{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:39:00Z","timestamp":1787323140358,"version":"3.56.0"},"reference-count":72,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1738003"],"award-info":[{"award-number":["DMS-1738003"]}],"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":[[2020,1]]},"abstract":"<jats:p>This paper deals with tensor completion for the solution of multidimensional inverse problems arising in nuclear magnetic resonance (NMR) relaxometry. We study the problem of reconstructing an approximately low-rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, nonuniform sampling strategies, and parameter selection methods are developed in this context. In particular, we derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property-based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by nonuniform sampling from a Parseval tight frame. The proposed algorithm is then applied to the setting of nuclear magnetic resonance relaxometry, for both simulated and experimental data. We compare our results with basis pursuit as well as with the state-of-the-art nonsubsampled data acquisition and reconstruction approach. Our experiments indicate that tensor recovery promises to significantly accelerate $N$-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments to be performed. Our methods could also be applied to other similar inverse problems arising in machine learning, signal and image processing, and computer vision.<\/jats:p>","DOI":"10.1137\/18m1193037","type":"journal-article","created":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T14:33:58Z","timestamp":1582122838000},"page":"176-213","source":"Crossref","is-referenced-by-count":3,"title":["$N$-Dimensional Tensor Completion for Nuclear Magnetic Resonance Relaxometry"],"prefix":"10.1137","volume":"13","author":[{"given":"Ariel","family":"Hafftka","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wojciech","family":"Czaja","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hasan","family":"Celik","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard G.","family":"Spencer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,2,19]]},"reference":[{"key":"atypb1","first-page":"380","volume":"48","author":"Arns C. H.","year":"2007","journal-title":"Petrophyics"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmr.2015.04.002"},{"key":"atypb3","unstructured":"R. Baraniuk, M. Davenport, M. Duarte, and C. Hegde,\n                      An Introduction to Compressive Sensing\n                      , OpenStax CNX, 2014."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2013.2278516"},{"key":"atypb5","first-page":"247","author":"Benedetto J. J.","year":"1994","journal-title":"Boca Raton, FL"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/cmr.a.21263"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1186\/1754-6834-6-55"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1021\/jf103384d"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1002\/mrm.25111"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1002\/mrm.25457"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2015.10.034"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"R. W. Brown, Y.C. N. Cheng, E. M. Haacke, M. R. Thompson, and R. Venkatesan,\n                      Magnetic Resonance Imaging: Physical Principles and Sequence Design\n                      , Wiley, Hoboken, NJ, 2014.","DOI":"10.1002\/9781118633953"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/23\/3\/008"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.862083"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.858979"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.885507"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmr.2013.07.008"},{"key":"atypb18","unstructured":"A. Cloninger,\n                      Exploiting Data-Dependent Structure for Improving Sensor Acquisition and Integration\n                      , Ph.D. Thesis, University of Maryland, College Park, MD, 2014."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/130932168"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01404567"},{"key":"atypb21","first-page":"1","author":"Davenport M. A.","year":"2012","journal-title":"Cambridge"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-015-9280-x"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.871582"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"Y. Eldar and G. Kutyniok,\n                      Compressed Sensing: Theory and Applications\n                      , Cambridge University Press, Cambridge, 2012.","DOI":"10.1017\/CBO9780511794308"},{"key":"atypb25","unstructured":"R. R. Ernst, G. Bodenhausen, and A. Wokaun,\n                      Principles of Nuclear Magnetic Resonance in One and Two Dimensions\n                      , Oxford University Press, Oxford, 1988."},{"key":"atypb26","first-page":"1043","author":"Fazel M.","year":"2008","journal-title":"NJ"},{"key":"atypb27","first-page":"241","volume":"45","author":"Freedman R.","year":"2004","journal-title":"Petrophysics"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/27\/2\/025010"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.1979.10489751"},{"key":"atypb30","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Tensor Spaces and Numerical Tensor Calculus\n                      , Springer, Berlin, 2012.","DOI":"10.1007\/978-3-642-28027-6"},{"key":"atypb31","unstructured":"A. Hafftka,\n                      Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry\n                      , Ph.D. Thesis, University of Maryland, College Park, MD, 2016."