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. 668681. ISBN 0769524680. https://resolver.caltech.edu/CaltechAUTHORS:20110609075242535

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Abstract
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 mdimensional 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_1minimization 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  

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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  
Series Name:  Annual IEEE Symposium on Foundations of Computer Science  
DOI:  10.1109/SFCS.2005.5464411  
Record Number:  CaltechAUTHORS:20110609075242535  
Persistent URL:  https://resolver.caltech.edu/CaltechAUTHORS:20110609075242535  
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.668681, 2525 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.  
ID Code:  23952  
Collection:  CaltechAUTHORS  
Deposited By:  INVALID USER  
Deposited On:  09 Jun 2011 17:11  
Last Modified:  09 Nov 2021 16:19 
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