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A Bayesian framework for spectral reprojection

Tongtong Li, Anne Gelb

TL;DR

A new Bayesian framework for spectral reprojection is proposed, which allows a greater understanding of the impact of noise on the reprojection method from a statistical point of view and is able to improve the robustness with respect to the Gegenbauer polynomials parameters.

Abstract

Fourier partial sum approximations yield exponential accuracy for smooth and periodic functions, but produce the infamous Gibbs phenomenon for non-periodic ones. Spectral reprojection resolves the Gibbs phenomenon by projecting the Fourier partial sum onto a Gibbs complementary basis, often prescribed as the Gegenbauer polynomials. Noise in the Fourier data and the Runge phenomenon both degrade the quality of the Gegenbauer reconstruction solution, however. Motivated by its theoretical convergence properties, this paper proposes a new Bayesian framework for spectral reprojection, which allows a greater understanding of the impact of noise on the reprojection method from a statistical point of view. We are also able to improve the robustness with respect to the Gegenbauer polynomials parameters. Finally, the framework provides a mechanism to quantify the uncertainty of the solution estimate.

A Bayesian framework for spectral reprojection

TL;DR

A new Bayesian framework for spectral reprojection is proposed, which allows a greater understanding of the impact of noise on the reprojection method from a statistical point of view and is able to improve the robustness with respect to the Gegenbauer polynomials parameters.

Abstract

Fourier partial sum approximations yield exponential accuracy for smooth and periodic functions, but produce the infamous Gibbs phenomenon for non-periodic ones. Spectral reprojection resolves the Gibbs phenomenon by projecting the Fourier partial sum onto a Gibbs complementary basis, often prescribed as the Gegenbauer polynomials. Noise in the Fourier data and the Runge phenomenon both degrade the quality of the Gegenbauer reconstruction solution, however. Motivated by its theoretical convergence properties, this paper proposes a new Bayesian framework for spectral reprojection, which allows a greater understanding of the impact of noise on the reprojection method from a statistical point of view. We are also able to improve the robustness with respect to the Gegenbauer polynomials parameters. Finally, the framework provides a mechanism to quantify the uncertainty of the solution estimate.
Paper Structure (25 sections, 1 theorem, 81 equations, 5 figures, 3 algorithms)

This paper contains 25 sections, 1 theorem, 81 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Resolution of the Gibbs phenomenon
  4. Spectral reprojection
  5. Gegenbauer polynomials
  6. Model discretization
  7. A Bayesian framework for spectral reprojection
  8. Statistical formulation for spectral reprojection
  9. Formulating the likelihood
  10. Formulating the prior
  11. Formulating the hyperpriors
  12. Bayesian inference
  13. The block-coordinate descent approach
  14. Updating the unknown signal $\mathbf{f}$
  15. Updating the hyperparameters ${\gamma}$ and ${\beta}$
  16. ...and 10 more sections

Key Result

Theorem 2.1

\newlabelthm:GSSV920 Consider an analytic and non-periodic function $f(x)$ on $[-1,1]$, satisfying Assume that the first $N$ Fourier coefficients in eq:fourcoeff1D are known. Let $\widehat{g}^\lambda_l$, $0 \leq l \leq m$, be the Gegenbauer reprojection coefficients in eq:pmsnf where $S_Nf$ is the Fourier partial sum given by eq:partial_four_sum1, $\delta_{0l}$ is the Kronecker delta function, $

Figures (5)

  • Figure 1: Recovery of $f(x) = e^x\sin{(5x)}$. (left) Fourier reconstruction $\mathbf{f}_N$ in \ref{['eq:partial_four_sum1 mv2']}; (center-left) Spectral reprojection $\mathbf{f}_{m,N}^{\lambda}$ in \ref{['alg:gegapprox']}; (center-right) $\mathbf{f}_{BSR}$ in \ref{['alg:bayesapprox']}; and (right) $\mathbf{f}_{GBSR}$ in \ref{['alg:bayesapprox2']}.
  • Figure 2: (Top-left) Fourier reconstruction $\mathbf{f}_N$ in \ref{['eq:partial_four_sum1 mv2']}; (top-middle) Spectral reprojection $\mathbf{f}_{m,N}^{\lambda}$ computed via \ref{['alg:gegapprox']}; (top right) pointwise errors for noiseless data for $\lambda=4$ and $\lambda=2$. (Bottom) Same experiments with SNR $= 10$.
  • Figure 3: Log error plots for varying SNR: (left) $l_2$ error in $[-1,1]$, (middle-left) at $x=-1$; (middle-right) at $x=-0.8$; and (right) $l_2$ error in $[-.5,.5]$ for $\mathbf{f}_N$, $\mathbf{f}_{m,N}^\lambda$$\mathbf{f}_{BSR}$, and $\mathbf{f}_{GBSR}$.
  • Figure 4: Log error plots for varying $\lambda$: $l_2$ error in $[-1,1]$ with SNR value 2 (left), 10 (middle) and 30 (left) for $\mathbf{f}_N$, $\mathbf{f}_{m,N}^\lambda$$\mathbf{f}_{BSR}$, and $\mathbf{f}_{GBSR}$.
  • Figure 5: Credible intervals with $\mathbf{f}_{BSR}$ and $\mathbf{f}_{GBSR}$ for SNR value 2 (left), 10 (middle) and 30 (left) and $\lambda=2$ (top and middle-top), $\lambda=4$ (middle-bottom and bottom).

Theorems & Definitions (14)

  • Definition 2.1
  • Theorem 2.1: Removal of the Gibbs Phenomenon
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Remark 4.1
  • Remark 4.2: Choice of $\lambda$
  • Remark 4.3: Choice of $m$
  • Remark 4.4: Sparse priors
  • ...and 4 more