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

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2009, RANDOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5687))

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.

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Authors and Affiliations

  1. School of Computer Science, Georgia Institute of Technology, USA

    Karthekeyan Chandrasekaran & Santosh Vempala

  2. Microsoft Research, India

    Amit Deshpande

Authors
  1. Karthekeyan Chandrasekaran
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  2. Amit Deshpande
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  3. Santosh Vempala
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Editor information

Editors and Affiliations

  1. Dept. of Applied Math and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Irit Dinur

  2. Institute for Computer Science and Applied Mathematics, University of Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Technion, Computer Science Department, 32000, Haifa, Israel

    Joseph Naor

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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© 2009 Springer-Verlag Berlin Heidelberg

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Chandrasekaran, K., Deshpande, A., Vempala, S. (2009). Sampling s-Concave Functions: The Limit of Convexity Based Isoperimetry. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_32

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  • DOI: https://doi.org/10.1007/978-3-642-03685-9_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03684-2

  • Online ISBN: 978-3-642-03685-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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