Learning from DPPs via Sampling: Beyond HKPV and symmetry

TL;DR

This paper introduces a novel, scalable method for sampling from general DPPs by approximating the distribution of linear statistics using Laplace transforms, surpassing traditional HKPV algorithms especially for non-symmetric kernels.

Contribution

The authors develop a new approach that leverages Laplace transform inversion and low-rank approximations to efficiently sample linear statistics of DPPs, extending applicability beyond symmetric kernels.

Findings

Method effectively approximates distribution functions of linear statistics.

Approach is scalable and applicable to general, non-symmetric DPPs.

Enables hypothesis testing and sampling of linear statistics.

Abstract

Determinantal point processes (DPPs) have become a significant tool for recommendation systems, feature selection, or summary extraction, harnessing the intrinsic ability of these probabilistic models to facilitate sample diversity. The ability to sample from DPPs is paramount to the empirical investigation of these models. Most exact samplers are variants of a spectral meta-algorithm due to Hough, Krishnapur, Peres and Vir\'ag (henceforth HKPV), which is in general time and resource intensive. For DPPs with symmetric kernels, scalable HKPV samplers have been proposed that either first downsample the ground set of items, or force the kernel to be low-rank, using e.g. Nystr\"om-type decompositions. In the present work, we contribute a radically different approach than HKPV. Exploiting the fact that many statistical and learning objectives can be effectively accomplished by only sampling certain key observables of a DPP (so-called linear statistics), we invoke an expression for the Laplace transform of such an observable as a single determinant, which holds in complete generality. Combining traditional low-rank approximation techniques with Laplace inversion algorithms from numerical analysis, we show how to directly approximate the distribution function of a linear statistic of a DPP. This distribution function can then be used in hypothesis testing or to actually sample the linear statistic, as per requirement. Our approach is scalable and applies to very general DPPs, beyond traditional symmetric kernels.