Abstract
We highlight the challenge of proving correlation bounds between boolean functions and real-valued polynomials, where any non-boolean output counts against correlation.
We prove that real-valued polynomials of degree 1 2 lg2 lg2 n have correlation with parity at most zero. Such a result is false for modular and threshold polynomials. Its proof is based on a variant of an anti-concentration result by Costello et al. [2006].
- James Aspnes, Richard Beigel, Merrick Furst, and Steven Rudich. 1994. The expressive power of voting polynomials. Combinatorica 14, 2, 135--148.Google ScholarCross Ref
- Richard Beigel. 1993. The polynomial method in circuit complexity. In Proceedings of the 8th Structure in Complexity Theory Conference. 82--95.Google ScholarCross Ref
- Kevin P. Costello. 2009. Bilinear and quadratic variants on the littlewood-offord problem. Israel J. Math. (To appear). http://arxiv.org/abs/0902.1538.Google Scholar
- Kevin P. Costello, Terence Tao, and Van Vu. 2006. Random symmetric matrices are almost surely nonsingular. Duke Math. J. 135, 2, 395--413.Google ScholarCross Ref
- Paul Erdos. 1945. On a lemma of littlewood and offord. Bull. Amer. Math. Soc. 51, 898--902.Google ScholarCross Ref
- Frederic Green. 2004. The correlation between parity and quadratic polynomials mod 3. J. Comput. Syst. Sci. 69, 1, 28--44. Google ScholarDigital Library
- John Littlewood and Albert Offord. 1943. On the number of real roots of a random algebraic equation. III. Rec. Math. Mat. Sbornik N. S. 12, 277--286.Google Scholar
- Saburo Muroga, Iwao Toda, and Satoru Takasu. 1961. Theory of majority decision elements. J. Franklin Inst. 271, 376--418.Google ScholarCross Ref
- Saburo Muroga. 1971. Threshold Logic and its Applications. Wiley-Interscience, New York.Google Scholar
- Hoi H. Nguyen and Van Vu. 2013. Small ball probability, inverse theorems, and applications. In Erdos Centennial, Springer.Google Scholar
- Alexander A. Sherstov. 2008. Communication lower bounds using dual polynomials. Bull. Euro. Assoc. Theor. Comput. Sci. 95, 59--93.Google Scholar
- Rocco A. Servedio and Emanuele Viola. 2012. On a special case of rigidity. http://www.ccs.neu.edu/home/viola/.Google Scholar
- Leslie G. Valiant. 1977. Graph-theoretic arguments in low-level complexity. In Proceedings of the 6th Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 53, Springer, 162--176.Google Scholar
- Emanuele Viola. 2009. On the power of small-depth computation. Foundat. Trends Theor. Comput. Sci. 5, 1, 1--72. Google ScholarDigital Library
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- Real Advantage
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