Document Open Access Logo

Dimension Reduction for Polynomials over Gaussian Space and Applications

Authors Badih Ghazi, Pritish Kamath, Prasad Raghavendra

PDF
Thumbnail PDF

File

LIPIcs.CCC.2018.28.pdf
  • Filesize: 0.77 MB
  • 37 pages

Document Identifiers

Author Details

Badih Ghazi
  • Google Research, 1600 Amphitheatre Parkway Mountain View, CA 94043, USA
Pritish Kamath
  • Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
Prasad Raghavendra
  • University of California Berkeley, Berkeley, CA, USA

Funding

  • Ghazi, Badih: The work was done while the author was a student at MIT. Supported in parts by NSF CCF-1650733 and CCF-1420692.
  • Kamath, Pritish: Supported in parts by NSF CCF-1420956, CCF-1420692, CCF-1218547 and CCF-1650733.
  • Raghavendra, Prasad: Research supported by Okawa Research Grant and NSF CCF-1408643.

Cite As Get BibTex

Badih Ghazi, Pritish Kamath, and Prasad Raghavendra. Dimension Reduction for Polynomials over Gaussian Space and Applications. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 28:1-28:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CCC.2018.28

Abstract

We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016].
Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
Keywords
  • Dimension reduction
  • Low-degree Polynomials
  • Noise Stability
  • Non-Interactive Simulation

