Time-Embedded Algorithm Unrolling for Computational MRI
Authors: Junno Yun
Yasar Utku Alcalar
Mehmet Akcakaya
Venue: NeurIPS 2025 poster ()
Keywords: Algorithm unrolling
Time-embedding
Computational Imaging
Fast MRI
Approximate Message Passing
Primary area: machine_learning_for_sciences

Abstract: Algorithm unrolling methods have proven powerful for solving the regularized least squares problem in computational magnetic resonance imaging (MRI). These approaches unfold an iterative algorithm with a fixed number of iterations, typically alternating between a neural network-based proximal operator for regularization, a data fidelity operation and auxiliary updates with learnable parameters. While the connection to optimization methods dictate that the proximal operator network should be shared across unrolls, this can introduce artifacts or blurring. Heuristically, practitioners have shown that using distinct networks may be beneficial, but this significantly increases the number of learnable parameters, making it challenging to prevent overfitting. To address these shortcomings, by taking inspirations from proximal operators with varying thresholds in approximate message passing (AMP) and the success of time-embedding in diffusion models, we propose a time-embedded algorithm unrolling scheme for inverse problems. Specifically, we introduce a novel perspective on the iteration-dependent proximal operation in vector AMP (VAMP) and the subsequent Onsager correction in the context of algorithm unrolling, framing them as a time-embedded neural network. Similarly, the scalar weights in the data fidelity operation and its associated Onsager correction are cast as time-dependent learnable parameters. Our extensive experiments on the fastMRI dataset, spanning various acceleration rates and datasets, demonstrate that our method effectively reduces aliasing artifacts and mitigates noise amplification, achieving state-of-the-art performance. Furthermore, we show that our time-embedding strategy extends to existing algorithm unrolling approaches, enhancing reconstruction quality without increasing the computational complexity significantly. Code available at https://github.com/JN-Yun/TE-Unrolling-MRI.

Algorithm Unrolling for Computational MRI
Junno Yun
Yaşar Utku Alçalar University of Minnesota alcal0290umn. edu
Mehmet Akçakaya' University of Minnesota akcakayadumn. edu
University of Minnesota yun000490umn. edu
Abstract
Algorithm unrolling methods have proven powerful for solving the regularized least squares problem computational magnetic resonance imaging (MRI). These approaches unfold an iterative algorithm with fixed number of iterations, typically alternating between a neural network-based proximal operator for regularization, data fidelity operation and auxiliary updates with learnable parameters. While the connection to optimization methods dictate that the proximal operator network should be shared across unrolls, this can introduce artifacts or blurring. Heuristi- cally, practitioners have shown that using distinct networks may be beneficial, but this significantly increases the number of learnable parameters, making it challeng- ing to prevent overfitting. To address these shortcomings, by taking inspirations from proximal operators with varying thresholds in approximate message passing (AMP) and the success of tineeenbe in diffusion models, we propose a time-embedded algorithm unrolling scheme for inverse problems. Specifically, we introduce a novel perspective on the iteration-dee geration-dependent proximal operation vector AMP (VAMP) and the subsequent Onsager correction in the context of algo- rithm unrolling, framing them as a time-embedded neural network. Similarly, the scalar weights in the data fidelity operation and its associated Onsager correction are cast as time-dependent learnable parameters. Our extensive experiments on the fastMRI dataset, spanning various acceleration rates and datasets, demonstrate that our method effectively reduces aliasing artifacts and mitigates noise amplification, achieving state-of-the-ate performance. Furthermore, embedding strategy extends to existing algorithm unrolling approaches, enhancing reconstruction quality without increasing the computational complexity signifi- cantly. Code available at utthttps:// battps://githbb http//tgttttu..... com JN- TTE/TE-ae Unrolling Mirolling-MRII MRI
show that our time-
Introduction
Algorithm unrolling/unfodinng has emerged as an effective method for addressing inverse problems in computational MRI 40
In this framework, traditional iterative optimization problems are unrolled for a fixed number of steps, with the network alternating between enforcing data fidelity based on the known physics-based forward operator and applying implicit regularization via a neural network based proximal operator. This unrolled network is trained end- to-end to jointly optimize the weight(s) for data fidelity and the neural network parameters for the proximal operator. Several different optimization methods have been explored for algorithm unrolling MRI 44
37
25 34 63 33), including gradient descent (GD) (28, proximal gradient descent
(PGD) [5 69 33 43, variable splitting with quadratic penalty (VSQP) [ [4] 64 and alternating
direction method multipliers (ADMM) (88 (66 among others
*Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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(a) Existing Unrolled Architecture
Inputi Sensitivity mape, Mask sharedlunshared (x)
BMy
N4
PFp
R$
D1
R4
D2
R
D2
DF
Sharedlunshared weights
(b) Propused Time-Embedded Unrolled Architecture
Sensibivity maps, Mask, time dime-dependent parameters for DF (ut)
Time-dapendent Onsagericoreection parameters
2, BMy
DF
Ng
R
DF
R
DF
DF
scale, Features el.p'
Shared weights 0
scale, Features g²'.
Time scall,
at,
E Encoding matrix
Data Fidelity
Regulartzer
Dual update (In ADMM)
Onsaager correction tarm
Figure I: Descriptions of (a) the existing unrolled architecture and (b) the proposed tine-embedded unrolled architecture.
Beyond the choice of optimization framework, another critical design decision in algorithm unrolling whether to share the proximal operator for the learned regularizer across unrolls. While theoretical connections optimization theory suggest that the proximal operator should remain fixed maintain consistency with traditional methods [3] 46), this may lead to unwanted artifacts. To address this, many practitioners instead allow the proximal operator vary across iterations, effectively using distinct networks to learn iteration-specifi regularization [28] 38 47). This empirical strategy often enhances reconstruction quality but comes with practical trade-offs, such larger number of trainable parameters which can heighten the risk of over-fitting especially for applications with limited training data 66 63 17]. Such limited data settings, which are the focus of this study, are especially important in many translational applications, where new sequences are being implemented or higher resolutions being pursued, as is often feasible curate databases with thousands of slices.
related perspective on iterative reconstruction emerges from approximate message passing (AMP) methods, which have been developed an iterative Bayesian estimator for recovering sparse signal for certain classes of measurement matrices in the context of compressed sensing [23] 22 AMP adapts the proximal operator each iteration based the prior distribution and includes Onsager correction term to stabilize the process and accelerate signal recovery, A notable extension, vector AMP (VAMP) (52). improves this approach by introducing vector-valued variable nodes, and estimating each node using Minimum Mean Square Error (MMSE) and Linear MMSE (LMMSE) estimators while preserving the properties of AMP. This enables VAMP to remain effective for broader class of measurement matrices, extending the applicability of AMP to broader class of measurement matrices.
Similar to the iteration-dependent proximal operator in AMP methods, a time-dependent denoiser has shown be highly effective in the context of diffusion models [31 57 32 35). In diffusion- based approaches, the denoiser adapts dynamically at each time step to better preserve structure and enhance signal recovery [56] 31). This time-dependent adjustment has been shown outperform static denoisers, particularly in tasks requiring high fidelity and sharpness [1]] as it allows the model better handle varying noise levels throughout the diffusion process.
Building on these principles from AMP methods and diffusion models, we propose a novel time- embedded unrolling of optimization algorithms, theoretically motivated from VAMP. Our main contributions are:
We introduce time (or deeeeden unrolling of optimization algorithms by incorporating inn-minformttion into the proximal operator, theoretically motivated by VAMP formalism. To the best of our knowledge, our approach is the first attempt bring in the time information into

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the algorithm unrolling to further improve performance with minimal increase in computational complexity.
Our method also learns the guidance scale (i.e., the data fidelity weight in our case) in a time- dependent manner during training, which is major deviation from commonly used guidance methods in diffusion models 21 [22
We demonstrate that our time-embedding strategy can be extended to various optimization algo- rithms, such VSQP and ADMM, and applied to different neural network architectures for the proximal operator.
We showcase the efficacy of incorporating the time information to the unrolling process through both quantitative and qualitative assessments on fastMRI dataset [39 68 Our approach performs on par with methods that use distinct proximal operator weights, which has substantially more learnable parameters and may face performance decrease small training database such as ours. Furthermore, our method consistently outperforms the baseline shared-regularizer approach across various acceleration rates, producing artifact-free reconstructions with minimal processing overhead.
2 Background and Related Work
2.1 Inverse Problems in Computational MRI
A canonical problem in computational MRI is to recover an image x E CN from noisy sub-sampled measurements Yn C The forward model in this case is given as
Yn Egx + n,
where Eg E CM X is a known encoding matrix that samples the Fourier domain (i.e. k-space)
locations specified by Q2 and includes coil sensitivities, and E CM is i.idd Gaussian measurement noise. The inverse problem corresponding to Eq. ( is typically ill-conditioned [8] 41
necessitating additional regularization to be incorporated into the objective function 29
arg min ||
En + R(x),
(2)
where the first term ensures data fidelity with the acquired measurements and R(-) regularizer.
2.2 Algorithm Unrolling
The optimization problem in Eq. can be solved using various methods (25, including VSQP and ADMM [4], all of which have been explored in algorithm unrolling. The unrolled network iterates between data fidelity and regularization, with the latter implicitly enforced via a neural
network, as illustrated in Fig. (a). VSQP unrolling
[ 19 24
65
64 62 solves Eq. ( via:
x'
(E En ud) (E yn uz²),
(3)
z ++1 arg min 1 2 ||xt 2| R(z) A (x"x
(4)
where the data fidelity parameter is learnable, and (r) is learned implicitly via a neu- ral network. While Eq. 33 has closed-form solution, it is numerically solved using the CG method [3]. contrast, ADMM a commonly used optimization approach with better convergence additional Lagrangian update, and has been popular in algorithm unrolling 666 58 26 20]
than VSQP [14, owing
x² ++1 (Ef En + 1xI)
-1 (E + (='z-=u)) -u').
(5)
z 1+1 = Proxx (x* + u²),
(6)
u *+1 = u' + X(x z 241),
where R(-), μ and X are learnable [66] 58
Although in of the cases, optimization theory 25 dictates that R(-) in Eq. 2 should fixed
across unrolls, researchers heuristically realized enabling R(.) to change across iterations yields better reconstructions [38 47). However, this increases the number of trainable parameters, and the risk of overfitting, particularly in data-limited settings.
3

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2.3
Approximate Message Passing
provides an alternative approach to solving Eq. when large i.i.d. (sub- Gaussian) matrix. The AMP algorithm uses ierationndependent proximal operator and Onsager correction term, which together enable faster convergence compared to PGD [18]. At iteration t, AMP applies the proximal operator with threshold proportional to of that represents estimate of the mean squared error of the current estimate. However, when the measurement matrix deviates from the i..dd sub- Gaussian regime, AMP methods often fail to converge 152
AMP 23
Vector AMP (VAMP) algorithm [52 is an alternative, offering convergence in the large limit for broader class of matrices E. It extends the AMP framework to vector- vector-valued nodes [51 48 [55] and has connections to the ADMM algorithm 42 48 while preserving the desirable properties AMP. These vector-valued operations lead to data fidelity operation based on linear MMSE estimation, and associated Onsager correction as:
x (E"
E + 1 I)
(E"x y + prr"),
(8)
1 3 N Tr [E followed by the proximal operator/denoising step with its Onsager correction:
x² U
u
/
(9)
z' PrOXR,, (u'),
(10)
1 (VProXR, (u')): u vf
(#= - nau') - /
0 1
u
(11)
Notably, both data fidelity and denoising steps have parameters, u² u², which are functions of the iteration number.
We note that AMP and its variants have also been explored in the context of algorithm unrolling.
In [12 and [3]. E and E" are reparameterized with tunable parameters E and E" C', where 3 and C' are trainable across unrollings via neural networks. This reparameterization influences the Onsager correction term and the denoising threshold, improving the robustness of E. Subsequent studies have explored training distinct, iee unshared in our previous terminology, proximal opera- tors 45 36) over iterations using neural networks, rather than reparameterizing the matrix and E" Other studies have also explored training both the system matrix and proximal operators using neural networks 70 36) across iterations.
2.4 Time-dmmedding in Neural Networks
Time-dependent processing plays crucial role in diffusion models as well [31 32]. In this context, information about the current diffusion step is encoded to guide the CNN model to capture sequential relationships effectively and reverse the noise process efficiently.
Feature- wise linear modulation (FiLM) 49 is widely utilized for transforming inputs with time- embedded features, illustrated in Fig. (a) and (b). The time information features are obtained through a sinusoidal encoder [60], followed by a learned function f-)). Subsequently, the functions gi and h, are adaptively learned generate of and B% respectively
a = (t((tt)))
B = hx()t)))
(12)
where af and (8; modulate the 1th features Fi of CNNs at the 4th iteration using FiLM, which applies scaling and shifting transformations using a; and 8 respectively. Moreover, [2 demonstrates that combining group normalization (61] with the FiL.M approach enhances the efficacy of time-embedded features, leading to improved model performance in diffusion model as follows:
H a @ GroupNorm| (F)) + 85
(13)
where Hi are features conditioned by time-embedded layers, C is feature-wise multiplication, @ is feature-wise addition. Each feature map in the network modulated independently by of and B For example, H3 is passed onto the next block as input Fl+1 and modulated by the next time-embedded scaling and shifting factors af af41 and 8%

