2,132 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
Decision Trees, Protocols, and the Fourier Entropy-Influence Conjecture
Given , define the \emph{spectral
distribution} of to be the distribution on subsets of in which the
set is sampled with probability . Then the Fourier
Entropy-Influence (FEI) conjecture of Friedgut and Kalai (1996) states that
there is some absolute constant such that . Here,
denotes the Shannon entropy of 's spectral distribution, and
is the total influence of . This conjecture is one
of the major open problems in the analysis of Boolean functions, and settling
it would have several interesting consequences.
Previous results on the FEI conjecture have been largely through direct
calculation. In this paper we study a natural interpretation of the conjecture,
which states that there exists a communication protocol which, given subset
of distributed as , can communicate the value of using
at most bits in expectation.
Using this interpretation, we are able show the following results:
1. First, if is computable by a read- decision tree, then
.
2. Next, if has and is computable by a
decision tree with expected depth , then .
3. Finally, we give a new proof of the main theorem of O'Donnell and Tan
(ICALP 2013), i.e. that their FEI conjecture composes.
In addition, we show that natural improvements to our decision tree results
would be sufficient to prove the FEI conjecture in its entirety. We believe
that our methods give more illuminating proofs than previous results about the
FEI conjecture
A composition theorem for parity kill number
In this work, we study the parity complexity measures
and .
is the \emph{parity kill number} of , the
fewest number of parities on the input variables one has to fix in order to
"kill" , i.e. to make it constant. is the depth
of the shortest \emph{parity decision tree} which computes . These
complexity measures have in recent years become increasingly important in the
fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and
pseudorandomness \cite{BK12, Sha11, CT13}.
Our main result is a composition theorem for .
The -th power of , denoted , is the function which results
from composing with itself times. We prove that if is not a parity
function, then In other words, the parity kill number of
is essentially supermultiplicative in the \emph{normal} kill number of
(also known as the minimum certificate complexity).
As an application of our composition theorem, we show lower bounds on the
parity complexity measures of and . Here is the sort function due to Ambainis \cite{Amb06},
and is Kushilevitz's hemi-icosahedron function \cite{NW95}. In
doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which
had applications to communication complexity and computational learning theory.
In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and
\cite{TWXZ13}
Szego limit theorem for operators with discontinuous symbols and applications to entanglement entropy
The main result in this paper is a one term Szego type asymptotic formula
with a sharp remainder estimate for a class of integral operators of the
pseudodifferential type with symbols which are allowed to be non-smooth or
discontinuous in both position and momentum. The simplest example of such
symbol is the product of the characteristic functions of two compact sets, one
in real space and the other in momentum space. The results of this paper are
used in a study of the violation of the area entropy law for free fermions in
[18]. This work also provides evidence towards a conjecture due to Harold
Widom.Comment: 18 pages, major revision, to appear in Int. Math. Res. No
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
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