arxiv: 2604.19698 · v1 · submitted 2026-04-21 · 💻 cs.LG · math.ST· stat.TH

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On two ways to use determinantal point processes for Monte Carlo integration

Guillaume Gautier , R\'emi Bardenet , Michal Valko

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Pith reviewed 2026-05-10 03:14 UTC · model grok-4.3

classification 💻 cs.LG math.STstat.TH

keywords determinantal point processesMonte Carlo integrationvariance reductionrepulsive samplingcontinuous DPPunbiased quadraturesampling algorithms

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The pith

Replacing independent Monte Carlo samples with a determinantal point process produces consistent estimators whose variance rate improves when the process is adapted to the integrand and measure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard Monte Carlo estimates the integral of f with respect to measure ω by averaging f at N points drawn independently from ω, giving variance of order 1/N. The paper examines two replacements that draw the points from a determinantal point process, a distribution that reduces the chance of selecting nearby points together. One replacement uses a fixed DPP and reaches variance of order N to the power of minus one minus one over dimension when f is smooth. The other adapts the DPP to f, keeps the estimator unbiased, and retains the 1/N variance rate. Both approaches are extended to continuous domains and equipped with explicit sampling algorithms.

Core claim

The paper shows that a determinantal point process can replace independent sampling in Monte Carlo integration while preserving consistency. The resulting variance decay depends on the degree of adaptation of the DPP to f and ω: a fixed DPP yields an improved rate O(N^{-(1+1/d)}) for smooth f, whereas an f-tailored DPP keeps the classical rate of 1/N together with unbiasedness. The authors generalize both estimators from discrete to continuous settings and supply algorithms for drawing the required DPP samples.

What carries the argument

Determinantal point processes as a repulsive sampling mechanism whose inclusion probabilities are controlled by a kernel, with the kernel's alignment to f and ω determining the variance improvement.

If this is right

The fixed-DPP estimator improves the convergence speed relative to Monte Carlo whenever the integrand is smooth.
The adapted-DPP estimator matches the unbiasedness and 1/N rate of classical Monte Carlo while still exploiting repulsion.
Explicit sampling procedures now exist that make both estimators usable on continuous domains.
The variance reduction scales with how closely the DPP kernel is matched to the particular f and ω under study.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same repulsive construction could be combined with control variates or importance sampling to compound variance gains.
In very high dimensions the cost of adapting the DPP may grow faster than the variance benefit, pointing toward dimension-reduction pre-processing.
Repulsive point processes of this kind might also lower variance in other Monte Carlo tasks such as expectation estimation or particle filtering.

Load-bearing premise

That determinantal point processes adapted to an arbitrary continuous integrand and measure can be sampled efficiently enough to realize the stated variance rates.

What would settle it

A simulation in a continuous domain that measures the empirical variance of either estimator and finds it no smaller than the independent-sampling baseline after accounting for sampling cost.

Figures

Figures reproduced from arXiv: 2604.19698 by Guillaume Gautier, Michal Valko, R\'emi Bardenet.

**Figure 1.** Figure 1: (a) A sample from a 2D Jacobi ensemble with N = 1000 points. (b)-(c) a i , bi = −1/2, the colors and numbers correspond to the dimension. For d = 1, the tridiagonal model (tri) of Killip and Nenciu offers tremendous time savings. (c) The total number of rejections grows as 2 dN log(N). 4 Empirical investigation We perform three main sets of experiments to investigate the properties of the BH (7) and EZ (14… view at source ↗

**Figure 2.** Figure 2: (a)-(d) cf. Section 4.1, the numbers in the legend are the slope and R2 (e)-(h) cf. Section 4.2. 4.1 The bump experiment Bardenet and Hardy [2019, Section 3] illustrate the behavior of IbBH N and its CLT (9) on a unimodal, smooth bump function; see Appendix B.1. The expected variance decay is of order 1/N1+1/d. We 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fd\omega$ relies on independent samples from $\omega$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $\omega$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper makes two older DPP-based Monte Carlo estimators practical for continuous spaces by adding explicit sampling algorithms and generalizations, though the variance gains stay modest outside low dimensions.

