Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
We introduce an efficient method to accurately quantify noise uncertainty in deep learning-based MRI reconstruction, achieving Monte-Carlo-like accuracy while significantly reducing computational and memory requirements.
摘要
评审与讨论
The authors propose a scheme for efficient voxel-wise noise estimation for Deep-learning based accelerated MRI algorithms. The method builds upon estimating the noise covariance from the DL-models Jacobian by using an unbiased estimator for the diagonal elements (of the cov-matrix) via the proposed Jacobian sketching approach.
给作者的问题
No further questions/remarks from my side.
论据与证据
The work includes claims to accurately reconstruct voxel-based noise levels in DL multi-coil based accelerated MRI reconstruction agnostic to the method and forward operator, which as to my opinion are fulfilled by the conducted experiments.
方法与评估标准
One notable weakness of the paper is that the presented method is only evaluated in the domain of MRI reconstruction, leaving its performance on other modalities like CT untested. This especially interesting since CT has Poisson distributed noise, which is a significant difference to the assumptions in MRI. Demonstrating the performance across different (medical) imaging techniques would significantly enhance the work’s contribution.
Furthermore, the evaluations conducted do not provide evidence that the estimated noise levels improve performance in downstream tasks. While the method efficiently and accurately reconstructs voxel-wise noise variance, it remains unproven whether these estimations actually translate into benefits for tasks like non-blind denoising or segmentation. Demonstrating an improvement in downstream applications is critical to validate the practical utility of the proposed technique.
However, despite this shortcoming, the method is evaluated on a broad range of reconstruction approaches, including supervised, self-supervised, fully data-driven, and physics-driven methods.
理论论述
The theoretical part of the manuscript seems sound and well-presented.
实验设计与分析
As mentioned above, for the task of mere noise estimation in MRI reconstruction, the experimental design appears valid. However, the evaluation lacks evidence that the estimated noise levels benefit downstream tasks or works on other modalities/noise distributions.
补充材料
The provided supplementary material contains the codebase for the presented work.
与现有文献的关系
The contribution to the broader scientific field of medical imaging seems somewhat limited due to the shortcomings addressed above.
遗漏的重要参考文献
The manuscript seems to include essential references.
其他优缺点
See above.
其他意见或建议
As mentioned above, the work conducted is solid. However, in my opinion, to justify a publication, the authors should consider extending the work as addressed above.
We appreciate the reviewer’s thoughtful comments and recognition of our extensive experimentation across diverse DL reconstruction methods and MR imaging scenarios. Below we explicitly address each concern, highlighting the relevance and impact of our approach in downstream medical imaging via extended discussion and new experiments. We then describe generalizability the other medical imaging tasks and modalities. We’d like to also refer the reviewer to 1. Novelty and Technical Contributions and 2. Clinical Motivation and Importance of Noise-Variance (NV) Estimation points in the rebuttal for reviewer SQro. We believe these points, holistically clarifies our study’s contributions to the broader scientific field of medical imaging.
1. Utility of Noise Variance Estimation on Downstream Tasks
Voxel-wise NV estimation has been extensively shown to enhance downstream imaging tasks such as denoising and segmentation across modalities including MRI, CT, and ultrasound. For example, classical adaptive filters like BM3D significantly benefit from NV information by effectively removing noise without compromising clinical details [1,2]. Similarly, adaptive filters that incorporate spatially varying noise variance improve diagnostic information in low-dose CT and ultrasound imaging [3,4]. In deep learning contexts, explicitly incorporating voxel-wise NV maps consistently improves performance by reducing false positives and increasing accuracy in segmentation and denoising networks [5,6].
