Scalable Sampling for Nonsymmetric Determinantal Point Processes

TL;DR

This paper introduces scalable algorithms for sampling from nonsymmetric determinantal point processes (NDPPs), enabling their application to large datasets and improving predictive performance in machine learning tasks.

Contribution

We develop a linear-time transformation of existing algorithms, a sublinear-time rejection sampling method, and analyze structural constraints to improve NDPP sampling scalability.

Findings

Linear-time algorithm for low-rank NDPP kernels

Sublinear-time rejection sampling proposal distribution

Structural constraints reduce rejection rate and improve scalability

Abstract

A determinantal point process (DPP) on a collection of $M$ items is a model, parameterized by a symmetric kernel matrix, that assigns a probability to every subset of those items. Recent work shows that removing the kernel symmetry constraint, yielding nonsymmetric DPPs (NDPPs), can lead to significant predictive performance gains for machine learning applications. However, existing work leaves open the question of scalable NDPP sampling. There is only one known DPP sampling algorithm, based on Cholesky decomposition, that can directly apply to NDPPs as well. Unfortunately, its runtime is cubic in $M$, and thus does not scale to large item collections. In this work, we first note that this algorithm can be transformed into a linear-time one for kernels with low-rank structure. Furthermore, we develop a scalable sublinear-time rejection sampling algorithm by constructing a novel proposal distribution. Additionally, we show that imposing certain structural constraints on the NDPP kernel enables us to bound the rejection rate in a way that depends only on the kernel rank. In our experiments we compare the speed of all of these samplers for a variety of real-world tasks.