11 research outputs found
Approaching Eldan’s and Lee & Vempala’s bounds for the KLS conjecture in a unified method
La principal idea de este artículo es revisar las pruebas de las mejores estimaciones conocidas para la conjetura KLS de salto espectral, demostradas por Eldan y Lee & Vempala, aplicando el esquema de localización de Eldan a dos sistemas de ecuaciones diferenciales estocásticas diferentes. Damos una prueba unificada de estas dos acotaciones obteniendo la estimación de Eldan desde el sistema de ecuaciones diferenciales estocásticas considerado por Lee & Vempala
Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures
In this paper, we study the asymptotic thin-shell width concentration for
random vectors uniformly distributed in Orlicz balls. We provide both
asymptotic upper and lower bounds on the probability of such a random vector
being in a thin shell of radius times the asymptotic value of
(as
), showing that in certain ranges our estimates are optimal. In
particular, our estimates significantly improve upon the currently best known
general Lee-Vempala bound when the deviation parameter goes down to
zero as the dimension of the ambient space increases. We shall also
determine in this work the precise asymptotic value of the isotropic constant
for Orlicz balls. Our approach is based on moderate deviation principles and a
connection between the uniform distribution on Orlicz balls and Gibbs measures
at certain critical inverse temperatures with potentials given by Orlicz
functions, an idea recently presented by Kabluchko and Prochno in [The maximum
entropy principle and volumetric properties of Orlicz balls, J. Math. Anal.
Appl. {\bf 495}(1) 2021, 1--19].Comment: 27 page
Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures
In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector being in a thin shell of radius times the asymptotic value of n^{-1/2}\left(\E\left[\Vert X_n\Vert_2^2\right]\right)^{1/2} (as ), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter goes down to zero as the dimension of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {\bf 495}(1) 2021, 1--19]
A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path
Arising from structural graph theory, treewidth has become a focus of study
in fixed-parameter tractable algorithms in various communities including
combinatorics, integer-linear programming, and numerical analysis. Many NP-hard
problems are known to be solvable in time, where is the treewidth of the input
graph. Analogously, many problems in P should be solvable in time; however, due to the lack of appropriate tools,
only a few such results are currently known. [Fom+18] conjectured this to hold
as broadly as all linear programs; in our paper, we show this is true:
Given a linear program of the form , and a width- tree decomposition of a graph related to , we
show how to solve it in time where is the number of variables and is
the relative accuracy. Combined with recent techniques in vertex-capacitated
flow [BGS21], this leads to an algorithm with run-time. Besides being the first of its
kind, our algorithm has run-time nearly matching the fastest run-time for
solving the sub-problem (under the assumption that no fast matrix
multiplication is used).
We obtain these results by combining recent techniques in interior-point
methods (IPMs), sketching, and a novel representation of the solution under a
multiscale basis similar to the wavelet basis