An elementary analysis of a procedure for sampling points in a convex body

@article{Bubley1998AnEA,
  title={An elementary analysis of a procedure for sampling points in a convex body},
  author={Russ Bubley and Martin E. Dyer and Mark Jerrum},
  journal={Random Struct. Algorithms},
  year={1998},
  volume={12},
  pages={213-235}
}
In this paper we describe a new method for proving the polynomial-time convergence of an algorithm for sampling (almost) uniformly at random from a convex body in high dimension. Previous approaches have been based on estimating conductance via isoperimetric inequalities. We show that a more elementary coupling argument can be used to give a similar result. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 213–235, 1998 

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References

SHOWING 1-10 OF 12 REFERENCES
Computing the volume of convex bodies : a case where randomness provably helps
  • M. Dyer
  • Computer Science, Mathematics
  • 1991
TLDR
The problem of computing the volume of a convex body K in R is discussed and worst-case results are reviewed and randomised approximation algorithms which show that with randomisation one can approximate very nicely are provided.
Approximating the Permanent
TLDR
A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...
Sampling and integration of near log-concave functions
TLDR
This work provides the first polynomial time algorithm to generate samples from a given log-concave distribution and proves a general isoperimetric inequality for convex sets and uses this together with recent developments in the theory of rapidly mixing Markov chains.
Markov chains and polynomial time algorithms
  • R. Kannan
  • Mathematics, Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial
Random walks on finite groups and rapidly mixing markov chains
© Springer-Verlag, Berlin Heidelberg New York, 1983, tous droits reserves. L’acces aux archives du seminaire de probabilites (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml),
Lectures on the Coupling Method
Preliminaries Discrete Theory Continuous Theory Inequalities Intensity-Governed Processes Diffusions Appendix Frequently Used Notation References Index.
Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains
TLDR
The general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomised approximate counting algorithms for these graphs with very good asymptotic behaviour.
A random polynomial-time algorithm for approximating the volume of convex bodies
TLDR
The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within K within Euclidean space.
...
...