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Time canee-eaeedeedddd B1 (FOO)
(c) Time- Tine-Enbreled Natworks
Deradky
UNet
Rasidual Block (RR,
Staff foctor D
(b) Time- Exnheddiing Module:
UlNel
ResNet
(RB.)
VN C.C.L.
of.0t
ef..0%
Reshtel
C.N.W
Canv Reaidual
ano--mbeedle dding module
Slip connection
Concetoration
R6.8
Pesture-stae Scating
Festurewitts Addition
Scatan-Fisture Muttptication
Figure 2: Illustrations of (a) Positional Encoder generating dimedeependet scaling (af) and shift (8) features, (b) Time-embedding module for ResNet and U-Net, and (c) Architectures of ResNet and U-Net showing how time-embedded features are applied.
3 Methodology
3.1 Proposed Time- Time-Embedding Strategies for Algorithm Unrolling
Building on Section 2.3 and Section 2.4 we propose a time-embedding framework for algorithm CNN
unrolling. Our time-dependent proximal operator is inspired by VAMP, and is implemented with the dime-embedding techniques from Section 2.4 This enables the proximal operator to exploit temporal dependencies across iterations, adapting its behavior dynamically, similar to the denoising in diffusion models. We note that, while the time step t explicitly models the noise level in the given stage in diffusion models, in operator's behavior across iterations by capturing the evolving distribution of intermediate features, similar the effect of the Onsager correction in VAMP.
with algorithm unrolling, it implicitly modulates the proximal
Time--mbedding in proximal operators We first consider the proximal operator step of VAMP in Eq. (TO) and its corresponding Onsager corrections in Eq. (TT). In the context of algorithm unrolling. these suggest that the learned proximal operator should be time- -ime-dependen, but with well-defined time schedule. Furthermore, the Onsager corrections in Eq. (TT) are also functions of time in the original VAMP setting, and r'+1 in Eq. (11) is effectively a function of u in Eq. (10) and t. To this end, an unrolled network setting, where intermediate parameters can also be learned, we propose to model this relationship directly with a time-embedded neural network. In other words, time-embedded neural network used to model steps (TO)-TTT). effectively capturing both the time-dependent denoising and the associated Onsager corrections implicitly to map r ++1 from u'. We represent this relationship as:
prox (u H+ 1-+1,a',,00 _a",3",) ((,))) l),
(14)
where at and 3r capture the time-embedding information described in Section 2.4
Time-embedding for data fidelity The data fidelity term in Eq. (8) is of the same form the data fidelity term in Eq. (3)
Algorithm 1 Time-embedded Unrolling Algorithms
Require: T. Eg, Yn 1: Initialize r and n1, 2 0 for f 1....
VSQP, with the notable distinction that the quadratic penalty µ² evolves in a time- dependent manner in the former. Thus, we implement μı as time-dependent learn- able parameter. Furthermore, the Onsager correction in (9) can be written as:
.....,T T do
x'+1 =(E En + p - + v'r²), u+t1 x
+ p (x++1 r'),
r+11 prox (utt1 tl a Bt.t)
6: end for
return (T)
u' === = + p'(' -r²).
(15)
5

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Table Shared R(-) weights, Unshared R(-) weights. Quantitative results are reported using limited data on the coronal PD, coronal PD-FS, and axial T2 datasets, with equispaced undersampling patterns at acceleration rates are highlighted.
4, 6, and The best and second-best results for each architecture
U-Net
ResNet
Ours
Ours
Ours
Ours
R
VSQP (6) VSQP ( ADDMM
ADMM (+
unrolls) unrolis) VSQP
YSOP
ADMM (
ADMM (
15 u1rolls) unrolls)
PSNR1
40.50
40.31
40.76
40.51
40.94
40.99 0.964
41.11
40.99 0.963
41.27 0.964
41.11 0.964
41.41
41.43
Coronal PD
x44
SSIM
0.962
0.960
0.964
0.963
0.964
0.965
0.966
0.965
PSNR1
38.12
38.02
38.85
38.52
39.08
38.93
39.54
39.18
39.61
39.60
39.65
39.66
x.6
SSIMP
0.945
0.942
0.950
0.949
0.952
0.950
0.954
0.950
0.953
0.953
0.954
0.954
PSNRT
35.98
35.61
36.31
35.71
36.45
36.34
36,46
36.04
36.72
36,41
36.76
36.87
SSIM?
0.920
0.914
0.924
0.917
0.925
0.923
0.924
0.919
0.926
0.921
0.925
0.929
Coronal PD-ES
PSNR
35.09
35,10
35,31
35,23
35,23
35.38
35,31
35.23
35.37
35.23
35,42
35,54
SSIM
0.849
0.847
0.851
0.848
0.847
0.851
0.851
0.847
0.848
0.849
0.847
0.849
PSNKT
34.17
34.05
34,26
34.27
34,29
34.44
34,48
34,25
34.53
34.33
34.54
34.59
SSIMT
0.821
0.817
0.821
0.824
0.822
0.825
0.823
0.820
0.822
0.823
0.822
0.822
33.11
32.86
33.21
33.06
33.27
33.36
33.09
32.71
33.35
33.09
33.48
33,50
88 SSIMI
0.794
0.791
0.795
0.796
0.797
0.797
0.796
0.785
0.796
0.789
0.794
0.794
PSNRT
36.37
36,42
36.60
36.54
36.59
36.60
36.63
36.53
36.81
36.75
36.77
36.81
0.927
0.926
0.928
0.928
0.925
0.928
0.926
0.923
0.925
0.926
0.926
0.927
Axial T2
PSNRT
34.53
34.69
35.05
34.91
35.03
35.09
35.07
34.94
35.35
35.10
35.37
35,44
SSIM
0.903
0.910
0.910
0.910
0.906
0.909
0.913
0.906
0.910
0.909
0.909
0.910
PSNR1
32.90
32,70
33.41
32.98
33.26
33.41
33.15
32.99
33.43
33.14
33.67
33.56
SSIM
0.889
0.889
0.893
0.890
0.890
0.892
0.894
0.885
0.890
0.889
0.891
0.891
where p² 1/0,-15 Thus, in the algorithm unrolling framework, the scalars in Eq. can be replaced with a time-dependent learnable parameter p² for a learned Onsager correction term. Thus, the full time-embedded unrolled network is summarized in Alg. [
3.2 Neural Network Architectures for Time-Embedded Proximal Operators
The U-Net architecture
53 has been widely used, especially in the context of diffusion models, as
time-embedded network by integrating time-embedding features into different layers [3]] 21 32] as shown in Fig. 2 (c). We follow the time-embedded U-Net design based ADM from 21
which employs group normalization and the FiLM method, as formulated in Eq. [2) and Eq. (3)
with modifications in the number of channels and up/down sampling. We also note that U-Net has
connections with message passing through belief propagation 44
We additionally propose novel time-embeddiee module for ResNet [30] which is commonly used
as a proximal operator in unrolling algorithms for MR reconstruction 64 63), as shown in Fig. (b)
and (c). The time-embedding module in ResNet is designed as:
H F
(af @ GroupNom(FF)
(16)
where of and 39 are as in Eq. 12). and is scaling factor. Instead of directly applying the trans- formed features from af and p. this module utilizes T as a scaling factor to indirectly influence the features. This approach ensures stable integration of time information into the ResNet architecture.
Experiments and Results
4.1 Experimental Setup
We carried out an in-depth assessment of our approach, analyzing its effectiveness quantitatively and visually through multiple acceleration rates and datasets. The data included fully sampled coronal proton density (PD) and PD with fat-suppression (PD-FS) knee MRI scans, as well as axial T2-weighted brain MRI scans. These scans were obtained from the New York University (NYU) fastMRI database
33 68), and were acquired with appropriate institutional review board approvals.

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UNet
Reference
VSQP (+)
Unshared-VSQP (4)
ADMM (+)
Unsharod-ADMM (4)
Ours
Coronal PD-P8
Knos MRI (R==)
Knoe MRI (RI(F=)) (R=6)
Coronal PD
Asial T2 Brain MRI (R--))
Figure 3: Qualitative comparisons between the standard shared and unshared R(-)) opti- mization methods (VS, ADMM) and the proposed embeddedd unrolled algorithms with T= 10 unrolls in U-Net. Top: Results for R = 4 using PD-FS data. Middle: Results for R = 6 using PD data. Bottom: Results for R using Axial T2-W data. The proposed methods reduce artifacts (yellow arrow) that the shared and unshared methods fail to eliminate (pink arrow).
All datasets were retrospectively undersampled using uniform/equiddsttat undersampling at accel- eration factors of R 4, 6, and 8 with 24 central kspace lines kept. For this study, we focused on uniform/equidistant undersampling patterns, as they are more commonly used in clinical practice and produce coherent artifacts that are more challenging to remove [29]. We additionally evaluate the generalization performance of our method on random undcrsampling patterns in Section For knee datasets, model training carried out on 380 slices from a separate set of 10 subjects [28]. For the brain dataset, training and testing were performed using 300 slices each.
conducted using 300 slices from 10 subjects, while testing was
4.2 Implementation Details
We compared our method with conventional algorithm unrolling based on VSQP and ADMM, which were implemented with both shared 3 [ and unshared 288
47 weights across iterations. We
note that multiple variants of ADMM and VSQP unrolling 664 66 67 24 have been proposed with different names, primarily differing in their choice of network architectures for the proximal step and their training strategies. In this work, our focus is not comparing these variations, but rather on analyzing the effect of the outer algorithm unrolling itself with matching proximal operator network structures and training processes. For the least squares problem in the data fidelity of these approaches, conjugate gradient (CG) with 15 iterations was utilized [3]. For the proximal operators, we chose two distinct network architectures: 1) a ResNet model with 15 residual blocks, where each block consists of 3 x 3 convolutional layers with 64 channels [64], and 2) U-Net model, adapted from the ADM diffusion model (2I) with slight modifications to number of channels and up/down sampling layers. Details about model architectures and hyper- parameters are provided Appendix The
Table 2: The number of parameters for the shared ), unshared ($), and our proposed methods using different networks with T -= 10 unrolls.
comparisons were first divided by the prox- imal network architecture, i.e. ResNet vs U- Net based. For each of these two proximal network architectures, we trained five un- rolled networks from scratch: the proposed
Networks VSQP
YSQP ADMM (4) ADMM (+) Ours
U-Net
1,724,035
17,240,341
1,724,036
17,240,342 | 1,993,,479
ResNet
592,129
5,921,281
592,130
5.921.282
866,581
7

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ResNet Reference
VSaP (+)
Unshared-VSOP (+)
ADMM (+)
Unshared-ADMM (+)
Ours
Coronal
Kowe
Krese (R--6)
Coronal
MRI (R=4)
Figure 4: Qualitative comparisons using ResNet (instead of U-Net in Fig. 3 The proposed methods reduce artifacts (yellow arrow) that the shared and unshared methods fail to eliminate (pink arrow).
time-embedded unrolling, VSQP with shared parameters and unshared parameters, ADMM with shared and unshared parameters. Note that incorporating time embedding into the shared proximal networks leads modest increase in parameters, shown in Tab. which is considerably smaller than the increase caused by using a distinct unshared regularizer at each unroll. This shows the efficacy of our proposed approach, which adapts the regularizer over time without significantly increasing network size.
4.3 Performance of Time-Embedded Unrolling versus Existing Methods
Quantitative Results Tab. D depicts the performance of different approaches on the Coronal PD, PD-FS, and Axial T2 datasets across different acceleration rates. The shared and unshared baselines trained with T = 10 unrolls, while our proposed method is trained with both T 5 and 10 unrolls. In almost all cases, the unshared baselines perform worse than their shared counterparts for both U-Net and ResNet in this limited data setting. Though unshared methods are known generalize well in large data regimes [47), in our limited data setting, they exhibit performance degradation due to the high number of trainable parameters, further detailed in Appendix B with experiments fine- tuning from pretrained shared baselines. Our proposed method with T unrolls outperforms both shared and unshared methods, achieving the best second-best performance across all acceleration rates and datasets. The only exceptions are SSIM on Coronal PD at R = 8 and Axial T2 at = 6 when using the U-Net proximal operator, and SSIM Coronal PD-FS R 6 and R when using the ResNet proximal operator. These results demonstrate that our proposed method performs best in the limited training data regime for fixed number of unrolls.
Remarkably, even with 5 unrolls, our proposed method achieves performance comparable to the shared baselines with T'= 10 unrolls in most cases. The only notable exception is the Axial T2 dataset, where a small performance gap remains compared to the top-performing methods. This shows that method can deliver strong performance while halving the number of network computations and reducing inference time by ~50%, offering a substantial advantage in clinical applications.
Overall, Restebb unrolling networks demonstrate stronger quantitative performance than U- Net-based ones. Additionally, there are a few cases where our performance does not fall within the second-best range. Nevertheless, note that PSNR and SSIM do not necessarily capture finer