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. OpenQIProblemsWiki - All the Bell Inequalities. http://qig.itp.uni-hannover.de/qiproblems/1. Accessed: 2016-07-12.
  2. Rudolf Ahlswede and Imre Csiszár. Common randomness in information theory and cryptography. part i: secret sharing. IEEE Transactions on Information Theory, 39(4), 1993. Google Scholar
  3. Rudolf Ahlswede and Imre Csiszár. Common randomness in information theory and cryptography. ii. cr capacity. Information Theory, IEEE Transactions on, 44(1):225-240, 1998. Google Scholar
  4. Noga Alon and Eyal Lubetzky. The shannon capacity of a graph and the independence numbers of its powers. Information Theory, IEEE Transactions on, 52(5):2172-2176, 2006. Google Scholar
  5. Boaz Barak, Moritz Hardt, Ishay Haviv, Anup Rao, Oded Regev, and David Steurer. Rounding parallel repetitions of unique games. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 374-383. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.55.
  6. Mohammad Bavarian, Dmitry Gavinsky, and Tsuyoshi Ito. On the role of shared randomness in simultaneous communication. In Automata, Languages, and Programming, pages 150-162. Springer, 2014. Google Scholar
  7. Salman Beigi. A new quantum data processing inequality. CoRR, abs/1210.1689, 2012. URL: http://arxiv.org/abs/1210.1689.
  8. Salman Beigi and Amin Gohari. On the duality of additivity and tensorization. arXiv preprint arXiv:1502.00827, 2015. Google Scholar
  9. Andrej Bogdanov and Elchanan Mossel. On extracting common random bits from correlated sources. Information Theory, IEEE Transactions on, 57(10):6351-6355, 2011. Google Scholar
  10. Christer Borell. Geometric bounds on the ornstein-uhlenbeck velocity process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 70(1):1-13, 1985. Google Scholar
  11. Gilles Brassard and Louis Salvail. Secret-key reconciliation by public discussion. In advances in Cryptology—EUROCRYPT’93, pages 410-423. Springer, 1994. Google Scholar
  12. Mark Braverman and Young Kun-Ko. Information value of two-prover games. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 12:1-12:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2018.12.
  13. Mark Braverman and Anup Rao. Information equals amortized communication. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 748-757. IEEE, 2011. Google Scholar
  14. Mark Braverman and Jon Schneider. Information complexity is computable. arXiv preprint arXiv:1502.02971, 2015. Google Scholar
  15. Clement Cannonne, Venkat Guruswami, Raghu Meka, and Madhu Sudan. Communication with imperfectly shared randomness. ITCS, 2014. Google Scholar
  16. Eric Chitambar, Runyao Duan, and Yaoyun Shi. Tripartite entanglement transformations and tensor rank. Physical review letters, 101(14):140502, 2008. Google Scholar
  17. Imre Csiszár and Prakash Narayan. Common randomness and secret key generation with a helper. Information Theory, IEEE Transactions on, 46(2):344-366, 2000. Google Scholar
  18. Sanjoy Dasgupta and Anupam Gupta. An elementary proof of a theorem of johnson and lindenstrauss. Random Struct. Algorithms, 22(1):60-65, 2003. URL: http://dx.doi.org/10.1002/rsa.10073.
  19. Anindya De, Elchanan Mossel, and Joe Neeman. Noise stability is computable and approximately low-dimensional. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 10:1-10:11. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.10.
  20. Anindya De, Elchanan Mossel, and Joe Neeman. Non interactive simulation of correlated distributions is decidable. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2728-2746. SIAM, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.174.
  21. Anindya De and Rocco A Servedio. Efficient deterministic approximate counting for low-degree polynomial threshold functions. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 832-841. ACM, 2014. Google Scholar
  22. Payam Delgosha and Salman Beigi. Impossibility of local state transformation via hypercontractivity. CoRR, abs/1307.2747, 2013. URL: http://arxiv.org/abs/1307.2747.
  23. Payam Delgosha and Salman Beigi. Impossibility of local state transformation via hypercontractivity. Communications in Mathematical Physics, 332(1):449-476, 2014. Google Scholar
  24. Peter Gács and János Körner. Common information is far less than mutual information. Problems of Control and Information Theory, 2(2):149-162, 1973. Google Scholar
  25. Hans Gebelein. Das statistische problem der korrelation als variations-und eigenwertproblem und sein zusammenhang mit der ausgleichsrechnung. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 21(6):364-379, 1941. Google Scholar
  26. Badih Ghazi and T. S. Jayram. Resource-efficient common randomness and secret-key schemes. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1834-1853. SIAM, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.120.
  27. Badih Ghazi, Pritish Kamath, and Prasad Raghavendra. Dimension reduction for polynomials over gaussian space and applications. Electronic Colloquium on Computational Complexity (ECCC), 24:125, 2017. URL: https://eccc.weizmann.ac.il/report/2017/125.
  28. Badih Ghazi, Pritish Kamath, and Madhu Sudan. Communication complexity of permutation-invariant functions. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1902-1921. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch134.
  29. Badih Ghazi, Pritish Kamath, and Madhu Sudan. Decidability of non-interactive simulation of joint distributions. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 545-554. IEEE Computer Society, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.65.
  30. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, 1995. URL: http://dx.doi.org/10.1145/227683.227684.
  31. Steven Heilman. Euclidean partitions optimizing noise stability. CoRR, abs/1211.7138, 2012. URL: http://arxiv.org/abs/1211.7138.
  32. Steven Heilman, Elchanan Mossel, and Joe Neeman. Standard simplices and pluralities are not the most noise stable. Israel Journal of Mathematics, 213(1):33-53, 2016. Google Scholar
  33. Hermann O. Hirschfeld. A connection between correlation and contingency. Mathematical Proceedings of the Cambridge Philosophical Society, 31(4):520–-524, 1935. URL: http://dx.doi.org/10.1017/S0305004100013517.
  34. Thomas Holenstein. Parallel repetition: Simplification and the no-signaling case. Theory of Computing, 5(1):141-172, 2009. URL: http://dx.doi.org/10.4086/toc.2009.v005a008.
  35. Marcus Isaksson and Elchanan Mossel. Maximally stable gaussian partitions with discrete applications. Israel Journal of Mathematics, 189(1):347-396, 2012. Google Scholar
  36. William Johnson and Joram Lindenstrauss. Extensions of lipschitz maps into a hilbert space. Contemporary Mathematics, 26:189-206, 01 1984. Google Scholar
  37. Sudeep Kamath and Venkat Anantharam. On non-interactive simulation of joint distributions. IEEE Trans. Information Theory, 62(6):3419-3435, 2016. URL: http://dx.doi.org/10.1109/TIT.2016.2553672.
  38. Julia Kempe, Hirotada Kobayashi, Keiji Matsumoto, Ben Toner, and Thomas Vidick. Entangled games are hard to approximate. SIAM J. Comput., 40(3):848-877, 2011. URL: http://dx.doi.org/10.1137/090751293.
  39. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? SIAM Journal on Computing, 37(1):319-357, 2007. Google Scholar
  40. László Lovász. On the shannon capacity of a graph. Information Theory, IEEE Transactions on, 25(1):1-7, 1979. Google Scholar
  41. Ueli M Maurer. Secret key agreement by public discussion from common information. Information Theory, IEEE Transactions on, 39(3):733-742, 1993. Google Scholar
  42. Elchanan Mossel. Gaussian bounds for noise correlation of functions. Geometric and Functional Analysis, 19(6):1713-1756, 2010. Google Scholar
  43. Elchanan Mossel and Ryan O'Donnell. Coin flipping from a cosmic source: On error correction of truly random bits. arXiv preprint math/0406504, 2004. Google Scholar
  44. Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. In Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on, pages 21-30. IEEE, 2005. Google Scholar
  45. Elchanan Mossel, Ryan O'Donnell, Oded Regev, Jeffrey E Steif, and Benny Sudakov. Non-interactive correlation distillation, inhomogeneous markov chains, and the reverse bonami-beckner inequality. Israel Journal of Mathematics, 154(1):299-336, 2006. Google Scholar
  46. Ilan Newman. Private vs. common random bits in communication complexity. Information processing letters, 39(2):67-71, 1991. Google Scholar
  47. Michael A Nielsen. Conditions for a class of entanglement transformations. Physical Review Letters, 83(2):436, 1999. Google Scholar
  48. Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM J. Comput., 40(6):1871-1891, 2011. URL: http://dx.doi.org/10.1137/080734042.
  49. Ran Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803, 1998. URL: http://dx.doi.org/10.1137/S0097539795280895.
  50. Ran Raz. A counterexample to strong parallel repetition. SIAM J. Comput., 40(3):771-777, 2011. URL: http://dx.doi.org/10.1137/090747270.
  51. Renato Renner and Stefan Wolf. Simple and tight bounds for information reconciliation and privacy amplification. In Advances in cryptology-ASIACRYPT 2005, pages 199-216. Springer, 2005. Google Scholar
  52. Alfréd Rényi. On measures of dependence. Acta mathematica hungarica, 10(3-4):441-451, 1959. Google Scholar
  53. Claude E Shannon. The zero error capacity of a noisy channel. Information Theory, IRE Transactions on, 2(3):8-19, 1956. Google Scholar
  54. Hans S Witsenhausen. On sequences of pairs of dependent random variables. SIAM Journal on Applied Mathematics, 28(1):100-113, 1975. Google Scholar
  55. Pawel Wolff. Hypercontractivity of simple random variables. Studia Mathematica, 180(3):219-236, 2007. Google Scholar
  56. Aaron D. Wyner. The common information of two dependent random variables. IEEE Trans. Information Theory, 21(2):163-179, 1975. URL: http://dx.doi.org/10.1109/TIT.1975.1055346.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact