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On Approximating the Eigenvalues of Stochastic Matrices in Probabilistic Logspace

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Abstract

We show that approximating the second eigenvalue of stochastic operators is BPL-complete, thus giving a natural problem complete for this class. We also show that approximating any eigenvalue of a stochastic and Hermitian operator with constant accuracy can be done in BPL. This work together with related work on the subject reveal a picture where the various space-bounded classes (e.g., probabilistic logspace, quantum logspace and the class DET) can be characterized by algebraic problems (such as approximating the spectral gap) where, roughly speaking, the difference between the classes lies in the kind of operators they can handle (e.g., stochastic, Hermitian or arbitrary).

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References

  • Romas Aleliunas, Richard M. Karp, R.J. Lipton, Laszlo Lovasz & C. Rackoff (1979). Random walks, universal traversal sequences, and the complexity of maze problems. In 20th Annual Symposium on Foundations of Computer Science, 218–223.

  • Alvarez Carme, Greenlaw Raymond (2000) A compendium of problems complete for symmetric logarithmic space. Computational Complexity 9(2): 123–145

    Article  MathSciNet  MATH  Google Scholar 

  • Stephen A. Cook (1985). A taxonomy of problems with fast parallel algorithms. Information and Control 64(13). International Conference on Foundations of Computation Theory.

  • Csansky L. (1976) Fast parallel matrix inversion algorithms. SIAM Journal of Computing 5(6): 618–623

    Article  MathSciNet  Google Scholar 

  • Diakonikolas Ilias, Gopalan Parikshit, Jaiswal Ragesh, A Servedio Rocco, Viola Emanuele (2010) Bounded independence fools halfspaces. SIAM Journal on Computing 39(8): 3441–3462

    Article  MathSciNet  MATH  Google Scholar 

  • Dean Doron & Amnon Ta-Shma (2015a). On the de-randomization of space-bounded approximate counting problems. Information Processing Letters 115(10), 750–753

  • Dean Doron & Amnon Ta-Shma (2015b). On the Problem of Approximating the Eigenvalues of Undirected Graphs in Probabilistic Logspace. In International Colloquium on Automata, Languages, and Programming, 419–431. Springer.

  • Eremenko Alexandre, Yuditskii Peter (2007) Uniform approximation of sgn(x) by polynomials and entire functions. Journal d’Analyse Mathématique 101(1): 313–324

    Article  MathSciNet  MATH  Google Scholar 

  • Bill Fefferman & Cedric Yen-Yu Lin (2016). A Complete Characterization of Unitary Quantum Space. arXiv:1604.01384.

  • James Allen Fill (1991). Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. The Annals of Applied Probability 62–87.

  • Aram W. Harrow, Avinatan Hassidim & Seth Lloyd (2009). Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150–502.

  • K. Hoffman (2007). Banach Spaces of Analytic Functions. Dover Books on Mathematics Series. Dover Publications, Incorporated. ISBN 9780486458748.

  • Russell Impagliazzo & Avi Wigderson (1997). P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, STOC ’97, 220–229. ACM, New York, NY, USA.

  • Mark Jerrum, Alistair Sinclair & Eric Vigoda (2004). A Polynomial-time Approximation Algorithm for the Permanent of a Matrix with Nonnegative Entries. J. ACM 51(4), 671–697. ISSN 0004-5411.

  • Adam R Klivans., Dieter van Melkebeek (2002) Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing 31(5): 1501–1526

    Article  MathSciNet  MATH  Google Scholar 

  • Swastik Kopparty, Shubhangi Saraf & Amir Shpilka (2014). Equivalence of polynomial identity testing and deterministic multivariate polynomial factorization. In 2014 IEEE 29th Conference on Computational Complexity (CCC), 169–180. IEEE.

  • François Le Gall (2016). Solving Laplacian Systems in Logarithmic Space. arXiv:1608.01426.

  • Harry R. Lewis., Christos H. Papadimitriou (1982) Symmetric space-bounded computation. Theoretical Computer Science 19(2): 161–187

    Article  MathSciNet  MATH  Google Scholar 

  • Dieter van Melkebeek & Thomas Watson (2010). Time-Space Efficient Simulations of Quantum Computations. Electronic Colloquium on Computational Complexity (ECCC) 17, 147.

  • Milena Mihail (1989). Conductance and convergence of Markov chains-A combinatorial treatment of expanders. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 526–531. IEEE Comput. Soc. Press.

  • Nisan Noam (1992) Pseudorandom generators for space-bounded computation. Combinatorica 12(4): 449–461

    Article  MathSciNet  MATH  Google Scholar 

  • Noam Nisan (1994) RL\({\subseteq}\) SC. Computational Complexity 4(1): 1–11

    Article  MathSciNet  Google Scholar 

  • Noam Nisan., Avi Wigderson (1994) Hardness vs randomness. Journal of computer and System Sciences 49(2): 149–167

    Article  MathSciNet  MATH  Google Scholar 

  • Omer Reingold (2008). Undirected connectivity in log-space. J. ACM 55(4).

  • Omer Reingold, Luca Trevisan & Salil Vadhan (2006). Pseudorandom walks on regular digraphs and the RL vs. L problem. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, 457–466. ACM.

  • Theodore J. Rivlin (1974). The Chebyshev polynomials. Pure and applied mathematics. Wiley. ISBN 9780471724704.

  • E. B. Saff & V. Totik (1989). Polynomial Approximation of Piecewise Analytic Functions. Journal of the London Mathematical Society s2-39(3), 487–498.

  • Michael E. Saks & Shiyu Zhou (1999). \({BP_{H}SPACE(S)}\) \({\subseteq}\) \({DSPACE(S^{3/2}}\)). J. Comput. Syst. Sci. 58(2), 376–403

  • Amnon Ta-Shma (2013). Inverting well conditioned matrices in quantum logspace. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC ’13, 881–890. ACM, New York, NY, USA. ISBN 978-1-4503-2029-0.

  • A. F. Timan (1963). Theory of Approximation of Functions of a Real Variable. Dover books on advanced mathematics. Pergamon Press. ISBN 9780486678306.

  • Leslie G Valiant (1979) The complexity of computing the permanent. Theoretical computer science 8(2): 189–201

    Article  MathSciNet  Google Scholar 

  • John Watrous (1999) Space-bounded quantum complexity. Journal of Computer and System Sciences 59(2): 281–326

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. The Blavatnik School of Computer Science, Tel-Aviv University, 6997801, Ramat Aviv, Israel

    Dean Doron, Amir Sarid & Amnon Ta-Shma

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  1. Dean Doron
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  2. Amir Sarid
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  3. Amnon Ta-Shma
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Correspondence to Dean Doron.

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Doron, D., Sarid, A. & Ta-Shma, A. On Approximating the Eigenvalues of Stochastic Matrices in Probabilistic Logspace. comput. complex. 26, 393–420 (2017). https://doi.org/10.1007/s00037-016-0150-y

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  • DOI: https://doi.org/10.1007/s00037-016-0150-y

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