{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T15:37:20Z","timestamp":1785253040556,"version":"3.55.0"},"reference-count":55,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T00:00:00Z","timestamp":1756857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Unrolled networks have become prevalent in various computer vision and imaging tasks. Although they have demonstrated remarkable efficacy in solving specific computer vision and computational imaging tasks, their adaptation to other applications presents considerable challenges. This is primarily due to the multitude of design decisions that practitioners working on new applications must navigate, each potentially affecting the network\u2019s overall performance. These decisions include selecting the optimization algorithm, defining the loss function, and determining the deep architecture, among others. Compounding the issue, evaluating each design choice requires time-consuming simulations to train, fine-tune the neural network, and optimize its performance. As a result, the process of exploring multiple options and identifying the optimal configuration becomes time-consuming and computationally demanding. The main objectives of this paper are (1) to unify some ideas and methodologies used in unrolled networks to reduce the number of design choices a user has to make, and (2) to report a comprehensive ablation study to discuss the impact of each of the choices involved in designing unrolled networks and present practical recommendations based on our findings. We anticipate that this study will help scientists and engineers to design unrolled networks for their applications and diagnose problems within their networks efficiently.<\/jats:p>","DOI":"10.3390\/e27090929","type":"journal-article","created":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T09:18:57Z","timestamp":1756977537000},"page":"929","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Comprehensive Examination of Unrolled Networks for Solving Linear Inverse Problems"],"prefix":"10.3390","volume":"27","author":[{"given":"Yuxi","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xi","family":"Chen","sequence":"additional","affiliation":[{"name":"Electrical and Computer Engineering Department, Rutgers University, New Brunswick, NJ 08854, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arian","family":"Maleki","sequence":"additional","affiliation":[{"name":"Department of Statistics, Columbia University, New York, NY 10027, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shirin","family":"Jalali","sequence":"additional","affiliation":[{"name":"Electrical and Computer Engineering Department, Rutgers University, New Brunswick, NJ 08854, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1182","DOI":"10.1002\/mrm.21391","article-title":"Sparse MRI: The application of compressed sensing for rapid MR imaging","volume":"58","author":"Lustig","year":"2007","journal-title":"Magn. Reson. Med."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1109\/MSP.2007.914731","article-title":"An Introduction To Compressive Sampling","volume":"25","author":"Candes","year":"2008","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1109\/MSP.2007.4286571","article-title":"Compressive sensing [lecture notes]","volume":"24","author":"Baraniuk","year":"2007","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"18914","DOI":"10.1073\/pnas.0909892106","article-title":"Message-passing algorithms for compressed sensing","volume":"106","author":"Donoho","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5117","DOI":"10.1109\/TIT.2016.2556683","article-title":"From denoising to compressed sensing","volume":"62","author":"Metzler","year":"2016","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Venkatakrishnan, S.V., Bouman, C.A., and Wohlberg, B. (2013, January 3\u20135). Plug-and-Play priors for model based reconstruction. Proceedings of the 2013 IEEE Global Conference on Signal and Information Processing, Austin, TX, USA.","DOI":"10.1109\/GlobalSIP.2013.6737048"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.acha.2015.03.003","article-title":"From compression to compressed sensing","volume":"40","author":"Jalali","year":"2016","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_9","first-page":"343","article-title":"An efficient algorithm for compression-based compressed sensing","volume":"8","author":"Beygi","year":"2019","journal-title":"Inf. Inference J. IMA"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Romano, Y., Elad, M., and Milanfar, P. (2017). The Little Engine that Could: Regularization by Denoising (RED). arXiv.","DOI":"10.1137\/16M1102884"},{"key":"ref_11","unstructured":"Lee, D., Sugiyama, M., Luxburg, U., Guyon, I., and Garnett, R. (2016). Deep ADMM-Net for Compressive Sensing MRI. Advances in Neural Information Processing Systems, Curran Associates, Inc."