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

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.