Signal reconstruction using determinantal sampling

TL;DR

This paper investigates function approximation from finite samples using determinantal point processes, demonstrating improved convergence rates and instance optimality in the context of RKHS, with theoretical guarantees in L^2 norm.

Contribution

It introduces a novel approach combining determinantal sampling with RKHS-based regularity, providing mean-square guarantees and superconvergence insights.

Findings

Determinantal sampling achieves faster convergence rates.

Mixtures of determinantal processes enhance approximation efficiency.

Guarantees of instance optimality with fewer samples than i.i.d. methods.

Abstract

We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in $L^2$ norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling generalizes i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.