Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zhouchen Lin, Minming Chen, Yi Ma

TL;DR

This work introduces scalable augmented Lagrange multiplier methods (Exact ALM and Inexact ALM) for robust principal component analysis and extends the approach to matrix completion. By formulating the problems with nuclear-norm and L1-regularization, the ALM framework achieves exact convergence with practical efficiency, often outperforming state-of-the-art APG and SVT-based methods in both speed and accuracy. The authors provide detailed implementation guidelines, convergence proofs, and public Matlab code, making the methods readily usable for large-scale low-rank recovery tasks. The results suggest significant practical impact for applications in image processing, web data analysis, and bioinformatics where large corrupted datasets must be recovered efficiently.

Abstract

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.html

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

TL;DR

This work introduces scalable augmented Lagrange multiplier methods (Exact ALM and Inexact ALM) for robust principal component analysis and extends the approach to matrix completion. By formulating the problems with nuclear-norm and L1-regularization, the ALM framework achieves exact convergence with practical efficiency, often outperforming state-of-the-art APG and SVT-based methods in both speed and accuracy. The authors provide detailed implementation guidelines, convergence proofs, and public Matlab code, making the methods readily usable for large-scale low-rank recovery tasks. The results suggest significant practical impact for applications in image processing, web data analysis, and bioinformatics where large corrupted datasets must be recovered efficiently.

Abstract

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 -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.html

Paper Structure

This paper contains 26 sections, 8 theorems, 66 equations, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Previous Algorithms for Matrix Recovery
  3. The Iterative Thresholding Approach
  4. The Accelerated Proximal Gradient Approach
  5. The Dual Approach
  6. The Methods of Augmented Lagrange Multipliers
  7. Two ALM Algorithms for Robust PCA (Matrix Recovery)
  8. An ALM Algorithm for Matrix Completion
  9. Implementation Details
  10. Predicting the Dimension of Principal Singular Space.
  11. Order of Updating $A$ and $E$.
  12. Memory Saving for Algorithm \ref{['alg:MC-IALM']}.
  13. Stopping Criteria.
  14. Updating $\mu_k$.
  15. Choosing Parameters.
  16. ...and 11 more sections

Key Result

theorem 1

For Algorithm alg:exact_ALM, any accumulation point $(A^*,E^*)$ of $(A_k^*,E_k^*)$ is an optimal solution to the RPCA problem and the convergence rate is at least $O(\mu_k^{-1})$ in the sense that where $f^*$ is the optimal value of the RPCA problem.

Theorems & Definitions (19)

  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • theorem 4
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 9 more