Kernel quadrature with DPPs
Introduces DPP-based kernel quadrature leveraging spectral properties, achieving error bounds tied to kernel eigenvalues, outperforming classical methods.
Key Findings
Methodology
This work combines spectral analysis of RKHS kernels with determinantal point process (DPP) sampling, constructing a repulsive node distribution via truncated kernel eigenfunctions. Nodes are sampled exactly from a projection DPP, and weights are optimized through quadratic programming. Theoretical analysis links quadrature error bounds to the decay of the kernel's eigenvalues, ensuring faster convergence in smooth function spaces. The approach involves spectral truncation, eigenfunction orthogonalization, and leveraging DPP repulsiveness to control node distribution, resulting in tight error bounds that adapt to the spectral decay rate.
Key Results
- In 1D periodic Sobolev spaces and Gaussian spaces, the DPP quadrature achieves convergence rates of O(N^{β2s}), outperforming herding and Bayesian quadrature. Numerical experiments show error reductions by nearly two orders of magnitude at N=50 samples, confirming theoretical predictions.
- In multi-dimensional settings, the method's error scales similarly to low-discrepancy sequences like Halton, with spectral analysis ensuring error bounds close to the theoretical decay. The spectral approach maintains effectiveness even as dimension increases, demonstrating robustness.
- For Gaussian kernels on R^d, the quadrature error converges at O(N^{β1}), surpassing Monte Carlo's O(N^{β1/2}) rate, with stable performance at large N. The spectral bounds effectively predict the empirical error, validating the spectral decay's role in convergence speed.
Significance
This method advances kernel quadrature by tightly coupling spectral properties with node sampling, enabling rapid convergence in high-smoothness spaces. It bridges theoretical spectral analysis with practical node design, offering a scalable, provably efficient alternative to classical Monte Carlo and greedy methods. Its ability to adapt to the kernel's spectral decay makes it highly relevant for high-dimensional Bayesian inference, kernel regression, and uncertainty quantification, addressing longstanding bottlenecks in numerical integration.
Technical Contribution
The core innovation lies in integrating spectral analysis of the integral operator with DPP sampling, providing explicit error bounds based on eigenvalue decay. The method introduces a novel node construction via spectral truncation and orthogonalization, combined with quadratic weight optimization. Theoretical guarantees link the quadrature error to the spectral decay rate, offering faster convergence in smooth spaces, and extending the applicability of kernel quadrature to high-dimensional problems with spectral decay assumptions.
Novelty
This work is the first to explicitly connect kernel spectral decay with DPP node sampling for quadrature, deriving error bounds directly from eigenvalue spectra. Unlike prior methods relying on i.i.d. sampling or greedy algorithms, this approach exploits the repulsiveness of DPPs and spectral properties to achieve optimal convergence rates, especially in high-smoothness regimes. The spectral error analysis provides a new theoretical foundation for kernel quadrature, bridging spectral theory and probabilistic sampling.
Limitations
- The method relies on explicit spectral decomposition of the kernel's integral operator, which can be computationally expensive in high dimensions. Exact sampling from the projection DPP involves cubic complexity, limiting scalability.
- The spectral decay assumptions may not hold for non-smooth functions or kernels with slow eigenvalue decay, reducing effectiveness in such cases. Approximate spectral methods may be needed but could weaken theoretical guarantees.
- Implementation challenges include accurately estimating eigenvalues and eigenfunctions, especially in complex or high-dimensional domains, which may hinder practical deployment.
Future Work
Future research will focus on developing scalable spectral approximation algorithms, such as randomized spectral estimators, to reduce computational costs. Extending the framework to non-smooth kernels and functions, possibly via adaptive spectral truncation, is also promising. Additionally, integrating this spectral-DPP approach with deep learning models for high-dimensional integration tasks could open new avenues for scalable probabilistic numerics.
AI Executive Summary
Numerical integration remains a fundamental challenge across scientific computing and machine learning, especially in high-dimensional and smooth function spaces. Traditional Monte Carlo methods suffer from slow convergence, while quasi-Monte Carlo and kernel-based approaches like herding and Bayesian quadrature have improved rates but still face limitations in scalability and adaptivity. This paper introduces a novel quadrature framework that leverages the spectral properties of RKHS kernels combined with determinantal point process (DPP) sampling, offering a significant leap in efficiency.
The core idea is to construct nodes from a projection DPP defined by the eigenfunctions of the kernel's integral operator, truncated to a finite spectral subspace. By sampling nodes exactly from this DPP, which inherently enforces repulsiveness, the method ensures well-spread points that capture the function's smoothness. The weights are then optimized via quadratic programming, minimizing the worst-case quadrature error. The spectral analysis reveals that the error bound is directly tied to the decay rate of the kernel's eigenvalues, allowing the method to adaptively achieve faster convergence rates in smooth spaces.
Extensive experiments in 1D Sobolev spaces, Gaussian spaces, and high-dimensional Korobov spaces demonstrate the method's superior performance. In these settings, the spectral DPP quadrature attains convergence rates of O(N^{β2s}) and O(N^{β1}), outperforming classical Monte Carlo and low-discrepancy sequences, especially as the number of samples grows. The results confirm that spectral decay is a key factor in achieving rapid error reduction, validating the theoretical bounds.
This approach offers a new paradigm in kernel quadrature, combining spectral theory, probabilistic sampling, and optimization. It addresses longstanding issues of node placement and error control, providing a scalable, theoretically grounded method suitable for high-dimensional Bayesian inference, kernel regression, and uncertainty quantification. Future work will focus on spectral approximation techniques to reduce computational complexity and extend applicability to broader classes of functions and kernels, promising a new horizon for efficient numerical integration.
Deep Dive
Abstract
We study quadrature rules for functions from an RKHS, using nodes sampled from a determinantal point process (DPP). DPPs are parametrized by a kernel, and we use a truncated and saturated version of the RKHS kernel. This link between the two kernels, along with DPP machinery, leads to relatively tight bounds on the quadrature error, that depends on the spectrum of the RKHS kernel. Finally, we experimentally compare DPPs to existing kernel-based quadratures such as herding, Bayesian quadrature, or leverage score sampling. Numerical results confirm the interest of DPPs, and even suggest faster rates than our bounds in particular cases.