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mousavi, A., and Baraniuk, R.G. (2017, January 5\u20139). Learning to invert: Signal recovery via Deep Convolutional Networks. Proceedings of the 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), New Orleans, LA, USA.","DOI":"10.1109\/ICASSP.2017.7952561"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chang, J.R., Li, C.L., Poczos, B., Vijaya Kumar, B., and Sankaranarayanan, A.C. (2017, January 22\u201329). One Network to Solve Them All\u2014Solving Linear Inverse Problems Using Deep Projection Models. Proceedings of the 2017 IEEE International Conference on Computer Vision (ICCV), Venice, Italy.","DOI":"10.1109\/ICCV.2017.627"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Mousavi, A., Dasarathy, G., and Baraniuk, R.G. (2017). DeepCodec: Adaptive Sensing and Recovery via Deep Convolutional Neural Networks. arXiv.","DOI":"10.1109\/ALLERTON.2017.8262812"},{"key":"ref_15","unstructured":"Metzler, C.A., Mousavi, A., and Baraniuk, R.G. (2017). Learned D-AMP: Principled Neural Network based Compressive Image Recovery. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1109\/MSP.2017.2739299","article-title":"Convolutional Neural Networks for Inverse Problems in Imaging: A Review","volume":"34","author":"McCann","year":"2017","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhang, J., and Ghanem, B. (2018). ISTA-Net: Interpretable Optimization-Inspired Deep Network for Image Compressive Sensing. arXiv.","DOI":"10.1109\/CVPR.2018.00196"},{"key":"ref_18","unstructured":"Diamond, S., Sitzmann, V., Heide, F., and Wetzstein, G. (2018). Unrolled Optimization with Deep Priors. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1109\/TMI.2017.2760978","article-title":"A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction","volume":"37","author":"Schlemper","year":"2018","journal-title":"IEEE Trans. Med Imaging"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gilton, D., Ongie, G., and Willett, R. (2019). Neumann Networks for Inverse Problems in Imaging. arXiv.","DOI":"10.1109\/TCI.2019.2948732"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1109\/TMI.2018.2865356","article-title":"MoDL: Model-Based Deep Learning Architecture for Inverse Problems","volume":"38","author":"Aggarwal","year":"2019","journal-title":"IEEE Trans. Med Imaging"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1109\/JSAIT.2020.2991563","article-title":"Deep Learning Techniques for Inverse Problems in Imaging","volume":"1","author":"Ongie","year":"2020","journal-title":"IEEE J. Sel. Areas Inf. Theory"},{"key":"ref_23","unstructured":"Veen, D.V., Jalal, A., Soltanolkotabi, M., Price, E., Vishwanath, S., and Dimakis, A.G. (2020). Compressed Sensing with Deep Image Prior and Learned Regularization. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1109\/TCI.2021.3118944","article-title":"Deep Equilibrium Architectures for Inverse Problems in Imaging","volume":"7","author":"Gilton","year":"2021","journal-title":"IEEE Trans. Comput. Imaging"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1109\/TCI.2021.3094714","article-title":"Model Adaptation for Inverse Problems in Imaging","volume":"7","author":"Gilton","year":"2021","journal-title":"IEEE Trans. Comput. Imaging"},{"key":"ref_26","first-page":"13242","article-title":"Stochastic Solutions for Linear Inverse Problems using the Prior Implicit in a Denoiser","volume":"Volume 34","author":"Ranzato","year":"2021","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1109\/JSAIT.2022.3207109","article-title":"Denoising Generalized Expectation-Consistent Approximation for MR Image Recovery","volume":"3","author":"Shastri","year":"2022","journal-title":"IEEE J. Sel. Areas Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Rout, L., Chen, Y., Kumar, A., Caramanis, C., Shakkottai, S., and Chu, W.S. (2023). Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusion. arXiv.","DOI":"10.1109\/CVPR52733.2024.00905"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1109\/MSP.2022.3208394","article-title":"Physics-Inspired Compressive Sensing: Beyond deep unrolling","volume":"40","author":"Zhang","year":"2023","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1109\/MSP.2022.3199595","article-title":"Plug-and-play methods for integrating physical and learned models in computational imaging: Theory, algorithms, and applications","volume":"40","author":"Kamilov","year":"2023","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_31","unstructured":"Gan, W., Hu, Y., Liu, J., An, H., and Kamilov, U. (2023, January 10\u201316). Block coordinate plug-and-play methods for blind inverse problems. Proceedings of the NIPS\u201923: 37th International Conference on Neural Information Processing Systems, New Orleans, LA, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1109\/OJSP.2024.3375276","article-title":"PtychoDV: Vision Transformer-Based Deep Unrolling Network for Ptychographic Image Reconstruction","volume":"5","author":"Gan","year":"2024","journal-title":"IEEE Open J. Signal Process."},{"key":"ref_33","unstructured":"Hu, Y., Peng, A., Gan, W., Milanfar, P., Delbracio, M., and Kamilov, U.S. (2024). Stochastic Deep Restoration Priors for Imaging Inverse Problems. arXiv."},{"key":"ref_34","unstructured":"Chung, H., Lee, S., and Ye, J.C. (2024). Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems. arXiv."