$\mathbb{S}^2$, and we choose them with support in

$$ \mathbb{S}_+^2 = \{(x, y, z) \in \mathbb{S}^2 : z \ge 0\} $$

in order to avoid numerical errors that could happen in stereographic coordinates for points which are close to the South pole (corresponding to the point at infinity). We take

$$ f_1(x, y, z) = z^2 \mathbf{1}_{z \ge 0}, \quad f_2(x, y, z) = |x|^{3/2} yz \mathbf{1}_{z \ge 0}. $$

Both functions are $\mathcal{C}^1$ on $\mathbb{S}_+^2$, but $f_1$ is actually smooth. Besides, $f_2$ is supported in the image of the open square $(-1, 1)^2$ and satisfies therefore the assumptions of Bardenet and Hardy (2020), whereas $f_1$ is nonzero on the image of the boundary of the square. Both functions also naturally satisfy the assumptions of the CLT for the i.i.d. Monte Carlo method. In both cases, the variances of the estimators have the same rankings, and the slopes are quite close to their theoretical values. It is interesting to see that for $f_2$, on low values of $N$, the estimator for the Jacobi ensemble has a slightly lower variance than the spherical ensemble, although it decays more slowly when $N$ grows. It does not happen for $f_1$, which is not surprising because it is an edge case for the method by Bardenet and Hardy (2020). We also remark that the randomized spiral points seem to provide an overall better performing estimator than all other methods, although there is no theoretical result to support that, with a similar slope to the spherical ensemble.

# 7 Conclusion and perspectives

Building on Berman's seminal work that led to the central limit theorem in (Berman, 2018), we showed that Bergman ensembles can lead to fast Monte Carlo integration on compact complex manifolds, just like multivariate orthogonal polynomial ensembles (OPEs) yield fast quadrature on compacts of the Euclidean space (Bardenet and Hardy, 2020). The take-home message is that, like OPEs, Bergman ensembles come up with a *fast* central limit theorem for Monte Carlo integration that is also *universal*, in the sense that the asymptotic variance of the central limit theorem is invariant to a suitable change of the reference measure and the kernel. Unlike OPEs, however, the dimension in the rate of convergence is now the *complex* dimension of the manifold, which has the important consequence that Bergman ensembles outperform multivariate OPEs for integration in Euclidean spaces of even real dimension. Actually, the error rate for Bergman ensembles matches the optimal worst-case rate by Bakhvalov (1965) for functions of class $\mathcal{C}^1$.

We are currently working on further experiments, to include for instance competitive approaches such as the Dirichlet DPP of Mazoyer et al. (2020a). We also strive to include a reasonable implementation of a Bergman ensemble on the sphere that is *not* the spherical ensemble, by considering a non-uniform reference measure. We expect the performance to be similar to the uniform case.

Future work includes the following tasks, roughly ranked by increasing difficulty. We shall first investigate the influence of the smoothness of the integrand on the error decay, in line with (Belhadji et al., 2019, 2020). Then, if we want to make DPPs on arbitrary complex manifolds practical, we need to circumvent the fact that the Bergman kernel is usually only available in the form of asymptotic estimates. We should thus investigate the
statistical effect of working with an approximate DPP built using these estimates. Alternately, it might be possible to use importance sampling with a proposal DPP for which we know how to numerically evaluate the Bergman kernel, for instance, using the Kodaira embedding of a Kähler manifold in a complex projective space of higher dimension, where the Bergman kernel is explicit. Finally, while compact complex manifolds already include important practical settings, such as Bayesian quantum tomography, extending the results to more general manifolds, e.g. with a boundary, would further broaden the applicability of DPP-based Monte Carlo integration.

## Acknowledgements

TL thanks Yohann Le Floch and we both thank Martin Rouault for insightful discussions. We acknowledge support from ERC grant BLACKJACK (ERC-2019-STG-851866) and ANR AI chair BACCARAT (ANR-20-CHIA-0002).

## References

S. Amari and H. Nagaoka. *Methods of information geometry*, volume 191. American Mathematical Soc., 2000.

N. S. Bakhvalov. On calculating multiple integrals with automatic choice of the step. In *Comput. Methods and Programming (Comput. Center Moscow Univ. Collect. Works III) (Russian)*, pages 237–240. Izdat. Moskov. Univ., Moscow, 1965.

R. Bardenet and A. Hardy. Monte Carlo with determinantal point processes. *Ann. Appl. Probab.*, 30(1):368–417, 2020. ISSN 1050-5164. doi: 10.1214/19-AAP1504. URL https://doi.org/10.1214/19-AAP1504.

A. Belhadji. An analysis of Ermakov-Zolotukhin quadrature using kernels. In *Advances in Neural Information Processing Systems (NeurIPS)*, 2021.

A. Belhadji, R. Bardenet, and P. Chainais. Kernel quadrature with determinantal point processes. In *Advances in Neural Information Processing Systems (NeurIPS)*, 2019.

A. Belhadji, R. Bardenet, and P. Chainais. Kernel interpolation with continuous volume sampling. In *International Conference on Machine Learning (ICML)*, 2020.

R. J. Berman. Bergman kernels and equilibrium measures for line bundles over projective manifolds. *Amer. J. Math.*, 131(5):1485–1524, 2009a. ISSN 0002-9327. doi: 10.1353/ajm.0.0077. URL https://doi.org/10.1353/ajm.0.0077.

