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

TL;DR

This work develops a constructive link between $D$-optimal designs and frame theory to discretize $L_2$-norms on $n$-dimensional subspaces $V_n$, yielding a discrete measure with at most $N\le n^2+1$ atoms that provides exact $L_2$-Marcinkiewicz-Zygmund sampling. It then builds equal-norm tight frames through a $D$-optimal redesign, proving verifiable recovery for RKHS via a sampling operator $S^{k,\mu}_{n,N}$ with error bounded by $3\sum_{j\ge n+1}\sigma_j^2$, and demonstrates applicability to general domains such as the $d$-torus and $d$-sphere. The results include sharp bounds like $N\le \dim(\text{span}\{f\overline g\})\le n^2$ and $N\le \dim \Pi_m \le (9m/d)^d$ for sphere, along with numerical evidence of sharpness and dimensionality effects. A verifiable recovery framework with $O(n^3)$ evaluation cost extends the practical impact to quadrature, sampling, and reconstruction in RKHS settings, while highlighting cases where lattice-cardinality alone cannot bound required sample counts.

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.

Figures (2)

• Figure 1: Experiment 1: points and weights forming an exact $L_2$-MZ inequality for frequencies in an $\ell_1$-ball (left), a hyperbolic cross (middle), and $2$-dimensional bad frequencies (right).
• Figure 2: Depiction of the $L_2$-MZ constant $\varepsilon = \|\sum_{i=1}^{N+1}\alpha_i \bm \varphi(\bm x^i)\cdot\bm \varphi(\bm x^i)^\ast - \bm I_m\|_{2\to 2}$ for: experiment 2 in $d=1$, $d=2$, $d=3$, $d=4$, $d=5$, $d=6$, and $d=7$ (left) and experiment 3 for $d=1$with weights and without weights (right).

Theorems & Definitions (41)

• Proposition 2.1
• proof
• Remark 2.2
• Corollary 2.3
• proof
• Proposition 2.4
• proof
• Lemma 2.5: A version of Lemma 10 from KPUU24
• Lemma 2.6
• proof