Published - CANaos07.pdf

", "abstract": "In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y=X\u03b2+z, where \u03b2\u2208Rp is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n\u226ap, and the zi's are i.i.d. N(0, \u03c3^2). Is it possible to estimate \u03b2 reliably based on the noisy data y?", "date": "2007-12", "date_type": "published", "publication": "Annals of Statistics", "volume": "35", "number": "6", "publisher": "Institute of Mathematical Statistics", "pagerange": "2313-2351", "id_number": "CaltechAUTHORS:20100219-092002261", "issn": "0090-5364", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100219-092002261", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-01-40698" }, { "agency": "Alfred P. Sloan Fellowship" }, { "agency": "Packard Foundation" } ] }, "doi": "10.1214/009053606000001523", "primary_object": { "basename": "CANaos07.pdf", "url": "https://authors.library.caltech.edu/records/484gh-bde27/files/CANaos07.pdf" }, "resource_type": "article", "pub_year": "2007", "author_list": "Candes, Emmanuel and Tao, Terence" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/38b5r-q6w74", "eprint_id": 23952, "eprint_status": "archive", "datestamp": "2023-08-22 02:54:14", "lastmod": "2024-01-13 05:17:04", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Candes-E", "name": { "family": "Candes", "given": "Emmanuel" } }, { "id": "Rudelson-M", "name": { "family": "Rudelson", "given": "Mark" } }, { "id": "Tao-T", "name": { "family": "Tao", "given": "Terence" } }, { "id": "Vershynin-R", "name": { "family": "Vershynin", "given": "Roman" } } ] }, "title": "Error correction via linear programming", "ispublished": "unpub", "full_text_status": "public", "keywords": "linear codes; decoding of (random) linear codes; sparse solutions to underdetermined systems; \u2113_1- minimization; linear programming; restricted orthonormality; Gaussian random matrices", "note": "\u00a9 2005 IEEE.\n\nPublished - 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." } ]