R. J. Berman. Determinantal point processes and fermions on complex manifolds: large deviations and bosonization. *Comm. Math. Phys.*, 327(1):1–47, 2014. ISSN 0010-3616. doi: 10.1007/s00220-014-1891-6. URL https://doi.org/10.1007/s00220-014-1891-6.
statistical effect of working with an approximate DPP built using these estimates. Alternately, it might be possible to use importance sampling with a proposal DPP for which we know how to numerically evaluate the Bergman kernel, for instance, using the Kodaira embedding of a Kähler manifold in a complex projective space of higher dimension, where the Bergman kernel is explicit. Finally, while compact complex manifolds already include important practical settings, such as Bayesian quantum tomography, extending the results to more general manifolds, e.g. with a boundary, would further broaden the applicability of DPP-based Monte Carlo integration.

## Acknowledgements

TL thanks Yohann Le Floch and we both thank Martin Rouault for insightful discussions. We acknowledge support from ERC grant BLACKJACK (ERC-2019-STG-851866) and ANR AI chair BACCARAT (ANR-20-CHIA-0002).

## References

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N. S. Bakhvalov. On calculating multiple integrals with automatic choice of the step. In *Comput. Methods and Programming (Comput. Center Moscow Univ. Collect. Works III) (Russian)*, pages 237–240. Izdat. Moskov. Univ., Moscow, 1965.

R. Bardenet and A. Hardy. Monte Carlo with determinantal point processes. *Ann. Appl. Probab.*, 30(1):368–417, 2020. ISSN 1050-5164. doi: 10.1214/19-AAP1504. URL https://doi.org/10.1214/19-AAP1504.

A. Belhadji. An analysis of Ermakov-Zolotukhin quadrature using kernels. In *Advances in Neural Information Processing Systems (NeurIPS)*, 2021.

A. Belhadji, R. Bardenet, and P. Chainais. Kernel quadrature with determinantal point processes. In *Advances in Neural Information Processing Systems (NeurIPS)*, 2019.

A. Belhadji, R. Bardenet, and P. Chainais. Kernel interpolation with continuous volume sampling. In *International Conference on Machine Learning (ICML)*, 2020.

R. J. Berman. Bergman kernels and equilibrium measures for line bundles over projective manifolds. *Amer. J. Math.*, 131(5):1485–1524, 2009a. ISSN 0002-9327. doi: 10.1353/ajm.0.0077. URL https://doi.org/10.1353/ajm.0.0077.

R. J. Berman. Determinantal point processes and fermions on complex manifolds: large deviations and bosonization. *Comm. Math. Phys.*, 327(1):1–47, 2014. ISSN 0010-3616. doi: 10.1007/s00220-014-1891-6. URL https://doi.org/10.1007/s00220-014-1891-6.
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C. Bittante, S. De Marchi, and G. Elefante. A new quasi-Monte Carlo technique based on nonnegative least squares and approximate Fekete points. *Numer. Math. Theory Methods Appl.*, 9(4):640–663, 2016. ISSN 1004-8979. doi: 10.4208/nmtma.2016.m1516. URL https://doi.org/10.4208/nmtma.2016.m1516.

L. Brandolini, C. Choirat, L. Colzani, G. Gigante, R. Seri, and G. Travaglini. Quadrature rules and distribution of points on manifolds. *Ann. Sc. Norm. Super. Pisa Cl. Sci.* (5), 13(4):889–923, 2014. ISSN 0391-173X.

J. S. Brauchart, E. B. Saff, I. H. Sloan, and R. S. Womersley. QMC designs: optimal order quasi Monte Carlo integration schemes on the sphere. *Math. Comp.*, 83(290):2821–2851, 2014. ISSN 0025-5718. doi: 10.1090/S0025-5718-2014-02839-1. URL https://doi.org/10.1090/S0025-5718-2014-02839-1.

L. Charles and B. Estienne. Entanglement entropy and Berezin-Toeplitz operators. *Comm. Math. Phys.*, 376(1):521–554, 2020. ISSN 0010-3616. doi: 10.1007/s00220-019-03625-y. URL https://doi.org/10.1007/s00220-019-03625-y.

Y. Chen, M. Welling, and A. Smola. Super-samples from kernel herding. In *Proceedings of the conference on Uncertainty in Artificial Intelligence (UAI)*, 2010.

S. De Marchi and G. Elefante. Quasi-Monte Carlo integration on manifolds with mapped low-discrepancy points and greedy minimal Riesz $s$-energy points. *Appl. Numer. Math.*, 127:110–124, 2018. ISSN 0168-9274. doi: 10.1016/j.apnum.2017.12.017. URL https://doi.org/10.1016/j.apnum.2017.12.017.

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P. Diaconis, S. Holmes, and M. Shahshahani. Sampling from a manifold. In *Advances in modern statistical theory and applications: a Festschrift in honor of Morris L. Eaton*, volume 10 of Inst. Math. Stat. (IMS) Collect., pages 102–125. Inst. Math. Statist., Beachwood, OH, 2013.
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G. Gautier, R. Bardenet, and M. Valko. On two ways to use determinantal point processes for Monte Carlo integration. In *Advances in Neural Information Processing Systems (NeurIPS)*, 2019b.

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