# A Faster Sampler for Discrete Determinantal Point Processes

arXiv: https://arxiv.org/abs/2210.17358 · HTML: https://mlsanity.com/paper/2210.17358/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 31 Oct 2022.

## Builds on (most cited first, 9 of 9)

- [Monte Carlo with Determinantal Point Processes](https://mlsanity.com/paper/1605.00361/) (2016)
- [Exact sampling of determinantal point processes with sublinear time preprocessing](https://mlsanity.com/paper/1905.13476/) (2019)
- [Determinantal Point Processes for Coresets](https://mlsanity.com/paper/1803.08700/) (2018)
- [Exact Sampling of Determinantal Point Processes without Eigendecomposition](https://mlsanity.com/paper/1802.08429/) (2018)
- [Minimax experimental design: Bridging the gap between statistical and worst-case approaches to least squares regression](https://mlsanity.com/paper/1902.00995/) (2019)
- [Optimal Sublinear Sampling of Spanning Trees and Determinantal Point Processes via Average-Case Entropic Independence](https://mlsanity.com/paper/2204.02570/) (2022)
- [Scalable MCMC Sampling for Nonsymmetric Determinantal Point Processes](https://mlsanity.com/paper/2207.00486/) (2022)
- [Asymptotic Equivalence of Fixed-size and Varying-size Determinantal Point Processes](https://mlsanity.com/paper/1803.01576/) (2018)
- [Scalable Sampling for Nonsymmetric Determinantal Point Processes](https://mlsanity.com/paper/2201.08417/) (2022)

## Built on by (most cited first, 6 of 6)

- [Weighted least-squares approximation with determinantal point processes and generalized volume sampling](https://mlsanity.com/paper/2312.14057/) (2023)
- [On sampling determinantal and Pfaffian point processes on a quantum computer](https://mlsanity.com/paper/2305.15851/) (2023)
- [Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance](https://mlsanity.com/paper/2509.10166/) (2025)
- [Adaptive randomized pivoting and volume sampling](https://mlsanity.com/paper/2510.02513/) (2025)
- [Bypassing orthogonalization in the quantum DPP sampler](https://mlsanity.com/paper/2503.05906/) (2025)
- [Spectral Certificates and Projection-DPP Rounding for Determinantal MAP Selection](https://mlsanity.com/paper/2606.19411/) (2026)

## Abstract (arXiv)

Discrete Determinantal Point Processes (DPPs) have a wide array of potential applications for subsampling datasets. They are however held back in some cases by the high cost of sampling. In the worst-case scenario, the sampling cost scales as O(n3) where n is the number of elements of the ground set. A popular workaround to this prohibitive cost is to sample DPPs defined by low-rank kernels. In such cases, the cost of standard sampling algorithms scales as O(np2 + nm2) where m is the (average) number of samples of the DPP (usually m << n) and p the rank of the kernel used to define the DPP (m ≤ p ≤ n). The first term, O(np2), comes from a SVD-like step. We focus here on the second term of this cost, O(nm2), and show that it can be brought down to O(nm + m3 log m) without loss on the sampling's exactness. In practice, we observe very substantial speedups compared to the classical algorithm as soon as n > 1000. The algorithm described here is a close variant of the standard algorithm for sampling continuous DPPs, and uses rejection sampling. In the specific case of projection DPPs, we also show that any additional sample can be drawn in time O(m3 log m). Finally, an interesting by-product of the analysis is that a realisation from a DPP is typically contained in a subset of size O(m log m) formed using leverage score i.i.d. sampling.