Randomized least-squares with minimal oversampling and interpolation in general spaces

TL;DR

The paper addresses stable approximation from point evaluations in general spaces using randomized least-squares with minimal oversampling. It introduces two greedy sampling schemes—one based on effective resistance with Christoffel-density guidance and another with fixed barrier increments—to guarantee near-optimal $L^2$ error when $m$ exceeds the dimension by a constant ratio, and interpolation stability when $m=n$. A key contribution is rigorous probabilistic bounds on the smallest eigenvalue of the Gram matrix, enabling controlled conditioning and robust recovery with $m\approx n$. The results are complemented by practical algorithms, complexity considerations, and numerical experiments demonstrating improved conditioning and accuracy in high-dimensional polynomial spaces, making near-optimal recovery feasible with modest sampling budgets.

Abstract

In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected $L^2$ sense, as soon as the sample size $m$ is larger than the dimension $n$ of the approximation space by a constant ratio. On the other hand, for $m=n$, we obtain an interpolation strategy with a stability factor of order $n$. The proposed sampling algorithms are greedy procedures based on arXiv:0808.0163 and arXiv:1508.03261, with polynomial computational complexity.

Figures (3)

• Figure 1: Histograms of condition number of ${\boldsymbol A}_m$ for various random sampling strategies
• Figure 2: Average over 400 runs of the number of rejections in each iteration
• Figure 3: Ratio between the least-squares error and the best approximation error for $n=128$ and $m$ ranging between 128 and 208, when the points and weights are i.i.d according to the Christoffel measure, or generated by Algorithms \ref{['algo_effective_resistance']} and \ref{['algo_fixed_increments']}.

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Corollary 1.4
• Remark 1.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3