Mathematics > Optimization and Control
[Submitted on 2 Dec 2018 (v1), last revised 16 Dec 2020 (this version, v2)]
Title:An equivalence between critical points for rank constraints versus low-rank factorizations
View PDFAbstract:Two common approaches in low-rank optimization problems are either working directly with a rank constraint on the matrix variable, or optimizing over a low-rank factorization so that the rank constraint is implicitly ensured. In this paper, we study the natural connection between the rank-constrained and factorized approaches. We show that all second-order stationary points of the factorized objective function correspond to fixed points of projected gradient descent run on the original problem (where the projection step enforces the rank constraint). This result allows us to unify many existing optimization guarantees that have been proved specifically in either the rank-constrained or the factorized setting, and leads to new results for certain settings of the problem. We demonstrate application of our results to several concrete low-rank optimization problems arising in matrix inverse problems.
Submission history
From: Wooseok Ha [view email][v1] Sun, 2 Dec 2018 15:24:57 UTC (20 KB)
[v2] Wed, 16 Dec 2020 16:36:28 UTC (32 KB)
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