A Matrix Hyperbolic Cosine Algorithm and Applications

  • Anastasios Zouzias
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

In this paper, we generalize Spencer’s hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size n, it constructs an expanding Cayley graph of logarithmic degree in near-optimal \(\mathcal{O}(n^2\log^3 n)\) time. For the latter case, we present a fast deterministic algorithm for spectral sparsification of positive semi-definite matrices, which implies an improved deterministic algorithm for spectral graph sparsification of dense graphs. In addition, we give an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrix sparsification. As a consequence, we obtain improved element-wise sparsification algorithms for diagonally dominant-like matrices.

Keywords

Cayley Graph Symmetric Matrice Deterministic Algorithm Dense Graph Fast Multipole Method 
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.

References

  1. 1.
    Achlioptas, D., McSherry, F.: Fast Computation of Low-rank Matrix Approximations. SIAM J. Comput. 54(2), 9 (2007)MathSciNetGoogle Scholar
  2. 2.
    Ahlswede, R., Winter, A.: Strong Converse for Identification via Quantum Channels. IEEE Transactions on Information Theory 48(3), 569–579 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Alon, N., Roichman, Y.: Random Cayley Graphs and Expanders. Random Struct. Algorithms 5, 271–284 (1994)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arora, S., Hazan, E., Kale, S.: Fast Algorithms for Approximate Semidefinite Programming using the Multiplicative Weights Update Method. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS), pp. 339–348 (2005)Google Scholar
  5. 5.
    Arora, S., Hazan, E., Kale, S.: A Fast Random Sampling Algorithm for Sparsifying Matrices. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 272–279. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Arora, S., Kale, S.: A Combinatorial, Primal-Dual Approach to Semidefinite Programs. In: Proceedings of the Symposium on Theory of Computing (STOC), pp. 227–236 (2007)Google Scholar
  7. 7.
    d’Aspremont, A.: Subsampling Algorithms for Semidefinite Programming. In: Stochastic Systems, pp. 274–305 (2011)Google Scholar
  8. 8.
    Batson, J.D., Spielman, D.A., Srivastava, N.: Twice-ramanujan sparsifiers. In: Proceedings of the Symposium on Theory of Computing (STOC), pp. 255–262 (2009)Google Scholar
  9. 9.
    Benczúr, A.A., Karger, D.R.: Approximating s-t Minimum Cuts in \(\widetilde{\mathcal{O}}(n^2)\) Time. In: Proceedings of the Symposium on Theory of Computing (STOC) (1996)Google Scholar
  10. 10.
    Bhatia, R.: Matrix Analysis, 1st edn. Graduate Texts in Mathematics, vol. 169. Springer, Heidelberg (1996)MATHGoogle Scholar
  11. 11.
    Boman, E.G., Hendrickson, B., Vavasis, S.: Solving Elliptic Finite Element Systems in Near-Linear Time with Support Preconditioners. SIAM J. on Numerical Analysis 46(6), 3264–3284 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Carrier, J., Greengard, L., Rokhlin, V.: A Fast Adaptive Multipole Algorithm for Particle Simulations. SIAM J. on Scientific and Statistical Computing 9(4), 669–686 (1988)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Drineas, P., Zouzias, A.: A note on Element-wise Matrix Sparsification via a Matrix-valued Bernstein Inequality. Information Processing Letters 111(8), 385–389 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Estrada, E., Rodríguez-Velázquez, J.A.: Subgraph Centrality in Complex Networks. Phys. Rev. E 71 (May 2005)Google Scholar
  15. 15.
    Golden, S.: Lower Bounds for the Helmholtz Function. Phys. Rev. 137(4B), B1127–B1128 (1965)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gu, M., Eisenstat, S.C.: A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem. SIAM J. Matrix Anal. Appl. 15 (1994)Google Scholar
  17. 17.
    Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics (SIAM) (2008)Google Scholar
  18. 18.
    Joshi, P., Meyer, M., DeRose, T., Green, B., Sanocki, T.: Harmonic Coordinates for Character Articulation. ACM Trans. Graph. 26 (2007)Google Scholar
  19. 19.
    Kale, S.: Efficient Algorithms Using the Multiplicative Weights Update Method. PhD in Computer Science, Princeton University (2007)Google Scholar
  20. 20.
    Koutis, I., Miller, G.L., Tolliver, D.: Combinatorial Preconditioners and Multilevel Solvers for Problems in Computer Vision and Image Processing. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Kuno, Y., Wang, J., Wang, J.-X., Wang, J., Pajarola, R., Lindstrom, P., Hinkenjann, A., Encarnação, M.L., Silva, C.T., Coming, D. (eds.) ISVC 2009. LNCS, vol. 5875, pp. 1067–1078. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Landau, Z., Russell, A.: Random Cayley Graphs are Expanders: a simplified proof of the Alon-Roichman theorem. The Electronic J. of Combinatorics 11(1) (2004)Google Scholar
  22. 22.
    Magen, A., Zouzias, A.: Low Rank Matrix-Valued Chernoff Bounds and Approximate Matrix Multiplication. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1422–1436 (2011)Google Scholar
  23. 23.
    Naor, A.: On the Banach Space Valued Azuma Inequality and Small set Isoperimetry of Alon-Roichman Graphs. To Appear in Combinatorics, Probability and Computing (September 2010), arxiv:1009.5695Google Scholar
  24. 24.
    Naor, A.: Sparse Quadratic Forms and their Geometric Applications (after Batson, Spielman and Srivastava) (January 2011), arxiv:1101.4324Google Scholar
  25. 25.
    Peleg, D., Schäffer, A.A.: Graph Spanners. J. of Graph Theory 13(1), 99–116 (1989)MATHCrossRefGoogle Scholar
  26. 26.
    Recht, B.: A Simpler Approach to Matrix Completion. J. of Machine Learning Research, 3413–3430 (December 2011)Google Scholar
  27. 27.
    Rudelson, M.: Random Vectors in the Isotropic Position. J. Funct. Anal. 164(1), 60–72 (1999)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Rudelson, M., Vershynin, R.: Sampling from Large Matrices: An Approach through Geometric Functional Analysis. J. ACM 54(4), 21 (2007)MathSciNetCrossRefGoogle Scholar
  29. 29.
    So, A.M.C.: Moment Inequalities for sums of Random Matrices and their Applications in Optimization. Mathematical Programming, pp. 1–27 (2009)Google Scholar
  30. 30.
    Spencer, J.: Balancing Games. J. Comb. Theory, Ser. B 23(1), 68–74 (1977)MATHCrossRefGoogle Scholar
  31. 31.
    Spencer, J.: Six Standard Deviations Suffice. Transactions of The American Mathematical Society 289, 679–679 (1985)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Spencer, J.: Balancing Vectors in the max Norm. Combinatorica 6, 55–65 (1986)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Spencer, J.: Ten Lectures on the Probabilistic Method, 2nd edn. Society for Industrial and Applied Mathematics, SIAM (1994)Google Scholar
  34. 34.
    Spielman, D.A.: Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices. In: Proceedings of the International Congress of Mathematicians, vol. IV, pp. 2698–2722 (2010)Google Scholar
  35. 35.
    Spielman, D.A., Srivastava, N.: Graph Sparsification by Effective Resistances. In: Proceedings of the Symposium on Theory of Computing, STOC (2008)Google Scholar
  36. 36.
    Srivastava, N.: Spectral Sparsification and Restricted Invertibility. PhD in Computer Science, Yale University (2010)Google Scholar
  37. 37.
    Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory (Computer Science and Scientific Computing). Academic Press, London (1990)Google Scholar
  38. 38.
    Thompson, C.J.: Inequality with Applications in Statistical Mechanics. J. of Mathematical Physics 6(11), 1812–1813 (1965)CrossRefGoogle Scholar
  39. 39.
    Tropp, J.A.: User-Friendly Tail Bounds for Sums of Random Matrices. Foundations of Computational Mathematics, pp. 1–46 (2011)Google Scholar
  40. 40.
    Tsuda, K., Rätsch, G., Warmuth, M.K.: Matrix Exponentiated Gradient Updates for on-line Learning and Bregman Projections. JMLR 6, 995–1018 (2005)MATHGoogle Scholar
  41. 41.
    Wigderson, A., Xiao, D.: Derandomizing the Ahlswede-Winter Matrix-valued Chernoff Bound using Pessimistic Estimators, and Applications. Theory of Computing 4(1), 53–76 (2008)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zouzias, A.: A Matrix Hyperbolic Cosine Algorithm and Applications. Ver. 1. (March 2011), arxiv:1103.2793Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anastasios Zouzias
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoCanada

Personalised recommendations