Full Length Paper Series A

Mathematical Programming

, Volume 128, Issue 1, pp 321-353

Fixed point and Bregman iterative methods for matrix rank minimization

  • Shiqian MaAffiliated withDepartment of Industrial Engineering and Operations Research, Columbia University Email author 
  • , Donald GoldfarbAffiliated withDepartment of Industrial Engineering and Operations Research, Columbia University
  • , Lifeng ChenAffiliated withDepartment of Industrial Engineering and Operations Research, Columbia University

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The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems (the code can be downloaded from http://​www.​columbia.​edu/​~sm2756/​FPCA.​htm for non-commercial use). Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 min by sampling only 20% of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms.


Matrix rank minimization Matrix completion problem Nuclear norm minimization Fixed point iterative method Bregman distances Singular value decomposition

Mathematics Subject Classification (2000)

65K05 90C25 90C06 93C41 68Q32