Published - Candes2005.pdf

", "abstract": "Suppose we wish to transmit a vector f \u0404 R^n reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are corrupted in a completely arbitrary fashion. We do not know which entries are affected nor do we know how they are affected. Is it possible to recover f exactly from the corrupted m-dimensional vector y? This paper proves that under suitable conditions on the coding matrix A, the input f is the unique solution to the \u2113_1 -minimization problem (\u2016x\u2016\u2113_1: = \u2211_i |xi|) min \u2016y \u2212 Ag\u2016\u2113_1 g^\u2208Rn provided that the fraction of corrupted entries is not too large, i.e. does not exceed some strictly positive constant \u03c1 \u2217 (numerical values for \u03c1 ^\u2217 are given). In other words, f can be recovered exactly by solving a simple convex optimization problem; in fact, a linear program. We report on numerical experiments suggesting that \u2113_1-minimization is amazingly effective; f is recovered exactly even in situations where a very significant fraction of the output is corrupted.\nIn the case when the measurement matrix A is Gaussian,\nthe problem is equivalent to that of counting lowdimensional\nfacets of a convex polytope, and in particular\nof a random section of the unit cube. In this case we can\nstrengthen the results somewhat by using a geometric functional\nanalysis approach.", "date": "2005", "date_type": "published", "publisher": "IEEE Computer Society Press", "place_of_pub": "Los Alamitos, CA", "pagerange": "668-681", "id_number": "CaltechAUTHORS:20110609-075242535", "isbn": "0-7695-2468-0", "book_title": "46th Annual IEEE Symposium on Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110609-075242535", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1109/SFCS.2005.5464411", "primary_object": { "basename": "Candes2005.pdf", "url": "https://authors.library.caltech.edu/records/38b5r-q6w74/files/Candes2005.pdf" }, "resource_type": "book_section", "pub_year": "2005", "author_list": "Candes, Emmanuel; Rudelson, Mark; et el." } ]