[1] Hanchate, SN Appl Sci., 2020
[2] Li, Med Phys, 2014
[3] Hariharan, Phys Med Biol, 2020
[4] Yu, IEEE TIP, 2002
[5] Dou, NMR Biomed., 2025
[6] Wang, Eng Appl AI, 2022
2. Additional Downstream Task Experiments
To directly demonstrate the practical value of our method, we have added two new downstream proof-of-concept experiments in the revised manuscript:
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- Non-blind Denoising: A Noise2Noise network was trained on the reconstructed brain images (a) assuming uniform image noise and (b) leveraging our predicted voxel-wise variance maps as noise-level priors. The variance-informed network achieved a 0.9 dB gain in PSNR, highlighting direct improvements from accurate local variance estimation.
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- Variance-Guided Segmentation: A U-Net was trained for cartilage segmentation with the predicted variance maps included as additional input channels, resulting in a 1.1% improvement in Dice coefficient and fewer false positives, demonstrating how noise-aware segmentation enhances clinical accuracy.
These experiments provide clear evidence that our analytical voxel-wise NV estimates are not merely theoretical metrics but actively enhance clinically relevant imaging tasks.
3. Extension Beyond MRI for Other Medical Imaging Tasks and Modalities
We agree that extending our voxel-wise noise variance (NV) estimation approach to imaging techniques beyond MRI (particularly CT with its Poisson-like noise characteristics) would further demonstrate the generalizability and impact of our method. However, our primary goal in this study was to specifically address multi-coil accelerated MRI due to its complex acquisition model and the widespread adoption of DL-based reconstructions, where explicit noise propagation has been relatively understudied. Notably, our framework is fundamentally generalizable and is not limited solely to Gaussian noise assumptions. The method requires only: A known forward operator describing the measurement process (e.g., X-ray projections in CT). An appropriate noise covariance reflecting the modality-specific noise statistics. For instance, extending to CT could involve incorporating Poisson-Gaussian noise models or variance-stabilizing transformations directly into . Thus, while we consider such extensions an important future direction, the current manuscript lays the foundational theoretical and computational framework necessary to address these broader applications.
I thank the authors for addressing most of my concerns. As the suggested method appears to have a positive impact on downstream tasks, I will raise my score. However, I am still not fully convinced that the contribution is substantial enough to justify publication in its current form. Extending the work to CT, as suggested in the discussion, would significantly strengthen the contribution.
We appreciate the reviewer’s acknowledgement of our method’s positive impact on downstream tasks and welcome the continued interest in the broader applicability of our framework. While our approach is fundamentally extensible to any modality with a linear forward operator and known noise covariance, our present study specifically targets multi-coil accelerated MRI due to:
- Complex Acquisition Operator. Multi-coil MRI relies on coil sensitivity maps, Fourier encoding, and undersampling masks, creating a high-dimensional, correlated noise model not fully addressed by existing uncertainty quantification methods.
- Widespread Adoption of Deep Learning Reconstructions. Recent advances in deep MRI reconstruction have rapidly entered clinical and research pipelines, yet voxel-wise noise propagation in these networks remains relatively understudied.
- Page Limits and Scope. Given the complexity of multi-coil MRI and the page constraints of this submission, we chose to focus on a rigorous analysis and validation in this setting, rather than diluting the results across multiple imaging modalities.
That said, we argue that extending to CT, PET, or other modalities is a natural next step, not a fundamental limitation. As discussed in Theory section, our framework fundamentally depends on (1) a linear forward operator and (2) a noise covariance matrix, thus making it adaptable to imaging modalities beyond MRI.
In CT, for instance, the forward operator would represent the Radon transform (mapping attenuation coefficients to projection data). While CT measurements exhibit predominantly Poisson-like noise, one can employ a variance-stabilizing transform (e.g., the Anscombe transform) to approximately convert that noise into a Gaussian form. Specifically, if we transform the measured projections to , then our noise model can treat as a (potentially diagonal) covariance matrix reflecting the stabilized data’s variance.
Concretely, in step 2 of Algorithm 2 in Appendix, we transform the random matrix via:
For CT, would encode the (approximate) covariance of the transformed sinogram data, ensuring the subsequent Jacobian sketching step accurately captures how Poisson-like noise propagates through the deep reconstruction network.
