Title: System-Embedded Diffusion Bridge Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: diffusion

generative

inverse

problems

image

translation

restoration

linear

system

bridge
Abstract: Solving inverse problems—recovering signals from incomplete or noisy measurements—is fundamental in science and engineering. Score-based generative models (SGMs) have recently emerged as a powerful framework for this task. Two main paradigms have formed: unsupervised approaches that adapt pretrained generative models to inverse problems, and supervised bridge methods that train stochastic processes conditioned on paired clean and corrupted data. While the former typically assume knowledge of the measurement model, the latter have largely overlooked this structural information. We introduce System-embedded Diffusion Bridge Models (SDBs), a new class of supervised bridge methods that explicitly embed the known linear measurement system into the coefficients of a matrix-valued SDE. This principled integration yields consistent improvements across diverse linear inverse problems and demonstrates robust generalization under system misspecification between training and deployment, offering a promising solution to real-world applications.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work identifies that many existing supervised methods for inverse problem solving ignore the a priori knowledge of the degradation operator. Motivated by this findings, the authors develop a new type of Schrödinger bridge model that takes the degradation operator into account for choosing the form of the SDE applied in training. In several experiments, the authors demonstrate the superiority of their approach over previous methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The paper’s motivation is clearly established, and the writing is easy to follow. I find that system-embedded diffusion bridges are an insightful extension to Schrödinger bridge models, making the most of the a priori knowledge of the inverse problem.\n* The evaluations are sound, and the results shown in the paper are encouraging. Moreover, SDB’s generalization capabilities are non-trivial.\n\nWeaknesses:\n* While the empirical results clearly show an improvement, I believe the manuscript lacks a theoretical analysis, hypothesis or discussion for why methods that embed the measurement information into the sampling scheme are desired. Many fields in deep learning have shown that hand-crafted priors may offer only limited benefits over systems that purely model the training data, for fully converged networks with sufficient data. The authors should take extra care to demonstrate that the increase in performance is solely driven by the measurement-embedding technique, and disentangle all other differences.\n* I believe that a discussion of the Perception-Distortion Tradeoff [1] is warranted for the comparisons made in this work.\n* Minor: I believe the paper would be easier to follow if a training and sampling algorithm was included.\n\n[1] Blau, Yochai, and Tomer Michaeli. \"The perception-distortion tradeoff.\" Proceedings of the IEEE conference on computer vision and pattern recognition. 2018.\n\nQUESTIONS:\nPlease see weaknesses.\nAdditionally:\n* Could the proposed method be applied such that several degradations are handled by a single model?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces System-embedded Diffusion Bridge Models (SDBs), a framework for supervised diffusion bridge methods that explicitly embeds the known linear measurement system into the coefficients of matrix-valued stochastic differential equations. The key contribution lies in leveraging the range-nullspace decomposition of linear operators to design specialized diffusion processes that handle range and null space components differently. The authors propose three variants (SDB-SB, SDB-VP, SDB-VE) and demonstrate improvements over existing bridge methods (I2SB, IR-SDE, GOUB, DDBM) across four inverse problems: image inpainting, super-resolution, CT reconstruction, and MRI reconstruction.\n\nSTRENGTHS AND WEAKNESSES:\npros:\n1. The principled embedding of measurement systems into matrix-valued SDE coefficients represents a theoretical advancement beyond scalar SDEs. The connection to optimal transport theory through Theorem 2 provides solid mathematical foundation.\n2. The paper evaluates across diverse inverse problems with varying system complexities, from simple inpainting to complex medical imaging tasks (CT/MRI reconstruction), demonstrating broad applicability.\n3. The systematic evaluation under system misspecification (Section 5.2) addresses a critical practical concern, showing SDB's superior generalization when training and deployment parameters differ.\n4.  The range-nullspace decomposition provides intuitive understanding of how SDB handles known (range) and missing (null space) information differently, which is conceptually appealing for inverse problems.\n4. SDB consistently outperforms baselines across metrics, with particularly notable improvements in medical imaging tasks where the gains are substantial.\n\ncons:\n1. While Theorem 1 from Tivnan et al. [2025] provides the foundation, the paper lacks theory analysis of the novel matrix-valued formulation. The conditions under which the Magnus expansion approximation becomes equality (commuting matrices) are restrictive and not proper discussed.\n2. The framework is limited to linear Gaussian measurement models. While the authors acknowledge this limitation, many real-world inverse problems involve nonlinear degradations that cannot be addressed by this approach.\n3. The paper does not analyze the computational overhead of the matrix-valued SDE formulation compared to scalar methods. The range-nullspace decomposition and pseudoinverse computations may introduce significant computational costs and bias.