Many machine learning and signal processing problems can be formulated as linearly constrained convex programs, which could be efficiently solved by the alternating direction method (ADM). However, usually the subproblems in ADM are easily solvable only when the linear mappings in the constraints are identities. To address this issue, we propose a linearized ADM (LADM) method by linearizing the quadratic penalty term and adding a proximal term when solving the subproblems. For fast convergence, we also allow the penalty to change adaptively according a novel update rule. We prove the global convergence of LADM with adaptive penalty (LADMAP).

Many machine learning and signal processing problems can be formulated as linearly constrained convex programs, which could be efficiently solved by the alternating direction method (ADM). However, usually the subproblems in ADM are easily solvable only when the linear mappings in the constraints are identities. To address this issue, we propose a linearized ADM (LADM) method by linearizing the quadratic penalty term and adding a proximal term when solving the subproblems. For fast convergence, we also allow the penalty to change adaptively according a novel update rule. We prove the global convergence of LADM with adaptive penalty (LADMAP).

Being one of the most effective methods, Alternating Direction Method (ADM) has been extensively studied in numerical analysis for solving linearly constrained convex program. However, there are few studies focusing on the convergence property of ADM under nonconvex framework though it has already achieved well-performance on applying to various nonconvex tasks. In this paper, a linearized algorithm with penalization is proposed on the basis of ADM for solving nonconvex and nonsmooth optimization. We start from analyzing the convergence property for the classical constrained problem with two variables and then establish a similar result for multi-block case. To demonstrate the effectiveness of our proposed algorithm, experiments with synthetic and real-world data have been conducted on specific applications in signal and image processing.

Many machine learning and signal processing problems can be formulated as linearly constrained convex programs, which could be efficiently solved by the alternating direction method (ADM). However, usually the subproblems in ADM are easily solvable only when the linear mappings in the constraints are identities. To address this issue, we propose a linearized ADM (LADM) method by linearizing the quadratic penalty term and adding a proximal term when solving the subproblems. For fast convergence, we also allow the penalty to change adaptively according a novel update rule. We prove the global convergence of LADM with adaptive penalty (LADMAP). As an example, we apply LADMAP to solve low-rank representation (LRR), which is an important subspace clustering technique yet suffers from high computation cost. By combining LADMAP with a skinny SVD representation technique, we are able to reduce the complexity $O(n^3)$ of the original ADM based method to $O(rn^2)$, where $r$ and $n$ are the rank and size of the representation matrix, respectively, hence making LRR possible for large scale applications. Numerical experiments verify that for LRR our LADMAP based methods are much faster than state-of-the-art algorithms.