In Theorem 3.1 and Lemma 3.2, we show that our estimator of is unbiased when the random vectors satisfy and . These requirements are agnostic to the underlying physical noise distribution, as long as correctly models it. Therefore, whether the noise is approximately Gaussian (as in MRI) or stabilized Poisson (as in CT), our procedure—via random-phase or Gaussian vectors—remains statistically valid for diagonal variance estimation.
We agree with the reviewer that a demonstration on CT would indeed strengthen our contribution by verifying performance under Poisson-like conditions. However, our current manuscript focuses on establishing the theoretical foundation and computational framework in the (multi-coil, correlated) MRI setting as outlined above. The modular design of our approach—based on abstract linear algebraic principles—makes it naturally extendable to CT, PET, or other modalities. We are actively exploring these future directions, confident that the same Jacobian-based random sketching and diagonal estimation procedure will hold under the appropriate and for each modality."
In the revised manuscript, we will incorporate a dedicated subsection within the Discussion section titled 'Extension to Other Modalities: CT and Beyond,' where we will elaborate on the practical considerations and theoretical adaptations necessary for applying our method to CT. Specifically, we will detail how our randomized sketching algorithm and noise modeling adapt to different forward operators (e.g., the Radon transform for CT) and noise distributions (e.g., Poisson-like). This discussion will outline practical considerations such as using the Anscombe transform for approximate Gaussianization and incorporating the appropriate noise covariance. We believe these additional clarifications will demonstrate the generality of our core method.
The authors have developed a computational- and memory-efficient estimator of the voxel-wise variance in MRI reconstruction for uncertainty quantification in the reconstruction. The method is evaluated on MRI data and compared to Monte Carlo-simulations. The method is simple and easy to understand and computationally efficient while providing very similar results compared to Monte Carlo.
给作者的问题
Line 259, right column: What is R? Needs to be explained.
If you make the novelty of the contribution clear, especially in relation the existing works, I would consider increasing my score.
论据与证据
The claims made are proven in a clear and straight-forward way. The empirical evidence supports the utility of the proposed method.
方法与评估标准
I think the results are fairly clear, but the noise map estimates can be difficult to visiaully evaluate. I would encourage the authors to provide other means as well to illustrate the reconstruction and highlight when it works well and when/where it fails. For instance, why not make a scatter plot with the uncertainties as a function of different voxel intensities. This would highlight that the uncertainties (likely) increase as the image intensities increase, for instance.
You say that a t-test was performed on the "difference between noise distributions", but it is not clear what test was actually performed, and where are those results presented? In this case, it would be better to compare the distributions rather than the means (through the t-test). The uncertainty profiles could be quite different, even though the means be the same.
理论论述
The proofs were easy to follow and appears correct.
实验设计与分析
To increase the evidence in favour of your method, I would encourage you to include at least one more experiment. I would also add a quantification of the results, for instance a comparison between the estimated variances as mentioned above.
补充材料
The supplementary material contained much more details and additional results. This is nice, and in principle fine, but it would be good to outline in the main paper what is in the supplementary, so that it is clear to the reader that they (perhaps) should look there as well.
与现有文献的关系
I'm missing an exposition of and comparison to previous work in uncertainty quantification in MRI reconstruction. There is quite some work done in this field. It would also be good to compare the proposed method to other noise estimation methods, to further strengthen the evidence in favour of the proposed method.
遗漏的重要参考文献
As mentioned above, I think an exposition of and comparison to previous work in uncertainty quantification in MRI reconstruction is missing.
其他优缺点
The paper is well-written and clear.
It is said that the proposed method was used before, by Pruessmann et al. (1999), but this is not discussed under related methods. What are the differences, and what in the present work makes it novel compared to previous work?
其他意见或建议
Some minor comments:
- The distinction between \Sigma_x and \Sigma_k could be stated explicitly.
- Line 150, left column: Double ending parenthesis.
- Line 156, right column: Cross-reference should be equation (6) and not (8), right?
- Line 208, right column: It would be good to make it clear that the algorithm description is in the appendix.
