Volume 53, pp. 406-425, 2020.

A subspace-accelerated split Bregman method for sparse data recovery with joint $\ell_1$-type regularizers

Valentina De Simone, Daniela di Serafino, and Marco Viola

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, 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 for multi-period portfolio selection problems using real data sets show the effectiveness of the proposed method.

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Key words

split Bregman method, subspace acceleration, joint $\ell_1$-type regularizers, multi-period portfolio optimization

AMS subject classifications

65K05, 90C25

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