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Bounded Independence Plus Noise Fools Products

Authors Elad Haramaty, Chin Ho Lee, Emanuele Viola

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Elad Haramaty
Chin Ho Lee
Emanuele Viola

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Elad Haramaty, Chin Ho Lee, and Emanuele Viola. Bounded Independence Plus Noise Fools Products. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 14:1-14:30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CCC.2017.14

Abstract

Let D be a b-wise independent distribution over {0,1}^m. Let E be the "noise" distribution over {0,1}^m where the bits are independent and each bit is 1 with probability eta/2. We study which tests f: {0,1}^m -> [-1,1] are epsilon-fooled by D+E, i.e., |E[f(D+E)] - E[f(U)]| <= epsilon where U is the uniform distribution. We show that D+E epsilon-fools product tests f: ({0,1}^n)^k -> [-1,1] given by the product of k bounded functions on disjoint n-bit inputs with error epsilon = k(1-eta)^{Omega(b^2/m)}, where m = nk and b >= n. This bound is tight when b = Omega(m) and eta >= (log k)/m. For b >= m^{2/3} log m and any constant eta the distribution D+E also 0.1-fools log-space algorithms. We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed-Solomon codes of dimension m/k where k = O(1), communication Omega(eta m) - O(log m) is required to decode one message symbol from a codeword with eta m errors, and communication O(eta m log m) suffices. Second, we obtain pseudorandom generators. We can epsilon-fool product tests f: ({0,1}^n)^k -> [-1,1] under any permutation of the bits with seed lengths 2n + O~(k^2 log(1/epsilon)) and O(n) + O~(sqrt{nk log 1/epsilon}). Previous generators have seed lengths >= nk/2 or >= n sqrt{n k}. For the special case where the k bounded functions have range {0,1} the previous generators have seed length >= (n+log k)log(1/epsilon).
Keywords
  • ounded independence
  • Noise
  • Product tests
  • Error-correcting codes
  • Pseudorandomness

