A non-convex approach to low-rank and sparse matrix decomposition
This addresses matrix decomposition for applications like robust PCA, but it appears incremental as it builds on existing nonconvex approaches.
The paper tackles the problem of low-rank and sparse matrix decomposition by developing a nonconvex method that replaces the rank function and l0-norm with a non-convex fraction function, and it performs very well in recovering low-rank matrices heavily corrupted by large sparse errors in numerical experiments.
In this paper, we develop a nonconvex approach to the problem of low-rank and sparse matrix decomposition. In our nonconvex method, we replace the rank function and the $l_{0}$-norm of a given matrix with a non-convex fraction function on the singular values and the elements of the matrix respectively. An alternative direction method of multipliers algorithm is utilized to solve our proposed nonconvex problem with the nonconvex fraction function penalty. Numerical experiments on some low-rank and sparse matrix decomposition problems show that our method performs very well in recovering low-rank matrices which are heavily corrupted by large sparse errors.