},{"key":"atypb32","first-page":"367","author":"Hafftka A.","year":"2015","journal-title":"NJ"},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"P. C. Hansen,\n                      Discrete Inverse Problems\n                      , SIAM, Philadelphia, 2010.","DOI":"10.1137\/1.9780898718836"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1145\/2512329"},{"key":"atypb35","doi-asserted-by":"crossref","unstructured":"B. Hills,\n                      Relaxometry: Two-dimensional methods\n                      , in Encyclopedia of Magnetic Resonance, Online, Wiley, New York, 2009.","DOI":"10.1002\/9780470034590.emrstm1042"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmr.2008.03.003"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1002\/nbm.3083"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1006\/jmre.1997.1123"},{"key":"atypb39","unstructured":"J. Keeler,\n                      Understand NMR Spectroscopy\n                      , 2nd ed., Wiley, Chichester, England, 2010."},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"atypb41","first-page":"836","volume":"26","author":"Krishnamurthy A.","year":"2013","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"atypb42","doi-asserted-by":"crossref","unstructured":"A. J. Laub,\n                      Matrix Analysis for Scientists and Engineers\n                      , SIAM, Philadelphia, 2005.","DOI":"10.1137\/1.9780898717907"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2012.39"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2013.2245416"},{"key":"atypb45","first-page":"2551","volume":"27","author":"Liu Y.","year":"2015","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1109\/TCYB.2014.2374695"},{"key":"atypb47","first-page":"1638","volume":"24","author":"Liu Y.-K.","year":"2011","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"atypb48","unstructured":"M. Lustig,\n                      SPARSE MRI\n                      , Ph.D. Thesis, Stanford University, Stanford, CA, 2008."},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1002\/mrm.21391"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2007.914728"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1002\/mrm.1910310614"},{"key":"atypb52","first-page":"73","volume":"32","author":"Mu C.","year":"2014","journal-title":"Proc. Mach. Learn. Res."},{"key":"atypb53","unstructured":"H. Rauhut, R. Schneider, and Z. Stojanac,\n                      Low rank tensor recovery via iterative hard thresholding\n                      , in Proceedings of the 10th International Conference on Sampling Theory and Applications, 2013."},{"key":"atypb54","first-page":"419","author":"Rauhut H.","year":"2015","journal-title":"Basel"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2017.02.028"},{"key":"atypb56","unstructured":"H. Rauhut and Z. Stojanac,\n                      Tensor Theta Norms And Low Rank Recovery\n                      , preprint,https:\/\/arxiv.org\/abs\/1505.05175, 2015."},{"key":"atypb57","doi-asserted-by":"publisher","DOI":"10.1137\/070697835"},{"key":"atypb58","doi-asserted-by":"publisher","DOI":"10.1002\/mrm.21926"},{"key":"atypb59","doi-asserted-by":"publisher","DOI":"10.1002\/mrm.22673"},{"key":"atypb60","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1522-2594(199907)42:1<150::AID-MRM20>3.0.CO;2-5"},{"key":"atypb61","doi-asserted-by":"publisher","DOI":"10.1002\/cmr.a.21427"},{"key":"atypb62","unstructured":"Z. Shi, J. Han, T. Zheng, S. Deng, and J. Li,\n                      Guarantees of augmented trace norm models in tensor recovery\n                      , in Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence, AIAA Press, Palo Alto, CA, 2013, pp. 1670-1676."},{"key":"atypb63","unstructured":"M. Talagrand,\n                      The Generic Chaining: Upper and Lower Bounds of Stochastic Processes\n                      , Springer, Berlin, 2006."},{"key":"atypb64","doi-asserted-by":"publisher","DOI":"10.1111\/j.2517-6161.1996.tb02080.x"},{"key":"atypb65","first-page":"1331","volume":"26","author":"Tomioka R.","year":"2013","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"atypb66","unstructured":"R. Tomioka, K. Hayashi, and H. Kashima,\n                      Estimation of low-rank tensors via convex optimization\n                      , preprint,https:\/\/arxiv.org\/abs\/1010.0789, (2010)."},{"key":"atypb67","unstructured":"R. Tomioka, T. Suzuki, K. Hayashi, and H. Kashima,\n                      Statistical performance of convex tensor decomposition\n                      , in Adv. Neural Inf. Process. Syst. 24, J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Weinberger, eds., Curran Associates, pp. 972-980."},{"key":"atypb68","doi-asserted-by":"publisher","DOI":"10.1109\/78.995059"},{"key":"atypb69","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2013.2254477"},{"key":"atypb70","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-015-9269-5"},{"key":"atypb71","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2015.01.099"},{"key":"atypb72","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2015.2392756"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1193037","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:43:43Z","timestamp":1787319823000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1193037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":72,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/18M1193037"],"URL":"https:\/\/doi.org\/10.1137\/18m1193037","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}