48 research outputs found
Optimal Concentration of Information Content For Log-Concave Densities
An elementary proof is provided of sharp bounds for the varentropy of random
vectors with log-concave densities, as well as for deviations of the
information content from its mean. These bounds significantly improve on the
bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543,
2011).Comment: 15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with
more general sufficient conditions for equality in Theorem 2.3. Also some
minor corrections and added reference
On the mean width of log-concave functions
In this work we present a new, natural, definition for the mean width of
log-concave functions. We show that the new definition coincide with a previous
one by B. Klartag and V. Milman, and deduce some properties of the mean width,
including an Urysohn type inequality. Finally, we prove a functional version of
the finite volume ratio estimate and the low-M* estimate.Comment: 15 page
Remarks on the KLS conjecture and Hardy-type inequalities
We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary
functions on a convex body , not necessarily
vanishing on the boundary . This reduces the study of the
Neumann Poincar\'e constant on to that of the cone and Lebesgue
measures on ; these may be bounded via the curvature of
. A second reduction is obtained to the class of harmonic
functions on . We also study the relation between the Poincar\'e
constant of a log-concave measure and its associated K. Ball body
. In particular, we obtain a simple proof of a conjecture of
Kannan--Lov\'asz--Simonovits for unit-balls of , originally due to
Sodin and Lata{\l}a--Wojtaszczyk.Comment: 18 pages. Numbering of propositions, theorems, etc.. as appeared in
final form in GAFA seminar note
Estimates for measures of sections of convex bodies
A estimate in the hyperplane problem with arbitrary measures has
recently been proved in \cite{K3}. In this note we present analogs of this
result for sections of lower dimensions and in the complex case. We deduce
these inequalities from stability in comparison problems for different
generalizations of intersection bodies
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change
Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems
We obtain new principles for transferring log-Sobolev and Spectral-Gap
inequalities from a source metric-measure space to a target one, when the
curvature of the target space is bounded from below. As our main application,
we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of
various conservative spin system models, consisting of non-interacting and
weakly-interacting particles, constrained to conserve the mean-spin. When the
self-interaction is a perturbation of a strongly convex potential, this
partially recovers and partially extends previous results of Caputo,
Chafa\"{\i}, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg
and Yau. When the self-interaction is only assumed to be (non-strongly) convex,
as in the case of the two-sided exponential measure, we obtain sharp estimates
on the system's spectral-gap as a function of the mean-spin, independently of
the size of the system.Comment: 57 page