method
R-2
R=1
NRMSE
SSIM
PSNR
NRMSK
SSIM
PSNR
|RMSS
SSIM
PSNR
NRMSE
SSIM
PSNR
FS-ALD
04.55
40.17
90.71
88.53
CS
91.10
36.13
0.143
82.08
31.58
77.07
27.74
75.04
Sipervised
R
McDL (R
0.036
97.56
43,00
0,076
94.54
36,05
87.56
31.84
81.15
29.01
MoDL (R
0.041
96.67
41.95
0,066
95.19
37.85
0.005
93.09
34.62
0.124
90.51
92.31
MaDL (R
0.045
95.37
41.04
0.067
94.53
37.65
0.090
93.41
35.04
0,111
91.87
33.21
MoDL (R
0.500
94.13
40.24
0.079
90.30
30.30
0.096
10.04
34.46
0.113
91.08
33.05
A-DES (our machhod)
0,058
94.83
38.96
0
93.99
36.82
0.023
92.90
34.84
0,113
91.63
33.19
AvALD (our sertood)
0.088
94.84
38,89
0.874
93.99
36.82
BJ8E3
55.53
34.79
0.136
91.52
33.07
R
1.I-DP
A-08
0.06
0836
97.54
M6-42
45.20
0.111
0.69
91:81
35.64
X3-19
0.511
0.119
12:37
10
10.44
0.205
0.196
64.85
38.09
38.25
SSDU
93-55
31.76
22:09
77.53
ID-Partitoreed SSDU
90.47
35.61
96.11
82.82
A (our menled)
92.31
92.82
35.67
91.58
34.24
91.06
33,05
A-ALDD sertood)
0.011
93.52
57.00
0.064
92.77
15.66
0.099
91.96
34.34
0,183
91.13
33.08
B m-4
L1-DPS
A-OS
0.108
0.040
96.59
91.25
51.95
42.45
0.105
0.112
89.98
91.83
33.58
30.91
0.167
0.136
88
54.58
29.0
31.60
0.211
0.162
80.26
67.24
27.77
30.43
SSDU
0072
97.84
37.20
0,088
92.82
15.36
0.131
51.11
0.167
87.01
29.73
Sell-Sapervised
ID-Paritioord SSDU
0.099
93.45
34.05
0.100
89.79
33.71
0.114
85.96
32.84
0.127
R2.96
31.91
A-DES (our nanhhe))
0.044
91.85
35.63
0
91.47
34.62
0.306
90.87
33
0.118
90.20
32.69
A-ALD (our serlood)
0.64
91.85
35.85
0.995
91:43
54.56
0.106
90:38
33445
0.119
90.12
12.60
A-06
0.127
39:83
31.95
0.116
90.25
30.81
0.136
89
31.60
88.16
41.73
0.112
33-72
99.42
87.20
ID-Pariiiitral SSDU
94.44
89.60
82.47
A-DES (our menlod)
0.059
90.51
31.15
0.307
98.47
33,46
0.117
89.94
32.71
0,126
89.50
32,10
A-ALD nurfand)
0.100
98.47
34,11
0.107
90.44
33.46
0.118
89.31
32.63
0.127
89.29
32.00
R m
LI-DPS
AO6
0.135
0.944
95.94
36.45
31.46
41.53
0.136
0.126
88.
97.00
31.46
32.41
0.152
0.185
96.95
R2.95
2900
30,53
0.164
0.225
96.54
78.94
27.24
29:85
SSDU
0.118
77.19
32.95
0.123
88. 15
30.42
0.151
87.95
30.55
0.177
86.16
29.17
ID-PutiSoned SSDU
0,083
94.07
35.59
0.114
85.99
32.85
0.134
84.75
31.41
0.148
$1.61
30.55
Table 2: Accelerated MRI reconstruction performance evaluation using Normalized Root Mean Squared Error (NRMSE L). Structural Similarity Index Measure (SSIM 1), and Peak Signal-to-None Ratio (PSNR 1) averaged across 100 test samples.
3.2 ACCELERATED MRI RECONSTRUCTION
We now evaluate the accelerated reconstruction performance of Ambient Diffusion models trained on subsampled MRI data. Our method, A-DPS was implemented by following Eq. [ 500 steps with likelihood weighting %
We consider the following baselines:
A3 denilll
Fully-Sampled Diffusion Posterior Sampling (FS-DPS), For models trained on clean data, we can use the Diffusion Posterior Sampling (DPS) (Chung et al., 2023) algorithm. Specifically, implemented the DPS algorithm with Diffusion model trained with fully sampled ground truth images (FS-DPS). Inference was run using the update rule of Eq. 2.3
for total of 500 steps and with 7/- 1ly-AE(as ano (ax Illa
Fully- Sampled Annealed Langevin Dynamics (FS-ALD). A popular method for solving
inverse problems via score-based priors is ALD Dalal al. 2021). We train caseebbaed model using the same fully sampled 10, 000 samples as above and run inference for 1,300
steps. This method acts another fully sampled comparison our technique.
L1-Wavelet Compressed Sensing (L.-CS). We use the same -.-Weaeleet reconstruction described previously as a standalone non-deep learning comparison.
Model Based Deep Learning Supervised (MoDL-Sup). We use a supervised end-to-end model using the MoDL architecture in a supervised fashion (Aggarwal et al.] 2019) for each acceleration factor. For inference, we pass the undersampled data through the trained MoDL network. Further details of this baseline are mentioned Section i the appendix.
Ambient Annealed Langevin Dynamies (A-ALD). We utilize our Ambient Diffusion MRI models with the inverse problem solver ALD (Jalal et al. 202I) for 1, 300 steps.
Ambient One Step (A-OS). Our method also admits one-step solution. This is outlined in Drain Atrain]. Thus we can
section 2.4 by noting that we train our model to estimate | use our models in a one-step fashion. The performance of the one-step prediction should only be expected to be good in in-distribution, i.e. when the model is evaluated at the same acceleration factor as the one during training.
