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Nonconvex Latent Optimally Partitioned Block-Sparse Recovery via Log-Sum and Minimax Concave Penalties

Takanobu Furuhashi, Hiroki Kuroda, Masahiro Yukawa, Qibin Zhao, Hidekata Hontani, Tatsuya Yokota

Abstract

We propose two nonconvex regularization methods, LogLOP-l2/l1 and AdaLOP-l2/l1, for recovering block-sparse signals with unknown block partitions. These methods address the underestimation bias of existing convex approaches by extending log-sum penalty and the Minimax Concave Penalty (MCP) to the block-sparse domain via novel variational formulations. Unlike Generalized Moreau Enhancement (GME) and Bayesian methods dependent on the squared-error data fidelity term, our proposed methods are compatible with a broad range of data fidelity terms. We develop efficient Alternating Direction Method of Multipliers (ADMM)-based algorithms for these formulations that exhibit stable empirical convergence. Numerical experiments on synthetic data, angular power spectrum estimation, and denoising of nanopore currents demonstrate that our methods outperform state-of-the-art baselines in estimation accuracy.

Nonconvex Latent Optimally Partitioned Block-Sparse Recovery via Log-Sum and Minimax Concave Penalties

Abstract

We propose two nonconvex regularization methods, LogLOP-l2/l1 and AdaLOP-l2/l1, for recovering block-sparse signals with unknown block partitions. These methods address the underestimation bias of existing convex approaches by extending log-sum penalty and the Minimax Concave Penalty (MCP) to the block-sparse domain via novel variational formulations. Unlike Generalized Moreau Enhancement (GME) and Bayesian methods dependent on the squared-error data fidelity term, our proposed methods are compatible with a broad range of data fidelity terms. We develop efficient Alternating Direction Method of Multipliers (ADMM)-based algorithms for these formulations that exhibit stable empirical convergence. Numerical experiments on synthetic data, angular power spectrum estimation, and denoising of nanopore currents demonstrate that our methods outperform state-of-the-art baselines in estimation accuracy.
Paper Structure (43 sections, 5 theorems, 57 equations, 11 figures, 1 table, 2 algorithms)

This paper contains 43 sections, 5 theorems, 57 equations, 11 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Review of Existing Methods
  3. Mathematical Notations
  4. Nonconvex Sparse Regularization
  5. Latent Optimal Partitioned (LOP)- b it sl $\ell\sb{2}/\ell\sb{1}$
  6. Generalized Moreau Enhancement (GME)-LOP- b it sl $\ell\sb{2}/\ell\sb{1}$
  7. Design of Proposed Regularizations
  8. General Variational Framework for Optimal Partitioning
  9. Logarithmic LOP- b it sl $\ell\sb{2}/\ell\sb{1}$ (LogLOP- b it sl $\ell\sb{2}/\ell\sb{1}$ )
  10. Variational Formulation
  11. Definition of LogLOP- b it sl $\ell\sb{2}/\ell\sb{1}$
  12. Asymptotic Behavior
  13. Validity for Block Partitioning
  14. Adaptively Weighted LOP- b it sl $\ell\sb{2}/\ell\sb{1}$ (AdaLOP- b it sl $\ell\sb{2}/\ell\sb{1}$ )
  15. Variational Formulation
  16. ...and 28 more sections

Key Result

lemma 1

lem:var-rwl1 For any fixed $x \in \mathbb R$ and $\epsilon > 0$, $\phi_\epsilon(x,\tau)$ has a unique global minimizer $\tau^* = (\abs{x}/\epsilon + 1)^2$ over $\tau \in \mathbb R$, and the minimum value is:

Figures (11)

  • Figure 1: Comparison of sparsity-inducing penalties. Our proposed methods extend the concept of nonconvex log-sum and MC penalties by integrating a block partitioning mechanism to handle unknown block sparsity.
  • Figure 2: Comparison of proximal operators. Unlike LOP- b it sl $\ell\sb{2}/\ell\sb{1}$ , the proposed LogLOP- b it sl $\ell\sb{2}/\ell\sb{1}$ and AdaLOP- b it sl $\ell\sb{2}/\ell\sb{1}$ mitigate underestimation bias for large amplitudes (e.g., $y_2=51, y_4=71$) while preserving block partitions, outperforming element-wise penalties.
  • Figure 3: Asymptotic behaviors of LOP, LogLOP (thm:var-rwl1), and AdaLOP (thm:ada-lop-l2l1) variants for b it sl $\ell\sb{2}/\ell\sb{1}$ regularization.
  • Figure 4: Experimental setup for compressive sensing. (Top) True block-sparse signal $\bm x_0 \in \mathbb{R}^{250}$ with four nonzero blocks. (Bottom) Initial pseudo-inverse recovery $\bm A^\dagger \bm y$ (SNR: 7.6 dB).
  • Figure 5: Quantitative evaluation results. (Top) SNR vs. $\lambda$. (Bottom) F1 score vs. $\lambda$. AdaLOP- b it sl $\ell\sb{2}/\ell\sb{1}$ achieves the highest SNR, maintaining robustness across a wide $\lambda$ range. The proposed methods maintain near-perfect support recovery (F1 $\approx$ 1.0) significantly better than convex baselines.
  • ...and 6 more figures

Theorems & Definitions (13)

  • definition 1: General LOP-type Penalty
  • lemma 1: Minimum Value of $\phi_\epsilon(x,\tau)$
  • proof
  • theorem 1: Asymptotic Behavior of LogLOP- b it sl $\ell\sb{2}/\ell\sb{1}$
  • proof
  • proposition 1: Block Decomposition Property
  • proof
  • lemma 2: Minimum Value of $\phi_{\gamma, w}(x, \tau)$
  • proof
  • theorem 2: Asymptotic Behavior of AdaLOP- b it sl $\ell\sb{2}/\ell\sb{1}$
  • ...and 3 more