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Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle

Congpei An, Mou Cai

TL;DR

This work develops a closed-form, $\ell^2$-regularized least squares framework for approximating noisy periodic functions on the unit circle using trigonometric polynomials. It combines Tikhonov regularization with a rotationally invariant penalty and trapezoidal sampling, yielding explicit expressions for the regularized coefficients and introducing a regularized barycentric interpolation variant. The authors perform rigorous $L_2$ and uniform error analyses, derive stability bounds, and propose a regularity condition guiding parameter choice. They compare Morozov's discrepancy principle, the L-curve, and generalized cross-validation for selecting regularization parameters, showing Morozov's principle is regular under the condition while the others are not, and validate the theory with numerical experiments on representative periodic functions.

Abstract

This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific cases. With the aid of the de la Vallée-Poussin approximation, we derive a uniform error bound and a concrete $L_2$ error bound. These error estimates demonstrate the effectiveness of Tikhonov regularization in the denoising process. A new regularity condition for the selection of regularization parameters is proposed. We investigate three strategies for choosing regularization parameters: Morozov's discrepancy principle, the L-curve, and generalized cross-validation, by explicitly combining these error bounds of the approximating trigonometric polynomial. We show that Morozov's discrepancy principle satisfies the proposed regularity condition, while the other two methods do not. Finally, numerical examples are provided to illustrate how the aforementioned methodologies, when applied with well-chosen parameters, can significantly improve the quality of approximation.

Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle

TL;DR

This work develops a closed-form, -regularized least squares framework for approximating noisy periodic functions on the unit circle using trigonometric polynomials. It combines Tikhonov regularization with a rotationally invariant penalty and trapezoidal sampling, yielding explicit expressions for the regularized coefficients and introducing a regularized barycentric interpolation variant. The authors perform rigorous and uniform error analyses, derive stability bounds, and propose a regularity condition guiding parameter choice. They compare Morozov's discrepancy principle, the L-curve, and generalized cross-validation for selecting regularization parameters, showing Morozov's principle is regular under the condition while the others are not, and validate the theory with numerical experiments on representative periodic functions.

Abstract

This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific cases. With the aid of the de la Vallée-Poussin approximation, we derive a uniform error bound and a concrete error bound. These error estimates demonstrate the effectiveness of Tikhonov regularization in the denoising process. A new regularity condition for the selection of regularization parameters is proposed. We investigate three strategies for choosing regularization parameters: Morozov's discrepancy principle, the L-curve, and generalized cross-validation, by explicitly combining these error bounds of the approximating trigonometric polynomial. We show that Morozov's discrepancy principle satisfies the proposed regularity condition, while the other two methods do not. Finally, numerical examples are provided to illustrate how the aforementioned methodologies, when applied with well-chosen parameters, can significantly improve the quality of approximation.
Paper Structure (16 sections, 15 theorems, 110 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 16 sections, 15 theorems, 110 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of solutions
  4. Regularized least squares approximation
  5. Regularized barycentric trigonometric interpolation
  6. Error Analysis
  7. $L_2$ error
  8. Uniform error
  9. Parameter choice strategies
  10. Laplace operator $\mathcal{R}_L$
  11. Morozov's discrepancy principle
  12. L-curve
  13. Generalized Cross-Validation
  14. Algorithms
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Lemma 3.1

Assume that the point set $\mathcal{X}_{N}=\{x_1,\ldots,x_{N}\}$ satisfies the trapezoidal rule (see exactness) with $2L+1\leq N.$ Then

Figures (3)

  • Figure 1: $L_2$ errors (left) and uniform errors (right) as a function of $\lambda$ for the trigonometric polynomial \ref{['soltutionp']} approximation to $f_1$ and $f_2$ with $N=501,\;L=250$ and 20 dB noise.
  • Figure 2: L-curve method: log-log plot of $K(\lambda)$ against $J(\lambda)$ for $f_1,\;f_2$ with $N=501,\;L=250$ and 20 dB noise.
  • Figure 3: Recovery efficiency of gallery periodic function 'tsunami' comes from CHEBFUN with 10 dB Gaussian white noise by approximation scheme $p_{\lambda,L,N}^\beta$\ref{['soltutionp']} with parameter $\lambda_{\rm random},\;\lambda_{\rm gcv},\;\lambda_{\rm corner}.$

Theorems & Definitions (23)

  • Definition 2.1
  • Lemma 3.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.3
  • Proposition 4.1
  • Lemma 4.1
  • ...and 13 more