# Kernel quadrature with DPPs

## TL;DR

This paper introduces a novel quadrature method for functions in RKHS using determinantal point processes, providing theoretical error bounds and demonstrating superior empirical performance over existing methods.

## Contribution

It establishes a new kernel quadrature approach with DPPs, linking kernels and spectra to achieve tighter error bounds and improved sampling efficiency.

## Key findings

- DPP-based quadrature outperforms existing methods in experiments.
- Theoretical bounds relate quadrature error to the spectrum of the RKHS kernel.
- Numerical results suggest DPPs can achieve faster convergence rates.

## 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.