arXiv:math/0502327v1 [math.MG] 15 Feb 2005 Decoding by Linear Programming Emmanuel Candes��� and Terence Tao��� ��� Applied and Computational Mathematics, Caltech, Pasadena, CA 91125 ��� Department of Mathematics, University of California, Los Angeles, CA 90095 December 2004 Abstract This paper considers the classical error correcting problem which is frequently dis- cussed in coding theory. We wish to recover an input vector f ��� Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the ���1-minimization problem (bardblxbardbl���1 := ��� i |xi|) min g���Rn bardbly ��� Agbardbl���1 provided that the support of the vector of errors is not too large, bardblebardbl���0 := |{i : ei = 0}| ��� �� �� m for some �� 0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underde- termined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work [5]. Finally, underlying the suc- cess of ���1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail. Keywords. Linear codes, decoding of (random) linear codes, sparse solutions to under- determined systems, ���1 minimization, basis pursuit, duality in optimization, linear pro- gramming, restricted orthonormality, principal angles, Gaussian random matrices, singular values of random matrices. Acknowledgments. E. C. is partially supported by National Science Foundation grants DMS 01-40698 (FRG) and ACI-0204932 (ITR), and by an Alfred P. Sloan Fellowship. T. T. is supported in part by a grant from the Packard Foundation. Many thanks to Rafail Ostrovsky for pointing out possible connections between our earlier work and the decod- ing problem. E. C. would also like to acknowledge inspiring conversations with Leonard Schulmann and Justin Romberg. 1