Sampling from a $k$-DPP without looking at all items

TL;DR

This paper introduces an efficient algorithm for sampling from a $k$-DPP that requires observing only a small subset of all items, significantly reducing computational costs while maintaining exact distributional guarantees.

Contribution

The authors develop a novel adaptive sampling algorithm that efficiently generates $k$-DPP samples without examining all items, improving scalability for large datasets.

Findings

Achieves several orders of magnitude faster sampling compared to previous methods.

Produces exact $k$-DPP samples by observing only a small fraction of data.

Empirically validated on large datasets with high accuracy.

Abstract

Determinantal point processes (DPPs) are a useful probabilistic model for selecting a small diverse subset out of a large collection of items, with applications in summarization, stochastic optimization, active learning and more. Given a kernel function and a subset size $k$, our goal is to sample $k$ out of $n$ items with probability proportional to the determinant of the kernel matrix induced by the subset (a.k.a. $k$-DPP). Existing $k$-DPP sampling algorithms require an expensive preprocessing step which involves multiple passes over all $n$ items, making it infeasible for large datasets. A na\"ive heuristic addressing this problem is to uniformly subsample a fraction of the data and perform $k$-DPP sampling only on those items, however this method offers no guarantee that the produced sample will even approximately resemble the target distribution over the original dataset. In this paper, we develop an algorithm which adaptively builds a sufficiently large uniform sample of data that is then used to efficiently generate a smaller set of $k$ items, while ensuring that this set is drawn exactly from the target distribution defined on all $n$ items. We show empirically that our algorithm produces a $k$-DPP sample after observing only a small fraction of all elements, leading to several orders of magnitude faster performance compared to the state-of-the-art.