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References

  1. Miklós Ajtai. Σ^1_1-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983. Google Scholar
  2. Miklós Ajtai, János Komlós, and Endre Szemerédi. Deterministic simulation in logspace. In 19th ACM Symp. on the Theory of Computing (STOC), pages 132-140, 1987. Google Scholar
  3. Miklos Ajtai and Avi Wigderson. Deterministic simulation of probabilistic constant-depth circuits. Advances in Computing Research - Randomness and Computation, 5:199-223, 1989. Google Scholar
  4. Noga Alon, Alexandr Andoni, Tali Kaufman, Kevin Matulef, Ronitt Rubinfeld, and Ning Xie. Testing k-wise and almost k-wise independence. In ACM Symp. on the Theory of Computing (STOC), pages 496-505, 2007. URL: http://dx.doi.org/10.1145/1250790.1250863.
  5. Noga Alon, László Babai, and Alon Itai. A fast and simple randomized algorithm for the maximal independent set problem. Journal of Algorithms, 7:567-583, 1986. Google Scholar
  6. Noga Alon, Oded Goldreich, Johan Håstad, and René Peralta. Simple constructions of almost k-wise independent random variables. Random Structures &Algorithms, 3(3):289-304, 1992. Google Scholar
  7. Noga Alon, Oded Goldreich, and Yishay Mansour. Almost k-wise independence versus k-wise independence. Inf. Process. Lett., 88(3):107-110, 2003. Google Scholar
  8. Roy Armoni, Michael E. Saks, Avi Wigderson, and Shiyu Zhou. Discrepancy sets and pseudorandom generators for combinatorial rectangles. In 37th IEEE Symp. on Foundations of Computer Science (FOCS), pages 412-421, 1996. Google Scholar
  9. Ziv Bar-Yossef, Omer Reingold, Ronen Shaltiel, and Luca Trevisan. Streaming through combinatorial objects. In Seventeenth IEEE Conference on Computational Complexity. IEEE Computer Soc., Los Alamitos, CA, 2002. Google Scholar
  10. Louay Bazzi and Sanjoy K. Mitter. Encoding complexity versus minimum distance. IEEE Transactions on Information Theory, 51(6):2103-2112, 2005. Google Scholar
  11. Louay M. J. Bazzi. Polylogarithmic independence can fool DNF formulas. SIAM J. Comput., 38(6):2220-2272, 2009. Google Scholar
  12. Anja Becker, Antoine Joux, Alexander May, and Alexander Meurer. Decoding random binary linear codes in 2^n/20: How 1 + 1 = 0 improves information set decoding. In Int. Conf. on the Theory and Applications of Cryptographic Techniques (EUROCRYPT), pages 520-536, 2012. URL: http://dx.doi.org/10.1007/978-3-642-29011-4_31.
  13. Andrej Bogdanov, Periklis A. Papakonstantinou, and Andrew Wan. Pseudorandomness for read-once formulas. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 240-246, 2011. Google Scholar
  14. Andrej Bogdanov, Periklis A. Papakonstantinou, and Andrew Wan. Pseudorandomness for linear length branching programs and stack machines. In Workshop on Randomization and Computation (RANDOM), pages 447-458, 2012. Google Scholar
  15. Andrej Bogdanov and Emanuele Viola. Pseudorandom bits for polynomials. SIAM J. on Computing, 39(6):2464-2486, 2010. Google Scholar
  16. Mark Braverman. Polylogarithmic independence fools AC^0 circuits. J. of the ACM, 57(5), 2010. Google Scholar
  17. J. Lawrence Carter and Mark N. Wegman. Universal classes of hash functions. J. of Computer and System Sciences, 18(2):143-154, 1979. Google Scholar
  18. Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. Improved algorithms via approximations of probability distributions. J. Comput. System Sci., 61(1):81-107, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1695.
  19. Sitan Chen, Thomas Steinke, and Salil P. Vadhan. Pseudorandomness for read-once, constant-depth circuits. CoRR, abs/1504.04675, 2015. URL: http://arxiv.org/abs/1504.04675.
  20. Benny Chor and Oded Goldreich. On the power of two-point based sampling. Journal of Complexity, 5(1):96-106, 1989. Google Scholar
  21. Anindya De. Beyond the central limit theorem: Asymptotic expansions and pseudorandomness for combinatorial sums. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 883-902, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.59.
  22. Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. Bounded independence fools halfspaces. SIAM J. on Computing, 39(8):3441-3462, 2010. Google Scholar
  23. Ilias Diakonikolas, Daniel Kane, and Jelani Nelson. Bounded independence fools degree-2 threshold functions. In 51th IEEE Symp. on Foundations of Computer Science (FOCS). IEEE, 2010. Google Scholar
  24. Guy Even, Oded Goldreich, Michael Luby, Noam Nisan, and Boban Velickovic. Approximations of general independent distributions. In ACM Symp. on the Theory of Computing (STOC), pages 10-16, 1992. Google Scholar
  25. Guy Even, Oded Goldreich, Michael Luby, Noam Nisan, and Boban Velickovic. Efficient approximation of product distributions. Random Struct. Algorithms, 13(1):1-16, 1998. Google Scholar
  26. Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984. Google Scholar
