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

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.Comment: Manuscript accepted by NIPS 201

The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices

This paper proposes scalable and fast algorithms for solving the Robust PCA problem, namely recovering a low-rank matrix with an unknown fraction of its entries being arbitrarily corrupted. This problem arises in many applications, such as image processing, web data ranking, and bioinformatic data analysis. It was recently shown that under surprisingly broad conditions, the Robust PCA problem can be exactly solved via convex optimization that minimizes a combination of the nuclear norm and the $\ell^1$-norm . In this paper, we apply the method of augmented Lagrange multipliers (ALM) to solve this convex program. As the objective function is non-smooth, we show how to extend the classical analysis of ALM to such new objective functions and prove the optimality of the proposed algorithms and characterize their convergence rate. Empirically, the proposed new algorithms can be more than five times faster than the previous state-of-the-art algorithms for Robust PCA, such as the accelerated proximal gradient (APG) algorithm. Moreover, the new algorithms achieve higher precision, yet being less storage/memory demanding. We also show that the ALM technique can be used to solve the (related but somewhat simpler) matrix completion problem and obtain rather promising results too. We further prove the necessary and sufficient condition for the inexact ALM to converge globally. Matlab code of all algorithms discussed are available at http://perception.csl.illinois.edu/matrix-rank/home.htmlComment: Please cite "Zhouchen Lin, Risheng Liu, and Zhixun Su, Linearized Alternating Direction Method with Adaptive Penalty for Low Rank Representation, NIPS 2011." (available at arXiv:1109.0367) instead for a more general method called Linearized Alternating Direction Method This manuscript first appeared as University of Illinois at Urbana-Champaign technical report #UILU-ENG-09-2215 in October 2009 Zhouchen Lin, Risheng Liu, and Zhixun Su, Linearized Alternating Direction Method with Adaptive Penalty for Low Rank Representation, NIPS 2011. (available at http://arxiv.org/abs/1109.0367