115 research outputs found
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 , we show how to
approximately sample uniformly random spanning trees from in
time per sample after an initial
time preprocessing. For a determinantal point
process on subsets of size of a ground set of elements, we show how to
approximately sample in time after an initial
time preprocessing, where 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 to
.
In our main technical result, we achieve the optimal limit on domain
sparsification for strongly Rayleigh distributions. In domain sparsification,
sampling from a distribution on is reduced to sampling
from related distributions on for . We show that for
strongly Rayleigh distributions, we can can achieve the optimal
. Our reduction involves sampling from
domain-sparsified distributions, all of which can be produced efficiently
assuming convenient access to approximate overestimates for marginals of .
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
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