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

TL;DR

This paper introduces a near-optimal least-squares approximation method in general spaces that requires minimal oversampling and offers stable, efficient sampling algorithms based on greedy procedures, balancing accuracy and computational cost.

Contribution

It demonstrates that near-optimal approximation can be achieved with minimal oversampling and develops efficient greedy sampling algorithms with polynomial complexity.

Findings

Near-optimal approximation with sample size just above the dimension

Stable interpolation with order n stability factor at m=n

Greedy algorithms with polynomial computational complexity

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.