Polynomial approximation of noisy functions

TL;DR

This paper introduces NoisyChebtrunc, a stable, fast polynomial approximation algorithm for noisy functions that combines Chebyshev interpolation with statistical criteria, achieving spectral convergence and robustness against noise.

Contribution

The study presents a novel $O(N ext{log}N)$ algorithm, NoisyChebtrunc, that improves stability and efficiency in polynomial approximation of noisy functions using Chebyshev interpolation and statistical degree selection.

Findings

Achieves spectral convergence until noise limits accuracy.

Error bound proportional to noise level and polynomial degree.

Performs well with subgaussian or subexponential noise distributions.

Abstract

Approximating a univariate function on the interval $[-1,1]$ with a polynomial is among the most classical problems in numerical analysis. When the function evaluations come with noise, a least-squares fit is known to reduce the effect of noise as more samples are taken. The generic algorithm for the least-squares problem requires $O(Nn^2)$ operations, where $N+1$ is the number of sample points and $n$ is the degree of the polynomial approximant. This algorithm is unstable when $n$ is large, for example $n\gg \sqrt{N}$ for equispaced sample points. In this study, we blend numerical analysis and statistics to introduce a stable and fast $O(N\log N)$ algorithm called NoisyChebtrunc based on the Chebyshev interpolation. It has the same error reduction effect as least-squares and the convergence is spectral until the error reaches $O(\sigma \sqrt{{n}/{N}})$, where $\sigma$ is the noise level, after which the error continues to decrease at the Monte-Carlo $O(1/\sqrt{N})$ rate. To determine the polynomial degree, NoisyChebtrunc employs a statistical criterion, namely Mallows' $C_p$. We analyze NoisyChebtrunc in terms of the variance and concentration in the infinity norm to the underlying noiseless function. These results show that with high probability the infinity-norm error is bounded by a small constant times $\sigma \sqrt{{n}/{N}}$, when the noise {is} independent and follows a subgaussian or subexponential distribution. We illustrate the performance of NoisyChebtrunc with numerical experiments.