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A first impulse to find the sparsest solution to an underdetermined system of linear equations might be to solve the combinatorial problem

# Error correction via linear programming

Pittsburgh, PA, pp.668-681, (2005)

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Abstract

Suppose we wish to transmit a vector f 2 Rn 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 aected nor do we know how they are aected. Is it possible to recover ...More

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Introduction
• 1.1 The error correction problem.
• This paper considers the model problem of recovering an input vector f ∈ Rn from corrupted measurements y = Af + e.
• In its abstract form, the problem is equivalent to the classical error correcting problem which arises in coding theory as the authors may think of A as a linear code; a linear code is a given collection of codewords which are vectors a1, .
• The question is : given the coding matrix A and Af + e, can one recover f exactly?
Highlights
• 1.1 The error correction problem

This paper considers the model problem of recovering an input vector f ∈ Rn from corrupted measurements y = Af + e
• We report on numerical experiments suggesting that 1minimization is amazingly effective; f is recovered exactly even in situations where a very significant fraction of the output is corrupted
• In its abstract form, our problem is equivalent to the classical error correcting problem which arises in coding theory as we may think of A as a linear code; a linear code is a given collection of codewords which are vectors a1, . . . , an ∈ Rm—the columns of the matrix A
• A first impulse to find the sparsest solution to an underdetermined system of linear equations might be to solve the combinatorial problem (P0 )
Results
• The authors' experiments show that the linear program recovers the input vector all the time as long as the fraction of the corrupted entries is less or equal to 22.5% in the case where m = 2n and less than about 35% in the case where m = 4n
Conclusion
• The paper establishes deterministic results showing that exact decoding occurs provided the coding matrix A obeys the conditions of Theorem 1.1
• It is of interest because the own work [8, 10] shows that the condition of Theorem 1.1 with large values of r for many other types of matrices, and especially matrices obtained by sampling rows or columns of larger Fourier matrices.
• The paper links solutions to sparse underdetermined systems to a linear programming problem for error correction, which the authors believe is new
Funding
• C. is partially supported in part by a National Science Foundation grant DMS 01-40698 (FRG) and by an Alfred P
• R. is partially supported by the NSF grant DMS 0245380
• T. is supported by a grant from the Packard Foundation
• Sloan Research Fellow He was also partially supported by the NSF grant DMS 0401032 and by the Miller Scholarship from the
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