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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 8

### Density matrix minimization with ${\ell}_1$ regularization

Pages: 2097 – 2117

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n8.a6

#### Authors

#### Abstract

We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with ${\ell}_1$ regularization. The minimization problem can be efficiently solved by a split Bregman iteration type algorithm. We further prove that from any initial condition, the algorithm converges to a minimizer of the variational principle.

#### Keywords

density matrix, ${\ell}_1$ regularization, eigenspace

#### 2010 Mathematics Subject Classification

65F30, 65K10