A multi-stage convex relaxation approach to noisy structured low-rank matrix recovery

  • This paper concerns with a noisy structured low-rank matrix recovery problem which can be modeled as a structured rank minimization problem. We reformulate this problem as a mathematical program with a generalized complementarity constraint (MPGCC), and show that its penalty version, yielded by moving the generalized complementarity constraint to the objective, has the same global optimal solution set as the MPGCC does whenever the penalty parameter is over a certain threshold. Then, by solving the exact penalty problem in an alternating way, we obtain a multi-stage convex relaxation approach. We provide theoretical guarantees for our approach under a mild restricted eigenvalue condition, by quantifying the reduction of the error and approximate rank bounds of the first stage convex relaxation in the subsequent stages and establishing the geometric convergence of the error sequence in a statistical sense. Numerical experiments are conducted for some structured low-rank matrix recovery examples to confirm our theoretical findings. Our code can be achieved from https://doi.org/10.5281/zenodo.3600639.

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Metadaten
Author:Shujun Bi, Shaohua Pan, Defeng Sun
DOI:https://doi.org/10.1007/s12532-020-00177-4
ISSN:1867-2949
Parent Title (English):Mathematical Programming Computation
Publisher:Springer Science and Business Media LLC
Document Type:Article
Language:English
Year of Completion:2020
Tag:Software; Theoretical Computer Science
Volume:12
Issue:4
Page Number:34
First Page:569
Last Page:602
Mathematical Programming Computation :MPC 2020 - Issue 4
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