Monte Carlo uniform sampling of high-dimensional
convex polytopes: reducing the condition number with
applications in metabolic network analysis
Center for life nanoscience CLNS-IIT, P.le
A.Moro 2, 00815, Rome, Italy
arXiv, Feb 18, 2014
Mathematics-Statistics
Presented by Chao Wang
Introduction
• From a theoretical viewpoint it leads to polynomial-time approximate
algorithms for the calculation of the volume of a convex body, whose
exact determination is a #P-hard problem.
• On the other hand general problems of inference from linear
constraints require an uniform sampling of the points inside a
convex polytope: examples include metabolic network analysis,
compressed sensing, freezing transition of hard spheres and density
reconstruction from gravitational lensing in astrophysics.
• The knowledge of all the vertices characterizes completely a
polytope but deterministic algorithms that perform an exhaustive
enumeration can be infeasible in high dimensions since the number
of such vertices could scale exponentially with the dimension.
• The faster and most popular algorithm in order to sample points
inside convex bodies is the Hit-and-Run Markov Chain Monte Carlo
Hit-And-Run
Building the ellipsoid with PCA
Building the ellipsoid with LP
Lovazs ellipsoid method
Conclusion