LI-DDifusion Posterior Sampling (LI-DDS). We utilize the diffusion models trained on the I.I-Wavelet reconstructions of data from measurement sets at R = 2, 4, 6, For reconstruction, we use DPS for total of 500 steps with 7/- TV-A3 æ
Self-Supervised learning via Data Under-sampling (SSDU). We train SSDU (Yaman) et al. 2020) models using the end-to-end restoration network by splitting the available
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Compressed sersing LPIPS results AFHQ
Compressed sensing MSE results AFHO
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Figure 5: Compressed Sensing results, AFHQ: performance metric and standard deviation. As shown, the model trained with clean data (p 0.0) only outperforms the models trained with corrupted data for more than 1000 measurements, in both LPIPS and MSE.
Downscaling LPIPS results
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Figure 6: Super-resolution results, AFHQ: Performance metric and standard deviation. The model trained with clean data (p- 0.0) performs worse, except downscaling factor 2.
measurements (9) of each sample yf into training sets (9) and loss sets (A) or measurements y! and v²A Further SSDU training details are given in Section C
ID-Partitioned SSDU. We train ID-Partitioned SSDU (Millard & Chiew 2023 which stems from the SSDU framework and introduces partitioning of the sampling set to ensure each subset has distribution similar to the original mask. The loss function, network architecture, and training hyperparameters were set according to Millard & Chiew (023)).
Results Quantitative metrics for the listed methods are included in Table under the NRMSE, SSIM and PSNR metrics respectively. Furthermore, we report beasere-based metrics including LPIPS and DISTS in Table 3 At low acceleration levels (Rt = 2, 4) FS-DPS outperforms most methods, including A-DPS. However, the validation acceleration increases to higher ratios R, A-DPS outperforms other models, including those trained on fully sampled data.
We also trained Ambient Diffusion model on undersampled abdominal MRI scans, where ground truth data is available (Zhang et al., 2013). We show visual examples from the prior and posterior sampling experiments in the Appendix (Figs 8
We visualize reconstructions for a subset of the methods listed at various accelerations in Fig. 3 Here, see that FS-DPS outperforms A-DPS lower accelerations (R = 4). However, at higher acceleration (R- = 8) we can see that FS-DPS introduces significant artifacts into the reconstruction while our A-DPS reconstructions maintain high degree of visual fidelity. We also report the

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distribution metrics (Conditional FID) between samples from the posterior and the ground-trute distribution for A-DPS and FS-DPS methods in Table [ Similar to our findings from posterior reconstruction metrics, we find that in the higher acceleration (R 8) regime, A-DPS outperforms FS-DPS. We also include graphical trends for NRMSE, SSIM, and PSNR for A-DPS, A-OS, FS-DPS, L1-DPS, FS-ALD, L1-CS, MoDL-Sup, and SSDU in the Appendix (Figs 9 10
13
3.3 AMBIENT DIFFUSION WITH PRE-TRAINED MODELS ON DATASETS OF NATURAL IMAGES
We use the models from the Ambient Diffusion DDras et al. 2023b) that are trained on randomly inpainted data with different erasure probabilities. Specifically, for AFHQ use the Ambient Models with erasure probability p E {0.2, 0.4, 0.6, J.8} and for Celeb-A we use the pre-trained models with p 0.8, 0.9}. We underline that all the Ambient Models have worse performance for unconditional generation than those trained with clean data (i.e. the models trained with p = 0.0). This work aims to explore the conditional generation performance of Ambient Models, where the conditioning is in the measurements yint. and compare it with models trained on uncorrupted data. To ensure that Ambient Models do not have an unfair advantage, we test only on restoration tasks that differ from those encountered in their training. Specifically, we use models trained on random inpainting and we evaluate Gaussian Compressed Sensing (Baraniuk) 2007) and Super Resolution.
Results. In Figure 4 we show visual examples from our experiment on the noisy super-resolution task for natural images using model: (1) trained with fully-sampled data (FS-DPS), (2) trained with randomly inpainted data at p = 0.6 (A-DPS). F presents Gaussian Compressed Sensing reconstruction results (i.e. reconstructing signal from Gaussian random projections). We show MSE and LPIPS performance metrics for the AFHQ dataset across varying number of measurements. The results are given for models that are trained with impainted images at different levels of corruption. indicated by the erasure probability p. The model trained with clean data outperforms the models trained with corrupted data when the number of measurements is high. However, as we reduce the number of measurements, Ambient Models outperform the models trained with clean data in the very low measurements regime. To the best of our knowledge, there is no known theoretical argument that explains this performance cross-over, and understanding this further is interesting research direction. Similar results are presented in Figure 6 for the task of super-resolution at AFHQ. The model trained on clean data (p- 0) slightly outperforms the Ambient Models in both LPIPS and MSE for reconstructing a 2x downsampled image, as expected. Yet, as the resolution decreases, there is again cross-over in performance and models trained corrupted data start to outperform the models trained on uncorrupted data. We include results for LPIPS and MSE for Compressed Sensing and Downsampling FFHQ and Celeb-A in the Appendix (Figs 15
16
17 18.
We compute MSE in other inverse problem settings for FFHQ: Box Inpainting, Additive Gaussian Noise, and Super Resolution (with Gaussian Noise o = 0.05). We find that trends in this experiment are consistent with previous findings. The results across multiple downsampling factors and noise levels in the Appendix (Figs [9] 20 pT). Finally, we examine how the number of sampling steps affects the performance. The MSE results for Compressed Sensing with 4000 measurements on AFHQ are shown in Figure As shown, the higher the erasure probability p during training, the better the Compressed Sensing performance of the model for low Number of Function Evaluations (NFEs). Models trained with higher corruption are faster since they require fewer steps for similar performance. For increased NFEs, the models trained on clean(er) data outperform. This result is consistent across different datasets (AFHQ, FFHQ CelebA), reconstruction tasks (Compressed Sensing, Downsampling), and metries (MSE, LPIPS) (Figs
25 26 in the Appendix).
