Decoding by linear programming

This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup n/ 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 exact...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 51; no. 12; pp. 4203 - 4215
Main Authors: Candes, E.J., Tao, T.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.12.2005
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Applied sciences
Basis pursuit
Codes
Coding
Coding, codes
Communications networks
Decoding
decoding of (random) linear codes
Detection, estimation, filtering, equalization, prediction
duality in optimization
Equations
Error correction
Error correction codes
Errors
Exact sciences and technology
Gaussian random matrices
Information systems
Information theory
Information, signal and communications theory
Linear code
linear codes
Linear programming
Mathematical analysis
Mathematics
principal angles
restricted orthonormality
Signal and communications theory
Signal, noise
singular values of random matrices
Sparse matrices
sparse solutions to underdetermined systems
Telecommunications and information theory
Vectors
Vectors (mathematics)
Singularity
linear codes
Uncertainty principle
Gaussian random matrices
Linear programming
Decoding
Random coding
Random codes
Convex programming
1 minimization
sparse solutions to underdetermined systems
Optimization
Linear code
restricted orthonormality
Linear system
singular values of random matrices
principal angles
Signal processing
Signal reconstruction
duality in optimization
Error correction
Index Terms-Basis pursuit
decoding of (random) linear codes
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup n/ 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 /spl lscr//sub 1/-minimization problem (/spl par/x/spl par//sub /spl lscr/1/:=/spl Sigma//sub i/|x/sub i/|) min(g/spl isin/R/sup n/) /spl par/y - Ag/spl par//sub /spl lscr/1/ provided that the support of the vector of errors is not too large, /spl par/e/spl par//sub /spl lscr/0/:=|{i:e/sub i/ /spl ne/ 0}|/spl les//spl rho//spl middot/m for some /spl rho/>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 underdetermined 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. Finally, underlying the success of /spl lscr//sub 1/ is a crucial property we call the uniform uncertainty principle that we shall describe in detail.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2005.858979