415 research outputs found
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
A Note on the Entropy/Influence Conjecture
The entropy/influence conjecture, raised by Friedgut and Kalai in 1996, seeks
to relate two different measures of concentration of the Fourier coefficients
of a Boolean function. Roughly saying, it claims that if the Fourier spectrum
is "smeared out", then the Fourier coefficients are concentrated on "high"
levels. In this note we generalize the conjecture to biased product measures on
the discrete cube, and prove a variant of the conjecture for functions with an
extremely low Fourier weight on the "high" levels.Comment: 12 page
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
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
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