4 CONCLUSION
We present Ambient Diffusion Posterior Sampling (A-DPS), a simple framework based on DPS for solving inverse problems with Ambient Diffusion models. We show that Ambient Diffusion models trained on corrupted data can be better suited for handling ill-posed inverse problems under severe corruption. By being exposed to corrupted training data, A-DPS exhibits robust priors that generalize better to high-acceleration regimes, unlike FS-DPS. which may overfit clean data and fail to adapt effectively to severe undersampling. Our framework fully unlocks the potential of Ambient Diffusion models that are critical in applications where access to full data is impossible or undesirable.
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A THEORETICAL RESULTS
A.1 THEORY PROOF
In this section, we provide the proof of Theorem 2.1 that was stated in the main paper. We begin by giving the formal statement of the theorem.
Theorem A.1 (Formal Statement of Theorem 2.1 Let a: E C be an unknown signal from distribution that admits density p(æ). Let (S.) be a random set of diagonal matrices E C that satisfies: Σ, 5 S Inxns Assume sample access to linearly corrupted measurements of x given by:
Yerain SH x,
(A.1)
Arron
where the discrete Fourier matrix and P is a random inpainting matrix, i.e. a diagonal matrix with either zeros or ones in the diagonal, such that Pr(Pa: 1] > 0, for all entries of the diagonal. Define:
Durain -( PFS: x,
(A.2)
Aurain
where P a further corrupted version of the sense that with some non-zero probability p. diagonal element that was in becomes O in P. Define also Dt, Mt,train Ut the noisy versions of x, Utrain, Derain respectively, in:
Øt = x + a. Yt,train Atrain (xc o.n), Wt,train Aerain (æc + o.n)
(A.3)
Then, the minimizer of the objective:
Aussiato (B P) yo,eroii 17.
J(0) E
(A.4)
Mo. crain Dr. crain Areain
is: ho (ft, P)
co tratin Pl.
Proof. We adapt the proof Theorem 4.1 in Ambient Diffusion (Daras et al. 2023b). To avoid notation clutter, we will be omitting the subscript "train", when necessary.
Let hg- (ge, P) = Ei Te. P]+ f(J,, P) be optimal solution. The value of the objective for optimal solution becomes:
J(0") = No. A.P || |AA wo Wi, P1 Af P) yo0 20
(A.5)
AB[zol I.. L yo 1| P)" PT A" (E(zolli. p1- =)0=20) 20) A" AS P)
E A.
Irseducible
(A.6)
will now work with the last term.
E E S... (1Sooo (0l0ol00s PI- =20)" AT ATA P)
(A.7)
( (E(zoolčee P| xo) AT Af P)
E A.
E Baoldn...
(A.8)
(IL We., 10xo Dr, A, PI) T AT AT P)
-Eg.A
(A.9)
0,
(A.10)
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since E(zol P] -Ezol A, PI, i.. the value of æy does not depend on À given P and yr.
We will now work with the middle term. We have that:
E B.A.A. [ P)T AF P)]
(A.11)
B. B BAADLL [f P\T AY P)]
(A.12)
= E. P [5(Er, AAP
(A.13)
From the last equation, it is evident that if E[A" P| is full-rank, and hence the minimizer is E[zo Dt. P] as needed. Since our A matrix is square, it suffices to show that TE| [[|] P] is full-rank. By Corollary A.I. in Ambient Diffusion, we have that: E[P[P] is full-rank. We will now show that [ is also full-rank. This can be easily proved with contradiction. Assume that PPP|(P is not full-rank. Then, there exists a vector * 0 such that:
P =0 +
(A.14)
E[PIP] Fw =0 4 z40
(A.15)
B[P|P|=
(A.16)
which is a contradiction, since = * and [[P[P] is full-rank.
So far, we have established that F E[ full-rank. By linearity of expectation, we further have that:
F =
(A.17)
and hence, the latter is also full-rank. Finally, since E, S. for any full-rank matrix C, have that >; S CS, is also full-rank. This is for any given set St and hence it is also true for the expectation over the sets {. Putting everything together, we have that:
[(S...P S P B[A|P),
(A.18)
is full-rank and the proof is complete.
Note on alternative Ambient Diffusion designs. Our Ambient Diffusion MRI training approach can be extended to use other linear operators to aggregate the measurements from different coils (c.g. we could have used the pseudoinverse). Alternatively, we can also further condition on the sensitivity maps, or directly feed all the coil measurements as input to the network without aggregation. We opted for a simple and fast method (aggregation through the adjoint). However, these other approaches can lead to improved performance because they fully leverage coil information.
B DATASET PREPARATION
We follow standard practices in the MRI literature. We download the FastMRI dataset, randomly select 2,000 T2- 12-Weighted scans, and pick 5 center slices from each scan, giving us a dataset of 10,000 T2-weighted brain scans (k-space measurements). We pass each multi-channel k-space sample through a noise pre- pre-whitening filter to transform the noise into standard white Gaussian. Then, we normalize each s-space sample by the absolute value of 99-th percentile of the root sum-of-son -fuuuu-quuares reconstructoon of the autocalibration region (24x x 24 pixels from the center). Given the fully sampled k-space samples, we estimate sensitivity maps using the ESPIRIT calibration method (Uecker et al.) 2014). These measurements serve fully sampled reference dataset (i.e., R = 1).
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C DETAILS FOR BASELINES.
L1 Methods, We used the BART implementation (Uecker et al. 2016 Blumenthal et al. 2022) and searched for the best regularization parameter over the training set. Specifically, we used regularization weighting of 0.001 with 100 iterations.
