We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include as special cases 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 $\widetilde{O}(\lvert V\rvert)$\footnote{Throughout, $\widetilde{O}(\cdot)$ hides polylogarithmic factors in $n$.} time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. This is the first nearlylinear runtime in the output size, which is clearly optimal. For a determinantal point process on $k$sized subsets of a ground set of $n$ elements, defined via an $n\times n$ kernel matrix, we show how to approximately sample in $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega1})$ time preprocessing, where $\omega<2.372864$ is the matrix multiplication exponent. The time to compute just the weight of the output set is simply $\simeq k^\omega$, a natural barrier that suggests our runtime might be optimal for determinantal point processes as well. As a corollary, we even improve the state of the art for obtaining a single sample from a determinantal point process, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega1})$.
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, the domain size can be reduced to nearly linear in the output size $t=\widetilde{O}(k)$, improving the state of the art from $t= \widetilde{O}(k^2)$ for general strongly Rayleigh distributions and the more specialized $t=\widetilde{O}(k^{1.5})$ for spanning tree distributions. Our reduction involves sampling from $\widetilde{O}(1)$ domainsparsified distributions, all of which can be produced efficiently assuming approximate overestimates for marginals of $\mu$ are known and stored in a convenient data structure. Having access to marginals is the discrete analog of having access to the mean and covariance of a continuous distribution, or equivalently knowing ``isotropy'' for the distribution, the key behind optimal samplers in the continuous setting based on the famous KannanLov\'aszSimonovits (KLS) conjecture. We view our result as analogous in spirit to the KLS conjecture and its consequences for sampling, but rather for discrete strongly Rayleigh measures.
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Parallel Discrete Sampling via Continuous Walks
We develop a framework for sampling from discrete distributions $\mu$ on the hypercube $\{\pm 1\}^n$ by sampling from continuous distributions supported on $\mathbb{R}^n$ obtained by convolution with spherical Gaussians. We show that for wellstudied families of discrete distributions $\mu$, convolving $\mu$ with Gaussians yields wellconditioned logconcave distributions, as long as the variance of the Gaussian is above an $O(1)$ threshold. We then reduce the task of sampling from $\mu$ to sampling from Gaussianconvolved distributions. Our reduction is based on a stochastic process widely studied under different names: backward diffusion in diffusion models, and stochastic localization. We discretize this process in a novel way that allows for high accuracy and parallelism.
As our main application, we resolve open questions Anari, Hu, Saberi, and Schild raised on the parallel sampling of distributions that admit parallel counting. We show that determinantal point processes can be sampled via RNC algorithms, that is in time $\log(n)^{O(1)}$ using $n^{O(1)}$ processors. For a wider class of distributions, we show our framework yields QuasiRNC sampling, i.e., $\log(n)^{O(1)}$ time using $n^{O(\log n)}$ processors. This wider class includes nonsymmetric determinantal point processes and random Eulerian tours in digraphs, the latter nearly resolving another open question raised by prior work. Of potentially independent interest, we introduce and study a notion of smoothness for discrete distributions that we call transport stability, which we use to control the propagation of error in our framework. Additionally, we connect transport stability to constructions of optimally mixing local random walks and concentration inequalities.
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 Award ID(s):
 2045354
 NSFPAR ID:
 10488668
 Publisher / Repository:
 ACM
 Date Published:
 ISBN:
 9781450399135
 Page Range / eLocation ID:
 103 to 116
 Format(s):
 Medium: X
 Location:
 Orlando FL USA
 Sponsoring Org:
 National Science Foundation
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