Sampling s-Concave Functions: The Limit of Convexity Based Isoperimetry

  • Authors:

  • Karthekeyan Chandrasekaran;Amit Deshpande;Santosh Vempala

  • Affiliations:

  • School of Computer Science, Georgia Institute of Technology,;Microsoft Research, India;School of Computer Science, Georgia Institute of Technology,

  • Venue:
  • APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
  • Year:
  • 2009

  • On sampling from multivariate distributions

    APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques

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Abstract

Efficient sampling, integration and optimization algorithms for logconcave functions [BV04, KV06, LV06a] rely on the good isoperimetry of these functions. We extend this to show that *** 1/(n *** 1)-concave functions have good isoperimetry, and moreover, using a characterization of functions based on their values along every line, we prove that this is the largest class of functions with good isoperimetry in the spectrum from concave to quasi-concave. We give an efficient sampling algorithm based on a random walk for *** 1/(n *** 1)-concave probability densities satisfying a smoothness criterion, which includes heavy-tailed densities such as the Cauchy density. In addition, the mixing time of this random walk for Cauchy density matches the corresponding best known bounds for logconcave densities.