Regularization Parameter Selection for the Low Rank Matrix Recovery

• Published in: Journal of optimization theory and applications Vol. 189; no. 3; pp. 772 - 792
• Main Authors: Shang, Pan, Kong, Lingchen
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computing time; Engineering; Heuristic; Machine learning; Mathematics; Mathematics and Statistics; Noise; Operations Research/Decision Theory; Optimization; Parameters; Recovery; Regularization; Signal processing; Theory of Computation; Upper bounds; 65K05; 90C46; 90C25; Regularization parameter selection rule; Nuclear norm regularized minimization; 90C06; Duality theory; Low rank matrix recovery; 62J99
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-021-01852-9

A popular approach to recover low rank matrices is the nuclear norm regularized minimization (NRM) for which the selection of the regularization parameter is inevitable. In this paper, we build up a novel rule to choose the regularization parameter for NRM, with the help of the duality theory. Our result provides a safe set for the regularization parameter when the rank of the solution has an upper bound. Furthermore, we apply this idea to NRM with quadratic and Huber functions, and establish simple formulae for the regularization parameters. Finally, we report numerical results on some signal shapes by embedding our rule into the cross validation, which state that our rule can reduce the computational time for the selection of the regularization parameter. To the best of our knowledge, this is the first attempt to select the regularization parameter for the low rank matrix recovery.