An invariance principle for polytopes

Journal of the ACM

Volumn 59, Issue 6, 2012, Pages

  Harsha, Prahlad ^a   Klivans, Adam ^b   Meka, Raghu ^c  

^a TATA INSTITUTE OF FUNDAMENTAL RESEARCH   (India)

^b University of Texas at Austin   (United States)

^c RUTGERS UNIVERSITY   (United States)

Author keywords

Agnostic learning; Intersection of halfspaces; Invariance; Noise sensitivity; Polytopes; Pseudorandomness

Indexed keywords

AGNOSTIC LEARNING; HALF SPACES; NOISE SENSITIVITY; POLYTOPES; PSEUDORANDOMNESS;

INTEGER PROGRAMMING; INVARIANCE; LEARNING ALGORITHMS; RANDOM PROCESSES;

TOPOLOGY;

EID: 84872440079     PISSN: 00045411     EISSN: 1557735X     Source Type: Journal    
DOI: 10.1145/2395116.2395118     Document Type: Conference Paper

References (57)

1. AILON, N. AND CHAZELLE, B. 2006. Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. In Proceedings of the 38th ACM Symposium on Theory of Computing (STOC). 557-563. (Pubitemid 44306590)
2. AUSTRIN, P. 2007. Balanced max 2-sat might not be the hardest. In Proceedings of the 39th ACMSymposium on Theory of Computing (STOC), D. S. Johnson andU. Feige, Eds. ACM, 189-197. http://doi.acm.org.10.1145 1250790.1250818. (Pubitemid 47630735)
3. AUSTRIN, P. 2010. Towards sharp inapproximability for any 2-CSP. SIAM J. Comput 39, 6, 2430-2463. http://dx.doi.org/10.1137/07011670.
4. BANSAL, N. AND KHOT, S. 2010. Inapproximability of hypergraph vertex cover and applications to scheduling problems. In Proceedings of the International Colloquim on Automata, Languages, and Programming (ICALP), S. Abramsky, C. Gavoille, C. Kirchner, F. M. auf der Heide, and P. G. Spirakis, Eds. Lecture Notes in Computer Science Series, vol. 6198. Springer, 250-261. http://dx.doi.org/10.1007/978-3-642- 14165-2.
5. BARVINOK, A. AND VEOMETT, E. 2008. The computational complexity of convex bodies. In Surveys on Discrete and Computational Geometry: Twenty Years Later, J. E. Goodman, J. Pach, and R. Pollack, Eds., Contemporary Mathematics Series, vol. 453. AMS, 117-137. http://www.ams.org/bookstore-getitem/ item=COMM-453, arXiv:math/0610325.
6. BECK, M. AND ROBINS, S. 2007. Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra 1st Ed. Undergraduate Texts in Mathematics, Springer. http://math.sfsu.edu/beck/ccd.html.
7. BENJAMINI, I., KALAI, G., AND SCHRAMM, O. 1999. Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes É tudes Sci. Publ. Math. 90, 1, 5-43.
8. BENTKUS, V. K. 1990. Smooth approximations of the norm and differentiable functions with bounded support in Banach space lk∞ Lithuan. Math. J. 30, 3, 223-230.
9. BENTKUS, V. K. 2003. On the dependence of the Berry-Esseen bound on dimension. J. Stat. Plan. Inf. 113, 2, 385-402.
10. BLAIS, E. AND O'DONNELL, R. 2010. Lower bounds for testing function isomorphism. In Proceedings of the IEEE Conference on Computational Complexity. IEEE Computer Society, 235-246. http://doi. ieeecomputersociety.org/10.1109/CCC. 2010.30.
11. CHATTERJEE, S. 2005. A simple invariance theorem. arXiv:math/0508213.
12. CRYAN, M. AND DYER, M. E. 2003. A polynomial-time algorithm to approximately count contingency tables when the number of rows is constant. J. Comput. Syst. Sci. 67, 2, 291-310.
13. DE WOLF, R. 2008. A brief introduction to Fourier analysis on the Boolean cube. Theory of Comput. Graduate Sur. 1, 1-20.
14. DIAKONIKOLAS, I., HARSHA, P., KLIVANS, A., MEKA, R., RAGHAVENDRA, P., SERVEDIO, R. A., AND TAN, L.-Y. 2010a. Bounding the average sensitivity and noise sensitivity of polynomial threshold functions. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), ACM, 533-542. http://doi.acm.org/ 10.1145/1806689.1806763.
