On the worst-case error of least squares algorithms for $L_2$-approximation with high probability

TL;DR

This paper improves the probabilistic guarantees for a weighted least squares algorithm in $L_2$-approximation, showing it works with very high probability, thus strengthening previous results on the power of function values in Hilbert spaces.

Contribution

It provides a refined proof that the weighted least squares algorithm succeeds with probability at least 1 minus an arbitrarily small polynomial decay, enhancing the reliability of the method.

Findings

The algorithm works with probability at least 1 - n^{-c} for any c > 0.

It confirms the effectiveness of random sampling in $L_2$-approximation.

The results strengthen the probabilistic guarantees of previous methods.

Abstract

It was recently shown in [4] that, for $L_2$-approximation of functions from a Hilbert space, function values are almost as powerful as arbitrary linear information, if the approximation numbers are square-summable. That is, we showed that \[ e_n \,\lesssim\, \sqrt{\frac{1}{k_n} \sum_{j\geq k_n} a_j^2} \qquad \text{ with }\quad k_n \asymp \frac{n}{\ln(n)}, \] where $e_n$ are the sampling numbers and $a_k$ are the approximation numbers. In particular, if $(a_k)\in\ell_2$, then $e_n$ and $a_n$ are of the same polynomial order. For this, we presented an explicit (weighted least squares) algorithm based on i.i.d. random points and proved that this works with positive probability. This implies the existence of a good deterministic sampling algorithm. Here, we present a modification of the proof in [4] that shows that the same algorithm works with probability at least $1-{n^{-c}}$ for all $c>0$.