A subspace-accelerated split Bregman method for sparse data recovery with joint l1-type regularizers

TL;DR

This paper introduces a subspace-accelerated split Bregman method for efficiently solving sparse data recovery problems with joint l1-type regularizers, applicable to various fields like finance and neuroimaging.

Contribution

The paper presents a novel subspace-accelerated Bregman approach tailored for linearly constrained problems with structured sparsity regularizers, improving computational efficiency.

Findings

Effective in portfolio optimization with real data

Accelerates convergence for structured sparsity problems

Demonstrates practical benefits in neuroimaging applications

Abstract

We propose a subspace-accelerated Bregman method for the linearly constrained minimization of functions of the form $f(\mathbf{u})+\tau_1 \|\mathbf{u}\|_1 + \tau_2 \|D\,\mathbf{u}\|_1$, where $f$ is a smooth convex function and $D$ represents a linear operator, e.g. a finite difference operator, as in anisotropic Total Variation and fused-lasso regularizations. Problems of this type arise in a wide variety of applications, including portfolio optimization and learning of predictive models from functional Magnetic Resonance Imaging (fMRI) data, and source detection problems in electroencephalography. The use of $\|D\,\mathbf{u}\|_1$ is aimed at encouraging structured sparsity in the solution. The subspaces where the acceleration is performed are selected so that the restriction of the objective function is a smooth function in a neighborhood of the current iterate. Numerical experiments on multi-period portfolio selection problems using real datasets show the effectiveness of the proposed method.