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46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) (2005)
Pittsburgh, Pennsylvania, USA
Oct. 23, 2005 to Oct. 25, 2005
ISBN: 0-7695-2468-0
pp: 295-308
Emmanuel Candes , Caltech
Mark Rudelson , University of Missouri
Roman Vershynin , University of California, Davis
Suppose we wish to transmit a vector f \varepsilon 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 by an error e. 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^1 = {\rm A}f + e? This paper proves that under suitable conditions on the coding matrix A, the input f is the unique solution to the (||x||\ell 1: = \Sigma i|xi|) \min ||y - {\rm A}f||\ell 1f\varepsilon R^n provided that the fraction of corrupted entries is not too large, i.e. does not exceed some strictly positive constant p^ * ( numerical values for p^ * are actually 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 \ell 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.

T. Tao, E. Candes, R. Vershynin and M. Rudelson, "Error Correction via Linear Programming," 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)(FOCS), Pittsburgh, Pennsylvania, USA, 2005, pp. 295-308.
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