6 research outputs found
The Entropy Influence Conjecture Revisited
In this paper, we prove that most of the boolean functions, satisfy the Fourier Entropy Influence (FEI) Conjecture
due to Friedgut and Kalai (Proc. AMS'96). The conjecture says that the Entropy
of a boolean function is at most a constant times the Influence of the
function. The conjecture has been proven for families of functions of smaller
sizes. O'donnell, Wright and Zhou (ICALP'11) verified the conjecture for the
family of symmetric functions, whose size is . They are in fact able
to prove the conjecture for the family of -part symmetric functions for
constant , the size of whose is . Also it is known that the
conjecture is true for a large fraction of polynomial sized DNFs (COLT'10).
Using elementary methods we prove that a random function with high probability
satisfies the conjecture with the constant as , for any constant
.Comment: We thank Kunal Dutta and Justin Salez for pointing out that our
result can be extended to a high probability statemen
A Lower Bound on the Constant in the Fourier Min-Entropy/Influence Conjecture
We describe a new construction of Boolean functions. A specific instance of
our construction provides a 30-variable Boolean function having
min-entropy/influence ratio to be which is presently
the highest known value of this ratio that is achieved by any Boolean function.
Correspondingly, is also presently the best known lower bound on the
universal constant of the Fourier min-entropy/influence conjecture
{Improved Bounds on Fourier Entropy and Min-entropy}
Given a Boolean function , the Fourier distribution assigns probability to . The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai asks if there exist a universal constant C>0 such that , where is the Shannon entropy of the Fourier distribution of and is the total influence of . 1) We consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture. This asks if , where is the min-entropy of the Fourier distribution. We show , where is the minimum parity certificate complexity of . We also show that for every , we have , where is the approximate spectral norm of . As a corollary, we verify the FMEI conjecture for the class of read- s (for constant ). 2) We show that , where is the average unambiguous parity certificate complexity of . This improves upon Chakraborty et al. An important consequence of the FEI conjecture is the long-standing Mansour's conjecture. We show that a weaker version of FEI already implies Mansour's conjecture: is ?, where are the 0- and 1-certificate complexities of , respectively. 3) We study what FEI implies about the structure of polynomials that 1/3-approximate a Boolean function. We pose a conjecture (which is implied by FEI): no "flat" degree- polynomial of sparsity can 1/3-approximate a Boolean function. We prove this conjecture unconditionally for a particular class of polynomials