},{"key":"ref_35","unstructured":"Chen, B., and Zhang, J. (2024). Practical Compact Deep Compressed Sensing. arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Chen, B., Zhang, X., Liu, S., Zhang, Y., and Zhang, J. (2024). Self-Supervised Scalable Deep Compressed Sensing. arXiv.","DOI":"10.1007\/s11263-024-02209-1"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s10334-024-01207-1","article-title":"MRI recovery with self-calibrated denoisers without fully-sampled data","volume":"38","author":"Shafique","year":"2024","journal-title":"Magn. Reson. Mater. Phys. Biol. Med."},{"key":"ref_38","unstructured":"Chen, Y., Chen, X., Jalali, S., and Maleki, A. (August, January 28). Deep Memory Unrolled Networks for Solving Imaging Linear Inverse Problems. Proceedings of the 15th International Conference on Sampling Theory and Applications, Vienna, Austria."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Kulkarni, K., Lohit, S., Turaga, P., Kerviche, R., and Ashok, A. (2016, January 27\u201330). ReconNet: Non-Iterative Reconstruction of Images from Compressively Sensed Measurements. Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.55"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.1109\/TBME.2020.3020741","article-title":"Do CNNs Solve the CT Inverse Problem?","volume":"68","author":"Sidky","year":"2021","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_41","unstructured":"Gregor, K., and LeCun, Y. (2010, January 24\u201326). Learning fast approximations of sparse coding. Proceedings of the 27th International Conference on Machine Learning, Madison, WI, USA. ICML\u201910."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Monga, V., Li, Y., and Eldar, Y.C. (2020). Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing. arXiv.","DOI":"10.1109\/MSP.2020.3016905"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"6735","DOI":"10.1109\/TIT.2017.2726549","article-title":"Compression-based compressed sensing","volume":"63","author":"Rezagah","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_44","unstructured":"Mardani, M., Sun, Q., Vasawanala, S., Papyan, V., Monajemi, H., Pauly, J., and Donoho, D. (2018). Neural Proximal Gradient Descent for Compressive Imaging. arXiv."},{"key":"ref_45","unstructured":"Li, Y., Bar-Shira, O., Monga, V., and Eldar, Y.C. (2021). Deep Algorithm Unrolling for Biomedical Imaging. arXiv."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","article-title":"A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems","volume":"2","author":"Beck","year":"2009","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"80373","DOI":"10.1007\/s11042-024-18724-9","article-title":"GSISTA-Net: Generalized structure ISTA networks for image compressed sensing based on optimized unrolling algorithm","volume":"83","author":"Zeng","year":"2024","journal-title":"Multimed. Tools Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1487","DOI":"10.1109\/TIP.2020.3044472","article-title":"AMP-Net: Denoising-Based Deep Unfolding for Compressive Image Sensing","volume":"30","author":"Zhang","year":"2021","journal-title":"IEEE Trans. Image Process."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"49112","DOI":"10.1109\/ACCESS.2020.2980266","article-title":"Convolutional Neural Networks With Intermediate Loss for 3D Super-Resolution of CT and MRI Scans","volume":"8","author":"Georgescu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Mou, C., Wang, Q., and Zhang, J. (2022, January 18\u201324). Deep Generalized Unfolding Networks for Image Restoration. Proceedings of the 2022 IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR), New Orleans, LA, USA.","DOI":"10.1109\/CVPR52688.2022.01688"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Szegedy, C., Vanhoucke, V., Ioffe, S., Shlens, J., and Wojna, Z. (2016, January 27\u201330). Rethinking the inception architecture for computer vision. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.308"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"3142","DOI":"10.1109\/TIP.2017.2662206","article-title":"Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising","volume":"26","author":"Zhang","year":"2017","journal-title":"IEEE Trans. Image Process."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"He, K., Zhang, X., Ren, S., and Sun, J. (2015). Deep Residual Learning for Image Recognition. arXiv.","DOI":"10.1109\/CVPR.2016.90"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"He, K., Zhang, X., Ren, S., and Sun, J. (2016). Identity Mappings in Deep Residual Networks. arXiv.","DOI":"10.1007\/978-3-319-46493-0_38"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1007\/s13244-018-0639-9","article-title":"Convolutional neural networks: An overview and application in radiology","volume":"9","author":"Yamashita","year":"2018","journal-title":"Insights Imaging"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/9\/929\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:38:58Z","timestamp":1760035138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/9\/929"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,3]]},"references-count":55,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["e27090929"],"URL":"https:\/\/doi.org\/10.3390\/e27090929","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,3]]}}}