A nonconvex approach to low-rank and sparse matrix decomposition with application to video surveillance

In this paper, we develop a new nonconvex approach to the problem of low-rank and sparse matrix decomposition. In our nonconvex method, we replace the rank function and the $\ell_{0}$-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 nonconvex problem with the non-convex fraction function penalty. Numerical experiments on video surveillance show that our method performs very well in separating the moving objects from the static background.