Monte Carlo uniform sampling of high-dimensional convex polytopes: reducing the condition number with applic

ations in metabolic network analysis

Center for life nanoscience CLNS-IIT, P.le A.Moro 2, 00815, Rome, Italy

arXiv, Feb 18, 2014Mathematics-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