# Monte Carlo with Determinantal Point Processes

arXiv: https://arxiv.org/abs/1605.00361 · HTML: https://mlsanity.com/paper/1605.00361/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 May 2016.

## Built on by (most cited first, 22 of 22)

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- [A Faster Sampler for Discrete Determinantal Point Processes](https://mlsanity.com/paper/2210.17358/) (2022)
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- [Uphill Roads to Variational Tightness: Monotonicity and Monte Carlo Objectives](https://mlsanity.com/paper/2201.10989/) (2022)
- [Signal reconstruction using determinantal sampling](https://mlsanity.com/paper/2310.09437/) (2023)
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- [On sampling determinantal and Pfaffian point processes on a quantum computer](https://mlsanity.com/paper/2305.15851/) (2023)
- [Policy Gradient with Kernel Quadrature](https://mlsanity.com/paper/2310.14768/) (2023)
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- [Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance](https://mlsanity.com/paper/2509.10166/) (2025)
- [A novel sampler for Gauss-Hermite determinantal point processes with application to Monte Carlo integration](https://mlsanity.com/paper/2203.08061/) (2022)
- [Duality induced by an embedding structure of determinantal point process](https://mlsanity.com/paper/2404.11024/) (2024)
- [Naturally Private Recommendations with Determinantal Point Processes](https://mlsanity.com/paper/2405.13677/) (2024)
- [Quenched large deviations for Monte Carlo integration with Coulomb gases](https://mlsanity.com/paper/2508.01392/) (2025)
- [On two ways to use determinantal point processes for Monte Carlo integration](https://mlsanity.com/paper/2604.19698/) (2026)
- [Fast determinantal sampling on general spaces and diffusion geometry](https://mlsanity.com/paper/2607.06644/) (2026)
- [Beyond Worst-Case Coreset Bounds for k-Clustering via Determinantal Sampling](https://mlsanity.com/paper/2609.06394/) (2026)

## Abstract (arXiv)

We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical N-1/2, where N is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as N-(1+1/d)/2, where d is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition, a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which, at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term.