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We study a class of non-convex and non-smooth problems with \textit{rank}regularization to promote sparsity in optimal solution. We propose to apply theproximal gradient descent method to solve the problem and accelerate theprocess with a novel support set projection operation on the singular values ofthe intermediate update. We show that our algorithms are able to achieve aconvergence rate of $O(\frac{1}{t^2})$, which is exactly same as Nesterov'soptimal convergence rate for first-order methods on smooth and convex problems.Strict sparsity can be expected and the support set of singular values duringeach update is monotonically shrinking, which to our best knowledge, is novelin momentum-based algorithms.

Strictly Low Rank Constraint Optimization -- An Asymptotically $\mathcal{O}(\frac{1}{t^2})$ Method