Abstract
Let f: {-1, 1}n → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L 1 influence of f can also be bounded as a function of d. Bačkurs and Bavarian answered this question in the affirmative, providing a bound of O(d 3) for general functions and O(d 2) for homogeneous functions. We improve on their results by providing a bound of d 2 for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Aaronson and A. Ambainis, The need for structure in quantum speedups, in Innovations in Computer Science, Beijing, China, 2011, Tsinghua University Press, Beijing, 2011, pp. 338–352.
S. Aaronson and A. Ambainis, The need for structure in quantum speedups, Theory of Computing 10 (2014), 133–166.
A. Backurs and M. Bavarian, On the sum of L1 influences, in IEEE 29th Conference on Computational Complexity, IEEE Computer Society, Los Alamitos, CA, 2014, pp. 132–143.
I. Dinur, E. Friedgut, G. Kindler and R. O’Donnell, On the Fourier tails of bounded functions over the discrete cube, Israel Journal of Mathematics 160 (2007), 389–412.
P. Erdos, M. Goldberg, J. Pach and J. Spencer, Cutting a graph into two dissimilar halves, Journal of Graph Theory 12 (1988), 121–131.
L. A. Harris, A Bernstein–Markov theorem for normed spaces, Journal Mathematical Analysis and Applications 208 (1997), 476–486.
L. A. Harris, Coefficients of polynomials of restricted growth on the real line, Journal of Approximation Theory 93 (1998), 293–312.
L. A. Harris, Optimal oscillation points for polynomials of restricted growth on the real line, in Approximation Theory and Applications, Hadronic Press, Palm Harbor, FL, 1998, pp. 85–106.
O. Klurman, V. Markov’s problem for monotone polynomials, Tech. report, arxiv.org/pdf/1205.0846v1.pdf., 2012.
P. Nevai, Orthogonal polynomials, Memoirs of the American Mathematical Society 213 (1979).
R. O’Donnell, Open problems in analysis of Boolean functions, Tech. report, arxiv.org/pdf/1204.6447v1.pdf, 2012.
R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, Cambridge, 2014.
S. Gy. Révész and Y. Sarantopoulos, On Markov constants of homogeneous polynomials over real normed spaces, East Journal on Approximations 9 (2003), 277–304.
T. J. Rivlin, The Chebyshev Polynomials, John Wiley and Sons, New York, 1975.
Y. Sarantopoulos, Bounds on the derivatives of polynomials on Banach spaces, Mathematical Proceedings of the Cambridge Philosophical Society 307 (1991), 307–312.
D. Scheder and L.-Y. Tan, On the average sensitivity and density of k-CNF formulas, in Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Computer Science, Vol. 8096, Springer, Heidelberg, 2013, pp. 683–698.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research conducted at the Simons Institute for the Theory of Computing during the 2013 fall semester on Real Analysis in Computer Science and at the Institute for Advanced Study.
Research supported in part by an NSERC, and an FQRNT grant.
Research supported in part by I.S.F. grant 402/13 and by the Alon fellowship.
Rights and permissions
About this article
Cite this article
Filmus, Y., Hatami, H., Keller, N. et al. On the sum of the L 1 influences of bounded functions. Isr. J. Math. 214, 167–192 (2016). https://doi.org/10.1007/s11856-016-1355-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-016-1355-0