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Random Construction of Interpolating Sets for High-Dimensional Integration

  • Mark Huber (a1) and Sarah Schott (a2)

Abstract

Computing the value of a high-dimensional integral can often be reduced to the problem of finding the ratio between the measures of two sets. Monte Carlo methods are often used to approximate this ratio, but often one set will be exponentially larger than the other, which leads to an exponentially large variance. A standard method of dealing with this problem is to interpolate between the sets with a sequence of nested sets where neighboring sets have relative measures bounded above by a constant. Choosing such a well-balanced sequence can rarely be done without extensive study of a problem. Here a new approach that automatically obtains such sets is presented. These well-balanced sets allow for faster approximation algorithms for integrals and sums using fewer samples, and better tempering and annealing Markov chains for generating random samples. Applications, such as finding the partition function of the Ising model and normalizing constants for posterior distributions in Bayesian methods, are discussed.

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Copyright

Corresponding author

Postal address: Department of Mathematical Sciences, Claremont McKenna College, 850 Columbia Avenue, Claremont, CA 91711, USA. Email address: mhuber@cmc.edu
∗∗ Postal address: Department of Mathematics, Duke University, 117 Physics Building, Science Drive, Durham, NC 27708, USA. Email address: schott@math.duke.edu

References

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Random Construction of Interpolating Sets for High-Dimensional Integration

  • Mark Huber (a1) and Sarah Schott (a2)

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