- Results: It would be better if you presented standard errors instead of standard deviations, so that the results can be directly compared.
- Line 368, left column: "4.1" should perhaps be "Section 4.1" instead.
- I would recommend that you clean up the references.
- Line 1362: Lower case t after full stop.
We thank the reviewer for the positive feedback. Thanks to the reviewer for thorough examination of the manuscript, we implemented all suggested corrections (typos, cross-refs, terminology) and cleaned up the references. We also added an appendix overview outlining its structure for improved readability. We addressed your valuable suggestions, and clarifed novelty and contributions of our study, especially in relation the existing works which we have discussed in great detail below, with newly included references. For a detailed breakdown of our contributions, please also see rebuttal to reviewer SQro: 1. Novelty and Technical Contributions. Per your recommendation, we also included new additional experiments outlined in the rebuttal to reviewer viwd: 2. Additional Downstream Task Experiments.
1. Extended Related Work and Clarification of our contributions
While Pruessmann [8] analytically computed noise variance for linear SENSE with Cartesian sampling, our method addresses nonlinear deep reconstructions with complex operators and random undersampling—where noise propagation is significantly more challenging.
Unlike our approach, Wen [1] uses a conformal prediction framework that constructs distribution-free uncertainty intervals at the level of downstream task outputs, without modeling voxel-level uncertainty or incorporating the physics of MRI acquisition. Their approach treats the reconstruction pipeline as a black box and requires calibration data to guarantee finite-sample statistical coverage. They do not estimate how acquisition noise propagates through multi-coil systems or nonlinear reconstructions.
Edupuganti [2] introduced a VAE-based probabilistic framework that models epistemic reconstruction uncertainty in a latent space and utilizes SURE-based estimators—relying on approximate Jacobian traces—and Monte Carlo (MC) sampling to compute uncertainty maps under the assumption of uncorrelated, i.i.d, single-coil noise. In contrast, our approach analytically propagates correlated multi-coil MRI noise through nonlinear deep reconstruction networks using network’s Jacobian as an operator via randomized sketching. In addition to these, in our revised submission, we will include references to additional studies [2–7] that were discussed under Uncertainty Quantification in MRI in Related Work section but inadvertently omitted from the bibliography. Our work uniquely closes a longstanding gap by providing the first analytical framework to quantify how acquisition noise in multi-coil k-space propagates through nonlinear deep MRI reconstructions. Existing methods do not analytically model this process, and thus do not solve the same problem. Consequently, direct comparisons are not appropriate; our MC baseline sufficiently represents these methods’ performance within the multi-coil setting, while our analytical estimator addresses a distinct, previously unresolved challenge in aleatoric uncertainty quantification.
2. Additional Visualizations
We performed additional visualizations of voxel-wise noise maps (link):
- A histogram of the relative estimation error reveals the full distribution of voxelwise deviations, showing that most errors cluster near zero.
- A scatter plot of the calculated noise maps vs. voxel intensity demonstrates a moderate positive correlation, consistent with typical MRI acquisition where noise can scale with signal magnitude.
- A scatter plot of the absolute estimation error vs voxel intensity likewise shows a mild positive correlation, confirming the reviewer’s hypothesis that noisier regions often coincide with brighter intensities.
As the reviewer suggested, these visualizations indeed offer an improved quantitative perspective on how closely our method aligns with empirical references and where noise tends to concentrate, supplementing direct map comparisons.
3. Statistical Tests for Noise Variance Distribution Comparisons
We initially reported the results of a two-sample t-test result on page 5, line 264. Following the reviewer’s suggestion, we expanded our analysis to include further statistical tests:
- Normality Check A Shapiro–Wilk test yielded a p=0.01<0.05, indicating that the distribution of voxelwise variance differences does not follow a normal distribution.
- Since normality was violated, we employed a Wilcoxon signed-rank test to check whether the variance maps from our method and the MC reference originate from the same distribution. We found no statistically significant difference between these two distributions (p=0.75>>0.05). Hence, our variance estimates align with the MC reference at both the mean and distributional levels, despite the non-normality in voxelwise differences.
[1] Wen ECCV 2024
[2] Edupuganti TMI 2020
[3] Hoppe ECCV 2024
[4] Edupuganti TMI 2021
[5] Tezcan TMI 2020
[6] Narnhofer TMI 2021
[7] Küstner MRM 2024
[8] Pruessmann MRM 1999
Thank you for the clear response and edits. I am satisfied with the additions you've made and will increase my score.
We are very grateful to Reviewer q9G3 for their positive feedback and for raising their score to 4. We thank them for their detailed review and for acknowledging that our clarifications regarding novelty, related work, and the additional visualizations and statistical tests have been satisfactory. We appreciate their time and effort in helping us improve the manuscript.
Authors propose a technique to calculate voxel-wise variance for quantifying uncertainty that stems from acquisition noise in accelerated MRI reconstructions. Authors propose to estimate the noise covariance using an approximation to the Jacobian of the neural network. The approximation is done through an unbiased estimator for the diagonal of the covariance by a sketching via random-phase vectors. The sketching algorithm is evaluated on knee and brain MRI datasets for data and physics driven networks trained in supervised and unsupervised manners. Method is shown to be robust against varying input noise levels, acceleration factors, and diverse under-sampling schemes.
给作者的问题
- Authors mention that "... even if global metrics indicate high reconstruction quality, a locally elevated noise variance in diagnostically relevant ROIs could compromise clinical interpretation.". Is there a concrete example of such a case? How is this work alleviating the issue? While I understand the interpretation of the noise maps as uncertainty quantification, my main concern is that this work does not go beyond trying out existing matrix sketching ideas in the context of MRI reconstruction.
- How does sketching with random-phase vectors compare against random Gaussian vectors? In practice do you observe significant enough difference in the estimation error?
- Why is the noise map in Figure 6 for the MoDL method look significantly different than the others?
- Are there bounds on the sketching error? How does it depend on the properties of the linear operator and the Jacobian?
论据与证据
Several claims do not have convincing evidence.
- In the conclusion section authors mention "... even if global metrics indicate high reconstruction quality, a locally elevated noise variance in diagnostically relevant ROIs could compromise clinical interpretation.". However, there is no concrete evidence provided in the paper to support these reasoning.
- Although in real-valued matrix sketching Rademacher vectors are known to have lower estimator variance compared to Gaussian vectors, the choice of using complex Rademacher vectors is not ablated. It is not clear whether it is strictly better than complex Gaussian random vectors for estimating noise maps both empirically and theoretically.
方法与评估标准
Benchmark datasets and the evaluation criteria makes sense for the problem at hand.
理论论述
Yes, verified the proofs for Theorem 3.1 and Lemma 3.2. Although they are correct, they are unnecessarily long.
实验设计与分析
As mentioned in the "Claims and Evidence" section, I have some concerns with the experimental analysis. Please refer there to avoid duplication.
补充材料
Yes, I have reviewed Appendix B (existence of network Jacobian) and F-to-J to check various experimental configurations (undersampling patterns and different network architectures)
与现有文献的关系
The estimation of the noise variance serves as a measure of uncertainty quantification. This is relevant to a line of work that uses network Jacobian and SURE estimator [1], and [2] which utilizes tools from conformal prediction to provide uncertainty quantification.
[1] Edupuganti, Vineet, et al. "Uncertainty quantification in deep MRI reconstruction." IEEE Transactions on Medical Imaging 40.1 (2020): 239-250.
[2] Wen, Jeffrey, Rizwan Ahmad, and Philip Schniter. "Task-Driven Uncertainty Quantification in Inverse Problems via Conformal Prediction." European Conference on Computer Vision. Cham: Springer Nature Switzerland, 2024.
遗漏的重要参考文献
Some of the missing citations are: [1] which quantifies the risk in the reconstructed images using Stein's Unbiased Risk Estimator (SURE) via the end-to-end network Jacobian. [2] using tools from conformal prediction to obtain risk maps.
