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

TL;DR

This paper introduces fast algorithms for sampling from strongly Rayleigh distributions, including spanning trees and determinantal point processes, achieving near-optimal domain sparsification and improved runtime for single samples.

Contribution

The paper develops the first near-linear time algorithms for approximate sampling from strongly Rayleigh distributions, with optimal domain sparsification bounds and improved runtimes for determinantal point processes.

Findings

Achieves $ ilde{O}(|V|)$ time for uniform spanning tree sampling after preprocessing.

Reduces determinantal point process sampling time to $ ilde{O}(nk^{ ext{w}-1})$ from previous $ ilde{O}( ext{min} obreak ext{ extasciitilde} obreak ext{O}(nk^2), n^ ext{w})$.

Establishes optimal domain sparsification limit $t= ilde{O}(k)$ for strongly Rayleigh distributions.

Abstract

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 $\widetilde{O}(\lvert V\rvert)$ time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ 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 $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega-1})$ time preprocessing, where $\omega<2.372864$ 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 $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega-1})$. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution $\mu$ on $\binom{[n]}{k}$ is reduced to sampling from related distributions on $\binom{[t]}{k}$ for $t\ll n$. We show that for strongly Rayleigh distributions, we can can achieve the optimal $t=\widetilde{O}(k)$. Our reduction involves sampling from $\widetilde{O}(1)$ domain-sparsified distributions, all of which can be produced efficiently assuming convenient access to approximate overestimates for marginals of $\mu$. 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.