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Matrix Sparsification and the Sparse Null Space Problem

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (RANDOM 2010, APPROX 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6302))

Abstract

We revisit the matrix problems sparse null space and matrix sparsification, and show that they are equivalent. We then proceed to seek algorithms for these problems: We prove the hardness of approximation of these problems, and also give a powerful tool to extend algorithms and heuristics for sparse approximation theory to these problems.

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References

  1. Amaldi, E., Kann, V.: The complexity and approximability of finding maximum feasible subsystems of linear relations. Theoretical computer science 147(1-2), 181–210 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes and linear equations. JCSS 54(2) (1997)

    Google Scholar 

  3. Berman, P., Karpinski, M.: Approximating minimum unsatisfiability of linear equations. Electronic Colloquium on Computational Complexity (ECCC) 8(25) (2001)

    Google Scholar 

  4. Berry, M., Heath, M., Kaneko, I., Lawo, M., Plemmons, R., Ward, R.: An algorithm to compute a sparse basis of the null space. Numer. Math. 47, 483–504 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brualdi, R.A., Friedland, S., Pothen, A.: The sparse basis problem and multilinear algebra. SIAM Journal on Matrix Analysis and Applications 16(1), 1–20 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Candès, E., Romberg, J., Tao, T.: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52, 489–509 (2006)

    Article  MathSciNet  Google Scholar 

  7. Frank Chang, S., Thomas McCormick, S.: A hierarchical algorithm for making sparse matrices sparser. Mathematical Programming 56(1-3), 1–30 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Coifman, R., Wickerhauser, M.: Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory 38(2), 713–718 (1992)

    Article  MATH  Google Scholar 

  9. Coleman, T.F., Pothen, A.: The null space problem I. complexity. SIAM Journal on Algebraic and Discrete Methods 7(4), 527–537 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dinur, I., Kindler, G., Raz, R., Safra, S.: Approximating CVP to within almost-polynomial factors is NP-hard. Combinatorica 23(2), 205–243 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Donoho, D.L.: For most large underdetermined systems of linear equations, the minimal l1-norm solution is also the sparsest (2004), http://www-stat.stanford.edu/~donoho/Reports/2004/l1l0EquivCorrected.pdf

  12. Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Oxford University Press, Oxford (1986)

    MATH  Google Scholar 

  13. Fuchs, J.J.: On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Th. 50(6), 1341–1344 (2004)

    Article  Google Scholar 

  14. Gilbert, A.C., Guha, S., Indyk, P., Muthukrishnan, S., Strauss, M.: Near-optimal sparse fourier representations via sampling. In: STOC (2002)

    Google Scholar 

  15. Gilbert, A.C., Muthukrishnan, S., Strauss, M.: Improved time bounds for near-optimal sparse fourier representations. In: Proc. SPIE Wavelets XI (2005)

    Google Scholar 

  16. Gilbert, J.R., Heath, M.T.: Computing a sparse basis for the null space. SIAM Journal on Algebraic and Discrete Methods 8(3), 446–459 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hoffman, A.J., McCormick, S.T.: A fast algorithm that makes matrices optimally sparse. In: Progress in Combinatorial Optimization. Academic Press, London (1984)

    Google Scholar 

  18. Johnson, D.S., Preparata, F.P.: The densent hemisphere problem. Theoret. Comput. Sci. 6, 93–107 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kann, V.: Polynomially bounded minimization problems that are hard to approximate. Nordic Journal of Computing 1(3), 317–331 (Fall 1994)

    MATH  MathSciNet  Google Scholar 

  20. Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: A faster algorithm for minimum cycle basis of graphs. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 846–857. Springer, Heidelberg (2004)

    Google Scholar 

  21. Matoušek, J., Gartner, B.: Understanding and Using Linear Programming. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  22. McCormick, S.T.: A combinatorial approach to some sparse matrix problems. PhD thesis, Stanford Univ., Stanford, California, Technical Report 83-5, Stanford Optimization Lab. (1983)

    Google Scholar 

  23. Muthukrishnan, S.: Nonuniform sparse approximation with haar wavelet basis. DIMACS TR:2004-42 (2004)

    Google Scholar 

  24. Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. on Comput. 24(2), 227–234 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Needell, D., Tropp, J.: Cosamp: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis 26(3), 301–322 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Neylon, T.: Sparse solutions for linear prediction problems. PhD thesis, New York University, New York, New York (2006)

    Google Scholar 

  27. Pothen, A.: Sparse null bases and marriage theorems. PhD thesis, Cornell University, Ithica, New York (1984)

    Google Scholar 

  28. Schmidt, E.: Zur thoerie der linearen und nichtlinearen integralgleichungen, i. Math. Annalen. 64(4), 433–476 (1906-1907)

    Article  Google Scholar 

  29. Singhal, S.: Amplitude optimization and pitch predictin in multi-pulse coders. IEEE Transactions on Acoustics, Speech and Signal Processing 37(3) (March 1989)

    Google Scholar 

  30. Smola, A., Schölkopf, B.: Sparse greedy matrix approximation for machine learning. In: Proc. 17th International Conf. on Machine Learning, pp. 911–918. Morgan Kaufmann, San Francisco (2000)

    Google Scholar 

  31. Temlyakov, V.: Nonlinear methods of approximation. Foundations of Comp. Math. 3(1), 33–107 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Transactions on Information Theory 50(10), 2231–2242 (2004)

    Article  MathSciNet  Google Scholar 

  33. Tropp, J.A.: Just relax: convex programming methods for subset selection and sparse approximation. IEEE Transactions on Information Theory 50(3), 1030–1051 (2006)

    Article  MathSciNet  Google Scholar 

  34. Tropp, J.A.: Recovery of short, complex linear combinations via ℓ1 minimization. IEEE Trans. Inform. Theory 51(1), 188–209 (2006)

    Article  MathSciNet  Google Scholar 

  35. Wang, X.: A simple proof of descartes’ rule of signs. Amer. Math. Monthly 111, 525–526 (2004)

    Article  MATH  Google Scholar 

  36. Zhao, X., Zhang, X., Neylon, T., Shasha, D.: Incremental methods for simple problems in time series: algorithms and experiments. In: Ninth International Database Engineering and Applications Symposium (IDEAS 2005), pp. 3–14 (2005)

    Google Scholar 

  37. Zhou, J., Gilbert, A., Strauss, M., Daubechies, I.: Theoretical and experimental analysis of a randomized algorithm for sparse fourier transform analysis. Journal of Computational Physics 211, 572–595 (2006)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Weizmann Institute of Science,  

    Lee-Ad Gottlieb

  2. Bynomial, Inc.,  

    Tyler Neylon

Authors
  1. Lee-Ad Gottlieb
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  2. Tyler Neylon
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Editor information

Editors and Affiliations

  1. Universitat Politecnica de Catalunya, Departament de Llenguatges i Sistemes Informatics, Campus Nord - Ed Omega, Desp. 235, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Maria Serna

  2. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

  3. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24098, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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© 2010 Springer-Verlag Berlin Heidelberg

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Gottlieb, LA., Neylon, T. (2010). Matrix Sparsification and the Sparse Null Space Problem. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_16

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  • DOI: https://doi.org/10.1007/978-3-642-15369-3_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15368-6

  • Online ISBN: 978-3-642-15369-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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