### Abstract

We consider the Linear Programming (LP) solution of the Compressed Sensing (CS) problem over reals, also known as the Basis Pursuit (BasP) algorithm. The BasP allows interpretation as a channel-coding problem, and it guarantees error-free reconstruction with a properly chosen measurement matrix and sufficiently sparse error vectors. In this manuscript, we examine how the BasP performs on a given measurement matrix and develop an algorithm to discover the sparsest vectors for which the BasP fails. The resulting algorithm is a generalization of our previous results on finding the most probable error-patterns degrading performance of a finite size Low-Density Parity-Check (LDPC) code in the error-floor regime. The BasP fails when its output is different from the actual error-pattern. We design a CS-Instanton Search Algorithm (ISA) generating a sparse vector, called a CS-instanton, such that the BasP fails on the CS-instanton, while the BasP recovery is successful for any modification of the CS-instanton replacing a nonzero element by zero. We also prove that, given a sufficiently dense random input for the error-vector, the CS-ISA converges to an instanton in a small finite number of steps. The performance of the CS-ISA is illustrated on a randomly generated 120 × 512 matrix. For this example, the CS-ISA outputs the shortest instanton (error vector) pattern of length 11.

Original language | English (US) |
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Title of host publication | IEEE International Symposium on Information Theory - Proceedings |

Pages | 1978-1982 |

Number of pages | 5 |

DOIs | |

State | Published - 2010 |

Event | 2010 IEEE International Symposium on Information Theory, ISIT 2010 - Austin, TX, United States Duration: Jun 13 2010 → Jun 18 2010 |

### Other

Other | 2010 IEEE International Symposium on Information Theory, ISIT 2010 |
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Country | United States |

City | Austin, TX |

Period | 6/13/10 → 6/18/10 |

### Fingerprint

### ASJC Scopus subject areas

- Applied Mathematics
- Modeling and Simulation
- Theoretical Computer Science
- Information Systems

### Cite this

*IEEE International Symposium on Information Theory - Proceedings*(pp. 1978-1982). [5513360] https://doi.org/10.1109/ISIT.2010.5513360

**Worst configurations (Instantons) for compressed sensing over reals : A channel coding approach.** / Chilappagari, Shashi Kiran; Chertkov, Michael; Vasic, Bane V.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*IEEE International Symposium on Information Theory - Proceedings.*, 5513360, pp. 1978-1982, 2010 IEEE International Symposium on Information Theory, ISIT 2010, Austin, TX, United States, 6/13/10. https://doi.org/10.1109/ISIT.2010.5513360

}

TY - GEN

T1 - Worst configurations (Instantons) for compressed sensing over reals

T2 - A channel coding approach

AU - Chilappagari, Shashi Kiran

AU - Chertkov, Michael

AU - Vasic, Bane V

PY - 2010

Y1 - 2010

N2 - We consider the Linear Programming (LP) solution of the Compressed Sensing (CS) problem over reals, also known as the Basis Pursuit (BasP) algorithm. The BasP allows interpretation as a channel-coding problem, and it guarantees error-free reconstruction with a properly chosen measurement matrix and sufficiently sparse error vectors. In this manuscript, we examine how the BasP performs on a given measurement matrix and develop an algorithm to discover the sparsest vectors for which the BasP fails. The resulting algorithm is a generalization of our previous results on finding the most probable error-patterns degrading performance of a finite size Low-Density Parity-Check (LDPC) code in the error-floor regime. The BasP fails when its output is different from the actual error-pattern. We design a CS-Instanton Search Algorithm (ISA) generating a sparse vector, called a CS-instanton, such that the BasP fails on the CS-instanton, while the BasP recovery is successful for any modification of the CS-instanton replacing a nonzero element by zero. We also prove that, given a sufficiently dense random input for the error-vector, the CS-ISA converges to an instanton in a small finite number of steps. The performance of the CS-ISA is illustrated on a randomly generated 120 × 512 matrix. For this example, the CS-ISA outputs the shortest instanton (error vector) pattern of length 11.

AB - We consider the Linear Programming (LP) solution of the Compressed Sensing (CS) problem over reals, also known as the Basis Pursuit (BasP) algorithm. The BasP allows interpretation as a channel-coding problem, and it guarantees error-free reconstruction with a properly chosen measurement matrix and sufficiently sparse error vectors. In this manuscript, we examine how the BasP performs on a given measurement matrix and develop an algorithm to discover the sparsest vectors for which the BasP fails. The resulting algorithm is a generalization of our previous results on finding the most probable error-patterns degrading performance of a finite size Low-Density Parity-Check (LDPC) code in the error-floor regime. The BasP fails when its output is different from the actual error-pattern. We design a CS-Instanton Search Algorithm (ISA) generating a sparse vector, called a CS-instanton, such that the BasP fails on the CS-instanton, while the BasP recovery is successful for any modification of the CS-instanton replacing a nonzero element by zero. We also prove that, given a sufficiently dense random input for the error-vector, the CS-ISA converges to an instanton in a small finite number of steps. The performance of the CS-ISA is illustrated on a randomly generated 120 × 512 matrix. For this example, the CS-ISA outputs the shortest instanton (error vector) pattern of length 11.

UR - http://www.scopus.com/inward/record.url?scp=77955699057&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955699057&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2010.5513360

DO - 10.1109/ISIT.2010.5513360

M3 - Conference contribution

AN - SCOPUS:77955699057

SN - 9781424469604

SP - 1978

EP - 1982

BT - IEEE International Symposium on Information Theory - Proceedings

ER -