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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