Given a Boolean function f:{ -1,1} ^{n}→ { -1,1, define the
Fourier distribution
to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f ˆ (S)
2
. The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(f
ˆ2
) ≤ C ⋅ Inf (f), where H
(fˆ2)
is the Shannon entropy of the Fourier distribution of
f
and Inf(f) is the total influence of
f
In this article, we present three new contributions toward the FEI conjecture:
(1)
Our first contribution shows that H(f
ˆ2
) ≤ 2 ⋅ aUC
⊕
(f), where aUC
⊕
(f) is the average unambiguous parity-certificate complexity of
f
. This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture.
(2)
We next consider the weaker
Fourier Min-entropy-influence
(FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H ∞
fˆ2)
≤ C ⋅ Inf(f), where H ∞
fˆ2)
is the min-entropy of the Fourier distribution. We show H
∞
(fˆ2)
≤ 2⋅C
min
⊕
(f), where C
min
⊕
(f) is the minimum parity-certificate complexity of
f
. We also show that for all ε≥0, we have H
∞
(fˆ2)
≤2 log(∥f
ˆ
∥1,ε/(1−ε)), where ∥f
ˆ
∥1,ε is the approximate spectral norm of
f
. As a corollary, we verify the FMEI conjecture for the class of read-
k
DNFs (for constant
k
).
(3)
Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no
flat polynomial
(whose non-zero Fourier coefficients have the same magnitude) of degree
d
and sparsity 2
ω(d)
can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI, and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.
Improved Bounds on Fourier Entropy and Min-entropy