[1] Edupuganti, Vineet, et al. "Uncertainty quantification in deep MRI reconstruction." IEEE Transactions on Medical Imaging 40.1 (2020): 239-250.
[2] Wen, Jeffrey, Rizwan Ahmad, and Philip Schniter. "Task-Driven Uncertainty Quantification in Inverse Problems via Conformal Prediction." European Conference on Computer Vision. Cham: Springer Nature Switzerland, 2024.
其他优缺点
Strengths:
- Paper is easy to read.
- Application of matrix sketching as a means to estimate noise variance in MRI reconstruction setting is interesting.
- Experimental results are conducted over a diverse set of configurations.
Weaknesses:
- Technical contribution is weak. The paper essentially boils down to applying matrix sketching ideas to MRI reconstruction networks.
- The theorems and lemmas (in chapter 3) are unnecessarily cluttered.
- The need for estimating the voxel-wise noise variance is not motivated enough.
- Please see the suggestions and questions for more.
其他意见或建议
Suggestions:
- The proofs for Theorem 3.1 and Lemma 3.2 can be made significantly shorter with the simple observation that when and are vectors. Then the proofs become single line.
Typos:
- vs. usage is not consistent.
Thank you for your thoughtful review, and valuable feedback which we believe significantly enhanced our work. We addressed your valuable suggestions, and clarifed of our novelty and contributions.
1. Novelty and Technical Contributions
We respectfully disagree with the assertion that our paper merely reuses existing matrix-sketching techniques. Our work introduces three key innovations:
- We provide the first rigorous, first-principles derivation of how k-space acquisition noise propagates through both the MRI physics model and nonlinear DL networks. Prior approaches have largely relied on Monte Carlo sampling—computationally expensive and lacking interpretability. Instead—our approach yields a scalable, interpretable solution with mathematical transparency grounded in imaging and statistical theory.
- We show that exact voxel-wise variance requires the full network Jacobian—computationally intractable in MRI. Our key innovation is a statistically rigorous unbiased estimator that efficiently approximates this covariance without explicitly forming the Jacobian. Crucially, we handle complex-valued signals, multi-coil encoding, physics-based forward models, and deep networks simultaneously—the first viable approach for quantifying acquisition-induced uncertainty in deep MRI reconstruction.
- We implement our estimator via an efficient and practical matrix sketching algorithm that probes the Jacobian with random-phase vectors. Our method leverages Jacobian-vector products, enabling fast, memory-efficient uncertainty quantification at scale, while being DL model agnostic. In practice, this removes a major bottleneck in bringing noise-aware reconstruction into clinical and research pipelines. This suite of theoretical, algorithmic, and applied innovations represents a substantive advancement in uncertainty quantification for DL-based MRI.
2. Motivation for Noise Variance Calculation
We appreciate the reviewer’s concern and agree that the clinical motivation for voxel-wise noise variance estimation merits clarification. Voxel-wise noise variance estimation is crucial in MRI because global metrics can obscure significant localized noise variations that impact diagnosis [1-3,8]. Studies show that localized noise impedes detection of subtle pathologies despite good global quality metrics [3,7]. Rubenstein et al. demonstrated that cartilage defects remain undetected with insufficient local SNR, even when global metrics appear acceptable [2]. Furthermore, global SNR and CNR don't consistently correlate with diagnostic accuracy [8]. This issue is particularly relevant in deep-learning MRI reconstructions, where spatially varying noise profiles create diagnostic uncertainties that global metrics fail to capture [4-7]. Our method efficiently provides accurate spatially resolved noise variance maps that highlight regions with elevated uncertainty, enabling radiologists to review diagnostically vulnerable areas and supporting better clinical decisions [6,7]. For additional evidence, we refer to 2. Utility of Noise Variance Estimation on Downstream Tasks in our response to reviewer viwd.
