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Hierarchical Bayesian Inverse Problems: A High-Dimensional Statistics Viewpoint

Daniel Sanz-Alonso, Nathan Waniorek

TL;DR

The paper develops a unified high-dimensional statistical framework for hierarchical Bayesian inverse problems with gamma hyperpriors, introducing approximate decomposability to bridge nondecomposable hierarchical penalties with decomposable norms. By tying MAP estimators to regularized M-estimators and imposing an RSC condition on the forward map, the authors derive non-asymptotic reconstruction error bounds that separate estimation, approximation, and approximation-decomposability contributions. They apply the theory to three core hierarchical models—sparsity, group sparsity, and sparse representations—proving linear convergence of MAP-optimization schemes (IAS, GS-IAS, O-IAS) and delivering deterministic and high-probability error rates that match minimax rates under suitable hyperparameters. The results provide a principled statistical justification for the practical efficacy of hierarchical Bayesian methods in high dimensions and open avenues for extending the theory to broader priors and forward-model classes. Overall, the work advances the theoretical understanding of when and how Bayesian hierarchical priors yield accurate, computationally tractable reconstructions in ill-posed, high-dimensional inverse problems.

Abstract

This paper analyzes hierarchical Bayesian inverse problems using techniques from high-dimensional statistics. Our analysis leverages a property of hierarchical Bayesian regularizers that we call approximate decomposability to obtain non-asymptotic bounds on the reconstruction error attained by maximum a posteriori estimators. The new theory explains how hierarchical Bayesian models that exploit sparsity, group sparsity, and sparse representations of the unknown parameter can achieve accurate reconstructions in high-dimensional settings.

Hierarchical Bayesian Inverse Problems: A High-Dimensional Statistics Viewpoint

TL;DR

The paper develops a unified high-dimensional statistical framework for hierarchical Bayesian inverse problems with gamma hyperpriors, introducing approximate decomposability to bridge nondecomposable hierarchical penalties with decomposable norms. By tying MAP estimators to regularized M-estimators and imposing an RSC condition on the forward map, the authors derive non-asymptotic reconstruction error bounds that separate estimation, approximation, and approximation-decomposability contributions. They apply the theory to three core hierarchical models—sparsity, group sparsity, and sparse representations—proving linear convergence of MAP-optimization schemes (IAS, GS-IAS, O-IAS) and delivering deterministic and high-probability error rates that match minimax rates under suitable hyperparameters. The results provide a principled statistical justification for the practical efficacy of hierarchical Bayesian methods in high dimensions and open avenues for extending the theory to broader priors and forward-model classes. Overall, the work advances the theoretical understanding of when and how Bayesian hierarchical priors yield accurate, computationally tractable reconstructions in ill-posed, high-dimensional inverse problems.

Abstract

This paper analyzes hierarchical Bayesian inverse problems using techniques from high-dimensional statistics. Our analysis leverages a property of hierarchical Bayesian regularizers that we call approximate decomposability to obtain non-asymptotic bounds on the reconstruction error attained by maximum a posteriori estimators. The new theory explains how hierarchical Bayesian models that exploit sparsity, group sparsity, and sparse representations of the unknown parameter can achieve accurate reconstructions in high-dimensional settings.
Paper Structure (17 sections, 17 theorems, 138 equations)

This paper contains 17 sections, 17 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Unified Theoretical Framework
  4. Approximate Decomposability
  5. General Bound on the Reconstruction Error
  6. Reconstruction Error for Sparsity-Promoting Models
  7. Hierarchical Models for Sparsity
  8. Hierarchical Models for Group Sparsity
  9. Hierarchical Models for Sparse Representations
  10. Proof of Reconstruction Error for Sparsity-Promoting Models
  11. Hierarchical Models for Sparsity
  12. Hierarchical Models for Group Sparsity
  13. Hierarchical Models for Sparse Representations
  14. Conclusions
  15. Computational Properties of GS-IAS and O-IAS
  16. ...and 2 more sections

Key Result

Theorem 2.1

For $\eta>0,$ the objective function ${\mathsf{J}}$ in eq:functional is strictly convex over $\mathbb{R}^d\times \mathbb{R}^d_+$ and thus admits a unique MAP estimator $(\hat{u},\hat{\theta}).$ Moreover, the $u$-iterates $u^\ell$ of the IAS algorithm eq:IASalgorithm converge at least linearly to $\h

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 26 more