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Stability of Least Square Approximation under Random Sampling

Zhiqiang Xu, Xinyue Zhang

TL;DR

This work analyzes the stability of the discrete least-squares projection $P_m^n$ onto the univariate polynomial space ${\mathbb P}_m$ using $n$ i.i.d. samples from a Jacobi-weighted measure on $[-1,1]$. It establishes that the stability threshold, measured by $\kappa_{2}(P_m^n)$, is governed by the sampling rate $n$ relative to the Jacobi parameters via $n \asymp m^{2(1+\gamma)}$ with $\gamma=\max\{\alpha,\beta\}$, and in the uniform case this reduces to $n \asymp m^2$ (up to a $\log n$ factor). The paper extends the classical impossibility theorem to random sampling with modified Jacobi weights, revealing a fundamental trade-off between stability and achievable convergence rates for analytic functions. Numerical experiments validate the phase transition in stability and illustrate the dependence on the weight parameters, underscoring practical guidelines for sampling in analytic-function recovery from random measurements.

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 measure $ρ_X$. 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 rates 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 integrating 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.

Stability of Least Square Approximation under Random Sampling

TL;DR

This work analyzes the stability of the discrete least-squares projection onto the univariate polynomial space using i.i.d. samples from a Jacobi-weighted measure on . It establishes that the stability threshold, measured by , is governed by the sampling rate relative to the Jacobi parameters via with , and in the uniform case this reduces to (up to a factor). The paper extends the classical impossibility theorem to random sampling with modified Jacobi weights, revealing a fundamental trade-off between stability and achievable convergence rates for analytic functions. Numerical experiments validate the phase transition in stability and illustrate the dependence on the weight parameters, underscoring practical guidelines for sampling in analytic-function recovery from random measurements.

Abstract

This paper investigates the stability of the least squares approximation within the univariate polynomial space of degree , denoted by . The approximation entails identifying a polynomial in that approximates a function over a domain based on samples of taken at randomly selected points, according to a specified measure . The primary goal is to determine the sampling rate necessary to ensure the stability of . Assuming the sampling points are i.i.d. with respect to a Jacobi weight function, we present the sampling rates that guarantee the stability of . Specifically, for uniform random sampling, we demonstrate that a sampling rate of is required to maintain stability. By integrating these findings with those of Cohen-Davenport-Leviatan, we conclude that, for uniform random sampling, the optimal sampling rate for guaranteeing the stability of is , up to a 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.
Paper Structure (19 sections, 15 theorems, 94 equations, 1 figure)

This paper contains 19 sections, 15 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. Problem setup.
  3. The Impact of Sampling Rate on Stability
  4. Impossibility theorem.
  5. Our contribution.
  6. Answer for the Question I
  7. Answer for Question II
  8. Organization.
  9. Preliminaries.
  10. Notations.
  11. Random sampling with respect to the modified Jacobi weight functions.
  12. Relation between $\kappa_{2}\left(P_m^n\right)$ and the quantity $D(n,m)$
  13. Maximal behavior of polynomials bounded on deterministic samples.
  14. The maximal behavior of polynomials bounded on random sampling points.
  15. The lower bound of $B(n,m)$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let where $\{L_j\}_{j=1}^{m}$ is an orthonormal basis of $V_m$ in the sense of $L^2(X,\rho_X)$. Then for any $r > 0$, if $m$ is such that we have and for any function $f:X\rightarrow \mathbb{R}$ with probability at least $1-2n^{-r}$.

Figures (1)

  • Figure 1: Values of $\log_{10}{\kappa_2(P_m^n)}$ for three choices of the sampling measure $\rho_X$ with $\alpha=\beta=1/2$ (left), $\alpha=\beta=0$ (middle), $\alpha=\beta=-1/4$ (right). Dashed lines follow $n = m^{3}$ (left), $n = m^{2}$ (middle), $n = m^{3/2}$ (right) respectively.

Theorems & Definitions (27)

  • Theorem 1.1: cohen2013stability
  • Theorem 1.2: platte2011impossibility
  • Theorem 1.3: adcock2019optimal
  • Theorem 1.4
  • Corollary 1.1
  • Theorem 1.5
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • ...and 17 more