Monte Carlo Markov Chain Algorithms for Sampling Strongly Rayleigh Distributions and Determinantal Point Processes

Strongly Rayleigh distributions are natural generalizations of product and determinantal probability distributions and satisfy the strongest form of negative dependence properties. We show that the "natural" Monte Carlo Markov Chain (MCMC) algorithm mixes rapidly in the support of a homogeneous strongly Rayleigh distribution. As a byproduct, our proof implies Markov chains can be used to efficiently generate approximate samples of a k-determinantal point process. This answers an open question raised by Deshpande and Rademacher which was studied recently by Kang, Li-Jegelka-Sra, and Rebeschini-Karbasi.