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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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
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