{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:32:18Z","timestamp":1787326338849,"version":"build-2736575974"},"reference-count":37,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-04-D-0003"],"award-info":[{"award-number":["W911NF-04-D-0003"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-16-2-0005"],"award-info":[{"award-number":["W911NF-16-2-0005"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100005712","name":"Missile Defense Agency","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100005712","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["ERC StG 335491 (RAPID)"],"award-info":[{"award-number":["ERC StG 335491 (RAPID)"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>Sparse models in dictionary learning have been successfully applied in a wide variety of machine learning and computer vision problems, and as a result have recently attracted increased research interest. Another interesting related problem based on linear equality constraints, namely the sparse null space (SNS) problem, first appeared in 1986 and has since inspired results on sparse basis pursuit. In this paper, we investigate the relation between the SNS problem and the analysis dictionary learning (ADL) problem, and show that the SNS problem plays a central role, and may be utilized to solve dictionary learning problems. Moreover, we propose an efficient algorithm of sparse null space basis pursuit (SNS-BP) and extend it to a solution of ADL. Experimental results on numerical synthetic data and real-world data are further presented to validate the performance of our method.<\/jats:p>","DOI":"10.1137\/15m1030376","type":"journal-article","created":{"date-parts":[[2016,8,9]],"date-time":"2016-08-09T11:24:11Z","timestamp":1470741851000},"page":"1107-1126","source":"Crossref","is-referenced-by-count":10,"title":["Sparsity and Nullity: Paradigms for Analysis Dictionary Learning"],"prefix":"10.1137","volume":"9","author":[{"given":"Xiao","family":"Bian","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hamid","family":"Krim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alex","family":"Bronstein","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liyi","family":"Dai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,8,9]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2006.881199"},{"key":"atypb2","first-page":"760","volume":"2012","author":"Bian X.","year":"2013","journal-title":"Computer Vision---ACCV"},{"key":"atypb3","unstructured":"X. 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Ma,\n                      The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices\n                      , preprint, https:\/\/arxiv.org\/abs\/1009.5055 arXiv:1009.5055, 2010."},{"key":"atypb24","unstructured":"Z. Lin, R. Liu, and Z. Su,\n                      Linearized alternating direction method with adaptive penalty for low-rank representation\n                      , in Advances in Neural Information Processing Systems 24, J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q. 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