Self-Supervised learning via Data Under-sampling (SSDU) (Yaman al. 2020) In supervised setting, end-to-end image restoration networks can be trained by passing measurements through a network like MoDL (Aggarwal et al.) 2019) or VarNet (Hammernik et al. 2018) and taking a loss directly on the output of the network with the ground truth image. In the self-supervised setting, however, this is not possible. SSDU, trains the end-to-end restoration network by splitting the available measurement set (I)) of each sample yf into training sets (0) and loss sets (A) or measurements v and y respectively where Ω = θ U A. Where the reconstruction network is given access to measurements in the training set to obtain an estimated image t ho(v,) and the loss is then defined over the loss measurements as
10
A A V
1
AA 1//"
L' 2
(C.I)
where AA is the forward operator with ones in the selection mask P at only the measurement locations in the loss set and zeros elsewhere. In this work, we selected ho(-) to have the MoDL architecture (Aggarwal et al. 2019). We trained individual models four different acceleration levels (R- = 2,4,6, 8). Each model was trained for ten epochs using the same training set (10, 000 slices) as previously described. We used uniform random sampling to separate the training and loss measurement groups for each sample and found that a split of p = [ performance and thus was used for reporting all subsequent metrics.
|Al
0.2 provided the best
MoDL Supervised (MoDL-Sup) To provide an upper bound on performance for supervised end- to-end methods we also trained the MoDL architecture in a supervised fashion (Aggarwal et al.) 2019). Again, we trained individual models at four different acceleration levels (R 2, 4, 6, 8). Each model was trained using the same 10, 000 slices as above. However, for these models, we used the normalized root mean squared error (NRMSH) as the loss function:
x Ez
L(x,, 2;,)
(C.2)
x
where o; are the ground truth images from our dataset and &; = by (y:: A₂) are the reconstructions provided by our network based on under-sampled measurements y;:
D HYPERPARAMETERS FOR MRI MODELS
Training Hy peperparasssrss. The EDM model trained on the fully-sampled (R 1) dataset, as well as the four EDM models trained on LI-Wavelet reconstructions at each accelcration factor R E (2, 4, 6, 8) were all trained with 65 million parameters. The Ambient Diffusion models on the other hand were trained with 36 million parameters, for faster training. All the Ambient Diffusion models were trained for 250 epochs. For the Ambient Diffusion training experiments, the further corrupted sampling masks are given as an input to the model by concatenating with the measurements along the channel dimension, as in the original Ambient Diffusion (Daras et al. 2023b paper.
Sampling Hyperparameters. The only tunable parameters for DPS (Eq. 2.3) and Ambient DPS (Eq. [1 are in the scheduling the magnitude the measurements likelihood term. In all the experiments in the DPS paper, this term is kept constant throughout the diffusion sampling trajectory and the authors recommend selecting a value in the range between [0.1, 10). We follow this recommendation and we keep this term constant. The value of the step size for each model is selected with hyperparameter search in the recommended range. For all our experiments, we follow exactly the DPS implementation provided in the official code repository of the paper. The other parameter
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that impacts performance is the number of steps going to each algorithm for, i.c. the
discretization level of the SDEs of Equations 2.3 2.11 Typically, the higher the number of steps the
better the performance since the discretization enror decreases (Chen al. 2022b 2023). For the performance results, we run each method for several steps E (50, 100, 150, 200, 250, 300
E ADDITIONAL MRI RESULTS
Table 3: Accelerated MRI reconstruction performance evaluation using LPIPS & and DISTS 4. averaged across 100 test samples.
Training
Training Data
Reconstruction Method
R
2
8
Method
LPIPS
DISTS
LPIPS
DISTS
LPIPS
DISTS
LPIPS
DISTS
Supervised
0.059
0.231
A-DPS
0,118
0.062
0.127
0.068
0.138
0.076
0.150
0.082
A-ALD
0.119
0.053
0.127
0.068
0.139
0.077
0.151
0.083
LI-DPS
0.067
0.032
0.131
0.075
0.201
0.115
0.262
0.143
ID-Partitioned SSDU
0.141
0.084
0.144
0.081
0.108
0.223
0.129
A-DPS
0.155
0.168
A-ALD
0.156
0.090
0.155
0.092
0.161
0.095
0.168
0.099
LI-DPS
0.096
0,051
0.193
0,112
0.258
0.147
0,301
0.167
Self Self- Supervised
Partitioned SSDU
0.169
0.105
0.169
0.092
0.187
0.104
0,214
0,123
A-DPS
0.180
0.110
0.178
0.110
0,181
0.111
0.186
0.114
A-ALD
0.179
0.109
0.179
0.111
0.181
0.111
0.186
0.114
R
ID-Partitioned SSDU
0.143
0.092
0.169
0.102
0.196
0.117
0.222
0.133
A-DPS
0.212
0.129
0,204
0.127
0,204
0.127
0.204
0.127
A-ALD
0,213
0.130
0.204
0,127
0.204
0.127
0.204
0.127
R
LI-DPS
0.114
0.134
ID Partitioned SSDU
0.142
0.094
0.176
0.109
0.206
0.127
0.231
0.140
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Training Dataset" Samples
Ambient Diffusion Unconditional Samples
Figure 7: Training samples (left) and prior samples (right) from ambient diffusion model trained on undersampled abdominal MRI scans, where no ground truth data is available.
Adjoint Reconstruction
L1 Wavelet Reconstruction
Amblent-OS Reconstruction
Ambient-DPS Reconstruction
Figure 8: Ambient diffusion posterior sampling (A-DPS) performance using ambient diffusion model
trained on undersampled abdominal MRI scans, where no ground truth data available.
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100
45.0
4
AOP5
AG15 (0
A0S (1
A-0P5 -8)
A.PSS (8
15.0
0.15
37.5
0.30
0.83
21.5
A-075
0.00
23.0
Figure 9: Ambient diffusion posterior sampling (A-DPS) multi-step performance metrics at R
2, 4, 6, 8 for models trained at R = 2, 4, 6, 8.
A0E 21
45.0
A05 =
A05 -1
MO R 41
A-00 01 41
A00 41
A-2 (3
A-05 -1
A-S (1-))
[8
405 -
4.05 -8
37.5
ARVISE
21.0
0.05
23.5
0.00
25.
Figure 10: Ambient diffusion one step (A-OS) performance metrics at = 2,,4,6 for models trained at R = 2,4, 6.
100
45.0
150P5
DPS
L1 1L005
41.5
1000
35.0
325
11.
