Linearized Alternating Direction Method with Adaptive Penalty for Low-Rank Representation

Abstract: Low-rank representation (LRR) is an effective method for subspace clustering and has found wide applications in computer vision and machine learning. The existing LRR solver is based on the alternating direction method (ADM). It suffers from $O(n^3)$ computation complexity due to the matrix-matrix multiplications and matrix inversions, even if partial SVD is used. Moreover, introducing auxiliary variables also slows down the convergence. Such a heavy computation load prevents LRR from large scale applications. In this paper, we generalize ADM by linearizing the quadratic penalty term and allowing the penalty to change adaptively. We also propose a novel rule to update the penalty such that the convergence is fast. With our linearized ADM with adaptive penalty (LADMAP) method, it is unnecessary to introduce auxiliary variables and invert matrices. The matrix-matrix multiplications are further alleviated by using the skinny SVD representation technique. As a result, we arrive at an algorithm for LRR with complexity $O(rn^2)$, where $r$ is the rank of the representation matrix. Numerical experiments verify that for LRR our LADMAP method is much faster than state-of-the-art algorithms. Although we only present the results on LRR, LADMAP actually can be applied to solving more general convex programs.