Exact Sampling of Determinantal Point Processes without Eigendecomposition

TL;DR

This paper introduces an exact sampling algorithm for determinantal point processes that avoids costly eigendecomposition by using Cholesky decompositions, making it more efficient for high-dimensional problems.

Contribution

The paper presents a novel exact sampling method for DPPs that replaces spectral decomposition with Cholesky-based techniques, improving computational efficiency.

Findings

The new algorithm is competitive with traditional methods.

It can be faster in certain high-dimensional scenarios.

The method maintains exact sampling accuracy.

Abstract

Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel $K$ that can be seen as a matrix storing the similarity between points. The diversity comes from the fact that the inclusion probability of a subset is equal to the determinant of a submatrice of $K$. The exact algorithm to sample DPPs uses the spectral decomposition of $K$, a computation that becomes costly when dealing with a high number of points. Here, we present an alternative exact algorithm in the discrete setting that avoids the eigenvalues and the eigenvectors computation. Instead, it relies on Cholesky decompositions. This is a two steps strategy: first, it samples a Bernoulli point process with an appropriate distribution, then it samples the target DPP distribution through a thinning procedure. Not only is the method used here innovative, but this algorithm can be competitive with the original algorithm or even faster for some applications specified here.