# High-performance sampling of generic Determinantal Point Processes

## TL;DR

This paper introduces efficient, high-performance algorithms for sampling from general Determinantal Point Processes (DPPs) using modified LU and LDL^H factorizations, avoiding costly spectral decompositions.

## Contribution

It demonstrates that simple modifications of LU and LDL^H factorizations enable direct, efficient sampling of both Hermitian and non-Hermitian DPP kernels, expanding practical applicability.

## Key findings

- Efficient DPP sampling schemes for non-Hermitian kernels.
- High-performance parallel implementations of DPP samplers.
- Open-source C++ software with dense and sparse DPP sampling methods.

## Abstract

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, https://hodgestar.com/catamari, 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.