Determinantal point processes (DPPs) were introduced by Macchi (Macchi 1975
Adv. Appl. Probab.
7
, 83–122) as a model for repulsive (fermionic) particle distributions. But their recent popularization is largely due to their usefulness for encouraging diversity in the final stage of a recommender system (Kulesza & Taskar 2012
Found. Trends Mach. Learn.
5
, 123–286). The standard sampling scheme for finite DPPs is a spectral decomposition followed by an equivalent of a randomly diagonally pivoted Cholesky factorization of an orthogonal projection, which is only applicable to Hermitian kernels and has an expensive set-up cost. Researchers Launay
et al.
2018 (
http://arxiv.org/abs/1802.08429
); Chen & Zhang 2018
NeurIPS
(
https://papers.nips.cc/paper/7805-fast-greedy-map-inference-for-determinantal-point-process-to-improve-recommendation-diversity.pdf
) have begun to connect DPP sampling to
LDL
H
factorizations as a means of avoiding the initial spectral decomposition, but existing approaches have only outperformed the spectral decomposition approach in special circumstances, where the number of kept modes is a small percentage of the ground set size. This article proves that trivial modifications of
LU
and
LDL
H
factorizations yield efficient direct sampling schemes for non-Hermitian and Hermitian DPP kernels, respectively. Furthermore, it is experimentally shown that even dynamically scheduled, shared-memory parallelizations of high-performance dense and sparse-direct factorizations can be trivially modified to yield DPP sampling schemes with essentially identical performance. The software developed as part of this research, Catamari (
hodgestar.com/catamari
) is released under the Mozilla Public License v.2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense and sparse-direct, Hermitian and non-Hermitian DPP samplers.
This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.
Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes