{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T02:58:08Z","timestamp":1649213888662},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2013,7]]},"abstract":"<jats:p> In correcting a real linear code y = Bx + w by \u2113<jats:sub>1<\/jats:sub> linear programming, where the encoding matrix B \u2208 \u211d<jats:sup>m \u00d7 n<\/jats:sup> has full rank with m \u2265 n and the noise w \u2208 \u211d<jats:sup>m<\/jats:sup> is a sparse random vector, it is numerically observed that the breakdown points of 50% successes in recovering the input vector x \u2208 \u211d<jats:sup>n<\/jats:sup> from the corrupted oversampled measurement y lie on the Donoho\u2013Tanner curves when reflected in their midpoint. The curves of 50% successes in solving underdetermined systems, z = Aw, by \u2113<jats:sub>1<\/jats:sub> linear programming with uniformly distributed compressed sensing matrices A \u2208 \u211d<jats:sup>d \u00d7 m<\/jats:sup>, where d &lt; m and w is a sparse vector, have been numerically observed and recently shown to coincide with the Donoho\u2013Tanner curves for normally-distributed compressed sensing matrices A derived from geometric combinatorics. When n \u2264 m\/2, correcting a linear code is faster if done directly by \u2113<jats:sub>1<\/jats:sub> linear programming. However, when n &gt; m\/2, to save computing time, this problem can be transformed into an underdetermined compressed sensing problem, Aw = z := Ay, for the syndrome z by a full rank matrix A \u2208 \u211d<jats:sup>d \u00d7 m<\/jats:sup>, d = m \u2013 n, such that AB = 0. For this purpose, to have equivalently high mean breakdown points by \u2113<jats:sub>1<\/jats:sub> linear programming, one can use uniformly distributed random matrices A \u2208 \u211d<jats:sup>(m-n) \u00d7 m<\/jats:sup> and matrices B \u2208 \u211d<jats:sup>m \u00d7 n<\/jats:sup> with orthonormal columns spanning the null space of A. Two exceptional cases have been found. Numerical results are collected in figures and tables. <\/jats:p>","DOI":"10.1142\/s0219691313600047","type":"journal-article","created":{"date-parts":[[2013,7,9]],"date-time":"2013-07-09T04:58:35Z","timestamp":1373345915000},"page":"1360004","source":"Crossref","is-referenced-by-count":0,"title":["PHASE TRANSITIONS IN ERROR CORRECTING AND COMPRESSED SENSING BY \u2113<sub>1<\/sub> LINEAR PROGRAMMING"],"prefix":"10.1142","volume":"11","author":[{"given":"RYUICHI","family":"ASHINO","sequence":"first","affiliation":[{"name":"Division of Mathematical Sciences, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R\u00c9MI","family":"VAILLANCOURT","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2013,7,8]]},"reference":[{"key":"rf1","first-page":"241","volume":"16","author":"Ashino R.","year":"2008","journal-title":"Can. 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