This work presents a general framework for solving the low rank and/or sparse
matrix minimization problems, which may involve multiple non-smooth terms. The
Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which
smooths the objective function and minimizes it by alternately updating the
variables and their weights. However, the traditional IRLS can only solve a
sparse only or low rank only minimization problem with squared loss or an
affine constraint. This work generalizes IRLS to solve joint/mixed low rank and
sparse minimization problems, which are essential formulations for many tasks.
As a concrete example, we solve the Schatten-$p$ norm and $\ell_{2,q}$-norm
regularized Low-Rank Representation (LRR) problem by IRLS, and theoretically
prove that the derived solution is a stationary point (globally optimal if
$p,q\geq1$). Our convergence proof of IRLS is more general than previous one
which depends on the special properties of the Schatten-$p$ norm and
$\ell_{2,q}$-norm. Extensive experiments on both synthetic and real data sets
demonstrate that our IRLS is much more efficient.

本文提出了一个解决低秩和 / 或稀疏矩阵最小化问题的一般框架，使用迭代重新加权最小二乘（IRLS）方法来解决混合低秩和稀疏最小化问题，例如用于解决 Schatten-p 规范和 ell_2,q-norm 规范的低秩表示问题，理论证明了所获得的解为静止点，并在合成和实际数据集上进行了广泛的实验以证明其有效性。