15. DIAKONIKOLAS, I.,KANE, D.M., ANDNELSON, J. 2010b. Bounded independence fools degree-2 threshold functions. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science (FOCS). IEEE, 11-20.
16. DINUR, I., FRIEDGUT, E., AND REGEV, O. 2008. Independent sets in graph powers are almost contained in juntas. Geomet. Funct. Anal. 18, 1, 77-97.
17. DINUR, I., MOSSEL, E., AND REGEV, O. 2009. Conditional hardness for approximate coloring. SIAM J. Comput 39, 3, 843-873. http://dx.doi.org/10.1137/ 07068062X.
18. DYER, M. E. 2003. Approximate counting by dynamic programming. In Proceedings of the 35th ACM Symposium on Theory of Computing (STOC). 693-699.
19. FELDMAN, V., GURUSWAMI, V., RAGHAVENDRA, P., AND WU, Y. 2009. Agnostic learning of monomials by halfspaces is hard. In Proceedings of the 50th IEEE Symposium on Foundation of Computer Science (FOCS). IEEE Computer Society, 385-394. http://doi.ieeecomputersociety.org/10.1109/FOCS.2009.
20. FELLER, W. 1968. An Introduction to Probability Theory and Its Applications, Volume 1 3rd Ed. Wiley.
21. FELLER, W. 1971. An Introduction to Probability Theory and Its Applications, Volume 2 2nd Ed. Wiley.
22. GOEMANS, M. X. AND WILLIAMSON, D. P. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 6, 1115-1145.
23. GOPALAN, P., KLIVANS, A., AND MEKA, R. 2010a. Polynomial-time approximation schemes for knapsack and related counting problems using branching programs. Electron. Colloq. Comput. Complex. (ECCC) 17, 133. http://eccc.hpi-web.de/report/2010/133.
24. GOPALAN, P., O'DONNELL, R., WU, Y., AND ZUCKERMAN, D. 2010b. Fooling functions of halfspaces under product distributions. In Proceedings of 25th IEEE Conference on Computational Complexity 223-234.
25. HARSHA, P., KLIVANS, A., AND MEKA, R. 2010. An invariance principle for polytopes. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC). ACM, 543-552.
26. HASTAD, J. 2001. Some optimal inapproximability results. J. ACM 48, 4, 798-859.
27. HAUSSLER, D. 1992. Decision theoretic generalizations of the PAC model for neural net and other learning applications. Inf. Comput. 100, 1, 78-150.
28. IMPAGLIAZZO, R., NISAN, N., ANDWIGDERSON, A. 1994. Pseudorandomness for network algorithms. In Proceedings of the 26th ACM Symposium on Theory of Computing (STOC). ACM, 356-364.
29. JERRUM, M. AND SINCLAIR, A. 1997. The Markov chain Monte Carlo method: An approach to approximate counting and integration. In Approximation Algorithms for NP-hard Problems, D. S. Hochbaum, Ed., PWS Publishing Company. http://www.ieor.berkeley.edu/-hochbaum/html/book-aanp.html.
30. KAHN, J., KALAI, G., AND LINIAL, N. 1988. The influence of variables on Boolean functions (extended abstract). In Proceedings of the 29th IEEE Symposium on Foundations of Computer Science (FOCS). 68-80.
31. KALAI, A. T., KLIVANS, A. R., MANSOUR, Y., AND SERVEDIO, R. A. 2008. Agnostically learning halfspaces. SIAM J. Comput. 37, 6, 1777-1805.
32. KALAI, G. 2005. Noise sensitivity and chaos in social choice theory. Tech. rep. 399, Center for Rationality and Interactive Decision Theory, Hebrew University of Jerusalem. http://www.ratio.huji.ac.il/dp files/dp-399.pdf.
33. KEARNS, M. J., SCHAPIRE, R. E., AND SELLIE, L. 1994. Toward efficient agnostic learning. Mach. Learn. 17, 2-3, 115-141.
34. KHOT, S., KINDLER, G.,MOSSEL, E., AND O'DONNELL, R. 2007. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput. 37, 1, 319-357. (Pubitemid 351321999)
35. KLIVANS, A. R., O'DONNELL, R., AND SERVEDIO, R. A. 2004. Learning intersections and thresholds of halfspaces. J. Comput. Syst. Sci. 68, 4, 808-840.