ALPS
Figure 11: Performance metrics for FS-DPS, L1-DPS, FS-ALD, and L1-CS R = 2,4.68
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HoBiSup -1
45.0
NcO1-Spp 09
HiBi-Sup
N101
Sup (6
N:0L (6
37.5
880006
35.0
0.10
0.03
NoBL-Sup 01
Noll-Sup --4
27.5
MoDL-Sup (8
NoBi-Sup 08 at
0.00
25.0
Figure 12: Fully supervised MoDL performance metrics R 2,4, 6, for models trained
R 2, 4, 6, 8.
100
500 10 21
45.0
5500 -1
SCBU -1
SS9J m1
S59U
SSBU
S6OJIR
42.
5550 0R
5500
22.0
32.5
73
0.00
250
Figure 13: SSDU performance metrics R =2
2, 4, 6, 6, for models trained at R= 2, 4, 6,8.
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F ADDITIONAL NATURAL IMAGE PERFORMANCE RESULTS
Compressed sersing
0.08
0.08
p-0.6
p-0.8
0.06
0-0
p-0.5
0.05
0.04
30
0.04
0.02
0.01
0.03
200
230
300
50
200
150
250
stup
Stap
(a) Compressed Sensing 1000 measurements per Number of Function Evaluations (NFEs).
(b) 2x Super-Resolution per Number Function
Evaluations (NFEs).
Figure 14: Speed performance plots for AFHQ.
Compressed sensing
FFHQ
Compressed sensing FFHQ
p=0.0
p=0.0
0-0.2
-e- p-0.2
p.0.6
0.020
D-0.9
p-0.9
0.30
0.015
0.20
0.10
0.005
0.000
500
1000
1500
2000
2500
3000
3500
4000
500
1000
15:00
2000
2500
3000
3500
4000
Number messurements
Number messurements
(a) LPIPS per Number Measurements.
(b) MSE per Number Measurements.
Figure 15: Compressed Sensing Results for FFHQ
Downscaling
0.40
0-0.0
0-0.2
p-0.2
p-0.5
0.030
p-0.6
p-0.8
0.30
0.025
p-0.9
0.20
0.01.
0.005
0.05
Downscaling Factor
Downscale scaling Factor
(a) LPIPS per Downscaling Factor.
(b) MSE per Downscaling Factor.
Figure 16: Downscaling Results for FFHQ.
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Compressed
Compressed
pm0.0
pm0.0
p=0.2
0.0175
pm0.2
p-0.6
p-0.6
0.0150
pw0.9
p=0.9
0.20
0.15
0.0075
0.10
0.0050
0.05
0.90
0.0000
500
1000
1500
2000
2500
3000
3500
4000
500
1000
15:00
2000
2500
3000
3500
4000
Number measurements
Number er of messurements
(a) LPIPS per Number Measurements.
(b) MSE per Number of Measurements.
Figure 17: Compressed Sensing Results for Celeb-A.
Downscaling CELEBA
Downscaling CELEBA
p-0.0
p=0.0
p-0.2
p-0.2
0-0.6
0.025
0-0.6
p-0.8
0-0.9
0.25
0.20
0.015
0.010
0.005
0.05
0.000
Dounscaling Factor
Dowrscaling Factor
LPIPS per Downscaling Factor.
(b) MSE per Downscaling Factor.
Figure 18: Downscaling Results for Celeb-A.
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For the following experiments, compare model trained without corruption to model trained with randomly inpainted data at = 0.6. The evaluation dataset is FFHQ. The training size and hyperparameters for both models the same. For both models, extensively tune the sampling hyperparameters to maximize their performance (starting from the recommended hyperparameters from the DPS work). The finding get in these additional experiments is that the model trained on corrupted data (inpainting) outperforms the model trained on clean data in the high corruption regime across all evaluation tasks.
Inpainting
FFHO
p-0.0
p-0.6
0.15
0.10
0.00
10
Figure 19: Box Inpainting results for FFHQ
Additive Caussian Noise MSE resuits on FFHO
0.20
p-0.0
p-0.6
0.50
0.00
Figure 20: Additive Gaussian Noise results for FFHQ
Super Resolction Gauscian (om0 05)- results FFHD
0.15
p=0.0
p-0.5
0.10
Downscaling Roctor
Figure 21: Super Resolution + Additive Gaussian Noise (a 0.05) results for FFHQ
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G
ADDITIONAL NATURAL IMAGE SPEED RESULTS
Compressed sensing AFHO
Downscaling
0.6
pm0.0
p=0.0
0-0.2
-c- p-0.2
p.0.4
p-0.6
0.4
- 0-0.8
0.4
-B- p-0.8
0.3
0.3
0.0
50
100
130
200
250
300
50
100
150
200
250
300
step
stop
(a) Compressed Sensing with 4000 measurements per Number of Function Evaluations (NFEs).
(b) 2x Super-Resolution per Number Function
Evaluations (NFEs).
Figure 22: Speed LPIPS performance plots for AFHQ.
Compressed sensing results FFHQ
Downscaling FFHQ
p-0.0
--- p-0.0
0-0.2
-s- p-0.2
0.08
p-0.6
p-0.8
-B 0-0.9
-#- p-0.9
0.96
M6E
0.00
0.80
50
100
150
200
250
300
50
100
150
200
250
300
slup
slep
(a) Compressed Sensing with 4000 measurements per Number of Function Evaluations (NFFs).
(b) 2x Super-Resolution per Number Function
Evaluations (NFFs).
Figure 23: Speed MSE performance plots for FFHQ.
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Compressed sensing LPIPS results on FRHQ
Downscaling LPIPS results en FFH
--- p-0.0
p-0.0
0.5
p-0.2
p-0.6
0-0.8
-#- p-0.8
0.4
0-0.9
p-0.9
0.3
0.2
0.1
0.0
100
stop
(a) Compressed Sensing with 1000 measurements per Number of Function Evaluations (NFEs).
(b) 2x Super-Resolution per Number of Function Evaluations (NFEs).
