Extremal properties of polynomial threshold functions

Proceedings of the Annual IEEE Conference on Computational Complexity

Volumn , Issue , 2003, Pages 3-12

  O'Donnell, Ryan ^a   Servedio, Rocco A ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b Columbia University ^*  (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOOLEAN FUNCTIONS; COMPUTATIONAL COMPLEXITY; COMPUTER SIMULATION; LOGIC CIRCUITS; BINARY DECISION DIAGRAMS; BINARY TREES; COMPUTER SCIENCE; DECISION THEORY; DECISION TREES; FUNCTIONS; MATHEMATICAL TECHNIQUES; NETWORKS (CIRCUITS); POLYNOMIALS;

POLYNOMIAL THRESHOLD FUNCTIONS (PTF);

POLYNOMIALS; BOOLEAN FUNCTIONS;

COMPLEXITY THEORY; EXTREMAL; LOWER ORDER TERMS; POLYNOMIAL THRESHOLD FUNCTIONS; UPPER BOUND; WILLIAMS;

EID: 0043016276     PISSN: 10930159     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (26)

1. N. Alon. Personal communication to M. Saks, reported in M. Saks, "Slicing the Hypercube", 1993.
2. M. Anthony. Classification by polynomial surfaces. Discrete Applied Mathematics, 61:91-103, 1995.
3. J. Aspnes, R. Beigel, M. Furst, and S. Rudich. The expressive power of voting polynomials. Combinatorica, 14(2):1-14, 1994.
4. R. Beigel. Perceptrons, PP, and the Polynomial Hierarchy. Computational Complexity, 4:339-349, 1994.
5. R. Beigel. Personal communication, 2000.
6. R. Beigel, N. Reingold, and D. Spielman. PP is closed under intersection. Journal of Computer and System Sciences, 50(2):191-202, 1995.
7. A. Blum. Rank-r decision trees are a subclass of r-decision lists. Information Processing Letters, 42(4): 183-185, 1992.
8. J. Bruck. Harmonic analysis of polynomial threshold functions. SIAM Journal on Discrete Mathematics, 3(2): 168-177, 1990.
9. 0 functions and spectral norms. SIAM Journal on Discrete Mathematics, 21:33-42, 1992.
10. T. Cover. Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Transactions on Electronic Computers, EC-14(3):326-334, 1965.
11. B. Fu. Separating PH from PP by relativization. Acta Math. Sinica, 8(3):329-336, 1992.
12. C. Gotsman. On boolean functions, polynomials and algebraic threshold functions. Technical Report TR-89-18, Department of Computer Science, Hebrew University, 1989.
13. R. L. Graham, K. Leeb, and B. L. Rothschild. Ramsey's theorem for a class of categories. Adv. Math., 8:417-433, 1972.
14. R. L. Graham, B. L. Rothschild, and J. L. Spencer. Ramsey Theory. Wiley-Interscience, New York, 1990.
15. W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13-30, 1963.
16. J. Jackson. An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. Journal of Computer and System Sciences, 55:414-440, 1997.
17. A. Klivans, R. O'Donnell, and R. Servedio. Learning intersections and thresholds of halfspaces. In Proceedings of the 43rd Annual Symposium on Foundations of Computer Science, 2002.
18. õ(n1/3). in Proceedings of the Thirty-Third Annual Symposium on Theory of Computing, pages 258-265, 2001.
19. F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes. North-Holland, New York, 1977.
20. M. Minsky and S. Papert. Perceptrons: an introduction to computational geometry. MIT Press, Cambridge, MA, 1968.
21. R. O'Donnell and R. Servedio. New degree bounds for polynomial threshold functions. Proceedings of the 35th ACM Symposium on Theory of Computing, 2003.
22. A. Razborov and S. Rudich. Natural proofs. Journal of Computer and System Sciences, 55(1):24-35, 1997.
23. M. Saks. Slicing the hypercube, pages 211-257. London Mathematical Society Lecture Note Series 187, 1993.
24. A. Samorodnitsky. Unpublished manuscript, 2000. Personal communication, 2003.
25. N. Vereshchagin. Lower bounds for perceptrons solving some separation problems and oracle separation of AM from PP. In Proceedings of the Third Annual Israel Symposium on Theory of Computing and Systems, 1995.
26. C. Wang and A.C. Williams. The threshold order of a boolean function. Discrete Applied Mathematics, 31:51-69, 1991.