36. KLIVANS, A. R., O'DONNELL, R., AND SERVEDIO, R. A. 2008. Learning geometric concepts via Gaussian surface area. In Proceedings of the Symposium on Foundations of Computer Science (FOCS) 541-550.
37. LINIAL, N., MANSOUR, Y., AND NISAN, N. 1993. Constant depth circuits, Fourier transform, and learnability. J. ACM 40, 3, 607-620. (Pubitemid 23707885)
38. MAHAJAN, S. AND HARIHARAN, R. 1999. Derandomizing approximation algorithms based on semidefinite programming. SIAM J. Comput. 28, 5, 1641-1663. (Pubitemid 129304146)
39. MANSOUR, Y. 1994. Learning Boolean functions via the Fourier transform. In Theoretical Advances in Neural Computation and Learning, V. P. Roychowdhury, K.-Y. Siu, and A. Orlitsky, Eds., Kluwer Academic Publishers, 391-424. http://www.springer.com/physics/complexity/book/978-0-7923-9478-5.
40. MEKA, R. AND ZUCKERMAN, D. 2010. Pseudorandom generators for polynomial threshold functions. In Proceedings of the ACM Symposium on Theory of Computing (STOC), 427-436.
41. MOSSEL, E. 2008. Gaussian bounds for noise correlation of functions and tight analysis of long codes. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), 156-165.
42. MOSSEL, E. 2012. A quantitative arrow theorem. Probab. Theory Relat. Fields 154, 49-88. DOI: 10.1007/s00440-011-0362-7.
43. MOSSEL, E., O'DONNELL, R., AND OLESZKIEWICZ, K. 2005. Noise stability of functions with low influences invariance and optimality. In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS). 21-30. (Pubitemid 44375716)
44. NAOR, J. AND NAOR, M. 1993. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput. 22, 4, 838-856. (Pubitemid 23677151)
45. NAZAROV, F. 2003. On the maximal perimeter of a convex set in Rn with respect to a Gaussian measure. In Geometric Aspects of Functional Analysis (Israel Seminar 2001-2002). Lecture Notes in Mathematics Series, vol. 1807/2003, Springer, 169-187.
46. O'DONNELL, R. 2004. Hardness amplification within NP. J. Comput. Syst. Sci. 69, 1, 68-94.
47. O'DONNELL, R. 2008. Some topics in analysis of Boolean functions. In Proceedings of the 40th ACMSymposium on Theory of Computing (STOC). 569-578.
48. O'DONNELL, R. AND WU, Y. 2009. Conditional hardness for satisfiable 3-CSPs. In Proceedings of the 41st ACM Symposium on Theory of Computing (STOC). 493-502. http://doi.acm.org/10.1145/1536414.1536482.
49. PAULAUSKAS, V. AND RACKAUSKAS, A. 1989. Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces. Kluwer Academic Publishers. (Translated from Russian).
50. PERES, Y. 2004. Noise stability of weighted majority. (arXiv:math/0412377.)
51. PINELIS, I. 1994. Extremal probabilistic problems and hotelling T2 test under a symmetry condition. Ann. Statist. 22, 1, 357-368.
52. RAGHAVENDRA, P. 2008. Optimal algorithms and inapproximability results for every CSP? In STOC, ACM, 245-254. http://doi.acm.org/10.1145/1374376.
53. ROTAR', V. I. 1979. Limit theorems for polylinear forms. J. Multivar. Anal. 9, 4, 511-530.
54. SHI, Y. 2000. Lower bounds of quantum black-box complexity and degree of approximating polynomials by influence of Boolean variables. Inf. Process. Lett. 75, 1-2, 79-83.
55. TAO, T. 2009. Talagrand's concentration inequality. (Post in Blog "What's new"). htt://terrytao.wordpress. com/2009/06/09/talagrands- concentration-inequality/.
56. WOLFF, P. 2007. Hypercontractivity of simple random variables. Studia Math 180, 3, 219-236.
57. ZIEGLER, G. M. 1995. Lectures on polytopes. Graduate texts in Mathematics Series, vol. 152. Springer. http://www.springer.com/math/geometry/book/978-0- 387-94365-7.