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Therefore, in the next section, we provide qualitative results to further demonstrate the effectiveness of our approach.
details, as noted in earlier studies [1
47
38 15
9
Qualitative Results Fig- S and Fig. depict representative reconstructions from different unrolling approaches with T = 10 unrolls using U-Net and ResNet based proximal operators, respectively. The shared VSQP and ADMM exhibit artifacts for both proximal operators in various cases. The unshared versions of these methods, which consist of 10 independent regularizers, cannot properly mitigate these artifacts (arrows). In contrast, our proposed method with the time-embedded proximal operators effectively addresses these artifacts across all acceleration rates and datasets. In addition to artifact reduction, our proposed method also enhances image sharpness, most clearly visible in the Axial T2 example in Fig. 3 More visual examples are provided in Appendix G
To further investigate the subtle and diagnostically important improvements afforded by our method, representative reconstructions were reviewed by an expert musculoskeletal radiologist, who was blinded to the reconstruction method. The reviewed images included the cases shown in Fig. and Fig. 4 as well as sample annotated pathology cases from the fastMRI+ [7]] dataset. The radiologist noted that our method was able to remove subtle artifacts that were observed with the other methods in all the reviewed cases. A detailed description of the readings and the artifacts, well as the pathological region assessments on fastMRI+ are provided in Appendix G These evaluations further confirm the strong performance of our approach beyond standard quantitative metrics.
4.4 Extension Other Unrolling Algorithms
Steps and 5 of Alg. [ are analogous to the VSQP updates in Eq. 33 4p except with time- dependent quadratic penalty parameter in the former and a time-dependent proximal operator in the latter. Thus, we set out to explore the effect of the Onsager correction in Step 4 of Alg. [1, specifically to empirically characterize whether the correction to the data fidelity out- put is minimal, i.e,, x x uz+ Indeed, we observed that the network made only minor dif- ferences between x and u ++1 over iterations.
Table 3: Comparison of shared baseline methods
and their time-embedded versions (baseline- TE>
using the U-Net proximal operator on coronal PD.
R
VSQP
VSQP-TE
ADMM
ADMM-TE
PSNR1 SSIM
40.50
40.92
40.76
40.87
X4
0.962
0.964
0.964
0.964
PSNRT
38.12
38.50
38.85
38.87
X6
SSIM
0.945
0.946
0.950
0.951
PSNR1
35.98
36.15
36.31
36.48
X8
SSIM
0.920
0.921
0.924
0.925
Details are provided in Appendix C Omitting the Onsager correction in Step 4 turns Alg. 1 to lime-embedded version of VSQP. Similarly. we can also unroll other algorithms, such as ADMM in the proposed time-embedded manner. As shown in Tab. time-embedded proximal units and data fidelity parameters improve the performance of VSQP and ADMM across all acceleration rates. Like our method in Section 4.3 other emmeddddddeed unrolled networks also exhibit superior performance in effectively reducing artifacts. Further details are presented in Appendix D
4.5 Ablation Studies
We conducted three ablation studies to evaluate the effects of varying hyperparameters of time- embedded unrolled networks.
Effect of Varying the Numbers of Unrolls We trained all the methods in Tab. I for T' E (5, 15). Our proposed method maintained stable performance regardless of the number unrolls, consis- tently reducing artifacts, while yielding sharp images, whereas other baselines exhibited varying performance depending on the number of iterations. Further details are given in Appendix F
Efficiency Analysis with Respect the Number of Parameters We investigated whether in- creasing the number of trainable parameters improves performance, in the shared and unshared baselines. Increasing parameters do not improve results in ResNet, whereas a slight quantitative gain is observed for U-Ner, though visual residual artifacts persist. These larger models also incur higher computational costs, whereas our method achieves better performance with only a marginal increase in parameters. Detailed experimental results are provided in Appendix Tab.

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Emneddinn Module with Different Hyperparameters We conducted experiments with time-embedded U-Net to evaluate key hyperparameters of the time-embedding module, including the sinusoidal encoding frequency, embedding dimension, and the number of hidden channels in the MLP layers. Performance was influenced by these hyperparameters, with optimal results achieved using a period of 10,000, an embedding dimension of 32, and 128 hidden channels. These settings were applied consistently across all experiments. Additional information is in Appendix E Tab.
4.6 Extended Experiments
Artifact Evolution Across Unrolls While the time step t in diffusion models explicitly determines the noise level each stage of the forward process, the time step in our setting plays different role. t implicitly governs the evolution of the proximal operator across iterations by accounting for the changing distribution of intermediate features, analogous to the Onsager correction term in VAMP, which stabilizes updates by compensating for iterative correlations. This enables time-embedded proximal operators with temporal information to adaptively apply varying levels of denoising at different stages, which is further illustrated in Appendix D Fig. 6
Validation Non-Uniform Sampling Masks To evaluate the effectiveness of our proposed method under non non-unifonn uniform (random) sampling patterns, we conducted experiments at various accel- eration rates, using both baseline methods and our approach with a U-Net architecture and 10 unrolls. These results confirm that the effectiveness of our method extends to non-uniform undersampling patterns. Results provided in Appendix F
Comparison with Diffusion-Based Models Since certusion-based reconstruction provides a promising approach for solving MR inverse problems [3
16 10 27 we compared our results with Decomposed Diffusion Sampling (DDS) [T6]. Our method outperforms DDS terms of both PSNR and SSIM. Details of the implementation and the corresponding results provided in Appendix
5 Limitations and Discussion
Limitations. As discussed in Section Section our experiments were conducted in limited data regime, using 300 slices per dataset. This setting is particularly relevant for translational applications, where new imaging sequences are being developed or when higher resolutions are targeted. While our method demonstrated strong performance and generalization in this regime, the performance gap between our approach and unshared baselines may narrow when training with more data samples, as the risk of overfitting will be lower for the latter.
Discussion. Through empirical evaluation, we examined whether second-moment matching holds in our time-embedded unrolling algorithms inspired by the Vector AMP framework, as well as the Lipschitz constants and stability of the time-embedded FiLM layers. Detailed discussions provided in Appendix
Conclusion
In this study, we introduced time-embedded algorithm unrolling framework inspired by AMP theory and time-embedding in diffusion models. Our unrolled networks used time-embedding in proximal operators, which performed both denoising and Onsager correction, well in data fidelity weights We extended these ideas to VSQP and ADMM- based unrolling, demonstrating the framework's s versatility. Our method outperformed both shared and unshared unrolling approaches under matched settings, producing sharper images with fewer artifacts, especially in limited data regime. Unlike unshared models, which showed signs of overfitting, method generalized better and remained robust across different unroll depths.
Acknowledgements
The authors gratefully acknowledge Dr. Jutta Ellermann for providing expert radiologist assessments during the rebuttal stage on short notice. This work was partially supported by NIH R0THL.15314 46, NIH ROI ROIEB032830, NIH P4 P41EB027061.
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Appendix
A Model Architectures and Relevant Hyperparameters
Neural Network Architectures. A ResNet and a U-Net architecture was used for proximal operators in the unrolled networks, as illustrated in Fig. (e). The ResNet used for the proximal operator [59] consists of 15 residual blocks, each containing x 3 convolutional layers with ReLUU activation and a scaling term. The scaling term is set to 1 x Ix10- [64] The U-Net used for the proximal operator, which is designed based on ADM from [2]] with slight modifications, has 2 downsampling layers. 2 upsampling layers, and a bottleneck. It uses residual blocks with 3 convolutional layers, normalization, and SiLU activation. The initial channel size is 32, which doubles during downsampling and recovered during upsampling. For time-embedded architectures, eeneeembddedd features are injected and modulated through group normalization and the FiLM method in each residual block, as shown in Fig. (b) and (c). All training processes are conducted using one NVIDIA A100-SXM4-400 SXM4-40GB GPU.
Shared/Unshared Baseline. For the data fidelity term, we use a shared ju initialized to 5x 10-2 in VSQP and 1.5 x 10-2 in ADMM. In ADMM, we also initialized the dual update parameter À to 1 x 10-1 For the shared baseline networks, both a ResNet and U-Net proximal operator as described above were used, without tine-embedding features. In the unshared case, there is a separate proximal operator for each unroll with no weight sharing or time-embedding. We train ResNet for 100 epochs and U-Net for 50 epochs, using a learning rate of 5 10 for coronal PD/PD-FS knee data and 2x 10 "0"for Axial T2 brain data with the Adam optimizer.
Time Embedded Unrolled Networks. For our proposed time-embedded unrolled networks, as described in Section and Section 3.2 we utilized time- dependent data fidelity. Onsager correction parameters. and time-embedded proximal operators. In particular, the data fidelity scalars oe were initialized to L.5x 10-2, and the dime-dependent Onsager correction parameters p² to x 10 For time-embedded neural networks, we apply the same optimization strategies as in the baseline, except for ResNet the coronal PD-FS case, where we use lcarning rate of 1.8x 10- 2. The scaling factor 7 of FiL.M in ResNet is set to 0.1. To extend our approach to other unrolling algorithms deseribed in Section 4.4 we replace the shared data fidelity term with the unshared data fidelity term p¹ and utilize the same proximal operators in our proposed methods.
Generalization to Diverse Datasets Our method incorporates a time-embedding module, which introduces additional hyperparameters. These include the frequency of the sinusoidal encoding, the embedding dimension, and the number hidden channels the MLP layers that process the time embeddings. We use the same contiguration across different datasets (Coronal PD. Coronal PD-FS, and Axial T2 brain), demonstrating the
Table 4: Comparison of the results from the fine-tuned Unshared (ADMM) method with those of shared and unshared baselines trained from scratch in limited data settings. FS: From Scratch; FT: Fine-Tuning. Quantitative results are reported across three datasets with varying undersampling patterns. The best values are highlighted in bold.
U-Net
ResNet
(Shared)
(Unshared)
(Unshared)
(Shured) FS (Undhared)
(Unsbared)
Epoch
100
100
10
20
30
40
S0
100
100
10
20
30
40
50
Coronal PD
X4
PSNRT
SSIM
40.76 0.964
40,51
0.963
40.96
0.964
0.964
40.89
0.964
40,82
0.963
40.78
0.963
40.69
0.965
41.27
0.964
41.11
0.965
41.45
41.37
0.965
0.965
41,28
0.964
41.25
0.964
41.18
X6 SSIM
PSNR
38.85
0.950
0.949
38.52
0.952
39.13
0.952
33
39.09
0.952
0.951
38.99
0.951
38.94
39.61
0.953
39.60
0.953
39.88
0.955
0.955
39.78
0.955
39.72
39.72
0.955
0.953
39.64
x8
PSNR
SSIM
0.924
36.31
0.917
35.71
0.925
36.51 36.29
0.923
0.923
36.29
0.922
36.25
0.920
36.11
0.926
36.72
0.921
36.41
36.97
0.927
0.927
36.93
0.925
36.76
0.925
39.74 36.63
0.925
Cornal PD-PS
x44
PSNRT
SSIM
0.851
35.31
0.848
35,23
0.850
35.44
35.32
0.849
35.22
0.847
35.12
0.844
35.00
0.841
35,37
0.848
0.849
35.23
38,59
0.850
0.851
35.57
0.852
35.54
38.51
0.852
0.852
35.50
XG
PSNR
SSIM?
0.821
34.26
0.824
34.45
0.822
34.33
0.822
0.820
34.26
34.17
0.819
34.11
0.819
0.822
34.53
34.33
0.823
34,69
0.823
0.823
34.64
0.825
34.61
0.825
0.825
34.58
PSNR
33.21
33.06
33.34
33,
33.05
32.96
32.92
33,35
33.09
33.61
33.57
33.53
33.44
33.38
SSIM
0.795
0.795
0.792
0.788
0.786
0.796
0.789
0.797
0.767
0.797
0.795
0.795
X4
PSNRT
SSIM
0.928
36.60
0.928
36.54
0.917
36.67
36.67
0.927
0.927
36.60
0.927
36.67
0.927
36.62
0.925
36.81
0.926
36.75
0.924
36.87
0.925
36.83
0.925
36.86
0.925
36.83
0.925
36.85
12
X6
PSNR
S
0.910
35.05
0.910
34,91
0.910
35.16
0.911
36.16
0.910
35.15
0.911
35.06
0.910
35.10
0.910
35.35
0.909
35.
0.907
35.41
0,908
35.41
0.909
35.47
0.908
35.44
0.907
35.33
PSNRT
33.41
32.98
33.52
33.40
33.36
33.39
33.42
33.43
33.14
33.91
33,93
33.90
33,83
33.78
X8
SSIM
0,893
0.890
0.893
0.892
0.892
0.892
0.892
0.890
0,889
0.890
0,890
0.890
0,890
0.890
21

---

robustness of our approach to diverse data. Similarly, apply identical hyperparameters to the neural networks used proximal operators across all datasets, on which the networks consistently perform well.
B Fine-tuned Unshared Methods in Limited Data Settings
To mitigate the overfitting tendency of unshared networks in limited data settings, we explore fine-tuning instead of training from scratch. Specifically, We initialize the unshared baseline unrolled network from the pre-trained shared baseline unrolled network. This unshared unrolled network is then fine-tuned for several epochs with learning rate of 10 for both ResNet and U-Net, and for knee and brain data comprising 300 slices each. As shown Tab. 4 the fine-tuned unshared methods generally oulperform both shared unshared methods trained from scratch. However, overfitting remains evident in the fine-tuned models as the number of training epochs increases under limited data conditions. Furthermore, although the fine-tuned unshared methods improve PSNR and SSIM scores, they still struggle suppress artifacts over iterations (see Fig. g for visual examples).
C Analysis of the Onsager Correction Term in Algorithm 1
unroll
2dd unroll
3rd unroll
unroll
unroll
unroll
7th unroll
unroll
9th unroli
10 unroll
uf
x
Figure 5: The intermediate visual results of the proposed method with ResNet- TE proximal operator each iteration in a coronal PD slice.
Table 5: Average normalized mean squared error between x' and u³ at each iteration of the unrolled network with ResNet-TE on the coronal PD test set (R 4).
Heration
Nemalllad MSE 1:01 5.48 9.02x10 a.11x10 6.75x10 1.20x10-
3
explore the effect of Onsager correction in Step 4 of Algorithm we evaluated whether the intermediate updates in the network satisfy x' z u'. We compared the outputs of the data fidelity, (x') and its Onsager correction term output, (u') across unrolls, which is shown in Fig. 5 The bottom row shows the scaled (x5) difference between them, which is minimal upon visual inspection. We further quantified this difference by calculating the normalized mean squared error between x' and u' each iteration. Tab. shows that the difference ranges from 1.01 x 10 3.35 x indicating no substantial variation.
D Qualitative Comparison of the Baseline and Time-Embedded Algorithm Unrolling
As discussed Section and view of Section C our sime-embedding approach can be extended other unrolled algorithms (VSQP and ADMM) by incorporating a time-embedding module into the proximal operators of these unrolled networks. The time-embedded versions of the unrolled algorithms are given below. For time-embedded VSQP:
x' E En + 1²t) (Enya + n'x').
(17)
zz prox (x', " "',,,,)),,))
(18)
time-embedded ADMM:
x+1 E En + n'1)
Enya + H' (d'-u')
(19)
[+] prox (x" (+| + u u',a",
(20)
(+1 u' + 1(x"+12 1-11-"" -1x 22111
(21)
22

---

VSQP
Zero-fled Drpa0
x²
X²
x
X
x²
x
Magystede Defeaso--E Maystid Difeeesee 1S Nagpital Diferesees Wagibaa 0
Astal MRI (R=44)
Rashet
RaeNNet-E
Garonal PD-FS, Knee MRI (R-45)
UNet
UNee-TEE
ADMM
Mayotae
Coronal Knee MRI (R=S)
ReeNet
Offerree SS Nayrhake D'fecese 15 Mayttads DevIRee Nightsk DrNATeNs DE
ResNee-TE
Coronal PD-FS, Knoee MRI (R-B) UNet-TE
Ulte
Figure 6: The estimates through different stages of the unrolled network with shared baseline VSQP, ADMM, and time-embedded VSQP, ADMM for different datasets with varying acceleration rates. The error maps illustrate the differences between the intermediate estimates, {x')[ and the reference. The shared basclines, utilizing both U-Net and ResNet proximal operators, exhibit persistent eπтоrs across unrolls, which are highlighted with pink arrows in the last unroll, and can be visualized at the same location in prior unrolls. contrast, the time-embedded networks, with proximal operators U-Net-TE and ResNet-TE, effectively reduce noise (yellow arrows).
where the data fidelity parameter y and the proximity operator prox (.) replaced with time-dependent parameters y² networks (", a","", 3., t).
respectively.
Fig. c presents examples how the reconstruction evolves when comparing shared VSQP and ADMM with its tineemmbedded counterparts. The shared methods exhibit persistent errors over iterations that the proximal operator fails to eliminate. However, integrating our proposed time- time-embedding methods into the baseline algorithms effectively addresses these issues, demonstrating gradual denoising, intended.
23

---

Table 6: Shared R(.)) weights, Unshared R(.) weights. Quantitative results on the coronal PD datasets using equispaced undersampling patterns at R = 4, 6, and 8 with and 15 unrolls. The best and second-best values for each architecture are highlighted along with their relative difference to 10 unrolls.
U-Net
ResNet
(b)
ADMM (4)
VS (4)
ADMM (4)
R
VS (a)
(Pine tuned)
ADMM (4)
Ours
VS (b))
(Fine tuned)
ADMM
(Fine cuned)
Ours
X4
PSNRT
40.58
40.71
40.86
40.93
40.94
40.66
41.19
41,16
41.38
41.41
5 Smrolli
SSIMF
0.963
0.963
0.964
0.964
0.964
0.963
0.963
0.964
0.965
0.966
PSNRT
38.01
38.57
38.49
38.92
39.08
39.25
39.57
39.44
39.76
39.65
X6
SSIM
0.943
0.948
0.947
0.95
0.952
0.952
0.952
0.954
0.955
0.954
X8
PSNRT
35.63
35.81
35.79
35,99
36.45
35.94
36.44
36.40
36.65
36.76 76
SSIM
0.916
0.918
0.917
0.920
0.925
0,919
0.922
0.924
0.925
0.925
PSNRT
40.36
40.84
40.59
41.01
40.88
41.16
41.36
41.44
41.48
41.52
15 uurrols
X4
SSIMF
0.962
0.964
0.963
0.965
0.964
0.964
0.964
0.966
0.965
0.966
PSNRT
38.77
39.01
38.94 14
39.61
39.78
39.80
39.83
SSIMT
0.944
0.950
0.948
0.952
0.951
0.955
0.953
0.952
0.954
0.954
X8
PSNRT
35.44
35.97
36,25
36.50
36,51
36.41
36.87
36,95
37.09)
37,09
SSIM
0.908
0.918
0.923
0.926
0.926
0,923
0.926
0.928
0.927
0.929
Table 7: The comparison results for different model sizes (T 10, R 4, and Coronal PD).
Table 8: The comparison results for time-
embedded unrolling networks with different
time-embedding hyperparameters (T = 10,
Method
Channel
Param.
PSNR1
SSIM1
R 6, and Coronal PDFS).
UNet-VSQQ ResNet- VSQP
Shared R
64
592,129
41.11
0.965
Unshared R
64
5.921.281
40.99
0.963
Freq-
Emb. dim.
Hidden layer dim.
PSNR
SSIM
Unshared R
Shared R
96
96
13,305,601
1,330,561
41.00
41.09
0.963
0.965
1,000
34,34
0.824
5,000
128
34.35
0.822
Ours Ours (7"
64
866,571
41.41
0,966
10,000
34.44
0.825
10)
64
866,581
41.43
0.965
Method
Channel
# Param,
PSNR1
SSIM1
32
34.44
0,825
10,000
64
128
34.32
0.824
Shared R
[32, 64, 128) [32. 64, 128
1,724.035
40.50
0.962
34.34
0.824
Unshared R
17,240,341
40.31
0.960
Shared
164, 128, 256]
6.878.467
40.77
0.964
34.38
0.824
Unshared R
[64, 128, 256]
68,784.661
40.55
0.962
10,000
32
128
34.44
0.825
196
34.37
0.824
Ours
64, 128)
1.963.469
40.94
0.964
Ours
10) [32, 64, 128]
1,963,479
0.964
E
Additional Details on the Ablation Studies
This section presents further implementation details and results for the experiments described Section 4.5
Time-embedding denoisers can recognize temporal sequence information, allowing them to adaptively apply varying degrees of denoising at different stages, as shown in Fig. 6 Based on these observations, we hypothesized that our proposed method can achieve stable performance even with reduced or increased number of unrolling iterations. We compared our approach with both shared and unshared unrolling methods, where each was trained with T 5 and 15 unrolls. In this experiment, the unshared networks were fine-tuned to improve performance; for a detailed rationale, please refer to Appendix [ Tab. Tab./o]reeeents quantitative reconstruction results for T 5 and 15 unrolls. With fewer iterations (T 5 5 unrolls), our approach exhibits greater flexibility and robustness compared to the shared baseline algorithms,, which experience performance degradation as the number of unrolls decreases. Notably, for R 8, both shared and unshared baselines for each architccture show sigmificant PSNR degradation when using T= unrolls. In contrast, our proposed method maintains performance even with fewer iterations, showing either slight improvement or only minimal degradation compared to T -= 10 unrolls, depending on the choice of the proximal operator architecture. Moreover, as the number of iterations are increased (T = 15 unrolls), our proposed method maintains its robustness and consistently improves performance against the baseline models with shared R(-), while introducing only minimal increase computational complexity.
Robust Eimeeed Unrolling with Different Numbers of Unrolls
The qualitative results in Fig. and Fig. 10 for T= 5 and unroll iterations, respectively, support these quantitative observations. Our proposed method effectively reduces artifacts and enhances image sharpness, while the shared and unshared baseline models struggle to achieve similar improvements, both with fewer and increased iterations.
24

---

Table Shared R(-) weights, d Unshared R(-) weights. Quantitative results are reported on the Coronal PD, Coronal PD-FS, and axial T2 datasets, with non-uniform undersampling patterns at acceleration rates R 4, 6, and The best result for each architecture are highlighted.
R
VSQP
VSQP (4)
ADMM
ADMM
Ours
HD
x4
PSNRT
SSIMT
40.10
0
40.26
40.20
40.13
0.962
40.43
PSNRT
38.41
38.59
38.64
38.72
38.73
SSIMT
0.947
x8
PSNRT
SSIMT
37.30
37.39
37.64
37.52
0.938
37.76
R
VSQP (0)
VSQP (4)
ADMM (0)
ADMM (4)
Ours
PSNRT
35.61
35.60
35.66
35.68
Cor. PDFS
X4
SSIMT
0.855
0.853
0.853
0.856
0. 0.854
PSNRT
34.59
34.55
34,70
34.61
34,74
X6
SSIMT
0.828
0.824
0.827
0.827
0.830
PSNRT
33.76
34,00 06
X8
SSIMT
0.809
0,808
0.810
0,810
0.812
R
VSQP (
VSQP (4)
ADMM (
ADMM (8)
Ours
PSNR
35.90
36.36
36.50
36.42
36.52
Axial
X4
SSIM
0.932
0,931
0.930
0
0.930
PSNRT
35.03
35.11
35.26
35.25
35.
X6
SSIM
0.917
0.916
0.915
0.905
0.914
PSNR1
34,09
34.31
34.37
34.39
34,51
X8
SSIM
0.908
0,908
0.903
0.906
0.904
Efficiency Relative to the Number of Parameters and l'mee-Emeedding Hyperparameters To assess efficiency with respect the number of parameters, explored the effect of increasing the number of parameters on performance, which resulted higher total parameter counts in both the shared and unshared baselines, As shown in Tab. 7 we use the following setups: (1) increasing the channels ResNet residual blocks from 64 to 96, (2) increasing the channels in U-Net up'downsampling blocks from [32, 64, 128] to [64, 128, 256), and (3) using T = 10, R 4, with the Coronal PD dataset.
For efficiency relative the sime-embedding hyperparameters, we evaluated (1) the frequency of the sinusoidal encoding, (2) the embedding dimension, and (3) the number of hidden channels in the MLP layers. The experiments were conducted using a U-Net architecture with 10 and on the Coronal PDFS dataset. Implementation details and results are provided in Tab,
F
Details on the Extended Experiments
This section provides additional implementation details and results for the experiments described in Section 4.6
Experiments Non-Uniform Undersampling Masks As shown in Tab. method consistently outperforms the baselines in terms PSNR all cases except the Axial T2 dataset at R=6. For SSIM, our method shows improvement in most cases for the PD and PD-FS datasets, although no improvement observed for the Axial T2 dataset.
Table 10: The comparison results with fiffusion-basee model (DDS).
Method
R
PSNRT
SSIM1
DDS (100)
PD
x4
37.41
0.920=000229
Ours (U-Net)
PD
x4
40.09:2.51
0.958
Comparison with Diffusion Model-Based Reconstruction Since DDS requires x20 320 inputs due its generative pre- pre-trained prior, we additionally evaluated PSNR and SSIM using the central x20 320 region. Note that this differs from the results reported in Tab. [ where evaluations were performed on images of size 320x368, 368, aligned with the original raw k-space data. We set the number of sampling steps to 100 for DDS. For method, we used T 10 unrolls. All experiments were conducted on the Coronal PD test dataset with an acceleration factor of R 4. As shown in Tab. TO method outperforms DDS in both PSNR and SSIM. Furthermore, diffusion based reconstruction requires tens hundreds of neural function evaluations (NFEs) during inference [16]. which remains far from the efficiency needed for large-scale or pall-imee applications. In contrast, our time-ent emmedddd unrolled networks achieve more promising results with substantially fewer NFEs (e.g.. 5-10), with smaller network architecture compared to diffusion based models.
25

