52,450 research outputs found
On the entropy of rectifiable and stratified measures
We summarize some results of geometric measure theory concerning rectifiable
sets and measures. Combined with the entropic chain rule for disintegrations
(Vigneaux, 2021), they account for some properties of the entropy of
rectifiable measures with respect to the Hausdorff measure first studied by
(Koliander et al., 2016). Then we present some recent work on stratified
measures, which are convex combinations of rectifiable measures. These
generalize discrete-continuous mixtures and may have a singular continuous
part. Their entropy obeys a chain rule, whose conditional term is an average of
the entropies of the rectifiable measures involved. We state an asymptotic
equipartition property (AEP) for stratified measures that shows concentration
on strata of a few "typical dimensions" and that links the conditional term of
the chain rule to the volume growth of typical sequences in each stratum.Comment: To appear in the proceedings of Geometric Science of Information
(GSI2023
Transport Inequalities. A Survey
This is a survey of recent developments in the area of transport
inequalities. We investigate their consequences in terms of concentration and
deviation inequalities and sketch their links with other functional
inequalities and also large deviation theory.Comment: Proceedings of the conference Inhomogeneous Random Systems 2009; 82
pages
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
Entropies, convexity, and functional inequalities
Our aim is to provide a short and self contained synthesis which generalise
and unify various related and unrelated works involving what we call
Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies
can be seen in particular as an inclusive interpolation between Poincare and
Gross logarithmic Sobolev inequalities. In addition to the known material,
extensions are provided and improvements are given for some aspects. Stability
by tensor products, convolution, and bounded perturbations are addressed. We
show that under simple convexity assumptions on Phi, such inequalities hold in
a lot of situations, including hyper-contractive diffusions, uniformly strictly
log-concave measures, Wiener measure (paths space of Brownian Motion on
Riemannian Manifolds) and generic Poisson space (includes paths space of some
pure jumps Levy processes and related infinitely divisible laws). Proofs are
simple and relies essentially on convexity. We end up by a short parallel
inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic
Gases and Information Theories.Comment: Formerly "On Phi-entropies and Phi-Sobolev inequalities". Author's
www homepage: http://www.lsp.ups-tlse.fr/Chafai
Concentration of the information in data with log-concave distributions
A concentration property of the functional is demonstrated,
when a random vector X has a log-concave density f on . This
concentration property implies in particular an extension of the
Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time
stochastic processes with log-concave marginals.Comment: Published in at http://dx.doi.org/10.1214/10-AOP592 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Cone-volume measures of polytopes
The cone-volume measure of a polytope with centroid at the origin is proved
to satisfy the subspace concentration condition. As a consequence a conjectured
(a dozen years ago) fundamental sharp affine isoperimetric inequality for the
U-functional is completely established -- along with its equality conditions.Comment: Slightly revised version thanks to the suggestions of the referees
and other readers; two figures adde
From Steiner Formulas for Cones to Concentration of Intrinsic Volumes
The intrinsic volumes of a convex cone are geometric functionals that return
basic structural information about the cone. Recent research has demonstrated
that conic intrinsic volumes are valuable for understanding the behavior of
random convex optimization problems. This paper develops a systematic technique
for studying conic intrinsic volumes using methods from probability. At the
heart of this approach is a general Steiner formula for cones. This result
converts questions about the intrinsic volumes into questions about the
projection of a Gaussian random vector onto the cone, which can then be
resolved using tools from Gaussian analysis. The approach leads to new
identities and bounds for the intrinsic volumes of a cone, including a
near-optimal concentration inequality.Comment: This version corrects errors in Propositions 3.3 and 3.4 and in Lemma
8.3 that appear in the published versio
- âŠ