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