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G Additional Qualitative Results
Pathological Region Inspection Using fastMRI+ We leveraged the annotations pathological regions provided by fastMRI+ ?1 to further validate the strengths of our method. As shown in Fig. 7 our approach produces clearer contrast in the pathological regions compared to other methods, which is further corroborated with radiologist assessments, detailed next.
Reference
Shared-VSQP
Shared-ADMI
Cartinge Partial Bittee Inadeffcc
Figure 7: Reconstruction results with annotated pathological regions.
Radiologist Readings For subset of all the data processed in this study, musculoskeletal radiologist with over 30 years of experience blindly reviewed the reconstructed images from the different methods. The radiologist's assessments highlight improvements achieved by our method that are critical diagnostic purposes. Details are provided below.
Fig. 3 (Middle) exhibits aliasing artifacts in the distal femoral metaphysis medially on PD PD-weighted images for VSQP. Unshared- ADMM, Unshared-ADMMM The artifacts effectively removed in the proposed method (ours).
Fig. (Bottom) shows blurring in the right occipital lobe in VSQP, Unshared- -SQ... ADMM, Unshared- ADMMM This is notably improved in the proposed method, where the gyri and sulci appear sharper.
Fig. 4 (Top) shows visible aliasing artifacts in the central aspect of the distal femoral condyle of the inset for VSQP, Unshared- ADMM, and Unshared- ADMM. This artifact is removed in the proposed method. The prominent penetrating intraosseous vessel at the lateral aspect of the proximal tibia (lower left the image) appears sharper in proposed methad, though overall image sharpness similar among methods.
Fig. 4 (Middle): There is an oblong, hypointense appearing aliasing artifact on the non-fat saturated PD- weighted images of the knee joint, just distal to the posteromedial femoral condyle, seen VSQP, Unshared- VSQP, ADMM, and Mushared-ADMM. when compared to the reference. This artifact is removed from the image the proposed method. Thus, only the proposed method accurately resembles the reference image.
Fig. (Bottom) reveals aliasing artifacts in VSQP, Unshared- VSQP, ADMM, Unshared-ADoe ADMM, depicted curvilincar, oblique hypointense signal in the occipital lobe. The artifact is nearly completely removed Unshared-ADMMN and only vaguely seen. The artifact completely removed ours, most accurately resembling reference image.
In Fig. 7. on the reference image there is focal area of T2-hyperintense signal, most consistent with partial/full thickness cartilage defect in the anteromedial trochlea. Images reconstructed by VSQP, Unshared- VSQP, ADMM, and Iishared-ADMMN the least amount of blurring compared to other methods, and shows the hyperintense region in the trochlear articular cartilage with the most fidelity compared to the reference data.
reveal significant blurring this area. Proposed method shows
Additional Qualitative Examples We provide additional representative reconstruction examples that demonstrate the visual superiority of our proposed method, since PSNR/SSIM do not necessarily align with perception, discussed in Section 4.3 Fig. Fig. T2] and Fig. 13 shows reconstruction results across all datasets and proximal operator architectures for 4. and 8. respectively, using the implementations described in the main text.
H Extended Quantitative Results with Standard Deviation
Tab. summarizes the standard deviation of PSNR and SSIM for the same settings as in Tab.
26

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Table 11:
Shared R(.) weights: + Unshared R(-) weights. Standard deviations of PSNR and
SSIM results for the same settings in Tab.
U-Net
ResNet
R
VSQI (4)
VSQP (4)
ADMM ())
ADMM (8)
Ours
VSQP (4)
VSQP (4)
ADMM (4)
ADMM (4) Ours
Coronal PD
X4
PSNR
SSIM
0.02
2.56
0.02
2.59
0.02
2.42
0.01
2.41
0.01
2,40
0.02
2.95
0.02
3,04
0.02
2.97
0.02
3,03
0.02
2.73
2.21
2.17
2.77
2,81
2.74
X-6 SSIM
0.02
0.03
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
8 SSIM
PSNR
0.03
0.04
2.22
0.04
0.03
2,00
0.03
0.04
2.60
0.04
2.70
0.04
2.66
2.67
0.04
0.03
Coronal PI-ES
x44 SSIM
0.10
0.10
2,80
0.10
0.10
0.09
2.76
0.10
2.78
0.10
2,80
0.10
0.10
2.79
2.88
PSNR
2.56
2.61
2.59
2.53
2.58
2.69
2.70
2.70
2.68
2.75
SSIM
PSNR
2.41
2.44
2.43
2.36
2.44
2.46
2,48
2.53
2.54
2.60
SSIM
0.11
0.12
0.12
Axtal 12-W
x4
PSNR
SSIM
3.01
2.88
0.05
2.99
2.91
2.92
0.06
0.06
3.18
0.06
3.17
3.29
0.06
0.06
3.12
3.19
PSNR
3.05
2.68
2,88
2.79
2.98
3.04
2.99
3.22
3.08
3,10
X6
SSIM
0.08
0.06
0.06
0.06
0.07
0.07
0.07
0.07
0.06
PSNR
2.40
2.35
2,55
2.57
2.61
2.60
2,53
2.77
2,55
2.82
SSIM
0.06
0.06
0.06
0.07
0.07
0.07
0.07
0.07
0.07
0.07
I
Discussions
Second-Moment Matching, Since our methods are inspired by the VAMP framework, an assessment of second moment matching for VAMP is desirable. Second-momee matching is typically evaluated by examining the estimated variances v and vi across iterations, as in 9 empirically observed in these estimates, this would not constitute formal proof. This further complicated in case, as hypothesize that time-embedded neural network models all update steps in [TO-(TT] and the learnable scalar parameter p' encapsulates the entire process described embedded within black-box neural modules and are not explicitly accessible.
and (T]. However, even if consistent trends are
D result, 2 and 2 are
Instead, to indirectly assess whether second-momen matching holds, we analyzed the relationships between intermediate estimates. Specifically, we examined the empirical differences between x' and u'. and between u' and r' in Alg. [ these pairs are intrinsically related to e and v respectively. If these differences remained consistently small across iterations, it provided evidence that underlying variance estimates are stable, The empirical difference between x" and the reconstructed uf was reported in Appendix C As shown in Tab, the difference (normalized MSE) ranges from 1.01 x 10 to 3.35 x 10 10-2, demonstrating stable behavior. These findings suggest that -oonntmmment matching is empirically preserved, despite the use of learned components.
L Lipschitz Constant or Gradient Explosion/Vannshing of Time-embedding (FiLM) Layers. Con- sider network where each time step E fI...., T). there K consecutive layers within the proximal operator networks composed of intermediate transformations followed by FiLM modulation. For each layer k: E K)....
x (c.k) (fe (t (.kk))))) )),
(22)
where f+)+)) denotes the intermediate layers (e.g.. convolution activation) preceding the FiLM block at layer k. Suppose each intermediate layer is Lipschitz continuous with constant Lt, and satisfies
[=][l & |x||=|| + 8c,
(23)
for some small & 2 0, allowing for nonzero bias or offset when fa((0) o O. Similarly, each FiLM block is Lipschitz continuous with constant Ir.,, satisfying
[FiLM( t)ll < Te.all=l| + B.
(24)
The composite function then satisfies:
[[Fi.M((((*) t)ll S Ta|p|| + (Te,6 + B.,))-
(25)
Thus, the overall Lipschitz constant of the composite layer at layer & and time is T,, T-g. As T grows, if each Tr. is strictly less than 1, their product decays exponentially, which may cause vanishing activations and hinder learning. If any 2 1, the product can grow exponentially, causing exploding activations and instability. Thus, controlling cumulative constant, I II 1 La is crucial for stable training.
While formal proof is not provided, as we do not analytically characterize the Lipschitz constants of individual components, empirical results support the stability of the proposed approach. In particular, we adopt the
27

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U-Net architecture and FiLM modules commonly used in diffusion denoising tasks. In the diffusson model literature, large number of diffusion steps (T' 2 0000 typically employed, which has been shown promote stable training. our unrolled networks, where a substantially smaller number steps is used (e.g- = 5-15), we observe that training remains stable, suggesting that the reduced does not compromisc empirical stability. These observations underscore the empirical nature of our Lipschitz constant bounds and their relevance to practical performance.
28

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Qualitative results of fine-tuned unshared networks with 10 unrolls
Reference
VSQP (+)
Unshared-VSQP (4)
(Fine-luned)
ADMM (+)
Unsharad-ADWM (4)
(Fine-tuned)
Ours
UNet
Magribute
Coronal PD-FS
MRI
Difoweee-00 25
Magrbaale
Coranal PD
Knes MRI
Difersse.o
Maystads
Azial 12 (Rts))
Ofterean
ResNet
Goronal PD-FS (R-4)-
Vagetado
Coronal PD
Vagasiao
Acial
Brain
CAFENEER AR
Figure 8: Shared R(-) weights, & Fine-tuned Unshared R(.) weights. Qualitative comparisons
of each unrolled network with T = 10 unrolls for U-Net and ResNet proximal operators. In each proximal operator, Top: Results for R 4 using PD data. Middle: Results for R = 6 using PD-FS data. Bottom:
Results for R using Axial T2-W data. The fine-tuned unshared networks still struggle to suppress artifacts over iterations, whereas the proposed methods perform well, effectively reducing artifacts.
29

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Qualitative results of each unrolled networks with 5 unrolls
Unshared-VSGP (4)
Unshared-DDMM (4)
Reforence
VSQP
(Fine-tuned)
ADMM (+)
(Fine-tuned)
Ours
UNet
Megasabo
Coronal PD-FS
Knoe MRI (R:4)
Megrlsd
Kino MRI (R:S)
Coronal PD
Axial T2 Brain MRI (R-b)
ResNet
Magebade
Coronal PD-FS Knee MRI (R=4)
Coronal PD Kise MRI (R=4)
Megriade
Adial T2 Brain MRI (R-88
Figure 9: Shared R(-) weights, Unshared R(-) weights. Qualitative comparisons of each unrolled network with =5 unrolls for U-Net and ResNet proximal operators. In cach proximal operator, Top: Results for R 4 using PD data. Middle: Results for R = 6 using PD-FS data. Bottom: Results for R = 8 using Axial T2-W data. The proposed methods still perform well with fewer iterations, effectively reducing artifacts.
30

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Qualitative results of each unrolled networks with 15 unrolls
Unshared-VS@P (4)
Unshared-DDM
Reference
VSaP
(Fine-tuned)
ADMM (+)
(Fins-luned)
Ours
UNet
Knnss MRI (R=4)
Coronal PD
Vageha
Coronal PD-FS Kice MRI (R=0) (R:6)
Axial T2 Braznn MRI (R-e)
ResNet
Coronal PD Knoe MRI (R=44
Coronal PD-FS Knee MRI (R=D)
Astal 72 Brain MRD (R=H))
Figure 10: Shared R(-) weights, Unshared R(.) weights. Qualitative comparisons of each
unrolled network with 7= 15 unrolls for U-Net and ResNet proximal operators. In each proximal operator, Top: Results for R 4 using PD data. Middle: Results for R 6 using PD-FS data. Bottom: Results for = 8 using Axial T2-W data. The proposed methods effectively reduce artifacts and sharpen images, whereas the baseline methods fail to achieve this, even with 15 unrolls.
31

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Additional qualitative results for R=4
Reference
VSQP (+)
Unshared-vSGP (+)
ADMM (+)
Unsharad-ADMM (+)
Oura
UNet
Magetiio
Coronal PD.F3
Kroee MRU
Sftucess
Magatads
Coronal PD Kinee MRI
Ofterres 15
Axial T2 Brain MRI
ResNet
Magyssal
Coronal PD-FS
Knee MR
Magrbakke
Coronal PD Knee MRI
Axial T2 Brain MRI
T
Figure 11: Shared R(-)) weights, Unshared R(-) weights. Qualitative comparisons for R 4
across datasets for each proximal operator (T 10 unrolls). Our proposed method consistently demonstrates superior performance by reducing artifacts.
32

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Additional qualitative results for R=6
Reference
VSQP (+)
Unshared-V3QP (+)
ADMM (+)
Unshared-ADMM (+)
Ours
UNet
Magriaut
Coronal PD-F3
Knee MN
Difoxeee.oo AS
Magriable
Coronal PD Kowee MRI
Difresse.c
Maystaas
Axial T2 Brain MRI
Offesean
ResNet
Caronal PD-FS
Knoe MRI
Magetado
Coronal PD Knoe MR)
Vagetii
Axcal T2 Brain MRI
Figure 12: Shared R(-) weights, Unshared R(-) weights. Qualitative comparisons for R across datasets for each proximal operator (T' = 10 unrolls). Our proposed method consistently demonstrates superior performance by reducing artifacts. Furthermore, it enhances image sharpness, as shown in the results for the axial T2 data with both U-Net and ResNet proximal operators.
33

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Additional qualitative results for R=8
Reference
VSQP (+)
Unshared-v3QP (+)
ADMM (e)
Unshared-ADMM (+)
Ours
UNet
Magrbaub
Coronal PD-F3
Knee MN
Difoneee.o AS
Magriabbe
Coronal PD Kowee MRI
Difresse. US
Magstaa
Axial T2 Brain MRI
ResNet
Vagebide
Caranal PD-FS
Knoee MRI
Magetada
Coronal PD Knoe MR]
Vagatii
Axcal T2 Brain MRI
Figure 13: Shared R(-) weights, Unshared R(-) weights. Qualitative comparisons for R
across datasets for each proximal operator (T' 10 unrolls). Similar to R=4 and R=6, R=8 also demonstrates artifact reduction and image sharpening. Through Fig. to Fig B our proposed method shows superior performance across all configurations.
34


