**Nov 29, 2016, 4pm**

SAV 131

**Yin-Tat Lee,** *Microsoft Research*

*Abstract:* We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes, a problem that is even more general than linear programming. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold. The resulting sampling algorithm for polytopes mixes in O*(mn^{3/4}) steps for a polytope in R^n specified by m inequalities. This is the first walk that breaks the quadratic barrier for mixing in high dimension, improving on the previous best bound of O*(mn) by Kannan and Narayanan for the Dikin walk. We also show that each step of the geodesic walk (solving an ODE) can be implemented efficiently, thus improving the time complexity for sampling polytopes.