Abstract: The task of dividing corrupted-data into their respective subspaces can be well
illustrated, both theoretically and numerically, by recovering low-rank and sparse-column
components of a given matrix. Generally, it can be characterized as a matrix and a
`2,1-norm involved convex minimization problem. However, solving the resulting problem
is full of challenges due to the non-smoothness of the objective function. One of the
earliest solvers is an 3-block alternating direction method of multipliers (ADMM) which
updates each variable in a Gauss-Seidel manner. In this paper, we present three variants
of ADMM for the 3-block separable minimization problem. More preciously, whenever
one variable is derived, the resulting problems can be regarded as a convex minimization
with 2 blocks, and can be solved immediately using the standard ADMM. If the inner
iteration loops only once, the iterative scheme reduces to the ADMM with updates in a
Gauss-Seidel manner. If the solution from the inner iteration is assumed to be exact, the
convergence can be deduced easily in the literature. The performance comparisons with a
couple of recently designed solvers illustrate that the proposed methods are effective and
competitive.

Key words:  Convex optimization, Variational inequality problem, Alternating direction method of multipliers, Low-rank representation, Subspace recovery