Kernel quadrature with DPPs
This work addresses quadrature approximation for kernel-based functions, which is incremental as it builds on existing kernel quadrature methods by introducing DPPs for improved error analysis.
The paper tackles the problem of quadrature rules for functions in reproducing kernel Hilbert spaces (RKHS) by using nodes sampled from determinantal point processes (DPPs), resulting in tight error bounds dependent on the kernel spectrum and experimental confirmation of DPPs' advantages over existing methods, with hints of faster rates in some cases.
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.