Monte Carlo methods on compact complex manifolds using Bergman kernels

Technical Summary: Monte Carlo on Compact Complex Manifolds using Bergman Kernels

Problem Statement
The paper addresses the problem of numerical integration (quadrature) on compact complex manifolds with respect to a continuous volume form. While Monte Carlo (MC) methods typically rely on independent and identically distributed (i.i.d.) sampling with a convergence rate of $O(N^{-1/2})$ for the mean squared error (MSE), and Quasi-Monte Carlo (QMC) methods offer deterministic worst-case error bounds, the authors seek a randomized method that combines the flexibility of MC with faster convergence rates. Specifically, they aim to generalize recent successes in Euclidean spaces—where Determinantal Point Processes (DPPs) based on orthogonal polynomials yield fast central limit theorems (CLTs)—to the setting of compact complex manifolds.

Methodology
The proposed method utilizes a specific class of DPPs known as Bergman ensembles as the source of quadrature nodes. The construction relies on the following geometric and probabilistic framework:

1. Geometric Setting: Let $M$ be a compact complex manifold of complex dimension $d$ (real dimension $d_R = 2d$). The method employs a positive holomorphic line bundle $L$ over $M$ equipped with a Hermitian metric $h = e^{-\phi}$ and a reference probability measure $\mu$.
2. Semiclassical Limit: The authors consider the "semiclassical" regime where the line bundle is replaced by its $k$-th tensor power $L^k$ as $k \to \infty$. The space of holomorphic sections $H^0(M, L^k)$ is finite-dimensional with dimension $N_k$.
3. Bergman Kernel and Ensemble: The quadrature nodes $(X_1, \dots, X_{N_k})$ are sampled from a DPP defined by the Bergman kernel $B(k\phi, \mu)$, which is the reproducing kernel of the Hilbert space of holomorphic sections equipped with the inner product induced by $(k\phi, \mu)$.
4. Unbiased Estimator Construction:
   • The empirical average of a function $f$ under the DPP converges to the integral of $f$ against the equilibrium measure $\mu_{eq}^\phi$, not the target measure $\mu$.
   • To obtain an unbiased estimator for $\int_M f d\mu$, the authors propose a diagonal reweighting scheme. The estimator is defined as:
     $$\hat{I}_k(f) = \frac{1}{N_k} \sum_{i=1}^{N_k} \frac{f(X_i)}{B(k\phi, \mu)(X_i, X_i)}$$
   • This reweighting compensates for the fact that the DPP samples are distributed according to the Bergman measure $d\beta_k = \frac{1}{N_k} B(k\phi, \mu)(x, x) d\mu(x)$.

Key Contributions and Results

1. Central Limit Theorem with Optimal Rate:
   The primary theoretical result (Theorem 1.2) establishes that for any $C^1$ function $f$, the estimator satisfies a CLT:
   $$N_k^{-1/2 - 1/(2d)} \left( \hat{I}_k(f) - \int_M f d\mu \right) \xrightarrow{d} \mathcal{N}\left(0, \frac{1}{2} \mathcal{E}_\phi(f w_\mu^\phi) \right)$$
   where $\mathcal{E}_\phi$ is a Dirichlet energy functional and $w_\mu^\phi$ is the equilibrium weight.

   • Convergence Rate: The mean squared error decays as $O(N_k^{-1 - 2/d_R})$. Since $d_R = 2d$, this rate is $O(N_k^{-1 - 1/d})$.
   • Optimality: This rate matches the optimal worst-case rate for randomized integration of $C^1$ functions in Euclidean spaces established by Bakhvalov (1965). It is strictly faster than the $O(N^{-1 - 1/d_R})$ rate achieved by previous DPP-based quadratures in Euclidean spaces (Bardenet and Hardy, 2020a) when the complex structure is exploited.

2. Universality and Importance Sampling:
   The paper demonstrates a universality property (Corollary 1.3). The same DPP, constructed with a specific kernel, can be used to estimate integrals with respect to a whole family of measures absolutely continuous with respect to the reference measure. By applying an inverse-Bergman reweighting combined with an importance sampling factor, the estimator remains unbiased, and the asymptotic variance is governed by the same Dirichlet energy functional, independent of the specific proposal density used for sampling.

3. Technical Extension of Berman's Work:
   The proofs rely heavily on the seminal work of Berman (2018), specifically the CLT for Bergman linear statistics. The authors' main technical contribution is extending Berman's asymptotic analysis to a setting where the test function in the linear statistic depends on $k$ (specifically, the inverse of the diagonal Bergman density). They prove that this weighted sequence converges in $C^1$ to the equilibrium weight, allowing the transfer of the CLT to the unbiased quadrature estimator.

4. Numerical Illustration:
   The authors provide numerical experiments on the Riemann sphere ($S^2 \simeq \mathbb{CP}^1$). In this case, the Bergman ensemble corresponds to the classical spherical ensemble. The experiments compare the spherical ensemble against:

   • Standard i.i.d. Monte Carlo.
   • A DPP based on Legendre polynomials (mapped from a square to the sphere).
   • Randomized spiral points (a QMC method).
     The results confirm the theoretical convergence rates, showing the spherical ensemble outperforming the Legendre DPP (which suffers from boundary effects in the mapped domain) and matching the performance of randomized spiral points, while providing a rigorous Gaussian fluctuation theory.

Significance and Claims
The paper claims that Bergman ensembles provide a natural geometric analogue to multivariate orthogonal polynomial ensembles (OPEs) in Euclidean spaces but with superior convergence properties due to the complex structure.

• Speed: The convergence rate $O(N^{-1-1/d})$ is faster than the $O(N^{-1-1/(2d)})$ rate typical of real Euclidean DPPs for the same real dimension, effectively halving the exponent in the dimension term.
• Optimality: The method achieves the optimal worst-case rate for randomized integration of $C^1$ functions, a benchmark previously established only for Euclidean spaces.
• Generality: The framework is applicable to any compact complex manifold with a positive line bundle, offering a unified approach to high-dimensional integration on curved spaces that leverages the intrinsic geometry via the Bergman kernel.

The authors note that while the theoretical framework is robust, practical implementation on arbitrary manifolds is currently limited by the difficulty of computing exact Bergman kernels, which are often only available via asymptotic expansions. Future work is suggested to address the statistical impact of using approximate kernels and to extend the results to manifolds with boundaries.