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SPP-SBL: Space-Power Prior Sparse Bayesian Learning for Block Sparse Recovery

Yanhao Zhang, Zhihan Zhu, Yong Xia

TL;DR

The paper addresses the problem of recovering block-sparse signals with unknown structural patterns. It introduces a unified variance transformation framework and a space-power prior on an undirected graph, enabling adaptive learning of space-coupling parameters via an EM-based SPP-SBL algorithm. A key insight is that the relative magnitudes of the coupling parameters $\vec{\boldsymbol{\beta}}$ drive substantial gains in recovery accuracy across complex patterns, including chain and multi-pattern data, as well as real-world audio and image signals. Empirical results demonstrate that SPP-SBL consistently outperforms existing pattern-based and block-sparse methods, mitigating boundary over-estimation and offering robust performance across diverse structured sparsity regimes, with potential for extension to richer graph-based priors.

Abstract

The recovery of block-sparse signals with unknown structural patterns remains a fundamental challenge in structured sparse signal reconstruction. By proposing a variance transformation framework, this paper unifies existing pattern-based block sparse Bayesian learning methods, and introduces a novel space power prior based on undirected graph models to adaptively capture the unknown patterns of block-sparse signals. By combining the EM algorithm with high-order equation root-solving, we develop a new structured sparse Bayesian learning method, SPP-SBL, which effectively addresses the open problem of space coupling parameter estimation in pattern-based methods. We further demonstrate that learning the relative values of space coupling parameters is key to capturing unknown block-sparse patterns and improving recovery accuracy. Experiments validate that SPP-SBL successfully recovers various challenging structured sparse signals (e.g., chain-structured signals and multi-pattern sparse signals) and real-world multi-modal structured sparse signals (images, audio), showing significant advantages in recovery accuracy across multiple metrics.

SPP-SBL: Space-Power Prior Sparse Bayesian Learning for Block Sparse Recovery

TL;DR

The paper addresses the problem of recovering block-sparse signals with unknown structural patterns. It introduces a unified variance transformation framework and a space-power prior on an undirected graph, enabling adaptive learning of space-coupling parameters via an EM-based SPP-SBL algorithm. A key insight is that the relative magnitudes of the coupling parameters drive substantial gains in recovery accuracy across complex patterns, including chain and multi-pattern data, as well as real-world audio and image signals. Empirical results demonstrate that SPP-SBL consistently outperforms existing pattern-based and block-sparse methods, mitigating boundary over-estimation and offering robust performance across diverse structured sparsity regimes, with potential for extension to richer graph-based priors.

Abstract

The recovery of block-sparse signals with unknown structural patterns remains a fundamental challenge in structured sparse signal reconstruction. By proposing a variance transformation framework, this paper unifies existing pattern-based block sparse Bayesian learning methods, and introduces a novel space power prior based on undirected graph models to adaptively capture the unknown patterns of block-sparse signals. By combining the EM algorithm with high-order equation root-solving, we develop a new structured sparse Bayesian learning method, SPP-SBL, which effectively addresses the open problem of space coupling parameter estimation in pattern-based methods. We further demonstrate that learning the relative values of space coupling parameters is key to capturing unknown block-sparse patterns and improving recovery accuracy. Experiments validate that SPP-SBL successfully recovers various challenging structured sparse signals (e.g., chain-structured signals and multi-pattern sparse signals) and real-world multi-modal structured sparse signals (images, audio), showing significant advantages in recovery accuracy across multiple metrics.
Paper Structure (24 sections, 2 theorems, 56 equations, 6 figures, 4 tables, 1 algorithm)

This paper contains 24 sections, 2 theorems, 56 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Variance Transformation Framework
  3. Revisit Pattern-coupled Priors
  4. A Unified Variance Transformation Framework
  5. The Symmetric Diversified Coupling Matrix $\boldsymbol{T}_{\rm SPP}$
  6. Space-Power Prior Bayesian model
  7. Space-Power Prior
  8. Posterior Estimation
  9. Bayesian Inference: SPP-SBL Algorithm
  10. Update $\boldsymbol{\alpha}$
  11. Update $\vec{\boldsymbol{\beta}}$
  12. Update $\gamma$
  13. Discussion
  14. Boundary Over-Estimation
  15. Complex Block Sparse Structure
  16. ...and 9 more sections

Key Result

Theorem 1

When $c > 1$, the equation cubic in $\beta_i$ ( $\forall i$) exists only one positive real root.

Figures (6)

  • Figure 1: Graph structures of pattern coupling. (A) The underlying graph structures corresponding to classical pattern-coupled models. (B) The proposed Space-Power-Prior (SPP) coupling model: (B1) uniform interactions between adjacent nodes, (B2) edge-specific parameters enabling boundary-aware learning, (B3) the resulting symmetric diversified coupling matrix $\boldsymbol{T}_{\text{SPP}}$, and (B4) the overall Bayesian hierarchical model.
  • Figure 2: The three possible boundary cases for $x_i$, where the shaded regions indicate nonzero positions and the white regions represent zero positions.
  • Figure 3: The impact of different hyperprior parameters $c$ in SPP-SBL and different $\beta$ choices in PC-SBL on success rate.
  • Figure 4: Heatmaps of the posterior variance and the $\boldsymbol{\beta}$ values estimated by the SPP-SBL algorithm across three datasets.
  • Figure 5: Phase transition diagram under different SNR and measurement ratios.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3