Splitting and linearizing augmented Lagrangian algorithm for subspace recovery from corrupted observations

Given a set of corrupted data drawn from a union of multiple subspace, the subspace recovery problem is to segment the data into their respective subspace and to correct the possible noise simultaneously. Recently, it is discovered that the task can be characterized, both theoretically and numerically, by solving a matrix nuclear-norm and a `2,1-mixed norm involved convex minimization problems. The minimization model actually has separable structure in both the objective function and constraint; it thus falls into the framework of the augmented Lagrangian alternating direction approach. In this paper, we propose and investigate an augmented Lagrangian algorithm. We split the augmented Lagrangian function and minimize the subproblems alternatively with one variable by fixing the other one. Moreover, we linearize the subproblem and add a proximal point term to easily derive the closed-form solutions. Global convergence of the proposed algorithm is established under some technical conditions. Extensive experiments on the simulated and the real data verify that the proposed method is very effective and faster than the sate-of-the-art algorithm LRR.