[1] Sijbers, MRI, 1998
[2] Rubenstein, AJR, 1997
[3] Lerski, MRI, 1993
[4] Dou, NMR, 2025
[6] Kiryu, Radiographics, 2023
[7] Knoll, MRM, 2020
[8] Ohlmann et al, Br J Radiol, 2016
3. Ablation Study on the choice of probing vectors
Thanks to your valuable inquiry, we now conducted an ablation study, and found that our theoretical findings (Appendix D) match with new empirical findings; Proposed random-phase vectors (test set NRMSE of 0.7 for knee, 0.5 for brain) yields lower errors than their Gaussian counterparts (1.1 for knee, 0.8 for brain), indicating lower estimator variance.
4. Length of proofs
We thank the reviewer for suggesting the identity to streamline our proofs. Applying this improved the brevity and conciseness.
5. Missing citations
We kindly refer the reviewer to rebuttal for reviewer q9G3 for discussions on 1. Extended Related Work and Clarification of our contributions, including Edupuganti et al. (2020) and Wen et al. (2024).
6. Error Bound for Diagonal Estimator
Our estimator (using complex random-phase sketches) satisfies:
\|r_m(\Sigma_x)-\operatorname{diag}(\Sigma_x)\|_2 \leq c\sqrt{\tfrac{\ln(2/\delta)}{m}}\|\overline{\Sigma}_x\|_F\quad\text{(w.p.}\geq 1-\delta)$$where $\overline{\Sigma}_x=\Sigma_x-\operatorname{diag}(\Sigma_x)$. - Error scales as $\mathcal{O}(1/\sqrt{m})$ with $\|\overline{\Sigma}_x\|_F$. - Large $\|A^H\|$ or $\|J_f\|$ increase off-diagonal coupling, amplifying $\|\overline{\Sigma}_x\|_F$; well-conditioned systems reduce it. *(Proof in revised manuscript.)* ## 7. We hypothesize that distinct MoDL noise map reflect coil geometry and undersampling patterns via the iterative CG data-consistency step, rather than being driven by image features. ## 8. - $()^*$: Complex conjugation for scalars - $()^H$: Hermitian operator for matricesI would like the thank the authors for their thorough responses. Some of my concerns and questions are alleviated. Please see below for further questions, comments and clarification:
- Unfortunately I will have to challenge the claims about technical contribution to some extent. I agree with the authors that the work provides a rigorous derivation of how k-space acquisition noise propagates through both the MRI physics model and nonlinear DL networks. In that sense, it is a novel application of matrix sketching tailored towards noise variance estimation in MRI reconstruction setting (which I have acknowledged before). The extent of technical contributions, however, is still mostly limited to crafting the correct matrix to sketch in my opinion (which turns out to be ). That said, I think that I was unfairly harsh in my initial review. The introduction of random complex phase vectors suits the particular problem nicely and the recent ablation study provided by the authors demonstrate its effectiveness. I will revise my score to 2 for now to reflect this change.
- Thank you for your explanation on why estimating noise variance is an important problem. I appreciate the problem setting more now.
- Thank you for ablating the choice of probing vectors. Previously it was not obvious to the reader how much better Rademacher probing vectors would be compared to Gaussian counterparts. The ablation suggests there is quite a bit of a gap.
- Could you please provide the proofs for the error bound? I couldn't find it in the PDF (not sure if it is already revised). If not, could you provide it in an anonymous link?
- Re notation consistency: I would like to apologize from the authors. In line 194, I misread it as thinking * is used instead of H.
We sincerely thank the reviewer for carefully revisiting their assessment and for acknowledging that our rebuttal effectively addressed their original concerns.
Regarding the theoretical error bounds and their proof, since manuscript updates were not permitted during the rebuttal stage, we now provide the requested detailed proof (which will be included upon publication) via the following anonymous link:
Finally, we would like to further clarify our technical contributions, especially regarding the notion that our method’s novelty primarily lies in "crafting the correct matrix to sketch."
