Stability of Least Squares Approximation under Random Sampling

TL;DR

This paper analyzes the stability of least squares polynomial approximation under random sampling, establishing the necessary sampling rates for stability and extending impossibility results to random samples.

Contribution

It determines the optimal sampling rate of n ~ m^2 for stable least squares approximation with uniform random sampling and extends existing impossibility theorems to random sampling scenarios.

Findings

Sampling rate n ~ m^2 guarantees stability.

Optimal sampling rate is n ~ m^2 up to a log factor.

Extended impossibility theorem to random samples.

Abstract

This paper investigates the stability of the least squares approximation $P_m^n$ within the univariate polynomial space of degree $m$, denoted by ${\mathbb P}_m$. The approximation $P_m^n$ entails identifying a polynomial in ${\mathbb P}_m$ that approximates a function $f$ over a domain $X$ based on samples of $f$ taken at $n$ randomly selected points, according to a specified probability measure $\rho_X$. The primary goal is to determine the sampling rate necessary to ensure the stability of $P_m^n$. Assuming the sampling points are i.i.d. with respect to a Jacobi weight function, we present the sampling rate that guarantee the stability of $P_m^n$. Specifically, for uniform random sampling, we demonstrate that a sampling rate of $n \asymp m^2$ is required to maintain stability. By combining these findings with those of Cohen-Davenport-Leviatan, we conclude that, for uniform random sampling, the optimal sampling rate for guaranteeing the stability of $P_m^n$ is $n \asymp m^2$, up to a $\log n$ factor. Motivated by this result, we extend the impossibility theorem, previously applicable to equally spaced samples, to the case of random samples, illustrating the balance between accuracy and stability in recovering analytic functions.