TY - JOUR
T1 - A structure theorem for almost low-degree functions on the slice
AU - Keller, Nathan
AU - Klein, Ohad
N1 - Publisher Copyright: © 2020, The Hebrew University of Jerusalem.
PY - 2020/10
Y1 - 2020/10
N2 - The Fourier-Walsh expansion of a Boolean function f: {0, 1}n → {0, 1} is its unique representation as a multilinear polynomial. The Kindler-Safra theorem (2002) asserts that if in the expansion of f, the total weight on coefficients beyond degree k is very small, then f can be approximated by a Boolean-valued function depending on at most O(2k) variables. In this paper we prove a similar theorem for Boolean functions whose domain is the ‘slice’ ([n]pn)={x∈{0,1}n:∑ixi=pn}, where 0 ≪ p ≪ 1, with respect to their unique representation as harmonic multilinear polynomials. We show that if in the representation of f:([n]pn)→{0,1}, the total weight beyond degree k is at most ϵ, where ϵ =min(p, 1 − p)O(k), then f can be O(ϵ)-approximated by a degree-k Boolean function on the slice, which in turn depends on O(2k) coordinates. This proves a conjecture of Filmus, Kindler, Mossel, and Wimmer (2015). Our proof relies on hypercontractivity, along with a novel kind of a shifting procedure. In addition, we show that the approximation rate in the Kindler-Safra theorem can be improved from ϵ + exp(O(k))ϵ5/4 to ϵ + ϵ2(2 ln(1/ϵ))k/k!, which is tight in terms of the dependence on ϵ and misses at most a factor of 2O(k) in the lower-order term.
AB - The Fourier-Walsh expansion of a Boolean function f: {0, 1}n → {0, 1} is its unique representation as a multilinear polynomial. The Kindler-Safra theorem (2002) asserts that if in the expansion of f, the total weight on coefficients beyond degree k is very small, then f can be approximated by a Boolean-valued function depending on at most O(2k) variables. In this paper we prove a similar theorem for Boolean functions whose domain is the ‘slice’ ([n]pn)={x∈{0,1}n:∑ixi=pn}, where 0 ≪ p ≪ 1, with respect to their unique representation as harmonic multilinear polynomials. We show that if in the representation of f:([n]pn)→{0,1}, the total weight beyond degree k is at most ϵ, where ϵ =min(p, 1 − p)O(k), then f can be O(ϵ)-approximated by a degree-k Boolean function on the slice, which in turn depends on O(2k) coordinates. This proves a conjecture of Filmus, Kindler, Mossel, and Wimmer (2015). Our proof relies on hypercontractivity, along with a novel kind of a shifting procedure. In addition, we show that the approximation rate in the Kindler-Safra theorem can be improved from ϵ + exp(O(k))ϵ5/4 to ϵ + ϵ2(2 ln(1/ϵ))k/k!, which is tight in terms of the dependence on ϵ and misses at most a factor of 2O(k) in the lower-order term.
UR - http://www.scopus.com/inward/record.url?scp=85091432648&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/s11856-020-2062-4
DO - https://doi.org/10.1007/s11856-020-2062-4
M3 - مقالة
SN - 0021-2172
VL - 240
SP - 179
EP - 221
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -