112 research outputs found
A Cubic Algorithm for Computing Gaussian Volume
We present randomized algorithms for sampling the standard Gaussian
distribution restricted to a convex set and for estimating the Gaussian measure
of a convex set, in the general membership oracle model. The complexity of
integration is while the complexity of sampling is for
the first sample and for every subsequent sample. These bounds
improve on the corresponding state-of-the-art by a factor of . Our
improvement comes from several aspects: better isoperimetry, smoother
annealing, avoiding transformation to isotropic position and the use of the
"speedy walk" in the analysis.Comment: 23 page
On the equivalence of modes of convergence for log-concave measures
An important theme in recent work in asymptotic geometric analysis is that
many classical implications between different types of geometric or functional
inequalities can be reversed in the presence of convexity assumptions. In this
note, we explore the extent to which different notions of distance between
probability measures are comparable for log-concave distributions. Our results
imply that weak convergence of isotropic log-concave distributions is
equivalent to convergence in total variation, and is further equivalent to
convergence in relative entropy when the limit measure is Gaussian.Comment: v3: Minor tweak in exposition. To appear in GAFA seminar note
Comments on the floating body and the hyperplane conjecture
We provide a reformulation of the hyperplane conjecture (the slicing problem)
in terms of the floating body and give upper and lower bounds on the
logarithmic Hausdorff distance between an arbitrary convex body \ and the convex floating body inside .Comment: 8 page
Pointwise Estimates for Marginals of Convex Bodies
We prove a pointwise version of the multi-dimensional central limit theorem
for convex bodies. Namely, let X be an isotropic random vector in R^n with a
log-concave density. For a typical subspace E in R^n of dimension n^c, consider
the probability density of the projection of X onto E. We show that the ratio
between this probability density and the standard gaussian density in E is very
close to 1 in large parts of E. Here c > 0 is a universal constant. This
complements a recent result by the second named author, where the
total-variation metric between the densities was considered.Comment: 17 page
Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
The convex set of quantum states of a composite system with
positive partial transpose is analysed. A version of the hit and run algorithm
is used to generate a sequence of random points covering this set uniformly and
an estimation for the convergence speed of the algorithm is derived. For this algorithm works faster than sampling over the entire set of states and
verifying whether the partial transpose is positive. The level density of the
PPT states is shown to differ from the Marchenko-Pastur distribution, supported
in [0,4] and corresponding asymptotically to the entire set of quantum states.
Based on the shifted semi--circle law, describing asymptotic level density of
partially transposed states, and on the level density for the Gaussian unitary
ensemble with constraints for the spectrum we find an explicit form of the
probability distribution supported in [0,3], which describes well the level
density obtained numerically for PPT states.Comment: 11 pages, 4 figure
Gibbs/Metropolis algorithms on a convex polytope
This paper gives sharp rates of convergence for natural versions of the
Metropolis algorithm for sampling from the uniform distribution on a convex
polytope. The singular proposal distribution, based on a walk moving locally in
one of a fixed, finite set of directions, needs some new tools. We get useful
bounds on the spectrum and eigenfunctions using Nash and Weyl-type
inequalities. The top eigenvalues of the Markov chain are closely related to
the Neuman eigenvalues of the polytope for a novel Laplacian.Comment: 21 pages, 1 figur
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