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Title:
Convergence of Hamiltonian Monte Carlo and Faster Polytope Volume Computation
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We explain the Hamiltonian Monte Carlo method and apply it to the problem of 1) generating uniform random points from polytopes, 2) computing the volume of polytopes. For polytopes in R^n specified by O(n) inequalities, the resulting algorithm for both problems takes merely O*(n^1.667) steps. For volume computation, this is a huge improvement over the previous best algorithm that requires O(n^4) steps. The key idea is to prove certain isoperimetric inequalities on manifolds defined by log barrier functions. Joint work with Santosh Vempala
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