Determinantal Point Processes (DPPs) were introduced by Macchi 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.
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 setup cost. Researchers 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. Further, 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,
this https URL, is released under the Mozilla Public License
v2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense
and sparse-direct, Hermitian and non-Hermitian DPP samplers.

这篇文章介绍了 $LU$ 和 $LDL^H$ 分解的修改方法，以更有效地直接对非 - Hermitian 和 Hermitian DPP 核进行抽样，并证明在高性能的稠密和稀疏直接分解的动态调度、共享内存并行化方面，也可以轻松修改以产生基本相同性能的 DPP 抽样方案，而 Catamari 则发布了相关软件。