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Generating Random Points in a Convex Body in High Dimensions

Khezeli, Ali | 2009

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 41772 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Alishahi, Kasra
  7. Abstract:
  8. “How can we generate a random point with uniform distribution over a convex body ?” According to it’s applications, it’s important for a solution to this problem to be applicable in high dimensions. Here, we are interested in algorithms with polynomial order with respect to the dimension. All existing methods for dealing with this problem are based on the Markov chain Monte Carlo method, i.e. a random walk is constructed in such that its stationary distribution is the uniform distribution over. Then, after simulating “enough” steps of this random walk, the distribution of the resulting point is “approximately” uniform. The real problem in Monte Carlo method is analyzing its “mixing time”, i.e. the number of steps needed to get close enough to the stationary distribution. This thesis focuses mostly on analyzing the mixing time of the Hit-and-run walk which is the fastest existing random walk for the mentioned problem. Moreover, we explain one of the most important applications of this problem: approximating the volume of a convex body. A fascinating point about this application is that no deterministic polynomial-time algorithm can approximate the volume of a convex body with bounded relative error.

  9. Keywords:
  10. Markov Process ; Convex Body ; Approximating Volume ; Mixing Time ; Conductance

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