Optimal Sublinear Sampling of Spanning Trees and Determinantal Point Processes via Average-Case Entropic Independence

We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include random spanning tree distributions and determinantal point processes. For a graph G=(V, E), we show how to approximately sample uniformly random spanning trees from G in widetildeO(lvert Vrvert) time per sample after an initial widetildeO(lvert Ervert) time preprocessing. For a determinantal point process on subsets of size k of a ground set of n elements, we show how to approximately sample in widetildeO(kω) time after an initial widetildeO(nkω-1) time preprocessing, where ω<2.372864 is the matrix multiplication exponent. We even improve the state of the art for obtaining a single sample from determinantal point processes, from the prior runtime of widetildeO(minnk2, nω) to widetildeO(nkω-1). In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution μ on binom[n]k is reduced to sampling from related distributions on binom[t]k for tll n. We show that for strongly Rayleigh distributions, we can can achieve the optimal t=widetildeO(k). Our reduction involves sampling from widetildeO(1) domain-sparsified distributions, all of which can be produced efficiently assuming convenient access to approximate overestimates for marginals of μ. Having access to marginals is analogous to having access to the mean and covariance of a continuous distribution, or knowing "isotropy" for the distribution, the key assumption behind the Kannan-Lov'asz-Simonovits (KLS) conjecture and optimal samplers based on it. We view our result as a moral analog of the KLS conjecture and its consequences for sampling, for discrete strongly Rayleigh measures.