{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T02:24:32Z","timestamp":1790648672843,"version":"4.1.0"},"reference-count":56,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1618485"],"award-info":[{"award-number":["1618485"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>Sparse dictionary learning is a popular method for representing signals as linear combinations of a few elements from a dictionary that is learned from the data. In the classical setting, signals are represented as vectors, and the dictionary learning problem is posed as a matrix factorization problem where the data matrix is approximately factorized into a dictionary matrix and a sparse matrix of coefficients. However, in many applications in computer vision and medical imaging, signals are better represented as matrices or tensors (e.g., images or videos). In such cases, instead of learning a large-scale dictionary tensor, it may be beneficial to exploit the multidimensional structure of the data to learn a more compact representation. One such approach is separable dictionary learning, where one learns separate dictionaries for different dimensions of the data (e.g., spatial and temporal dimensions of a video). However, while there has been significant recent work on separable dictionary learning, typical formulations involve solving a nonconvex optimization problem; thus, guaranteeing global optimality remains a challenge. In this work, we propose a framework that builds upon recent developments in matrix factorization to provide theoretical and numerical guarantees of global optimality for separable dictionary learning. Specifically, we prove that local minima are guaranteed to be global when some dictionary atoms and the corresponding coefficients are zero. We also propose an algorithm to find such a globally optimal solution, which alternates between following local descent steps and checking a certificate for global optimality. We illustrate our approach on diffusion magnetic resonance imaging (dMRI) data, a medical imaging modality that measures water diffusion along multiple angular directions in every voxel of a magnetic resonance imaging volume. State-of-the-art methods in dMRI either learn dictionaries only for the angular domain of the signals or in some cases learn spatial and angular dictionaries independently. In this work, we apply the proposed separable dictionary learning framework to learn spatial and angular dMRI dictionaries jointly and provide preliminary validation on denoising phantom and real dMRI brain data.<\/jats:p>","DOI":"10.1137\/18m121976x","type":"journal-article","created":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T14:44:10Z","timestamp":1574779450000},"page":"1967-2008","source":"Crossref","is-referenced-by-count":8,"title":["Global Optimality in Separable Dictionary Learning with Applications to the Analysis of Diffusion MRI"],"prefix":"10.1137","volume":"12","author":[{"given":"Evan","family":"Schwab","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Benjamin D.","family":"Haeffele","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ren\u00e9","family":"Vidal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicolas","family":"Charon","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,11,26]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1002\/mrm.22365","volume":"64","author":"Aganj I.","year":"2010","journal-title":"Magn. 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Magnin,\n                      Sparse representation based MRI denoising with total variation\n                      , in Proceedings of the 9th International Conference on Signal Processing, 2008, pp. 2154-2157.","DOI":"10.1109\/ICOSP.2008.4697573"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/mrm.24505"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/080738970"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.10.002"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-009-9045-5"},{"key":"atypb11","first-page":"639","author":"Cheng J.","year":"2013","journal-title":"Springer"},{"key":"atypb12","first-page":"174","author":"Cheng J.","year":"2015","journal-title":"Springer"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2017.2681159"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896305696"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2008.2004424"},{"key":"atypb16","first-page":"4610","author":"Dohmatob E.","year":"2016","journal-title":"Proceedings of the Advances in Neural Information Processing Systems Conference"},{"key":"atypb17","unstructured":"G. Duan, H. Wang, Z. Liu, J. Deng, and Y.W. Chen,\n                      K-CPD: Learning of overcomplete dictionaries for tensor sparse coding\n                      , in Proceedings of the 21st International Conference on Pattern Recognition (ICPR), 2012, pp. 493-496."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2006.881969"},{"key":"atypb19","first-page":"2443","author":"Engan K.","year":"1999","journal-title":"Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2011.01.032"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"M. Ghassemi, Z. Shakeri, A. D. Sarwate, and W. U. Bajwa,\n                      STARK: Structured Dictionary Learning Through Rank-One Tensor Recovery\n                      , preprint,arXiv:1711.04887, 2017.","DOI":"10.1109\/CAMSAP.2017.8313164"},{"key":"atypb22","first-page":"608","volume":"47","author":"Goh A.","year":"2011","journal-title":"Neuroimage"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.media.2013.08.006"},{"key":"atypb24","first-page":"35","author":"Gupta K.","year":"2016","journal-title":"Springer"},{"key":"atypb25","first-page":"453","author":"Gupta K.","year":"2017","journal-title":"Springer"},{"key":"atypb26","unstructured":"B. Haeffele and R. Vidal,\n                      Global Optimality in Tensor Factorization, Deep Learning, and Beyond\n                      ,arXiv:1506.07540, 2015."},{"key":"atypb27","first-page":"2007","author":"Haeffele B.","year":"2014","journal-title":"Proceedings of the International Conference on Machine Learning"},{"key":"atypb28","author":"Haeffele B. 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Charon,\n                      ($k, q$)-compressed sensing for dMRI with joint spatial-angular sparsity prior\n                      , in Proceedings of the MICCAI Workshop on Computational Diffusion MRI, 2017.","DOI":"10.1007\/978-3-319-73839-0_2"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1016\/j.media.2018.05.002"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1016\/j.media.2016.02.010"},{"key":"atypb47","first-page":"121","author":"Stevens A.","year":"2017","journal-title":"Proceedings of the Conference on Artificial Intelligence and Statistics"},{"key":"atypb48","first-page":"619","author":"Sun J.","year":"2013","journal-title":"Springer"},{"key":"atypb49","first-page":"182","author":"Trist\u00e1n-Vega A.","year":"2011","journal-title":"Springer"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1016\/S0896-6273(03)00758-X"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2010.2050625"},{"key":"atypb52","first-page":"940","author":"Ye W.","year":"2012","journal-title":"Proceedings of the IEEE International Symposium on Biomedical Imaging, Springer"},{"key":"atypb53","first-page":"650","author":"Yellin F.","year":"2017","journal-title":"Proceedings of the IEEE International Symposium on Biomedical Imaging"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1109\/ICSP.2016.7877957"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2016.09.099"},{"key":"atypb56","first-page":"1","author":"Zubair S.","year":"2013","journal-title":"Proceedings of the IEEE 18th International Conference on Digital Signal Processing (DSP)"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M121976X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:28:58Z","timestamp":1787322538000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M121976X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":56,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M121976X"],"URL":"https:\/\/doi.org\/10.1137\/18m121976x","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}