A Nonconvex Fractional Regularization Model in Robust Principal Component Analysis via the Symmetric Alternating Direction Method of Multipliers

This paper addresses the NP-hard problem of solving the rank of a matrix in Robust Principal Component Analysis (RPCA) by proposing a nonconvex fractional regularization approximation. Compared to existing convex regularization (which often yields suboptimal solutions) and nonconvex regularization (which typically requires parameter selection), the proposed model effectively avoids parameter selection while preserving scale invariance. By introducing an auxiliary variable, we transform the problem into a nonconvex optimization problem with a separable structure. We use a more flexible Symmetric Alternating Direction Method of Multipliers (SADMM) to arrive at a solution and provide a rigorous convergence proof. In numerical experiments involving synthetic data, image recovery, and foreground–background separation for surveillance video, the proposed fractional regularization model demonstrates high computational accuracy, and its performance is comparable to that of many state-of-the-art algorithms.