Worst configurations (Instantons) for compressed sensing over reals: A channel coding approach

Shashi Kiran Chilappagari, Michael Chertkov, Bane Vasic

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

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 languageEnglish (US)
Title of host publication2010 IEEE International Symposium on Information Theory, ISIT 2010 - Proceedings
Pages1978-1982
Number of pages5
DOIs
StatePublished - 2010
Event2010 IEEE International Symposium on Information Theory, ISIT 2010 - Austin, TX, United States
Duration: Jun 13 2010Jun 18 2010

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8103

Other

Other2010 IEEE International Symposium on Information Theory, ISIT 2010
CountryUnited States
CityAustin, TX
Period6/13/106/18/10

ASJC Scopus subject areas

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

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    Chilappagari, S. K., Chertkov, M., & Vasic, B. (2010). Worst configurations (Instantons) for compressed sensing over reals: A channel coding approach. In 2010 IEEE International Symposium on Information Theory, ISIT 2010 - Proceedings (pp. 1978-1982). [5513360] (IEEE International Symposium on Information Theory - Proceedings). https://doi.org/10.1109/ISIT.2010.5513360