Adaptive Block Sparse Regularization under Arbitrary Linear Transform

TL;DR

This work tackles block-sparse signal recovery when the underlying linear transform is arbitrary and potentially non-invertible. It introduces a convex LOP-$\ell_2$/$\ell_1$ under Arbitrary Linear Transform (ALT) formulation that jointly estimates latent block structure and enforces block sparsity via $\Psi_\alpha(\bm R\bm x)$, solvable by a primal-dual Loris–Verhoeven scheme. A key contribution is deriving convergence conditions and an expressive operator-norm bound to ensure algorithmic stability, enabling application to non-derivative transforms. Empirical results across synthetic block-derivative signals, nanopore ion currents, and natural images show competitive denoising performance and robustness to over-smoothing, along with practical guidance for parameter choices. Overall, the method broadens the reach of block-sparse regularization to a wider class of transforms, offering a convex and efficient tool for diverse signal-processing tasks.

Abstract

We propose a convex and fast signal reconstruction method for block sparsity under arbitrary linear transform with unknown block structure. The proposed method is a generalization of the similar existing method and can reconstruct signals with block sparsity under non-invertible transforms, unlike the existing method. Our work broadens the scope of block sparse regularization, enabling more versatile and powerful applications across various signal processing domains. We derive an iterative algorithm for solving proposed method and provide conditions for its convergence to the optimal solution. Numerical experiments demonstrate the effectiveness of the proposed method.

Figures (4)

• Figure 1: Illustration of block sparsity under non-invertible transform (here, differentiation) in Cantor function dovgosheyCantorFunction2006.
• Figure 2: Comparison of denoising results for ion currents. The red, green, and blue lines represent the estimated values for $\lambda = 1, 5, 10$, respectively.
• Figure 3: Comparison of block structure of ion current derivatives estimated by the proposed method.
• Figure 4: Comparison of denoising results for natural images.