37 research outputs found
Boolean functions with small second order influences on the discrete cube
Motivated by a recent paper of Kevin Tanguy, in which the concept of second
order influences on the discrete cube and Gauss space has been investigated in
detail, the present note studies it in a more specific context of Boolean
functions on the discrete cube. Some bounds which Tanguy obtained as
applications of his more general approach are extended and complemented
Noise stability of functions with low influences: invariance and optimality
In this paper we study functions with low influences on product probability
spaces. The analysis of boolean functions with low influences has become a
central problem in discrete Fourier analysis. It is motivated by fundamental
questions arising from the construction of probabilistically checkable proofs
in theoretical computer science and from problems in the theory of social
choice in economics.
We prove an invariance principle for multilinear polynomials with low
influences and bounded degree; it shows that under mild conditions the
distribution of such polynomials is essentially invariant for all product
spaces. Ours is one of the very few known non-linear invariance principles. It
has the advantage that its proof is simple and that the error bounds are
explicit. We also show that the assumption of bounded degree can be eliminated
if the polynomials are slightly ``smoothed''; this extension is essential for
our applications to ``noise stability''-type problems.
In particular, as applications of the invariance principle we prove two
conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer
science, which was the original motivation for this work, and the ``It Ain't
Over Till It's Over'' conjecture from social choice theory
Strong Contraction and Influences in Tail Spaces
We study contraction under a Markov semi-group and influence bounds for
functions in tail spaces, i.e. functions all of whose low level Fourier
coefficients vanish. It is natural to expect that certain analytic inequalities
are stronger for such functions than for general functions in . In the
positive direction we prove an Poincar\'{e} inequality and moment decay
estimates for mean functions and for all , proving the degree
one case of a conjecture of Mendel and Naor as well as the general degree case
of the conjecture when restricted to Boolean functions. In the negative
direction, we answer negatively two questions of Hatami and Kalai concerning
extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. That
is, we construct a function whose Fourier
coefficients vanish up to level , with all influences bounded by for some constants . We also construct a function
with nonzero mean whose remaining Fourier
coefficients vanish up to level , with the sum of the influences
bounded by for some constants
.Comment: 20 pages, two new proofs added of the main theore
On reverse hypercontractivity
We study the notion of reverse hypercontractivity. We show that reverse
hypercontractive inequalities are implied by standard hypercontractive
inequalities as well as by the modified log-Sobolev inequality. Our proof is
based on a new comparison lemma for Dirichlet forms and an extension of the
Strook-Varapolos inequality.
A consequence of our analysis is that {\em all} simple operators L=Id-\E as
well as their tensors satisfy uniform reverse hypercontractive inequalities.
That is, for all and every positive valued function for we have . This should
be contrasted with the case of hypercontractive inequalities for simple
operators where is known to depend not only on and but also on the
underlying space.
The new reverse hypercontractive inequalities established here imply new
mixing and isoperimetric results for short random walks in product spaces, for
certain card-shufflings, for Glauber dynamics in high-temperatures spin systems
as well as for queueing processes. The inequalities further imply a
quantitative Arrow impossibility theorem for general product distributions and
inverse polynomial bounds in the number of players for the non-interactive
correlation distillation problem with -sided dice.Comment: Final revision. Incorporates referee's comments. The proof of
appendix B has been corrected. A shorter version of this article will appear
in GAF
Lacunary matrices
We study unconditional subsequences of the canonical basis e_rc of elementary
matrices in the Schatten class S^p. They form the matrix counterpart to Rudin's
Lambda(p) sets of integers in Fourier analysis. In the case of p an even
integer, we find a sufficient condition in terms of trails on a bipartite
graph. We also establish an optimal density condition and present a random
construction of bipartite graphs. As a byproduct, we get a new proof for a
theorem of Erdos on circuits in graphs.Comment: 14 page