We develop new nonconvex approaches for matrix optimization problems involving sparsity. The heart of the methods is a new, nonconvex penalty function that is designed for efficient minimization by means of a generalized shrinkage operation. We apply this approach to the decomposition of video into low rank and sparse components, which is able to separate moving objects from the stationary background better than in the convex case. In the case of noisy data, we add a nonconvex regularization, and apply a splitting approach to decompose the optimization problem into simple, parallelizable components. The nonconvex regularization ameliorates contrast loss, thereby allowing stronger denoising without losing more signal to the residual.