Efficient and Guaranteed Rank Minimization by Atomic Decomposition

Abstract

Recht, Fazel, and Parrilo provided an analogy between rank minimization and ${\ell }_0$-norm minimization. Subject to the rank-restricted isometry property, nuclear norm minimization is a guaranteed algorithm for rank minimization. The resulting semidefinite formulation is a convex problem but in practice the algorithms for it do not scale well to large instances. Instead, we explore missing terms in the analogy and propose a new algorithm which is computationally efficient and also has a performance guarantee. The algorithm is based on the atomic decomposition of the matrix variable and extends the idea in the CoSaMP algorithm for ${\ell }_0$-norm minimization. Combined with the recent fast low rank approximation of matrices based on randomization, the proposed algorithm can efficiently handle large scale rank minimization problems.

submitted to ISIT 2009

References in corpus (6)

• Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
• A Singular Value Thresholding Algorithm for Matrix Completion
• Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization
• CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
• Subspace Pursuit for Compressive Sensing Signal Reconstruction
• ADMiRA: Atomic Decomposition for Minimum Rank Approximation

Cited by in corpus (1)

• ADMiRA: Atomic Decomposition for Minimum Rank Approximation