\n4. While the paper compares against recent bridge methods, it lacks comparison with state-of-the-art unsupervised methods like DPS, πGDM variants, or recent optimal transport approaches that don't require paired training data.\n5. The authors use simple linear scheduling for range and null space coefficients, missing opportunities for more sophisticated designs that could better leverage the dual-process structure.\n\nQUESTIONS:\nI would like to know following the cons:\n\n1. Thm1 applicability. The Magnus expansion approximation becomes exact when drift matrices commute. How restrictive is this condition for the proposed SDB variants? Do the specific choices of parameters in equations (8a-8b) satisfy this condition?\n2. Pseudoinverse stability. For rank-deficient measurement matrices A, how does numerical stability of the pseudoinverse computation affect SDB performance? Have the authors considered regularized pseudoinverses for ill-conditioned systems?\n3. What is the computational complexity of the matrix-valued SDE compared to scalar formulations? How does this scale with problem dimension d and measurement dimension m?\n4. The authors mention local linearization as a potential workaround for nonlinear measurement systems. Can you provide more details on how this would work in practice and its limitations? e.g., failure cases\n5. The current formulation treats range and null spaces independently. Could there be benefits to introducing coupling terms between these spaces?\n\nLIMITATIONS:\nplease refer to the questions and cons.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work proposes System-embedded Diffusion Bridge Models (SDBs) that embed the known linear measurement system into the coefficients of the SDE process for solving linear inverse problems. Experiments show the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength:**\n1. This work provides a theoretically grounded method for embedding measurement system into the inverse process, filling the gap of prior bridge methods;\n2. The proposed SDB is shown to be robust when the measurement system differs between training and deployment;\n3. Experiments shown the effectiveness of the proposed method across diverse domains, including natural images and medical images;\n\n**Weaknesses:**\n1. While the paper includes some evaluations for system parameter changes, these are controlled perturbations. It would be valuable to explore settings where the operator A or noise model are misspecified in more adversarial or realistic ways (e.g., using a wrong noise distribution, or a blurred operator);\n2. As the proposed SDB requires paired data for supervised training, which may not always be available, especially for rare measurements or privacy-sensitive domains. It does not address scenarios where only unpaired data is accessible, limiting applicability compared to plug-and-play or unsupervised diffusion approaches.\n3. There is little ablation on key design choices such as the choice of noise schedules, the effect of imperfect A, or the benefit of each SDB component.\n4. Some theoretically motivated variants (SDB-SB, SDB-VP, SDB-VE) are included, but the empirical results do not fully disentangle when each variant is preferable. For instance, SDB-SB generally performs best, but it is not entirely clear under what circumstances SDB-VP/VE might be better or whether their variance schedules are optimal for the tasks at hand.\n\nQUESTIONS:\n1. The experiments, while covering standard benchmarks, are all 2D, moderately sized, and with relatively controlled corruptions. It would strengthen the work to see results on more diverse and challenging real-world scenarios (e.g., more realistic noise, higher noise levels, extreme undersampling, or out-of-distribution corruptions).\n\nLIMITATIONS:\nThe authors have provided the limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a supervised framework for solving linear inverse problems using score-based generative models. The core motivation is to embedding known information about the linear measurement system into the diffusion SDEs, which contrasts with previous diffusion bridge methods that often remain agnostic to this structural information. Experimental results demonstrate the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The writing is clear and the motivations are promising.\n\n2. The construction of the system-embedded SDEs is reasonable and novel.\n\n3. Experimental results demonstrate the effectiveness of embedding measurement-system into diffusion SDEs.\n\nWeaknesses:\n\n1. The method relies on pseudo-inverse of the forward measurement matrix A. In the experiments the author use SVD to compute the pseudo-inverse.  However, for some linear measurement system, SVD is hard, e.g., motion deburing [R1]. This could limits the practicality of the proposed method.\n\n2. I believe that the conventional datasets for diffusion-based solvers (ImageNet, FFHQ) [R1，R2] should be included to better demonstrate the effectiveness of the proposed method.\n\n3. It is not immediately clear how the proposed model ensures sampling from the correct conditional distribution $p(x|y)$. The authors should provide further clarification or theoretical justification on this point.\n\nReference:\n\n[R1] Chung, Hyungjin, et al. \"Diffusion posterior sampling for general noisy inverse problems.\" arXiv preprint arXiv:2209.14687 (2022).\n\n[R2] Zhang, Bingliang, et al. \"Improving diffusion inverse problem solving with decoupled noise annealing.\" Proceedings of the Computer Vision and Pattern Recognition Conference. 2025.\n\nQUESTIONS:\nPlease refer to the section of Weaknesses.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}