An equivalence between critical points for rank constraints versus low-rank factorizations

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-constraine...

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Bibliographic Details
Published in:arXiv.org 2020-12
Main Authors: Ha, Wooseok, Liu, Haoyang, Rina Foygel Barber
Format: Article
Language:English
Subjects:
Constraints
Critical point
Fixed points (mathematics)
Inverse problems
Optimization
Online Access:Get full text
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Summary: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.
ISSN:2331-8422