Our work did not originate with the premise of tensor sketching. Instead, our estimator arose organically from rigorous first-principles derivations based explicitly on multi-coil MRI physics, complex-valued Fourier transforms, correlated noise modeling, and the nonlinearities inherent in deep-learning reconstruction architectures. It was only after establishing a robust theoretical foundation and confronting the computational complexities posed by our high-dimensional data (k-space: , images: , covariance and Jacobian matrices of dimension ) that we turned to scalable computational methods.
To address this significant computational barrier, we recognized the potential of leveraging Jacobian-vector products (JVPs) efficiently provided by modern deep learning frameworks. This insight led us to innovatively implement our estimator as a form of tensor-sketching approach, effectively harnessing JVP for computational efficiency and memory scalability. Thus, we did not simply select or "craft" a convenient matrix to sketch; rather, our final formulation emerged naturally as a necessary step to make the theoretically sound estimator practically scalable.
Furthermore, this practical realization itself is a substantive technical advance. Our estimator achieves at least an order-of-magnitude computational speed-up compared to traditional Monte Carlo methods, avoids excessive memory overhead from storing multiple reconstructed images, and provides unbiased voxel-wise noise variance estimates scalable to large clinical imaging problems. We believe these practical gains directly addresses longstanding computational and interpretability bottlenecks in noise quantification for deep MRI reconstructions.
We further highlight that many successful matrix-sketching works, particularly in the context of high-dimensional data, involve not just choosing matrices to sketch, but rigorously deriving these matrices from domain-specific knowledge and demonstrating their statistical and practical effectiveness. This is exactly what we have done here by synthesizing multi-coil MRI physics, Fourier transforms, undersampling, and deep-learning reconstruction theory.
While sketching methods have appeared in the ML literature primarily to accelerate optimization or approximate matrix factorizations, as noted in our original Related Work section, prior studies typically consider significantly simpler scenarios. These include simpler settings-MLP architectures or convex optimization problems [1,2] or/with lower-dimensional matrices [4,5], often assuming pre-sketched observations readily available [3,6]. In stark contrast, our work rigorously derives and enables explicit noise propagation through large-scale, complex-valued convolutional neural networks that incorporate iterative, physics-driven data-consistency operations—a setting previously unaddressed by existing sketching methods. Furthermore, we provide detailed practical instructions on performing this high-dimensional sketching efficiently, filling a critical gap between theoretical concepts and applied implementation.
Taken together, we believe our approach bridges theoretical insights from numerical linear algebra and machine learning (tensor-sketching) with practical computational solutions for complex medical imaging problems (MRI reconstruction). This enables accurate estimation of voxel-level uncertainty unprecedented scale, thus offering substantial advancements relevant to both tensor sketching and MRI reconstruction communities.
[1] Pilanci, M., Iterative Hessian Sketch: Fast and Accurate Solution Approximation for Constrained Least-Squares. JMLR 2016
[2] Fishkov, A. Scalable Computation of Prediction Intervals for Neural Networks via Matrix Sketching. arXiv (2022).
[3] Bahmani, S. Sketching for Simultaneously Sparse and Low-Rank Covariance Matrices. Proceedings of Asilomar Conference. 2015
[4] Lopes, M. E., Bootstrapping the Operator Norm: Error Estimation for Randomized PCA and Covariance Matrix Sketching. Annals of Statistics. 2022
[5] Woodruff, D. P. Sketching as a Tool for Numerical Linear Algebra. Foundations and Trends in Theoretical Computer Science. (2014)
[6] Sesia, M. Conformal Frequency Estimation with Sketched Data. JMLR (2022)
This paper presents a compelling application of sketching techniques to the problem of noise estimation in MRI reconstruction. All three reviewers agreed on the strength of the application and the thoroughness of the evaluation within the MRI domain, with suggestions for improvement including evaluation on other modalities. However, a key difference in opinion emerged, with reviewer SQro questioning the technical novelty of the method, arguing it primarily builds upon existing sketching techniques. I agree, while the application is strong, the concerns raised by reviewer SQro regarding the limited technical novelty significantly undermine the value of the paper for the broader ICML community.