Figure 24: Speed LPIPS performance plots for FFHQ.
Compressed sensing MSE results on CELEBA
Downscaling MSE results on CELEBA
p-0.0
p-0.0
pm0.2
-#6 p-0.8
-4-- p-0.8
pm0.9
pm0.9
368
0.04
0.02
0.00
1.00
155
200
300
1.00
150
200
250
.30p
(a) Compressed Sensing with 4000 measurements per Number of Function Evaluations (NFEs).
(b) 2x Super-Resolution per Number of Function Evaluations (NFEs).
Figure 25: Speed MSE performance plots for Celeb-A.
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Compressed sensing LPIPS results CELEBA
Downscaling LPIPS results CELEBA
0-0.0
p-0.0
0.5
2.0
p.0.2
--- p-0.6
0.4
-+- p=0.6
p-0.8
0.4
pm0.9
0.3
L905
0.2
0.1
0.1
100
150
250
300
50
130
200
step
(a) Compressed Sensing with 4000 measurements per Number of Function Evaluations (NFEs).
(b) 2x Super-Resolution per Number of Function
Evaluations (NFEs).
Figure 26: Speed LPIPS performance plots for Celeb-A.
H LIMITATIONS AND FUTURE DIRECTIONS.
Our work has several limitations. First, there is lack of theoretical understanding of the observed experimental phenomenon: we not have a good grasp of why models trained on highly corrupted data lead to better priors for inverse problems of another corruption type. Second, only tested the reconstruction performance of our models using the DPS algorithm. Several other recent algorithms have been developed for solving inverse problems with diffusion models and it unknown whether our findings generalize for these reconstruction algorithms. Finally, our method relies on the existence of Ambient Diffusion models. There are many more models available trained on clean data than models trained on highly corrupted data.
The auto-calibration signal (ACS) region shared across training samples, which could lead an over-eepresentatioo of certain k-space regions. While our current approach does not explicitly address this, introducing a diagonal weighting matrix the loss function, as suggested by Millard & Chiew (2023), is promising direction to mitigate this issue.
Currently, we define Atrain according to Eq 2.7 This is convenient because lets us use an image- to-image network architecture such as that used by EDM. An alternative could be to define Atrain directly as Eq in which case the network would go from multi-coil k-space to an image. This network could be preceded by an IFFT, which would effectively be multi-coil-to-image network, and could be standardized using coil compression (Zhang al. 2013). Such a network architecture could be interesting and potentially more expressive, through zero-
currently collapse the multi-coil information zero-filling filling adjoint. Another possibility is to define Atrain as the pseudo-inverse applied to A, which could be solved efficiently with the conjugate gradient algorithm.
I ACKNOWLEDGEMENTE
This research has been supported by NSF Grants CCF 1763702, NSF CCF-2239687 (CAREER), AF1901292, CNS 2148144 141, Tripods CCF 1934932, NSF AI Institute for Foundations of Machine Learning (IFML) 2019844, the Texas Advanced Computing Center (TACC) and research gifts by Western Digital, WNCG IAP, UT Austin Machine Learning Lab (MLL), Cisco and the Archie Straiton Endowed Faculty Fellowship. Giannis Daras has been supported by the Onassis Fellowship (Scholarship ID: F ZS 012- 11/202222223)) the Bodossaki Fellowship and the Leventis Fellowship.
28


Reviews:
[{"review_id": "w2GAyLE4GY", "reviewer": "Reviewer_323G", "summary": "This paper uses ambient diffusion models as a prior for solving ill-posed inverse problems. The ambient diffusion is trained on corrupted data acquired from a linear ill-posed operator. The authors also extend the ambient diffusions to use MRI k-space subsampled measurements as training data. The authors show the proposed framework can outperform DPS trained on clean images when dealing with severely ill-posed inverse problems.", "questions": "Please take a look at the comments under weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "1. The paper is motivated by the challenge of solving inverse problems without access to clean images, a common problem in real-world imaging applications.\n2. Experimental results show the proposed framework is effective for severely ill-posed problems, sometimes outperforming priors trained on clean images.", "weaknesses": "1. Despite promising experiments, the novelty of the paper seems limited; the authors primarily extend ambient diffusion to MRI and adapt it within the DPS framework for solving inverse problems.\n\n2. Below Eq. 2.11, the authors state, \"We remark that similar to DPS, the proposed algorithm is an approximation to sampling from the true posterior distribution\". However, the accuracy of this approximation should be rigorously compared to that of the original DPS, as it relies on estimating the score function from corrupted data.\n\n3. The authors should consider including additional, recent self-supervised learning approaches as baselines, such as the following paper:\nGan, Weijie, et al. \"Self-supervised deep equilibrium models with theoretical guarantees and applications to MRI reconstruction.\" IEEE Transactions on Computational Imaging (2023).\n\n4. The paper needs to be further polished, for example, Eq 2.9 seems incorrect.", "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": "XRbBB4lg29", "reviewer": "Reviewer_GxHM", "summary": "This paper applies the recent advancement in self-supervised learning for diffusion models, named ambient diffusion, to inverse problems. The effectiveness of this approach is shown through its application to MRI reconstruction and image inpainting.", "questions": "1) The further corruption ratio is simply R+1. How does this further corruption ratio impact the performance, says R+2, R+3, etc?\n2) Does the supervised DPS share the same network architecture with the proposed method?\n3) How does the coil sensitivity map in this training dataset and the L1-wavelet baseline be estimated?\n4) What is the time complexity of the methods?\n5) What is the actual under sampled ratio considering the sampling mask for different R has the same 20 lines of the auto calibration signal?", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1) The paper represents an early investigation into the application of self-supervised learning for diffusion models in the context of inverse problems.\n2) The paper is well-organized, and the presentation is easy to follow.", "weaknesses": "1) Although this paper introduces the first self-supervised diffusion method for inverse problems, the overall novelty is somewhat limited. It appears to be a straightforward integration of existing ambient diffusion with well-known diffusion posterior sampling, with application to MRI reconstruction. While the investigation holds value from a research perspective, the new insight/concept might be limited.