Exact Sampling of Determinantal Point Processes without Eigendecomposition

Agnès Desolneux (1, 2) , Claire Launay (3, 4, 5) , Bruno Galerne (6)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 CNRS - Centre National de la Recherche Scientifique
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
4 UPD5 - Université Paris Descartes - Paris 5
5 USPC - Université Sorbonne Paris Cité
6 IDP - Institut Denis Poisson

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 below.

Mots clés: Thinning; Determinantal point processes; General marginal; Cholesky decom-position; Exact Sampling; Cholesky decomposition

Domaines: Machine Learning [stat.ML]; Modélisation et simulation

Agnès Desolneux, Claire Launay, Bruno Galerne. Exact Sampling of Determinantal Point Processes without Eigendecomposition. 2019. ⟨hal-01710266v4⟩