Advertisement

Abstract

A central achievement of classical linear algebra is a thorough examination of the problem of solving systems of linear equations. The results – definite, timeless, and profound – give the subject a completely settled appearance. As linear systems of equations are the core engine in many engineering developments and solutions, much of this knowledge is practically and successfully deployed in applications. Surprisingly, within this well-understood arena there is an elementary problem that has to do with sparse solutions of linear systems, which only recently has been explored in depth; we will see that this problem has surprising answers, and it inspires numerous practical developments. In this chapter we shall concentrate on defining this problem carefully, and set the stage for its answers in later chapters.

Keywords

Convex Combination Sparse Representation Atomic Decomposition Sparse Solution Redundant Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Further Reading

  1. 1.
    D.P. Bertsekas, Nonlinear Programming, 2nd Edition, Athena Scientific, 2004.Google Scholar
  2. 2.
    S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.MATHGoogle Scholar
  3. 3.
    A.M. Bruckstein, D.L. Donoho, and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Review, 51(1):34–81, February 2009.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S.S. Chen, D.L. Donoho, and M.A. Saunders, Atomic decomposition by basis pursuit, SIAM Journal on Scientific Computing, 20(1):33–61 (1998).CrossRefMathSciNetGoogle Scholar
  5. 5.
    S.S. Chen, D.L. Donoho, and M.A. Saunders, Atomic decomposition by basis pursuit, SIAM Review, 43(1):129–159, 2001.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. Daniel and F.S. Wood, Fitting Equations to Data: Computer Analysis of Multifactor Data, 2nd Edition, John Wiley and Sons, 1980.MATHGoogle Scholar
  7. 7.
    G. Davis, S. Mallat, and Z. Zhang, Adaptive time-frequency decompositions, Optical-Engineering, 33(7):2183–91, 1994.CrossRefGoogle Scholar
  8. 8.
    G.H. Golub and C.F. Van Loan, Matrix Computations, Johns Hopkins Studies in Mathematical Sciences, Third edition, 1996.Google Scholar
  9. 9.
    R.A. Horn C.R. Johnson, Matrix Analysis, New York: Cambridge University Press, 1985.MATHGoogle Scholar
  10. 10.
    A.K. Jain, Fundamentals of Digital Image Processing, Englewood Cli_s, NJ, Prentice-Hall, 1989.MATHGoogle Scholar
  11. 11.
    D. Luenberger, Linear and Nonlinear Programming, 2nd Edition, Addison- Wesley, Inc., Reading, Massachusetts 1984.MATHGoogle Scholar
  12. 12.
    S. Mallat, A Wavelet Tour of Signal Processing, Academic-Press, 1998.MATHGoogle Scholar
  13. 13.
    S. Mallat and E. LePennec, Sparse geometric image representation with bandelets, IEEE Trans. on Image Processing, 14(4):423–438, 2005.CrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Mallat and Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Processing, 41(12):3397–3415, 1993.MATHCrossRefGoogle Scholar
  15. 15.
    M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Prindle, Weber & Schmidt, Dover, 1992.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Computer Science DepartmentThe Technion – Israel Institute of TechnologyHaifaIsrael

Personalised recommendations