1. 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

3. 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

7. Hit-And-Run

8. Building the ellipsoid with PCA

9. Building the ellipsoid with LP

10. Lovazs ellipsoid method

13. Conclusion