Reviews:
[{"review_id": "5z5CWtrsVZ", "reviewer": "Reviewer_D5F5", "summary": "The authors propose Time-Embedded Algorithm Unrolling (TE-Unroll) for DL-based inverse problems in MRI.\n\nBuilding on the link between vector AMP and diffusion models, they: (i) make the proximal network explicitly time‐conditioned via FiLM-style embeddings; (ii) treat the data-fidelity weight and Onsager term as learnable, time-varying scalars; and (iii) show how the idea plugs into multiple unrolling schemes (VSQP, ADMM) with both ResNet and U-Net proximals. \n\nExperiments on the fastMRI knee (PD / PD-FS) and brain (T2) data at R = 4/6/8 undersampling demonstrate consistent PSNR/SSIM gains over shared and unshared baselines while adding <15 % extra parameters (e.g., 1.96 M vs 1.72 M for U-Net) . Extensive ablations (unroll depth, alternative optimisers) and qualitative slices support the claims. \n\nThe paper is very interesting, in terms of embedding the # of step into the unrolled network, I do have some concerns regarding the evaluations, comparisons. I will elaborate later.", "questions": "I have put my questions in the Strengths And Weaknesses section.", "limitations": "1. little performance improvements compared to other methods.\n2. not comparing to updated literatures.\n3. pathologies region reconstruction analysis.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "For this paper i would like to put strengths and weaknesses together as follows:\n\n1. This paper is very well-written, clear, one small thing is that the figures are not in PDF format, probably png, which makes it look blurry when i zoomed in to see the results, a small thing.\n\n2. I agree embedding time information into unrolled framework is important and could benefit the efficiency with non-shared regularizer. However, its still unclear to me, in which situation, shared regularizer performs worse compared to non-shared regularizer. From my experience, it's not always true, depends on the model size, dataset size and problems. So Would appreciate if you could elaborate more on this part.\n\nWeaknesses:\n1. To be honest, from the evaluations, i really didn't see much differences compared to some of the other methods, one way to make this convincing is through inspecting the pathology regions, for example: https://github.com/microsoft/fastmri-plus - you can find a few examples with pathologies and see how your method performs. \n\n2. I don't think the methods you compared are very updated, try to compare with some diffusion-based models, starting from this paper: Robust Compressed Sensing MRI with Deep Generative Priors https://arxiv.org/abs/2108.01368 there are a bunch of works. e.g., Adaptive diffusion priors for accelerated MRI reconstruction.\n\n3. One point you are selling is the efficiency, but i didn't see many efficiency analysis, ablations in the paper, and you really want control the # of parameters when doing the ablation studies.", "was_revised": true, "final_justification": "I increased my score, thanks for reporting the radiologist opinions.", "rebuttal": "We thank you for your thoughtful feedback. We are encouraged that you found our method interesting and novel, appreciated the clarity of our exposition and the thoroughness of our experiments, as well as our time-embedded unrolling networks approach and the resulting performance improvements.\n\\\n\\\n**(W1, L3, and L1) Radiologist reading, pathology inspection and degree of improvement achieved by our method.**\n\nThank you for this valuable suggestion. While there was not enough time for a full reader study during the rebuttal period, we did work with a musculoskeletal radiologist (30+ years of experience), who *blindly reviewed* the reconstructions included in our paper. The radiologist's comments are included below:\n\n- Figure 3 (Middle) exhibits aliasing artifacts in the distal femoral metaphysis medially on PD-weighted images for VSQP, Unshared-VSQP, ADMM, Unshared-ADMM. The artifacts are effectively removed in the proposed method (ours).\n- Figure 3 (Bottom) shows blurring in the right occipital lobe in VSQP, Unshared-VSQP, ADMM, Unshared-ADMM. This is notably improved in the proposed method, where the gyri and sulci appear sharper.\n- Figure 4 (Top) shows visible aliasing artifacts in the central aspect of the distal femoral condyle of the inset for VSQP, Unshared-VSQP, ADMM, and Unshared-ADMM. This artifact is removed in the proposed method. The prominent penetrating intraosseous vessel at the lateral aspect of the proximal tibia (lower left on the image) appears sharper in proposed method, though overall image sharpness is similar among methods.\n- Figure 4 (Middle): There is an oblong, hypointense appearing aliasing artifact on the non-fat saturated PD-weighted images of the knee joint, just distal to the posteromedial femoral condyle, seen on VSQP, Unshared-VSQP, ADMM, and Unshared-ADMM, when compared to the reference. This artifact is removed from the image for the proposed method. Thus, only the proposed method accurately resembles the reference image.\n- Figure 4 (Bottom) reveals aliasing artifacts in VSQP, Unshared-VSQP, ADMM, Unshared-ADMM, depicted as curvilinear, oblique hypointense signal in the occipital lobe. The artifact is nearly completely removed in Unshared-ADMM, and only vaguely seen. The artifact is completely removed in ours, most accurately resembling the reference image.\n\nAdditionally, following your suggestion, we also reconstructed several **annotated pathology cases from the FastMRI+ dataset** (DOI:10.1038/s41597-022-01255-z), and the radiologist also reviewed these. Note that these were not included in the original submission. The radiologist noted the following:\n\n- On the reference image there is a focal area of T2-hyperintense signal, most consistent with a partial/full thickness cartilage defect in the anteromedial trochlea. Images reconstructed by VSQP, Unshared-VSQP, ADMM, and Unshared-ADMM reveal significant blurring in this area. Proposed method shows the least amount of blurring compared to other methods, and shows the hyperintense region in the trochlear articular cartilage with the most fidelity compared to the reference data.\n\nUnfortunately, we cannot include new figures here, due to NeurIPS's rebuttal policy, but representative examples and quotes will be included in our final manuscript. We will also add arrows to the figures to emphasize the differences highlighted by the radiologist.\n\nWe would like to emphasize that while our quantiative improvements in Table 1 may appear modest, we believe that **these radiologist readings really highlight the improvements made by our method** that are critical for diagnostic purposes. We thank you again for this suggestion that will substantially strengthen our work.\n\\\n\\\n**(W2, L2) Comparison with diffusion-based models.**  \nThank you for this insightful suggestion. We agree that diffusion-based reconstruction offers a compelling and promising approach for solving MR inverse problems. Moreover, since our method is inspired by the time-dependent denoising steps of diffusion models, we included a comparison with Decomposed Diffusion Sampling (DDS) (ICLR 2024, arXiv.2303.05754). Since DDS requires 320×320 inputs due to its generative pre-trained prior, we additionally evaluated PSNR and SSIM using the central 320×320 region. Note that this differs from the results reported in Table 1, where evaluations were performed on images of size 320×368, aligned with the original raw k-space data. We set the number of sampling steps to 100 for DDS. For our method, we used *T = 10* unrolls. All experiments were conducted on the Coronal PD test dataset with an acceleration factor of *R = 4*. The quantitative results are included below:\n\\\n\\\nTable: The comparison results with diffusion-based model (DDS)\n| **Method** | **Data** | **R** | **PSNR↑** | **SSIM↑** |\n|-----------|----------|-------|-----------|-----------|\n| **DDS** | PD | ×4 | 37.41±3.25 | 0.940 ± 0.029 |\n| **Ours (U-Net)** | PD | ×4 | 40.09±2.51 | 0.958±0.017 |\n\nAs shown in the table, our method achieves superior performance compared to the diffusion-based model DDS in both PSNR and SSIM. Due to NeurIPS's rebuttal policy, we cannot include new figures here, but the figures will be updated to include the DDS results in the final version.\n\\\n\\\n**(W3) Efficiency analysis with respect to the number of parameters.**\n\nThis is an important point and apologies for not making it clearer. Our initial submission emphasized efficiency not only in terms of the number of parameters but also in computational cost particularly when reducing the number of unrolls. However, we agree with the reviewer, and explored whether increasing the number of parameters may lead to improved performance. To this end, we conducted an ablation study under the following setups:  (1) increasing the number of channels in the residual blocks from 64 to 96 in ResNet,  (2) increasing the number of channels in the up/downsampling blocks from [32, 64, 128] to [64, 128, 256] in U-Net, (3) using the settings *T = 10*, *R = 4*, and the Coronal PD dataset.\n\nThese modifications increased the total number of parameters in both the shared and unshared baseline methods, as shown in the table below.\n\\\n\\\nTable: The comparison results for different model sizes ($T=10$, $R=4$, and PD)\n|ResNet-VSQP|Channel|#Param.|PSNR↑|SSIM↑|\n|--|--|--|--|--|\n|Shared 𝓡|64|592,129|41.11|0.965|\n|Unshared 𝓡|64|5,921,281|40.99|0.963|\n|Shared 𝓡|*96*|1,330,561|41.09|0.965|\n|Unshared 𝓡|*96*|13,305,601|41.00|0.963|\n|**Ours (*T=5*)**|64|866,571|**41.41**|**0.966**|\n|**Ours (*T=10*)**|64|866,581|**41.43**|**0.965**|\n\n|UNet-VSQP|Channel|#Param.|PSNR↑|SSIM↑|\n|--|--|--|--|--|\n|Shared 𝓡|[32,64,128]|1,724,035|40.50|0.962|\n|Unshared 𝓡|[32,64,128]|17,240,341|40.31|0.960|\n|Shared 𝓡|*[64,128,256]*|6,878,467|40.77|0.964|\n|Unshared 𝓡|*[64,128,256]*|68,784,661|40.55|0.962|\n|**Ours (*T=5*)**|[32,64,128]|1,963,469|**40.94**|**0.964**|\n|**Ours (*T=10*)**|[32,64,128]|1,963,479|**40.99**|**0.964**|\n\nAs shown in the table, in ResNet, although the shared and unshared models have significantly more parameters than both the baseline (2.25×) and our time-embedded network (1.54×), they do not achieve improved results. In U-Net, a slight improvement is observed after increasing the number of parameters by 3.99× compared to the baseline; however, we observed that visual residual artifacts remain. It is important to note that these larger models incur higher computational costs. In contrast, our method achieves better performance with only a marginal increase in parameters.\n\nAdditionally, our time-embedded unrolled network with just 5 unrolls already surpasses both the basic and larger baselines in performance, while reducing computation time by approximately 50% due to the reduced number of unrolls. These experiments and analyses will be included in our final manuscript.\n\\\n\\\n\\\n**Minor Comments (mentioned in Strengths).**\n\n**Blurring issue on uploaded images.**  \nWe thank the reviewer for pointing this out. We will ensure that the figures in the camera-ready version are saved in high-resolution PDF format to improve readability.\n\\\n\\\n**Case when shared regularizer performs worse than non-shared regularizer.**  \nWe would like to clarify that we embed time information into the *shared regularizer* within the unrolled network, as illustrated in Figure 1b. We will ensure this is clearly stated in the final manuscript. We agree that a shared regularizer does not always perform worse. Since this paper focuses on a limited training dataset (300 slices), the shared regularizer performs better than the unshared regularizer, since the latter is prone to overfitting in this case, as shown in Table 1. However, the performance gap between shared and unshared methods narrows when training with more data, as the risk of overfitting becomes less of a concern for the unshared variant. Additionally, if we initialize the unshared baseline unrolled network from the pretrained shared baseline and then fine-tune it, the resulting unshared model generally outperforms both the shared and unshared models trained from scratch, as discussed in Appendix B."}, {"review_id": "PZuko3nXqy", "reviewer": "Reviewer_Ed3h", "summary": "In this paper, the authors proposed a time-dependent unrolling-based neural network for fast MRI reconstruction. Simulation experiment demonstrates its effectiveness. Addressing the typical inverse problem of compressed sensing MRI, this paper could trigger the interest of a wide range of audiences in computational imaging.", "questions": "(1) \"Time\" in the \"time-embedded\" is referred to as the diffusion and denoising step in diffusion models, whereas it is the iteration step in unrolling-based methods in CS-MRI. The authors might want to clarify this point.\n(2) Following the last comment, it would be interesting to discuss the time-embedded networks in dynamic MRI.\n(3) In Figure 1, E is not undefined in this figure.\n(4) Authors mentioned the data-scarce condition in CS-MRI, which limits the unshared networks. Please detail such limitations.\n(5) In Table 1, what does the spades denote?\n(6) Please explain if the network in your method is shared except for the time condition.\n(7) Could you explain more about why TE works better on VSQP in Section 4.4?\n(8) In Equation 16, what is the value of \\tau?", "limitations": "(1) There are many hyperparameters in the proposed method, and I am curious if it could be well generalized to other datasets.\n(2) I would encourage the code to be publicized.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n(1) This paper borrows the advantages from the time-embedding design in diffusion model and vector AMP algorithm and successfully apply them in CS-MRI.\n(2) This paper is well organized and easy to read.\n(3) Experiments on NYU fastMRI dataset are adequate and evaluation is performed on different un-rolling baselines.\n\nWeaknesses:\n(1) Key motivation is vague.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your thoughtful feedback and reviewer’s careful reading of our paper. We are glad that you found our method both interesting, and that you valued our clear presentation, comprehensive experiments, the time-embedded unrolling network approach, and the observed performance gains.\n\\\n\\\n**(W1, Q4) Key motivation is vague and \"data-scarce\" conditions are not clearly explained.**\n\nWe apologize for not making this clearer. In unrolled networks, the proximal operator, implemented implicitly by a neural network, can either be shared (fixed) or unshared (distinct) across iterations/unrolls or unshared. The former aligns well with classical optimization methods but may underperform, while the latter often enhances reconstruction quality albeit significantly increasing the number of trainable parameters. This raises the risk of overfitting, especially in data-limited settings, where one needs to *train on tens or hundreds of datasets instead of thousands*, which is common in many translational applications. This setting is what we mean by \"data-scarce\" conditions, and our key motivation is to improve performance in this setting without substantially increasing trainable parameters and the risk of overfitting.  To elaborate further on this overfitting issue in the data-scarce regime, when the proximal operator is varied across iterations, as in unshared unrolled networks, the number of trainable parameters increases proportionally with the number of unrolls. For example, as shown in Table 2, the unshared model has approximately ten times more parameters than the shared model when $T=10$. Despite this, under limited training data conditions (300 samples in our setting), the unshared model performs worse than the shared model, as illustrated in Table 1. We also observed that the training loss of the unshared network was lower than that of the shared network, but its performance did not generalize well on the test set, indicating a tendency to overfit under data-scarce conditions.\n\nIn summary, our primary motivation is to tackle this data-scarce setup and enhance the performance unrolled algorithms by incorporating a time (iteration)-aware proximal operator inspired by VAMP, while not substantially increasing the number of trainable parameters. Indeed retaining the shared proximal operator but introducing a time-embedding module leads to only a marginal increase in parameters. Our method consistently outperforms the baseline shared/ushared regularizer approach across various acceleration rates, producing artifact-free reconstructions with minimal computational overhead. We will further extend on these points in the final version.\n\\\n\\\n**(Q1) Clarification needed on the meaning of \"time\".**  \nThank you for raising this point. In our work, \"time\" refers to the iteration index of the unrolled algorithm. We will add this clarification in the final manuscript.\n\\\n\\\n**(Q2) Suggestion on time-embedded networks in dynamic MRI.**  \nWe appreciate the reviewer’s insightful suggestion regarding the potential application of time-embedded networks to dynamic MRI. While our current work focuses on static MRI reconstruction from retrospectively undersampled k-space data and interprets \"time\" as the iteration index in algorithm unrolling, we agree that extending this concept to capture true temporal dynamics, such as those present in dynamic MRI, would be an exciting future direction. The temporal dimension in dynamic MRI naturally can be aligned with the idea of time embedding, and we believe our framework could be adapted to handle dynamic MRI tasks as well. We will consider adding a brief discussion of this point in the final manuscript.\n\\\n\\\n**(Q3) In Figure 1, E is not defined.**  \nThank you for noting this. E represents the encoding matrix, as in Section 2.1. We will ensure that E is clearly defined in the figure caption.\n\\\n\\\n**(Q5) Clarify the meaning of the spade symbol in Table 1.**  \nWe apologize for the confusion. The spade symbol indicates a *shared regularizer*. We will include this description in the final manuscript.\n\\\n\\\n**(Q6) Please explain whether the network is shared except for the time condition.**  \nWe embed time information into the *shared regularizer* within the unrolled network, as in Figure 1b. We will ensure that this point is clearly stated in the final manuscript.\n\\\n\\\n**(Q7) Further explanation requested on why TE performs better on VSQP in Section 4.4.**  \nAs discussed in W1, the original VSQP unrolling uses a proximal operator that is fixed across unrolls/iterations. In contrast, in our method, a time-embedded proximal operator (Eq. 14) is used, where the \"time\" information corresponds to the iteration index of the unrolled algorithm. This iteration-dependent information enables the networks to learn iteration-specific denoising behavior and implicitly account for the associated Onsager corrections, as described in Section 3.1. We observe that this simple modification to VSQP (and similarly to ADMM) improves performance across datasets and acceleration rates in terms of PSNR and SSIM, while also reducing coherent artifacts.\n\\\n\\\n**(Q8) Question regarding the value of $\\tau$ in Equation 16.**  \nThank you for pointing out this omission. We set $\\tau$ to 0.1, which we found heuristically to be optimal for ResNet. This information will be added to Appendix A.\n\\\n\\\n**(L1) Questions about hyperparameters and generalizability to other datasets.**\n\nWe thank the reviewer for raising this question. In the unrolled networks considered here, there are several *trainable parameters*, including data fidelity parameters, $\\mu$ in all methods, Lagrangian multiplier related weights, $\\lambda$ in ADMM, or the Onsager correction weights, $\\rho$ in our method. Thus, even though they appear as additional parameters in the algorithms, they are learned during training.\n\nThat being said, we agree that there are more hyperparameters in our method, since we use a time-embedding module. This time-embedding module adds more hyperparameters to the hyperparameters related to the neural networks used as proximal operators (ResNet and U-Net variants in our case). The hyperparameters associated with the time-embedding module include the frequency of the sinusoidal encoding, the embedding dimension, and the number of hidden channels in the MLP layers that process the time embedding. We use the same configuration across different datasets (Coronal PD, Coronal PD-FS, and Axial T2 brain), suggesting robustness across different data. Similarly, we apply the same hyperparameters across all datasets for the neural networks used as proximal operators, under which the networks consistently perform well. While these suggest generalizability, we will nonetheless clarify the increased number of hyperparameters and associated concerns in the final version.