Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising

Technical Summary: Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising

Problem Statement
The paper addresses the challenge of denoising one-dimensional signals, particularly in low signal-to-noise ratio (SNR) environments. Classical wavelet estimators (e.g., hard/soft thresholding, SURE, CV) and existing Bayesian methods often struggle to balance the attenuation of noise with the preservation of genuine signal features, such as isolated peaks and sharp transitions. A specific difficulty arises in low-SNR settings where moderate-sized coefficients may represent either genuine local features or noise. While Bayesian shrinkage offers a framework to incorporate uncertainty, many existing priors lack the flexibility to adapt their shape across different resolution levels or fail to provide a computationally tractable posterior mean estimator with bounded support.

Methodology
The author proposes a Bayesian wavelet shrinkage model utilizing a resolution-adaptive spike-and-slab prior. The model operates in the wavelet domain, assuming a Gaussian observation model $y = f + \epsilon$ where $\epsilon \sim N(0, \sigma^2)$.

1. The Prior Structure:

   • Spike: A point mass at zero ($\pi_j \delta_0$) to enforce sparsity.
   • Slab: A continuous mixture of two compactly supported densities, scaled by a resolution-specific support parameter $\beta_j$:
     • Wendland Component ($g_W$): A normalized polynomial kernel with a high-order zero at the support boundary. It concentrates mass near zero, representing small nonzero coefficients difficult to distinguish from noise.
     • Semicircle Component ($g_S$): A semicircle density that is more dispersed, placing relatively more mass on moderate coefficients.
   • Adaptive Mixture: The relative weight of the Wendland component ($\omega_j$) varies by resolution level $j$. This weight is modeled via a logistic function of the normalized resolution level, estimated via Empirical Bayes (EB).
   • Resolution-Dependent Hyperparameters: The spike probability $\pi_j$ increases with resolution (finer scales are more likely to be noise), and the support scale $\beta_j$ is determined by a robust upper-quantile rule of the empirical coefficients.

2. Likelihood and Estimation:

   • The primary specification uses a Gaussian likelihood.
   • An alternative Laplace working likelihood is considered to facilitate closed-form expressions for the Wendland component.
   • Posterior Mean Estimator: Under squared-error loss, the Bayes estimator is derived as a weighted sum of the posterior means of the continuous components.
   • Computation:
     • For the Wendland component under the Laplace likelihood, the posterior moments are derived as finite-sum expressions involving elementary exponential integrals, avoiding numerical integration.
     • For the Semicircle component, the moments are evaluated via stable one-dimensional integration using Gauss–Legendre quadrature.
   • Empirical Bayes Procedure: The mixture weight parameters ($\eta$) are estimated by maximizing the marginal log-likelihood of the observed wavelet coefficients.

Key Contributions

• Novel Prior Construction: The paper introduces the first use of a resolution-adaptive mixture of a Wendland-type polynomial kernel and a semicircle density as a slab prior for Bayesian wavelet shrinkage. This combination allows the prior shape to adapt from highly concentrated (Wendland) to more dispersed (Semicircle) across resolution levels.
• Analytical and Computational Tractability: The author derives explicit finite-sum expressions for the Wendland contribution under the Laplace likelihood and stable integration schemes for the semicircle component. This ensures the posterior-mean rule is computationally feasible.
• Theoretical Properties: The paper establishes that the resulting shrinkage rule is symmetric, continuous, bounded by the support scale $\beta_j$, and exhibits specific tail behaviors depending on the likelihood (approaching $\pm \beta_j$ for Gaussian likelihood).
• Comprehensive Evaluation: The method is evaluated against universal thresholding, FDR, CV, SURE, BAMS, and a Nonlocal Prior (NLP) method across four standard test signals (Bumps, Blocks, Doppler, HeaviSine) at varying sample sizes and SNR levels.

Results

• Low-SNR Performance: In the primary Gaussian-error simulation study, the WS–Gaussian (Wendland–Semicircle with Gaussian likelihood) version was the strongest non-NLP method in 26 of the 36 design cells, including all low-SNR cells (SNR = 0.2). It achieved the lowest Mean Squared Error (MSE) among non-NLP methods in every low-SNR cell.
• Comparison with NLP: The Nonlocal Prior (NLP) method achieved the best overall pooled reconstruction accuracy, particularly at moderate and favorable SNR levels. However, WS–Gaussian remained competitive, especially for signals with discontinuities (Bumps, Blocks) in low-SNR regimes.
• Computational Efficiency: WS–Gaussian demonstrated a substantially more favorable computational profile than both the Laplace-likelihood version (WS–Laplace) and the NLP method. WS–Gaussian had a median runtime of ~3.4 seconds, compared to ~128 seconds for WS–Laplace and ~92 seconds for NLP.
• Real-Data Application: Applied to a seismic acceleration trace from the 2008 Chino Hills earthquake, the WS–Gaussian method effectively attenuated rapid fluctuations while preserving the timing of the dominant acceleration event. Unlike classical thresholding methods which retained nearly 100% of the observed energy (and thus noise), the proposed method reduced the peak acceleration amplitude (from ~44.27 to ~37.49 cm/s²) while maintaining the pulse timing within 0.01 seconds.

Significance and Claims
The paper claims that the proposed WS–Gaussian method offers a robust balance between reconstruction accuracy and computational cost, particularly in challenging low-SNR environments where distinguishing noise from signal is difficult.

• Adaptability: The resolution-adaptive mixture allows the prior to express both strong concentration near zero (for noise) and the preservation of moderate coefficients (for signal features) within a single framework.
• No Universal Dominance: The author modestly concludes that no single kernel or method dominates universally. Instead, the utility of the Wendland–semicircle construction depends on the SNR, signal structure, and computational constraints.
• Practical Utility: The method provides a computationally efficient alternative to NLP-based methods for Gaussian noise, offering a favorable accuracy–runtime trade-off. The bounded support structure serves as a modeling choice to reduce the influence of extreme noisy coefficients, though the author notes this implies saturation for coefficients exceeding the truncation scale.

The paper does not claim to solve the denoising problem for all signal types or noise distributions universally but positions the WS–Gaussian estimator as a highly competitive, interpretable, and efficient tool for Bayesian wavelet shrinkage in practical applications involving Gaussian noise.