Candes, Emmanuel and Rudelson, Mark and Tao, Terence and Vershynin, Roman (2005) Error correction via linear programming. In: 46th Annual IEEE Symposium on Foundations of Computer Science. Annual IEEE Symposium on Foundations of Computer Science . IEEE Computer Society Press , Los Alamitos, CA, pp. 668-681. ISBN 0-7695-2468-0 http://resolver.caltech.edu/CaltechAUTHORS:20110609-075242535
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Suppose we wish to transmit a vector f Є 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 ℓ_1 -minimization problem (‖x‖ℓ_1: = ∑_i |xi|) min ‖y − Ag‖ℓ_1 g^∈Rn provided that the fraction of corrupted entries is not too large, i.e. does not exceed some strictly positive constant ρ ∗ (numerical values for ρ ^∗ 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 ℓ_1-minimization is amazingly effective; f is recovered exactly even in situations where a very significant fraction of the output is corrupted. In the case when the measurement matrix A is Gaussian, the problem is equivalent to that of counting lowdimensional facets of a convex polytope, and in particular of a random section of the unit cube. In this case we can strengthen the results somewhat by using a geometric functional analysis approach.
|Item Type:||Book Section|
|Additional Information:||© 2005 IEEE.|
|Subject Keywords:||linear codes; decoding of (random) linear codes; sparse solutions to underdetermined systems; ℓ_1- minimization; linear programming; restricted orthonormality; Gaussian random matrices|
|Official Citation:||Candes, E.; Rudelson, M.; Tao, T.; Vershynin, R.; , "Error correction via linear programming," Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on , vol., no., pp.668-681, 25-25 Oct. 2005 doi: 10.1109/SFCS.2005.5464411 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5464411&isnumber=32664|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||09 Jun 2011 17:11|
|Last Modified:||09 Jun 2011 17:14|
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