## The Annals of Probability

### A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions

Daniel Kane

#### Abstract

We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_{0}$, which can be decomposed as some function of polynomials $q_{1},\ldots,q_{m}$ with $q_{i}$ normalized and $m=O_{d}(1)$, so that if $X$ is a Gaussian random variable, the probability distribution on $(q_{1}(X),\ldots,q_{m}(X))$ does not have too much mass in any small box.

Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.

#### Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1612-1679.

Dates
Revised: December 2014
First available in Project Euclid: 15 May 2017

https://projecteuclid.org/euclid.aop/1494835227

Digital Object Identifier
doi:10.1214/16-AOP1097

Mathematical Reviews number (MathSciNet)
MR3650411

Zentralblatt MATH identifier
1377.60051

Subjects
Primary: 60G15: Gaussian processes
Secondary: 68R05: Combinatorics

#### Citation

Kane, Daniel. A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions. Ann. Probab. 45 (2017), no. 3, 1612--1679. doi:10.1214/16-AOP1097. https://projecteuclid.org/euclid.aop/1494835227

#### References

• [1] Bogachev, V. I. (1998). Gaussian Measures. Mathematical Surveys and Monographs 62. Amer. Math. Soc., Providence, RI.
• [2] Bonami, A. (1970). Étude des coefficients de Fourier des fonctions de $L^{p}(G)$. Ann. Inst. Fourier (Grenoble) 20 335–402 (1971).
• [3] Carbery, A. and Wright, J. (2001). Distributional and $L^{q}$ norm inequalities for polynomials over convex bodies in $\mathbb{R}^{n}$. Math. Res. Lett. 8 233–248.
• [4] Diakonikolas, I., Harsha, P., Klivans, A., Meka, R., Raghavendra, P., Servedio, R. A. and Tan, L.-Y. (2010). Bounding the average sensitivity and noise sensitivity of polynomial threshold functions. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC).
• [5] Diakonikolas, I., Raghavendra, P., Servedio, R. A. and Tan, L.-Y. (2014). Average sensitivity and noise sensitivity of polynomial threshold functions. SIAM J. Comput. 43 231–253.
• [6] Diakonikolas, I., Servedio, R., Tan, L.-Y. and Wan, A. (2010). A Regularity Lemma, and Low-Weight Approximators, for Low-Degree Polynomial Threshold Functions. In 25th Conference on Computational Complexity (CCC).
• [7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
• [8] Fulton, W. (1998). Intersection Theory, 2nd ed. Springer, Berlin.
• [9] Gotsman, C. and Linial, N. (1994). Spectral properties of threshold functions. Combinatorica 14 35–50.
• [10] Green, B. and Tao, T. (2009). The distribution of polynomials over finite fields, with applications to the Gowers norms. Contrib. Discrete Math. 4 1–36.
• [11] Harsha, P., Klivans, A. and Meka, R. (2014). Bounding the sensitivity of polynomial threshold functions. Theory Comput. 10 1–26.
• [12] Kane, D. M. (2010). The Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions. In 25th Annual IEEE Conference on Computational Complexity—CCC 2010 205–210. IEEE Computer Soc., Los Alamitos, CA.
• [13] Kane, D. M. (2011). A small PRG for polynomial threshold functions of Gaussians. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science—FOCS 2011 257–266. IEEE Computer Soc., Los Alamitos, CA.
• [14] Kane, D. M. (2013). The correct exponent for the Gotsman–Linial conjecture. In 2013 IEEE Conference on Computational Complexity—CCC 2013 56–64. IEEE Computer Soc., Los Alamitos, CA.
• [15] Kaufman, T. and Lovett, S. (2008). Worst Case to Average Case Reductions for Polynomials, In The 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008).
• [16] Lindeberg, J. W. (1922). Eine neue herleitung des exponential-gesetzes in der wahrscheinlichkeitsrechnung. Math. Z. 15 211–235.
• [17] Meka, R. and Zuckerman, D. (2010). Pseudorandom generators for polynomial threshold functions. In STOC’10—Proceedings of the 2010 ACM International Symposium on Theory of Computing 427–436. ACM, New York.
• [18] Mossel, E., O’Donnell, R. and Oleszkiewicz, K. (2010). Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. (2) 171 295–341.
• [19] Nelson, E. (1973). The free Markoff field. J. Funct. Anal. 12 211–227.
• [20] Paley, R. E. A. C. and Zygmund, A. (1932). A note on analytic functions in the unit circle. Math. Proc. Cambridge Philos. Soc. 28 266–272.