2,679 research outputs found
A Lower Bound on the Entropy Rate for a Large Class of Stationary Processes and its Relation to the Hyperplane Conjecture
We present a new lower bound on the differential entropy rate of stationary
processes whose sequences of probability density functions fulfill certain
regularity conditions. This bound is obtained by showing that the gap between
the differential entropy rate of such a process and the differential entropy
rate of a Gaussian process with the same autocovariance function is bounded.
This result is based on a recent result on bounding the Kullback-Leibler
divergence by the Wasserstein distance given by Polyanskiy and Wu. Moreover, it
is related to the famous hyperplane conjecture, also known as slicing problem,
in convex geometry originally stated by J. Bourgain. Based on an entropic
formulation of the hyperplane conjecture given by Bobkov and Madiman we discuss
the relation of our result to the hyperplane conjecture.Comment: presented at the 2016 IEEE Information Theory Workshop (ITW),
Cambridge, U
Invariances in variance estimates
We provide variants and improvements of the Brascamp-Lieb variance inequality
which take into account the invariance properties of the underlying measure.
This is applied to spectral gap estimates for log-concave measures with many
symmetries and to non-interacting conservative spin systems
The shape of a random affine Weyl group element and random core partitions
Let be a finite Weyl group and be the corresponding affine
Weyl group. We show that a large element in , randomly generated by
(reduced) multiplication by simple generators, almost surely has one of
-specific shapes. Equivalently, a reduced random walk in the regions of
the affine Coxeter arrangement asymptotically approaches one of -many
directions. The coordinates of this direction, together with the probabilities
of each direction can be calculated via a Markov chain on . Our results,
applied to type , show that a large random -core obtained
from the natural growth process has a limiting shape which is a
piecewise-linear graph. In this case, our random process is a periodic analogue
of TASEP, and our limiting shapes can be compared with Rost's theorem on the
limiting shape of TASEP.Comment: Published at http://dx.doi.org/10.1214/14-AOP915 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Thin shell implies spectral gap up to polylog via a stochastic localization scheme
We consider the isoperimetric inequality on the class of high-dimensional
isotropic convex bodies. We establish quantitative connections between two
well-known open problems related to this inequality, namely, the thin shell
conjecture, and the conjecture by Kannan, Lovasz, and Simonovits, showing that
the corresponding optimal bounds are equivalent up to logarithmic factors. In
particular we prove that, up to logarithmic factors, the minimal possible ratio
between surface area and volume is attained on ellipsoids. We also show that a
positive answer to the thin shell conjecture would imply an optimal dependence
on the dimension in a certain formulation of the Brunn-Minkowski inequality.
Our results rely on the construction of a stochastic localization scheme for
log-concave measures.Comment: 33 page
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