Journal
COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
Volume 81, Issue 1, Pages 67-90Publisher
SPRINGER
DOI: 10.1007/s10589-021-00328-w
Keywords
Rank-adaptive; Fixed-rank manifold; Bounded-rank matrices; Riemannian optimization; Low-rank; Matrix completion
Funding
- Fonds de la Recherche Scientifique - FNRS
- Fonds Wetenschappelijk Onderzoek - Vlaanderen under EOS [30468160]
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In this paper, a Riemannian rank-adaptive method is proposed to address the low-rank matrix completion problem on a set of bounded-rank matrices. Numerical experiments demonstrate its superior performance compared to state-of-the-art algorithms, and show that each aspect of this rank-adaptive framework can be separately incorporated into existing algorithms for performance improvement.
The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.
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