{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T21:53:06Z","timestamp":1774993986933,"version":"3.50.1"},"reference-count":11,"publisher":"MIT Press - Journals","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2004,6,1]]},"abstract":"<jats:p> In this letter, we analyze a two-stage cluster-then-l<jats:sup>1<\/jats:sup>-optimization approach for sparse representation of a data matrix, which is also a promising approach for blind source separation (BSS) in which fewer sensors than sources are present. First, sparse representation (factorization) of a data matrix is discussed. For a given overcomplete basis matrix, the corresponding sparse solution (coefficient matrix) with minimum l<jats:sup>1<\/jats:sup> norm is unique with probability one, which can be obtained using a standard linear programming algorithm. The equivalence of the l<jats:sup>1<\/jats:sup>\u2014norm solution and the l<jats:sup>0<\/jats:sup>\u2014norm solution is also analyzed according to a probabilistic framework. If the obtained l<jats:sup>1<\/jats:sup>\u2014norm solution is sufficiently sparse, then it is equal to the l<jats:sup>0<\/jats:sup>\u2014norm solution with a high probability. Furthermore, the l<jats:sup>1<\/jats:sup>\u2014norm solution is robust to noise, but the l0\u2014norm solution is not, showing that the l<jats:sup>1<\/jats:sup>\u2014norm is a good sparsity measure. These results can be used as a recoverability analysis of BSS, as discussed. The basis matrix in this article is estimated using a clustering algorithm followed by normalization, in which the matrix columns are the cluster centers of normalized data column vectors. Zibulevsky, Pearlmutter, Boll, and Kisilev (2000) used this kind of two-stage approach in underdetermined BSS. Our recoverability analysis shows that this approach can deal with the situation in which the sources are overlapped to some degree in the analyzed <\/jats:p>","DOI":"10.1162\/089976604773717586","type":"journal-article","created":{"date-parts":[[2004,4,21]],"date-time":"2004-04-21T12:01:57Z","timestamp":1082548917000},"page":"1193-1234","source":"Crossref","is-referenced-by-count":248,"title":["Analysis of Sparse Representation and Blind Source Separation"],"prefix":"10.1162","volume":"16","author":[{"given":"Yuanqing","family":"Li","sequence":"first","affiliation":[{"name":"Laboratory for Advanced Brain Signal Processing and RIKEN Brain Science Institute, Wako shi, Saitama, 3510198, Japan, and Automation Science And Engineering Institute, Southchina University of Technology, Guangzhou, China,"}]},{"given":"Andrzej","family":"Cichocki","sequence":"additional","affiliation":[{"name":"Laboratory for Advanced Brain Signal Processing and RIKEN Brain Science Institute, Wako shi, Saitama, 3510198, Japan, and The Department of Electrical Engineering, Warsaw University of Technology, Warsaw, Poland,"}]},{"given":"Shun-ichi","family":"Amari","sequence":"additional","affiliation":[{"name":"Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Wako shi, Saitama, 3510198, Japan,"}]}],"member":"281","reference":[{"key":"p_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-1684(01)00120-7"},{"key":"p_2","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596304010"},{"key":"p_4","doi-asserted-by":"publisher","DOI":"10.1162\/089976603762552951"},{"key":"p_5","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0437847100"},{"key":"p_6","doi-asserted-by":"publisher","DOI":"10.1162\/089976601753196003"},{"key":"p_8","doi-asserted-by":"publisher","DOI":"10.1038\/44565"},{"key":"p_9","doi-asserted-by":"publisher","DOI":"10.1109\/97.752062"},{"key":"p_10","doi-asserted-by":"publisher","DOI":"10.1162\/089976600300015826"},{"key":"p_11","doi-asserted-by":"publisher","DOI":"10.1126\/science.1066168"},{"key":"p_12","doi-asserted-by":"publisher","DOI":"10.1016\/S0042-6989(97)00169-7"},{"key":"p_15","doi-asserted-by":"publisher","DOI":"10.1162\/089976601300014385"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/089976604773717586","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T21:51:08Z","timestamp":1615585868000},"score":1,"resource":{"primary":{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/direct.mit.edu\/neco\/article\/16\/6\/1193-1234\/6836"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,6,1]]},"references-count":11,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2004,6,1]]}},"alternative-id":["10.1162\/089976604773717586"],"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1162\/089976604773717586","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,6,1]]}}}