\n\\\n\\\n**(L2) Suggestion to make the code publicly available.**  \nWe agree, and the code will be made public upon the acceptance of the paper.\n\\\n\\\n\\\n\\\n**Additional evaluations of our method: Radiologist reading and pathology inspection.**  \nWhile this was not a question raised by the reviewer, we wanted to mention an additional reader study we performed during the rebuttal to highlight the strengths of our method. A musculoskeletal radiologist (30+ years of experience) blindly reviewed the reconstructions included in our paper. Additionally, we also reconstructed several annotated pathology cases from the FastMRI+ dataset(DOI:10.1038/s41597-022-01255-z), and the radiologist also reviewed these. These evaluations demonstrate the strong performance of our method beyond the quantitative metrics reported in the original submission. The radiologist’s comments and details are provided in Reviewer D5F5 (W1, L3, and L1) and Reviewer Bi9Y (W1). Unfortunately, we cannot include new figures here, due to NeurIPS's rebuttal policy, but representative examples and quotes will be included in our final manuscript."}, {"review_id": "F23prOfTT4", "reviewer": "Reviewer_Ea5r", "summary": "The paper introduces a Time-Embedded Algorithm Unrolling framework for MRI inverse problems, integrating diffusion-model time embeddings with AMP/VAMP’s Onsager corrections and data-fidelity weights as learnable, iteration-dependent functions. Each unrolled iteration leverages step-specific FiLM networks and residual corrections to achieve high-fidelity reconstructions.", "questions": "- It would be better to include ablation studies on embedding dimension, sinusoidal frequencies, FiLM layer width, with learning curves in the appendix.\n- Validate on real multi-coil, non-Gaussian noise data or simulate Rician/Poisson noise and non-uniform sampling masks.\n- Please report mean ± std for PSNR/SSIM.. and perform significance tests (e.g. t-test) to confirm improvements.", "limitations": "The authors have not discussed potential reconstruction artefacts arising from noise-model mismatch (e.g. Rician or Poisson) or motion-induced errors. I'd like to recommend adding a discussion of these limitations and their possible diagnostic implications.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths:\n\n- Extending FiLM-based networks to learn iteration-dependent residual corrections is novel.\n- Recasting unrolling methods through an information-theoretic and free-energy perspective.\n- The framework seems generalizes to any iterative reconstruction algorithm. The experimental results are impressive!\n\n# Weaknesses:\n- Section 3 maps the diffusion-model Markov chain to a VAMP-inspired unrolling, assuming unbiased estimators and free-energy conservation in Onsager corrections, but omits a formal proof of second-moment matching or energy invariance.\n- Lacks formal convergence/error bounds. Stability of the deep FiLM layers with time embeddings is only empirically demonstrated; no bounds on the Lipschitz constant or gradient explosion/vanishing are provided for large T.", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your encouraging feedback. We are glad you found the extension of FiLM-based networks for iteration-dependent residual corrections to be novel, and our new interpretation of unrolling based on VAMP insightful. We are also encouraged that you found our framework to be general and the experimental results impressive.\n\\\n\\\n**(W1) Formal proof of second-moment matching or energy invariance missing.**  \nWe agree that a formal proof of second-moment matching is desirable. Second-moment matching is typically evaluated by examining the estimated variances $ \\upsilon_x^{t}$ and $ \\upsilon_z^{t} $ across iterations, as in Eq. (9) \\& (11). However, even if consistent trends are empirically observed in these estimates, this would not constitute a formal proof. This is further complicated in our case, as we hypothesize that a time-embedded neural network models all update steps in Eq. (10)-(11), and the learnable scalar parameter $\\rho^t$ encapsulates the entire process described in Eq. (9). As a result, $\\upsilon_x^{t}$ and $\\upsilon_z^{t}$ are embedded within black-box neural modules and are not explicitly accessible.\n\nInstead, to indirectly assess whether second-moment matching holds, we analyzed the relationships between intermediate estimates. Specifically, we examined the empirical differences between $\\mathbf{x}^t$ and $\\mathbf{u}^t$, and between $\\mathbf{u}^t $ and $\\mathbf{r}^{t}$ in Algorithm 1, as these pairs are intrinsically related to $\\upsilon_x^{t}$ and $\\upsilon_z^{t}$, respectively. If these differences remained consistently small across iterations, it provided evidence that the underlying variance estimates are stable. The empirical difference between $\\mathbf{x}^t$ and the reconstructed $\\mathbf{u}^t$ was reported in Appendix C. As shown in Table 5, the difference (normalized MSE) ranges from $1.01 \\times 10^{-9}$ to $3.35 \\times 10^{-2}$, demonstrating stable behavior. These findings suggest that second-moment matching is empirically preserved, despite the use of learned components. We will clarify these points in more detail in the final version, and clearly state the lack of a formal proof.\n\\\n\\\n**(W2) Lipschitz constant or gradient explosion/vanishing of time-embedding (FiLM) layers.**  \n\nConsider a network where at each time step $t \\in$ {$1, \\ldots, T$}, there are $K$ consecutive layers within the proximal operator networks composed of intermediate transformations followed by FiLM modulation. For each layer $k \\in$ {$1, \\ldots, K$},  \n\n$x^{(t,k)} = \\mathrm{FiLM}( f_{k}(x^{(t,k-1)}), t ),$\n\nwhere $f_{k}(\\cdot)$ denotes the intermediate layers (e.g., convolution + activation) preceding the FiLM block at layer $k$. Suppose each intermediate layer is Lipschitz continuous with constant $L_{k}$, and satisfies  \n\n$\\||f_{k}(x)\\|| \\leq L_{k} \\||x\\|| + \\delta_{k},$\n\nfor some small $\\delta_{k} \\geq 0$, allowing for nonzero bias or offset when $f_{k}(0) \\neq 0$. Similarly,  each FiLM block is Lipschitz continuous with constant $ \\Gamma_{t,k} $, satisfying  \n\n$\\||\\mathrm{FiLM}(z, t)\\|| \\leq \\Gamma_{t,k} \\||z\\|| + B_{t,k}.$\n\nThe composite function then satisfies:  \n\n$\\||\\mathrm{FiLM}(f_{k}(x), t)\\|| \\leq \\Gamma_{t,k} L_{k} \\||x\\|| + (\\Gamma_{t,k} \\delta_{k} + B_{t,k}).$\n\nThus, the overall Lipschitz constant of the composite layer at layer $k$ and time $t$ is $\\Gamma_{t,k} L_{k}$. As $T$ grows, if each $\\Gamma_{t,k} L_{k}$ is strictly less than 1, their product decays exponentially, which may cause vanishing activations and hinder learning. If any are $\\geq$ 1, the product can grow exponentially, causing exploding activations and instability. Thus, controlling cumulative constant, $\\prod_{t=1}^T \\prod_{k=1}^K \\Gamma_{t,k} L_{k}$ is crucial for stable training.  \n\nWith this in mind, we agree with the reviewer that this still does not constitute a formal proof, since we do not characterize the Lipschitz constants of individual components analytically. However, we do believe this lends more credence to our empirical results. In particular, we followed the U-Net architecture and FiLM module commonly used in diffusion denoising tasks. In the diffusion model community, a large $T$ ($\\geq$ 1,000) is typically used, which has been shown to result in stable training. In our unrolled networks, where we use a much smaller $T$ (e.g., 5–15), we expect that this does not lead to instability during training. We will include this discussion in the final version, and clearly state the empirical nature of our Lipschitz constant bounds.\n\\\n\\\n**(Q1) Ablation studies on time-embedding module with different hyperparameters including learning curves.**  \nWe thank the reviewer for raising this important point. We conducted additional ablation studies to assess the effectiveness of key hyperparameters in the time-embedding module. Specifically, we evaluated: 1) the frequency of the sinusoidal encoding, 2) the embedding dimension, and 3) the number of hidden channels in the MLP layers. The experiments were conducted using a U-Net architecture with $T=10$, $R=6$ on Coronal PDFS dataset.\n\nWe observed that variations in these hyperparameters can influence the performance of our proposed method. The best results were obtained with a period of 10,000, an embedding dimension of 32, and a hidden layer size of 128 channels, as shown in the table below. These are the same hyperparameters used throughout all of our experiments. We did not observe any noticeable differences in the learning curves across different settings. These will be added to the final Appendix.\n\\\n\\\nTable: Ablation study of time-embedded unrolling networks with different time-embedding hyperparameters.  \n|Freq.base(max period)|Emb.dim.|Hid.layer dim.|PSNR↑|SSIM↑|  \n|-|-|-|-|-|  \n|*1,000*|32|128|34.34|0.824|  \n|*5,000*|32|128|34.35|0.822|  \n|*10,000*|32|128|**34.44**|**0.825**|  \n|10,000|*32*|128|**34.44**|**0.825**|  \n|10,000|*64*|128|34.32|0.824|  \n|10,000|*96*|128|34.34|0.824|  \n|10,000|32|*64*|34.38|0.824|  \n|10,000|32|*128*|**34.44**|**0.825**|  \n|10,000|32|*196*|34.37|0.824|  \n\n\\\n**(Q2) Validate on real multi-coil, non-Gaussian noise data or simulate Rician/Poisson noise and non-uniform sampling masks.**  \n\nThank you for these suggestions. The noise in raw MRI data is modeled as Gaussian (doi:10.1118/1.595679). Rician noise is used to model the noise in magnitude MR images (doi: 10.1002/mrm.1910340618), but does not apply to our setup that works with raw MRI data. We are not aware of Poisson noise in MRI (this may be related to CT, which is not considered here). That being said, we made an egregious omission by not including non-uniform sampling masks in the original submission.\n\nTo evaluate the effectiveness of our proposed method under non-uniform sampling patterns, we conducted experiments at various acceleration rates, using both baseline methods and our approach with a U-Net architecture and 10 unrolls. As shown in the table below, our method consistently outperforms the baselines in terms of PSNR in all cases except for the Axial T2 dataset at R=6. For SSIM, our method shows improvement in most cases for the PD and PD-FS datasets, although no improvement is observed for the Axial T2 dataset. These results validate the effectiveness of our method even when applied to non-uniform sampling patterns.\n\\\n\\\nTable: Validation on non-uniform sampling masks (**S**:Shared, **U**:Unshared)  \n||Metric|VSQP(S)|VSQP(U)|ADMM(S)|ADMM(U)|Ours|  \n|-|-|-|-|-|-|-|  \n|**PD**|||||||  \n|×4|PSNR↑|40.10|40.26|40.20|40.13|**40.43**|  \n||SSIM↑|0.961|0.961|0.961|0.961|**0.962**|  \n|×6|PSNR↑|38.41|38.59|38.64|38.72|**38.73**|  \n||SSIM↑|0.947|0.947|0.948|0.949|**0.949**|  \n|×8|PSNR↑|37.30|37.39|37.64|37.52|**37.76**|  \n||SSIM↑|0.935|0.934|**0.939**|0.938|0.938|  \n|**PD-FS**|||||||  \n|×4|PSNR↑|35.61|35.60|35.62|35.66|**35.68**|  \n||SSIM↑|0.855|0.853|0.853|**0.856**|0.854|  \n|×6|PSNR↑|34.59|34.55|34.70|34.61|**34.74**|  \n||SSIM↑|0.828|0.824|0.827|0.827|**0.830**|  \n|×8|PSNR↑|33.76|33.97|34.06|34.07|**34.14**|  \n||SSIM↑|0.809|0.808|0.810|0.810|**0.812**|  \n|**Axial T2**|||||||  \n|×4|PSNR↑|35.90|36.36|36.50|36.42|**36.52**|  \n||SSIM↑|**0.932**|0.931|0.930|0.931|0.930|  \n|×6|PSNR↑|35.03|35.11|**35.26**|35.25|35.21|  \n||SSIM↑|**0.917**|0.916|0.915|0.905|0.914|  \n|×8|PSNR↑|34.09|34.31|34.37|34.39|**34.51**|  \n||SSIM↑|0.908|0.908|0.903|**0.906**|0.904|  \n\n\\\n**(Q3) Standard deviation for PSNR/SSIM.**  \nThank you for raising this important point. While we did not include the standard deviations in Table 1 to improve readability, we reported these in Table 7 in Appendix G. We will clarify this in the final version, and add statistical tests.\n\\\n\\\n**(L1) Discussion of motion-induced errors.**  \nThank you for the suggestion. We will include a discussion on motion-induced mismatches and their possible diagnostic implications in the final version.\n\\\n\\\n\\\n\\\n**Additional evaluations of our method: Radiologist reading and pathology inspection.**  \nWhile this was not a question raised by the reviewer, we wanted to mention an additional reader study we performed during the rebuttal to highlight the strengths of our method. A musculoskeletal radiologist (30+ years of experience) blindly reviewed the reconstructions included in our paper. Additionally, we also reconstructed several annotated pathology cases from the FastMRI+ dataset(DOI:10.1038/s41597-022-01255-z), and the radiologist also reviewed these. These evaluations demonstrate the strong performance of our method beyond the quantitative metrics reported in the original submission. The radiologist’s comments and details are provided in Reviewer D5F5 (W1, L3, and L1) and Reviewer Bi9Y (W1). Unfortunately, we cannot include new figures here, due to NeurIPS's rebuttal policy, but representative examples and quotes will be included in our final manuscript."}, {"review_id": "5TDgS96e7K", "reviewer": "Reviewer_Bi9Y", "summary": "This paper proposes to incorporate time information into the proximal operator within the algorithm unrolling scheme for accelerated MRI reconstruction, in order to address the trade-off in unrolled networks where sharing a proximal operator across iterations can cause artifacts, while using distinct networks per iteration leads to overfitting. The proposed method leads to on-par results with non-weight-sharing unrolling methods weight-sharing unrolling methods on uniformly 4x,6x,8x undersampled MRI data from fastMRI.", "questions": "The gains of the proposed methods over weight-sharing methods are not obvious either.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The idea of the time-dependent proximal operator to avoid using non-weight-sharing regularizers has some merits.\n- The experiments are comprehensive\n\nWeaknesses:\n- The gains brought the proposed method seem marginal both quantitatively and qualitatively (Tab 1, Fig 3). Even the performance between the weight-sharing regularizers and the non-weight-sharing regularizers is close already (Tab 1). \n-  The core technical contribution of the paper is the time-embedding applied to the proximal operator within a VAMP-inspired framework, which seems limited: AMP has been explored in algorithm unrolling as also noted by the authors, and the time is embeded using FiLM.", "was_revised": true, "final_justification": "I appreciate the additional evaluations provided by the authors during the rebuttle period, so I raised my score.", "rebuttal": "We sincerely thank you for your thoughtful feedback. We are pleased to hear that the idea of employing a time-dependent proximal operator was seen as meaningful, and that the comprehensiveness of our experiments was appreciated. We are also grateful for the opportunity to address your concerns and further elaborate on our contributions.\n\\\n\\\n**(W1) Degree of improvement achieved by our method.**  \n\nThank you for raising this important comment. Indeed the quantitative metrics reported in the tables do not necessarily do justice to the subtle improvements afforded by our method. Thus, to show the strength of our method beyond quantitative metrics reported in the text, we worked with a radiologist who provided their expert opinion. While there was not enough time for a full reader study during the rebuttal period, a radiologist (30+ years of experience), *blindly reviewed* the reconstructions included in our paper. The radiologist's comments are included below:\n\n- Figure 3 (Middle) exhibits aliasing artifacts in the distal femoral metaphysis medially on PD-weighted images for VSQP, Unshared-VSQP, ADMM, Unshared-ADMM. The artifacts are effectively removed in the proposed method (ours).\n\n- Figure 3 (Bottom) shows blurring in the right occipital lobe in VSQP, Unshared-VSQP, ADMM, Unshared-ADMM. This is notably improved in the proposed method, where the gyri and sulci appear sharper.\n\n- Figure 4 (Top) shows visible aliasing artifacts in the central aspect of the distal femoral condyle of the inset for VSQP, Unshared-VSQP, ADMM, and Unshared-ADMM. This artifact is removed in the proposed method. The prominent penetrating intraosseous vessel at the lateral aspect of the proximal tibia (lower left on the image) appears sharper in proposed method, though overall image sharpness is similar among methods.\n\n- Figure 4 (Middle): There is an oblong, hypointense appearing  aliasing artifact on the non-fat saturated PD-weighted images of the knee joint, just distal to the posteromedial femoral condyle, seen on VSQP, Unshared-VSQP, ADMM, and Unshared-ADMM, when compared to the reference. This artifact is removed from the image for the proposed method. Thus, only the proposed method accurately resembles the reference image. \n\n- Figure 4 (Bottom) reveals aliasing artifacts in VSQP, Unshared-VSQP, ADMM, Unshared-ADMM, depicted as curvilinear, oblique hypointense signal in the occipital lobe. The artifact is nearly completely removed in Unshared-ADMM, and only vaguely seen. The artifact is completely removed in ours, most accurately resembling the reference image.\n\nAdditionally, we also reconstructed several **annotated pathology cases from the FastMRI+ dataset** (DOI:10.1038/s41597-022-01255-z), and the radiologist also reviewed these. Note that these were not included in the original submission. The radiologist noted the following:  \n\n- On the reference image there is a focal area of T2-hyperintense signal, most consistent with a partial/full thickness cartilage defect in the anteromedial trochlea. Images reconstructed by VSQP, Unshared-VSQP, ADMM, and Unshared-ADMM reveal significant blurring in this area. Proposed method shows the least amount of blurring compared to other methods, and shows the hyperintense region in the trochlear articular cartilage with the most fidelity compared to the reference data.\n\nUnfortunately, we cannot include new figures here, due to NeurIPS's rebuttal policy, but representative examples and quotes will be included in our final manuscript. We will also add arrows to the figures to emphasize the differences highlighted by the radiologist. \n\nWe would like to emphasize that while our quantitative improvements in Table 1 may appear modest, we believe that **these radiologist readings really highlight the improvements made by our method** that are critical for diagnostic purposes. We thank you again for this suggestion that will substantially strengthen our work.\n\\\n\\\n**(W2) Core technical contribution of the paper.**  \nWe appreciate the opportunity to clarify this point. Indeed, AMP and its variants have been explored in the context of algorithm unrolling, as described in the text. One line of work ([11, 12] in the original submission) reparameterizes the encoding matrix ${\\bf E}$ and ${\\bf E}^H$, which affects the Onsager correction term. Another approach [33, 42] keeps the encoding matrix fixed, but trains distinct (unshared) proximal operators using neural networks at each iteration. These distinct proximal operators operate *without awareness of temporal (or equivalently iteration) information* (i.e., unshared in our terminology). Finally, some methods [33, 66] combine the above two strategies by jointly training both the system matrix and the proximal operators. \n\nOur method is technically distinct from these approaches in several key ways:\n\n1) We keep the system matrix ${\\bf E}$ fixed, since this is known from MR physics, and use a single shared proximal operator modeled by a neural network, but introduce iteration-awareness via a time-embedding technique, enabling the network to behave differently at each unrolling stage. This is distinct from the unshared proximal operators used in AMP-type algorithm unrolling approaches discussed earlier.\n\n2) Our method is inspired by the structure of VAMP, rather than AMP. To the best of our knowledge, our work represents the first attempt to integrate VAMP with algorithm unrolling to solve computational imaging inverse problems.\n\n3) We propose time-embedded ResNet and U-Net-based proximal operators for inverse problems in MRI. In particular, for the ResNet-based proximal operator that has been used in several MRI reconstruction works, we introduce a novel time-embedding module with a scaling factor, as described in Equation (16). For the U-Net-based proximal operator, we adapt the U-Net architecture from diffusion models, making several adjustments to reduce model size to make it suitable for inverse problems in MRI."}]

