1,785 research outputs found
Gaussian Bounds for Noise Correlation of Functions
In this paper we derive tight bounds on the expected value of products of
{\em low influence} functions defined on correlated probability spaces. The
proofs are based on extending Fourier theory to an arbitrary number of
correlated probability spaces, on a generalization of an invariance principle
recently obtained with O'Donnell and Oleszkiewicz for multilinear polynomials
with low influences and bounded degree and on properties of multi-dimensional
Gaussian distributions. The results derived here have a number of applications
to the theory of social choice in economics, to hardness of approximation in
computer science and to additive combinatorics problems.Comment: Typos and references correcte
Phase transitions in Phylogeny
We apply the theory of markov random fields on trees to derive a phase
transition in the number of samples needed in order to reconstruct phylogenies.
We consider the Cavender-Farris-Neyman model of evolution on trees, where all
the inner nodes have degree at least 3, and the net transition on each edge is
bounded by e. Motivated by a conjecture by M. Steel, we show that if 2 (1 - 2
e) (1 - 2e) > 1, then for balanced trees, the topology of the underlying tree,
having n leaves, can be reconstructed from O(log n) samples (characters) at the
leaves. On the other hand, we show that if 2 (1 - 2 e) (1 - 2 e) < 1, then
there exist topologies which require at least poly(n) samples for
reconstruction.
Our results are the first rigorous results to establish the role of phase
transitions for markov random fields on trees as studied in probability,
statistical physics and information theory to the study of phylogenies in
mathematical biology.Comment: To appear in Transactions of the AM
Mixing under monotone censoring
We initiate the study of mixing times of Markov chain under monotone
censoring. Suppose we have some Markov Chain on a state space with
stationary distribution and a monotone set . We
consider the chain which is the same as the chain started at some except that moves of of the form where and are {\em censored} and replaced by the move . If is
ergodic and is connected, the new chain converges to conditional on
. In this paper we are interested in the mixing time of the chain in
terms of properties of and . Our results are based on new connections
with the field of property testing. A number of open problems are presented.Comment: 6 page
Complete Characterization of Functions Satisfying the Conditions of Arrow's Theorem
Arrow's theorem implies that a social choice function satisfying
Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant
Alternatives (IIA) must be dictatorial. When non-strict preferences are
allowed, a dictatorial social choice function is defined as a function for
which there exists a single voter whose strict preferences are followed. This
definition allows for many different dictatorial functions. In particular, we
construct examples of dictatorial functions which do not satisfy Transitivity
and IIA. Thus Arrow's theorem, in the case of non-strict preferences, does not
provide a complete characterization of all social choice functions satisfying
Transitivity, the Pareto Principle, and IIA.
The main results of this article provide such a characterization for Arrow's
theorem, as well as for follow up results by Wilson. In particular, we
strengthen Arrow's and Wilson's result by giving an exact if and only if
condition for a function to satisfy Transitivity and IIA (and the Pareto
Principle). Additionally, we derive formulas for the number of functions
satisfying these conditions.Comment: 11 pages, 1 figur
Robust dimension free isoperimetry in Gaussian space
We prove the first robust dimension free isoperimetric result for the
standard Gaussian measure and the corresponding boundary measure
in . The main result in the theory of Gaussian
isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently
by Borell) states that if then the surface area of is
bounded by the surface area of a half-space with the same measure,
. Our results imply in particular that if
satisfies and
then there exists a half-space
such that for an absolute constant . Since the
Gaussian isoperimetric result was established, only recently a robust version
of the Gaussian isoperimetric result was obtained by Cianchi et al., who showed
that for some function with
no effective bounds. Compared to the results of Cianchi et al., our results
have optimal (i.e., no) dependence on the dimension, but worse dependence on .Comment: Published at http://dx.doi.org/10.1214/13-AOP860 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Approximation Resistant Predicates From Pairwise Independence
We study the approximability of predicates on variables from a domain
, and give a new sufficient condition for such predicates to be
approximation resistant under the Unique Games Conjecture. Specifically, we
show that a predicate is approximation resistant if there exists a balanced
pairwise independent distribution over whose support is contained in
the set of satisfying assignments to
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