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Monte Carlo methods on compact complex manifolds using Bergman kernels

Thibaut Lemoine, Rémi Bardenet

TL;DR

This work introduces a Bergman-ensemble Monte Carlo method for numerical integration on compact complex manifolds by sampling nodes from a determinantal point process with Bergman kernel. The estimator, formed by weighting function values with the inverse diagonal kernel, is unbiased for the target measure and satisfies a central limit theorem with a rate tied to the complex dimension, yielding a mean-square error decay of $N^{-1-2/d_{\real}}$ where $d_{\real}=2d$. The main contribution is a universal CLT for these DPP-based estimators, matching the Bak lower bound for $\mathscr{C}^1$-class functions in Euclidean spaces and showing improved performance over prior DPP-based quadratures. The paper provides detailed kernel estimates, a rigorous proof of the MC_main theorem, and a concrete Riemann-sphere application with numerical experiments, illustrating practical efficacy and potential extensions to broader manifolds.

Abstract

In this paper, we propose a new randomized method for numerical integration on a compact complex manifold with respect to a continuous volume form. Taking for quadrature nodes a suitable determinantal point process, we build an unbiased Monte Carlo estimator of the integral of any Lipschitz function, and show that the estimator satisfies a central limit theorem, with a faster rate than under independent sampling. In particular, seeing a complex manifold of dimension $d$ as a real manifold of dimension $d_{\mathbb{R}}=2d$, the mean squared error for $N$ quadrature nodes decays as $N^{-1-2/d_{\mathbb{R}}}$; this is faster than previous DPP-based quadratures and reaches the optimal worst-case rate investigated by [Bakhvalov 1965] in Euclidean spaces. The determinantal point process we use is characterized by its kernel, which is the Bergman kernel of a holomorphic Hermitian line bundle, and we strongly build upon the work of Berman that led to the central limit theorem in [Berman, 2018].We provide numerical illustrations for the Riemann sphere.

Monte Carlo methods on compact complex manifolds using Bergman kernels

TL;DR

This work introduces a Bergman-ensemble Monte Carlo method for numerical integration on compact complex manifolds by sampling nodes from a determinantal point process with Bergman kernel. The estimator, formed by weighting function values with the inverse diagonal kernel, is unbiased for the target measure and satisfies a central limit theorem with a rate tied to the complex dimension, yielding a mean-square error decay of where . The main contribution is a universal CLT for these DPP-based estimators, matching the Bak lower bound for -class functions in Euclidean spaces and showing improved performance over prior DPP-based quadratures. The paper provides detailed kernel estimates, a rigorous proof of the MC_main theorem, and a concrete Riemann-sphere application with numerical experiments, illustrating practical efficacy and potential extensions to broader manifolds.

Abstract

In this paper, we propose a new randomized method for numerical integration on a compact complex manifold with respect to a continuous volume form. Taking for quadrature nodes a suitable determinantal point process, we build an unbiased Monte Carlo estimator of the integral of any Lipschitz function, and show that the estimator satisfies a central limit theorem, with a faster rate than under independent sampling. In particular, seeing a complex manifold of dimension as a real manifold of dimension , the mean squared error for quadrature nodes decays as ; this is faster than previous DPP-based quadratures and reaches the optimal worst-case rate investigated by [Bakhvalov 1965] in Euclidean spaces. The determinantal point process we use is characterized by its kernel, which is the Bergman kernel of a holomorphic Hermitian line bundle, and we strongly build upon the work of Berman that led to the central limit theorem in [Berman, 2018].We provide numerical illustrations for the Riemann sphere.
Paper Structure (28 sections, 17 theorems, 162 equations, 4 figures)

This paper contains 28 sections, 17 theorems, 162 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Numerical integration on complex manifolds
  3. Main result
  4. Comparison with other methods
  5. Outline of the article
  6. Complex manifolds and Bergman kernels
  7. Basic definitions
  8. Operations on vector bundles
  9. Differential forms and integration on complex manifolds
  10. Line bundles and Bergman kernel
  11. Examples
  12. Bergman ensembles
  13. Definitions
  14. Convergence of the Bergman measures
  15. Laplace transform of linear statistics
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $L$ be a holomorphic line bundle over a compact complex manifold ${\mathcal{M}}$. Let $(\phi,\mu)$ be a strongly regular weighted measure and $\mu_\mathrm{eq}$ be the associated Monge--Ampère measure. For any $k\in{\mathbb N}^*$, let $(X_1,\ldots,X_{N_k})$ be a DPP with kernel $B_{(k\phi,\mu)}$. where $\Vert \mathrm{d} f\Vert_{\mathrm{d}\mathrm{d}^c\phi}^2$ is the Dirichlet norm

Figures (4)

  • Figure 1: The interaction between two charts and local coordinates.
  • Figure 2: A vector bundle of rank 1 over a complex manifold, made of copies of the complex plane over each point of ${\mathcal{M}}$.
  • Figure 3: Independent samples of size $N=500$ from four distributions on the sphere.
  • Figure 4: $\log$ variance of four integral estimators wrt. the number $N$ of quadrature nodes.

Theorems & Definitions (36)

  • Remark 1.1
  • Theorem 1.1: Ber7, Theorem 1.5
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: BH, Theorem 3
  • Theorem 1.5: BCCGST
  • Theorem 1.6: Ber8, Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 26 more