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Random tree Besov priors: Data-driven regularisation parameter selection

Hanne Kekkonen, Andreas Tataris

TL;DR

This work tackles data‑driven regularisation parameter selection in Bayesian inversion using random tree Besov priors. It introduces a hierarchical model that allows the wavelet density parameter β to vary across scales and uses levelwise MAP optimization to automatically identify β, enabling adaptive regularisation across resolutions. A recursive tree‑pruning algorithm is developed for both Gaussian and Laplace base priors, and the approach is demonstrated in nonparametric regression, deconvolution, and 2D image denoising. Results show that automatic β selection performs comparably to optimally tuned fixed β settings while removing manual tuning, with competitive denoising quality and potential plug‑and‑play applicability in broader inverse problems.

Abstract

We develop a data-driven algorithm for automatically selecting the regularisation parameter in Bayesian inversion under random tree Besov priors. One of the key challenges in Bayesian inversion is the construction of priors that are both expressive and computationally feasible. Random tree Besov priors, introduced in Kekkonen et al. (2023), provide a flexible framework for capturing local regularity properties and sparsity patterns in a wavelet basis. In this paper, we extend this approach by introducing a hierarchical model that enables data-driven selection of the wavelet density parameter, allowing the regularisation strength to adapt across scales while retaining computational efficiency. We focus on nonparametric regression and also present preliminary plug-and-play results for a deconvolution problem.

Random tree Besov priors: Data-driven regularisation parameter selection

TL;DR

This work tackles data‑driven regularisation parameter selection in Bayesian inversion using random tree Besov priors. It introduces a hierarchical model that allows the wavelet density parameter β to vary across scales and uses levelwise MAP optimization to automatically identify β, enabling adaptive regularisation across resolutions. A recursive tree‑pruning algorithm is developed for both Gaussian and Laplace base priors, and the approach is demonstrated in nonparametric regression, deconvolution, and 2D image denoising. Results show that automatic β selection performs comparably to optimally tuned fixed β settings while removing manual tuning, with competitive denoising quality and potential plug‑and‑play applicability in broader inverse problems.

Abstract

We develop a data-driven algorithm for automatically selecting the regularisation parameter in Bayesian inversion under random tree Besov priors. One of the key challenges in Bayesian inversion is the construction of priors that are both expressive and computationally feasible. Random tree Besov priors, introduced in Kekkonen et al. (2023), provide a flexible framework for capturing local regularity properties and sparsity patterns in a wavelet basis. In this paper, we extend this approach by introducing a hierarchical model that enables data-driven selection of the wavelet density parameter, allowing the regularisation strength to adapt across scales while retaining computational efficiency. We focus on nonparametric regression and also present preliminary plug-and-play results for a deconvolution problem.
Paper Structure (11 sections, 1 theorem, 57 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 1 theorem, 57 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Random tree Besov priors
  3. Besov random variables
  4. Random tree Besov variables
  5. Tree pruning with automatic $\boldsymbol{\beta}$ selection
  6. Tree pruning with a fixed $\beta$
  7. Automatic $\beta$ selection with Gaussian draws
  8. Automatic $\beta$ selection with Laplace draws
  9. Numerical Results
  10. Signal denoising and deconvolution
  11. Image denoising

Key Result

Proposition 1

Consider the denoising problem of estimating $f$ from a noisy measurement eq:DenoisingMain. We assume a truncated random tree Besov prior for $f$ as described in eq:DefPrior. Denote by $T^{1}_\beta(m_1)$ the denoised signal arising from tree pruning with a fixed $\beta$ when $w_i\sim {\mathcal{N}}(0 with $c=\sigma^2(\kappa^2+\sigma^2)/(2\kappa^2)>0$.

Figures (5)

  • Figure 1: The original signal in black and the measurement in red (left), denoised signal using Gaussian base prior and automatic $\beta$ selection (middle), and denoised signal using Laplace base prior and automatic $\beta$ selection (right).
  • Figure 2: Selected wavelet coefficient tree for the denoised signal under the Gaussian base prior (left) and the levelwise estimates $\log(\hat{\boldsymbol{\beta}})$ (right).
  • Figure 3: The original signal in black, measurement in blue and deconvolved signal in red.
  • Figure 4: The original image and the noisy measurement with $7\%$ Gaussian noise.
  • Figure 5: Denoised images using Gaussian base prior and automatic $\beta$ selection (top left), Gaussian base prior and optimal fixed $\beta$ (top right), Laplace base prior and automatic $\beta$ selection (bottom left), and Laplace base prior and optimal fixed $\beta$.

Theorems & Definitions (4)

  • Definition 1
  • Proposition 1
  • Remark 1
  • proof : Proof of Proposition \ref{['Prop:scaling']}