2,768 research outputs found
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
A composition theorem for the Fourier Entropy-Influence conjecture
The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96]
seeks to relate two fundamental measures of Boolean function complexity: it
states that holds for every Boolean function , where
denotes the spectral entropy of , is its total influence,
and is a universal constant. Despite significant interest in the
conjecture it has only been shown to hold for a few classes of Boolean
functions.
Our main result is a composition theorem for the FEI conjecture. We show that
if are functions over disjoint sets of variables satisfying the
conjecture, and if the Fourier transform of taken with respect to the
product distribution with biases satisfies the conjecture,
then their composition satisfies the conjecture. As
an application we show that the FEI conjecture holds for read-once formulas
over arbitrary gates of bounded arity, extending a recent result [OWZ11] which
proved it for read-once decision trees. Our techniques also yield an explicit
function with the largest known ratio of between and
, improving on the previous lower bound of 4.615
Spectral Norm of Symmetric Functions
The spectral norm of a Boolean function is the sum
of the absolute values of its Fourier coefficients. This quantity provides
useful upper and lower bounds on the complexity of a function in areas such as
learning theory, circuit complexity, and communication complexity. In this
paper, we give a combinatorial characterization for the spectral norm of
symmetric functions. We show that the logarithm of the spectral norm is of the
same order of magnitude as where ,
and and are the smallest integers less than such that
or is constant for all with . We mention some applications to the decision tree and communication
complexity of symmetric functions
A stability result for the cube edge isoperimetric inequality
We prove the following stability version of the edge isoperimetric inequality
for the cube: any subset of the cube with average boundary degree within of
the minimum possible is -close to a union of disjoint cubes,
where is independent of the dimension. This extends
a stability result of Ellis, and can viewed as a dimension-free version of
Friedgut's junta theorem.Comment: 12 page
{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
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