  27. Parikshit Gopalan, Daniek Kane, and Raghu Meka. Pseudorandomness via the discrete fourier transform. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 903-922, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.60.
  28. Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, and Salil Vadhan. Better pseudorandom generators from milder pseudorandom restrictions. In IEEE Symp. on Foundations of Computer Science (FOCS), 2012. Google Scholar
  29. Parikshit Gopalan, Raghu Meka, Omer Reingold, and David Zuckerman. Pseudorandom generators for combinatorial shapes. SIAM J. on Computing, 42(3):1051-1076, 2013. URL: http://dx.doi.org/10.1137/110854990.
  30. Parikshit Gopalan, Ryan O'Donnell, Yi Wu, and David Zuckerman. Fooling functions of halfspaces under product distributions. In 25th IEEE Conf. on Computational Complexity (CCC), pages 223-234. IEEE, 2010. Google Scholar
  31. Parikshit Gopalan and Amir Yehudayoff. Inequalities and tail bounds for elementary symmetric polynomials. Electronic Colloquium on Computational Complexity (ECCC), 21:19, 2014. URL: https://eccc.weizmann.ac.il/report/2014/019/.
  32. Andre Gronemeier. A note on the decoding complexity of error-correcting codes. Information Processing Letters, 100(3):116-119, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2006.06.006.
  33. Johan Håstad. Computational limitations of small-depth circuits. MIT Press, 1987. Google Scholar
  34. Johan Håstad. On the correlation of parity and small-depth circuits. SIAM J. on Computing, 43(5):1699-1708, 2014. Google Scholar
  35. Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for AC^0. In ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 961-972, 2012. Google Scholar
  36. Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from shrinkage. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 111-119, 2012. Google Scholar
  37. Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In 26th ACM Symp. on the Theory of Computing (STOC), pages 356-364, 1994. Google Scholar
  38. Chin Ho Lee and Emanuele Viola. Some limitations of the sum of small-bias distributions. To appear in Theory of Computing, pre-print available at URL: https://eccc.weizmann.ac.il/report/2015/005/.
  39. Shachar Lovett. Unconditional pseudorandom generators for low degree polynomials. Theory of Computing, 5(1):69-82, 2009. http://arxiv.org/abs/toc:v005/a003, URL: http://dx.doi.org/10.4086/toc.2009.v005a003.
  40. Chi-Jen Lu. Improved pseudorandom generators for combinatorial rectangles. Combinatorica, 22(3):417-433, 2002. Google Scholar
  41. Joseph Naor and Moni Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. on Computing, 22(4):838-856, 1993. Google Scholar
  42. Noam Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4):449-461, 1992. Google Scholar
  43. Noam Nisan and David Zuckerman. Randomness is linear in space. J. of Computer and System Sciences, 52(1):43-52, February 1996. Google Scholar
  44. Alexander A. Razborov. A simple proof of Bazzi’s theorem. ACM Transactions on Computation Theory (TOCT), 1(1), 2009. Google Scholar
  45. Omer Reingold, Thomas Steinke, and Salil P. Vadhan. Pseudorandomness for regular branching programs via Fourier analysis. In Workshop on Randomization and Computation (RANDOM), pages 655-670, 2013. Google Scholar
  46. Amir Shpilka. Constructions of low-degree and error-correcting epsilon-biased generators. Computational Complexity, 18(4):495-525, 2009. Google Scholar
  47. Thomas Steinke, Salil P. Vadhan, and Andrew Wan. Pseudorandomness and fourier growth bounds for width-3 branching programs. In Workshop on Randomization and Computation (RANDOM), pages 885-899, 2014. Google Scholar
  48. B. A. Subbotovskaya. Realizations of linear functions by formulas using +, *, -. Soviet Mathematics-Doklady, 2:110-112, 1961. Google Scholar
  49. Avishay Tal. Tight bounds on The Fourier Spectrum of AC⁰, 2014. Electronic Colloquium on Computational Complexity, Technical Report TR14-174. URL: https://eccc.weizmann.ac.il/report/2014/174/.
  50. Emanuele Viola. The sum of d small-bias generators fools polynomials of degree d. Computational Complexity, 18(2):209-217, 2009. Google Scholar
  51. Emanuele Viola. The complexity of distributions. SIAM J. on Computing, 41(1):191-218, 2012. Google Scholar
  52. Emanuele Viola. Randomness buys depth for approximate counting. Computational Complexity, 23(3):479-508, 2014. Google Scholar
  53. Thomas Watson. Pseudorandom generators for combinatorial checkerboards. Computational Complexity, 22(4):727-769, 2013. URL: http://dx.doi.org/10.1007/s00037-012-0036-6.
  54. Andrew Yao. Separating the polynomial-time hierarchy by oracles. In 26th IEEE Symp. on Foundations of Computer Science (FOCS), pages 1-10, 1985. Google Scholar
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