\n\n2) In the training phase (as described in Equation 2.10), the network takes \\tilde{A} as input, considering one type of imaging artifact. However, during testing (Equation 2.11), the network uses A as input, which features a different imaging artifact. This inconsistency raises concerns about potential suboptimal performance. Could the authors clarify this aspect?\n\n3) There is insufficient discussion regarding why the proposed A-DPS method outperforms FS-DPS, especially at high undersampling rates. Given that FS-DPS uses a diffusion model trained on fully sampled data, the results seem unexpectedly favorable.\n\n4) The manuscript lacks discussions with recent theoretical developments in self-supervised learning for MRI [1,2], which share similar proof strategy with Theorem 1 in this paper.\n\n5) The numerical validation could be improved, with certain experimental setups are not fully explored (see specific questions below). Additionally, the baseline methods for MRI reconstruction cited are well-known but somewhat dated (2019, 2020). Given the extensive literature on MRI reconstruction, these baselines may not represent the current state of the field. Comparing recent alternatives, such as [1-3] and self-supervised or supervised methods in survey [3] and [4], respectively, could improve this study. There is also no comparison between A-OS and other end-to-end methods. Despite that, I fully understand the time constraints of the rebuttal period and do not intend to suggest that the authors include comparisons with new baselines.\n\nI am willing to increase my score if the authors could address my concerns.\n\n[1] Millard, Charles, and Mark Chiew. \"A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise.\" IEEE transactions on computational imaging (2023).\n\n[2] Gan, Weijie, et al. \"Self-supervised deep equilibrium models with theoretical guarantees and applications to MRI reconstruction.\" IEEE Transactions on Computational Imaging (2023).\n\n[3] Sriram, Anuroop, et al. \"End-to-end variational networks for accelerated MRI reconstruction.\" Medical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd International Conference, Lima, Peru, October 4–8, 2020, Proceedings, Part II 23. Springer International Publishing, 2020.\n\n[4] Akçakaya, Mehmet, et al. \"Unsupervised deep learning methods for biological image reconstruction and enhancement: An overview from a signal processing perspective.\" IEEE Signal Processing Magazine 39.2 (2022): 28-44.\n\n[5] Heckel, Reinhard, et al. \"Deep learning for accelerated and robust MRI reconstruction: a review.\" arXiv preprint arXiv:2404.15692 (2024).", "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": "PGPvDT1A8e", "reviewer": "Reviewer_GFMn", "summary": "This paper proposes a novel method for learning a diffusion model from corrupted data, supported by a theoretical analysis of training the MMSE restoration estimator under such conditions. Empirical results indicate that the learned MMSE restoration estimator can operate similarly to an MMSE denoiser, enabling effective score-based generation and restoration tasks. Overall, this is a highly interesting paper with the potential to open many new research directions. However, certain aspects of the analysis and discussion could benefit from further improvement.", "questions": "The questions have been covered in the weakness part.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "(1) This paper presents a novel framework for learning a ‘diffusion model’ solely from corrupted data, which is especially valuable in the field of medical imaging reconstruction, where only undersampled data is often available. \n(2) A solid theoretical proof for learning MMSE restoration estimator from corrupted data under suitable assumption.\n(3) Based on this diffusion model, the authors demonstrate its effectiveness in both generative tasks and solving inverse problems, highlighting the efficiency and potential impact of this approach.", "weaknesses": "(1) This work introduces an intriguing approach to learning a diffusion model from corrupted data. However, the authors’ analysis assumes that the learned MMSE restoration estimator can approximate a learned MMSE denoising estimator to derive a score function for sampling—an assumption that could benefit from further clarification. In approaches like DPS, posterior sampling depends on the score derived from Tweedie’s formula, which is specifically suited for denoisers. In this study, however, the corresponding score function is not discussed in sufficient detail. A discussion of recent approaches such as DRP[1] and SHARP[2], which utilize the MMSE restoration estimator with a modified score function, could enhance the analysis and provide a more fitting framework for this model.\n\n(2) The theoretical approach in this work is compelling and well-suited for training with randomly inpainted images. However, in the MRI setting used here, different training samples may share the auto-calibration signal (ACS) region, potentially leading to an over-representation of corresponding k-space regions in the training data. A more solid approach might involve introducing a diagonal weighting matrix in the loss function to account for these oversampled regions. A similar work discussing this point for MRI reconstruction is [3].\n\n(3) Further explanation to how the FID value is calculated for MRI data will be helpful.\n\n(4) Some minor suggestions regarding the paper's presentation: (1) In Table 3, it would be beneficial to standardize the significant digits for the FID values to improve readability. (2) For Figures 7 and 8, adjusting the color bar could enhance visual clarity and make the figures easier to interpret.\n\nReference: \n[1] Y. Hu, M. Delbracio, P. Milanfar, and U. S. Kamilov, “A Restoration Network as an Implicit Prior,” Proc. Int. Conf. Learn. Represent. (ICLR 2024) (Vienna, Austria, May 7-11).\n[2] Y. Hu, A. Peng, W. Gan, P. Milanfar, M. Delbracio, and U. S. Kamilov, “Stochastic Deep Restoration Priors for Imaging Inverse Problems.”  [https://arxiv.org/abs/2410.02057]\n[3] C. Millard and M. Chiew, \"A Theoretical Framework for Self-Supervised MR Image Reconstruction Using Sub-Sampling via Variable Density Noisier2Noise,\" in IEEE Transactions on Computational Imaging, vol. 9, pp. 707-720,", "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": "At2vkIFyXA", "reviewer": "Reviewer_gieB", "summary": "The paper proposes a posterior sampling method for inverse problems called Ambient Diffusion Posterior Sampling (A-DPS), which utilizes a diffusion model trained only on corrupted images. They modify the existing Ambient Diffusion framework to work for the accelerated MRI problem, allowing the training of an unconditional diffusion model with only subsampled MRI measurements. Then, they integrate the Ambient Diffusion models into the DPS framework to form a posterior sampling approach. The experiments on accelerated MRI and natural image datasets consistently show better recovery performance than the state-of-the-art DPS when the conditioning is highly corrupted but not when the corruption level is low.", "questions": "- Would it work better to train a single network on multiple acceleration rates rather than a separate network for each acceleration rate?