Exact discretization, tight frames and recovery via D-optimal designs

TL;DR

This paper introduces a deterministic, optimal discretization method using D-optimal designs to construct tight frames and exact inequalities for function recovery in $L_2$, applicable to various domains including spheres and tori.

Contribution

It presents a direct, constructive approach for discretizing $L_2$ norms with at most $n^2+1$ atoms, improving deterministic sampling for function reconstruction.

Findings

Constructs discrete measures with at most $n^2+1$ atoms for $L_2$ norm subsampling.

Provides a deterministic method for tight frame construction and function recovery.

Numerical experiments confirm the sharpness and limitations of the proposed discretization approach.

Abstract

$D$-optimal designs originate in statistics literature as an approach for optimal experimental designs. In numerical analysis points and weights resulting from maximal determinants turned out to be useful for quadrature and interpolation. Also recently, two of the present authors and coauthors investigated a connection to the discretization problem for the uniform norm. Here we use this approach of maximizing the determinant of a certain Gramian matrix with respect to points and weights for the construction of tight frames and exact Marcinkiewicz-Zygmund inequalities in $L_2$. We present a direct and constructive approach resulting in a discrete measure with at most $N \leq n^2+1$ atoms, which discretely and accurately subsamples the $L_2$-norm of complex-valued functions contained in a given $n$-dimensional subspace. This approach can as well be used for the reconstruction of functions from general RKHS in $L_2$ where one only has access to the most important eigenfunctions. We verifiably and deterministically construct points and weights for a weighted least squares recovery procedure and pay in the rate of convergence compared to earlier optimal, however probabilistic approaches. The general results apply to the $d$-sphere or multivariate trigonometric polynomials on $\mathbb{T}^d$ spectrally supported on arbitrary finite index sets~$I \subset \mathbb{Z}^d$. They can be discretized using at most $|I|^2-|I|+1$ points and weights. Numerical experiments indicate the sharpness of this result. As a negative result we prove that, in general, it is not possible to control the number of points in a reconstructing lattice rule only in the cardinality $|I|$ without additional condition on the structure of $I$. We support our findings with numerical experiments.