read the letter

The main point of this paper is that it turns two older DPP-based Monte Carlo estimators into something usable in continuous spaces by providing the necessary sampling algorithms and generalizations. They look at the Bardenet and Hardy 2020 estimator, which uses a fixed DPP and achieves a variance rate of O(N^{-(1+1/d)}) for smooth functions, and the Ermakov and Zolotukhin 1960 one, which tailors the DPP to the integrand f for an unbiased estimator with the usual 1/N rate. The paper generalizes these to continuous domains and gives explicit ways to sample the points. The strength here is the practical side. Having sampling algorithms means others can actually try these out rather than just read about the theory. The math on consistency and the variance bounds seems to check out from the way the kernels are set up and the asymptotic analysis they do. One area that could use more attention is how well these perform when the dimension is not small. The improved rate for the fixed DPP case loses its edge as d grows, and even with their sampling methods, generating the DPP samples might carry overhead compared to plain independent sampling. They do address the continuous case, but real-world tests on non-toy problems would help show if the gains are worth it. This work is for specialists in Monte Carlo integration and variance reduction techniques. A reader who already follows DPP literature or works on numerical methods in ML will get the most out of the sampling procedures and the unified view of the two approaches. It is solid enough to go to peer review. The claims are grounded in prior work and the extensions are clear. I recommend sending it out for review.

Referee Report

0 major / 3 minor

Summary. The paper investigates the use of determinantal point processes (DPPs) as an alternative to independent sampling in Monte Carlo integration of an integrand f with respect to a measure ω. It analyzes two specific constructions: the Bardenet-Hardy estimator using a fixed DPP achieving a convergence rate of O(N^{-(1 + 1/d)}) for smooth functions in d dimensions, and the Ermakov-Zolotukhin estimator which is unbiased and achieves the standard Monte Carlo rate of O(1/N) but requires tailoring the DPP to f. The authors extend these methods to continuous domains and provide explicit sampling algorithms for both.

Significance. If the claims are substantiated by the proofs and algorithms in the manuscript, this paper offers a valuable contribution to variance reduction techniques in numerical integration by leveraging the repulsive properties of DPPs. The explicit provision of sampling procedures is particularly noteworthy as it facilitates reproducibility and practical application. The analysis of how the degree of adaptation affects the variance rates provides a clear framework for understanding the benefits and trade-offs of DPP-based estimators compared to standard Monte Carlo.

minor comments (3)

The abstract could more precisely state the dimensions d in which the rates hold and any assumptions on f and ω.
The manuscript would benefit from a dedicated section comparing the computational costs of the two sampling algorithms.
Notation for the DPP kernel K is introduced but its relation to the measure ω could be clarified earlier in the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The referee's summary accurately captures the paper's focus on generalizing the Bardenet-Hardy and Ermakov-Zolotukhin DPP-based estimators to continuous domains, along with the associated variance rates and sampling algorithms.

Circularity Check

0 steps flagged

No significant circularity; self-citation to prior work is non-load-bearing

full rationale

The paper revisits the Bardenet-Hardy (2020) and Ermakov-Zolotukhin (1960) constructions, extends both to continuous domains, and derives explicit sampling procedures. The central claims—consistency of the DPP-based estimators and variance rates depending on adaptation to f and ω—follow from the stated kernels, unbiasedness arguments, and asymptotic analysis in the generalizations. The self-citation to Bardenet & Hardy is present but does not reduce the new derivations to fitted inputs or self-definitions; the prior results serve as starting points for independent extensions. No steps reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the work appears to rest on standard properties of DPPs and Monte Carlo estimators from prior literature.

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discussion (0)

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