\n- Were the evaluations done on a single posterior sample for each method? Or do you generate many posterior samples and compute a posterior mean?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The paper addresses a significant problem in solving inverse problems when ground truth data is unavailable. By only requiring corrupted measurements for training, the method presents an avenue to solve a wider set of inverse problems.\n- The framework is novel as it incorporates the benefits of unconditional Ambient Diffusion with the conditional flexibility of DPS. \n- The experiments are adequately extensive and effectively demonstrate the benefits (and shortcomings) of their approach compared to several existing methods.", "weaknesses": "While the general methodology is explained well, the description of particular details could be modified to improve clarity and soundness\n- Eq: 2.9: Are these equations correct? It seems to me that it should be $y_{t,\\text{train}} = A_{\\text{train}}(y_{\\text{train}} + \\sigma_{t}\\eta)$ if $A_{\\text{train}}$ is an orthogonal projection matrix or $y_{t,\\text{train}} = A_{\\text{train}}(x_{\\text{train}} + \\sigma_{t}\\eta)$ if not\n- What is the input to the diffusion model $h_\\theta$? In Eq 2.10, it takes the additional subsampling matrix $\\tilde{P}$. In Eq 2.11, it takes the full forward operator $A_\\text{train}$. \n- An intuitive explanation of $h_{\\theta}(\\tilde{y}\\_{t, t_{\\text{train}}}, \\tilde{P})$ would be helpful for understanding the overall algorithm. Specifically, what does the training accomplish and why can this be directly plugged into Eq. 2.11?\n- What is the reason or intuition for why an A-DPS trained on one corruption can generate posterior samples for a different corruption? A good hypothesis could better sell this property of the method.\n\nMinor Points:\n- L93-94: This sentence is confusing because it sounds like the Ambient Diffusion framework is \"the first framework to solve inverse problems with diffusion models learned from linearly corrupted data\". I would reword this to avoid confusion.\n- Eq 2.4: It was not clear what $\\tilde{A}_\\text{train}$ represents. I would explain this matrix first before the equation, so the equation is more interpretable\n- L255: Typo with \"In that regards the multiplicity...\"\n- L364: Typo with \"Similar to the results in The authors of ...\"\n- The construction of Fig 3 is confusing with the labeling of \"Recon-FS\". Since it's vertically displaced, it is easy to misread. I would suggest changing this to \"Error\" or rotate it vertically so \"Recon-FS\" is on a single line", "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": "tY1kOaMGfW", "reviewer": "Reviewer_Q5u6", "summary": "In this work the authors provide a framework for solving inverse problems with diffusion models that are trained on corrupted data. The authors extend the existing ambient diffusion framework to enable Fourier domain-corrupted measurements and propose ambient diffusion posterior sampling, an algorithm that leverages generative models pre-trained on one type of corruption to solve other inverse problems. The authors show the performance of their method on accelerated MRI reconstruction and on various problems for natural images.", "questions": "See weaknesses. Although, I do have one actual question:\n\nQ1) Did you consider training your MRI diffusion model (ambient or other) with the multi-coil data? To be clear: I am not asking you to do this, as it is a fundamental departure from what you have here. That said, I wonder if you tried this. I know that in, e.g., fastMRI, the number of coils is inconsistent, but you can get around this issue with virtual coils [1]. I suspect that the overall performance could be further improved with a multi-coil diffusion process, rather than a coil-combined one.\n\n[1] T. Zhang, J. M. Pauly, S. S. Vasanawala, and M. Lustig, “Coil compression for accelerated imaging with\nCartesian sampling,” 2013", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "S1) The paper is very well written and provides an appropriate amount of detail. Similarly, all analysis is easy to follow.\n\nS2) The authors include a nice theoretical result in Appendix A.\n\nS3) The quantitative performance of A-DPS is impressive, and the selected baselines offer an easy way to contextualize it (particularly for MRI).\n\nS4) In my opinion, and based on the results, this is the first diffusion-MRI method to truly push the state-of-the-art forward since [1].\n\n[1] Ajil Jalal, Marius Arvinte, Giannis Daras, Eric Price, Alexandros G Dimakis, and Jon Tamir, \"Robust\ncompressed sensing mri with deep generative priors,\" 2021", "weaknesses": "I find this paper to be very impactful (hence my score). That said, it is very important to me that the following two weaknesses are addressed:\n\nW1) While the experimental performance of A-DPS is impressive, I think that the MRI results suffer from insufficient evaluation with all competitors. NRMSE being the only metric reported in Table 2 is a concern. It would be worthwhile to add some feature-based metrics like LPIPS/DISTS for all methods. You use VGG to compute FID, you can also use VGG for those metrics (since VGG agrees with radiologists' perceptions [1]). Alternatively, you could also use AlexNet as your feature extractor [1]. Either way, I think that it is important to see these results for your method and competitors. The lack of additional metrics holds this paper back. I understand the space concern given the size of Table 2, but even putting these results in an appendix would be useful to provide readers full context and make a convincing statement about A-DPS' superiority.\n\nW2) I am confused by the bolding in Table 2. For methods trained with clean data, no values are bolded. For the R=2 and R=4 sections, A-DPS has equal performance to A-ALD in some metrics. The A-ALD metrics that are equally performant should be bolded for fairness. Please revise this table so that it is consistent.\n\n[1] P. M. Adamson, A. D. Desai, J. Dominic, C. Bluethgen, J. P. Wood, A. B. Syed, R. D. Boutin, K. J. Stevens,\nS. Vasanawala, J. M. Pauly, A. S. Chaudhari, and B. Gunel, “Using deep feature distances for evaluating\nMR image reconstruction quality,” 2023", "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)