Multi-dimensional signal approximation with sparse structured priors using split Bregman iterations

TL;DR

This paper introduces a novel sparse decomposition method for multi-dimensional signals that incorporates spatio-temporal priors and employs split Bregman iterations for efficient optimization, demonstrating improved convergence and effectiveness.

Contribution

It presents a new regularization framework combining spatio-temporal priors with $ ext{l}_1$ regularization and an optimized split Bregman algorithm for structured sparse signal decomposition.

Findings

Achieves state-of-the-art accuracy with faster convergence.

Effectively recovers expected decompositions on artificial data.

Outperforms existing methods in EEG brainwave analysis.

Abstract

This paper addresses the structurally-constrained sparse decomposition of multi-dimensional signals onto overcomplete families of vectors, called dictionaries. The contribution of the paper is threefold. Firstly, a generic spatio-temporal regularization term is designed and used together with the standard $\ell_1$ regularization term to enforce a sparse decomposition preserving the spatio-temporal structure of the signal. Secondly, an optimization algorithm based on the split Bregman approach is proposed to handle the associated optimization problem, and its convergence is analyzed. Our well-founded approach yields same accuracy as the other algorithms at the state-of-the-art, with significant gains in terms of convergence speed. Thirdly, the empirical validation of the approach on artificial and real-world problems demonstrates the generality and effectiveness of the method. On artificial problems, the proposed regularization subsumes the Total Variation minimization and recovers the expected decomposition. On the real-world problem of electro-encephalography brainwave decomposition, the approach outperforms similar approaches in terms of P300 evoked potentials detection, using structured spatial priors to guide the decomposition.