Author rebuttal:

Meta-review: The paper introduces Time-Embedded Algorithm Unrolling (TE-Unroll) for deep learning–based MRI reconstruction, inspired by connections between vector AMP and diffusion models. The proposed method incorporates time-conditioned proximal networks (via FiLM embeddings), learnable time-varying scalars for data-fidelity weighting, and Onsager correction. It applies to multiple unrolling schemes (e.g., VSQP, ADMM) with ResNet or U-Net proximals. Extensive experiments on fastMRI knee and brain datasets at acceleration factors of 4/6/8 show consistent PSNR/SSIM improvements over shared and unshared baselines, with little parameter overhead. Ablation studies on unroll depth, optimizers, and qualitative results further validate the approach.

In the initial review, multiple concerns were raised by the reviewers, including questioning the utility and significance of the (improved) images reconstructed, the absence of comparison with the state-of-the-art baseline models, the lack of clarity around the notion of efficiency of the proposed method, and the poor motivation of the work. However, during the rebuttal period, the authors were able to sufficiently and convincingly address most of the reviewers' concerns, prompting the reviewers to update their scores. 

Keeping in mind the rebuttal posed by the authors and the subsequent changes promised in the final manuscript, this is a well-motivated, technically solid work backed up by extensive experiments and ablation studies on open-source datasets. Other strengths of the paper are: 
- Novel model that borrows the advantages from the time-embedding design in the diffusion model and the vector AMP algorithm
- Well-organized paper and easy to read
- Extensive experiments on the NYU fastMRI dataset are adequate, and evaluation is performed on different unrolling baselines
- The anecdotal human evaluation results gathered during the rebuttal period further strengthen the story
Decision: Accept (poster)