Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minim...

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Vydáno v:IEEE transactions on signal processing Ročník 67; číslo 20; s. 5239 - 5269
Hlavní autoři: Yuejie Chi, Lu, Yue M., Yuxin Chen
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 15.10.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Algorithms
Complexity theory
Computational modeling
Estimation
Factorization
First-order methods
Iterative methods
landscape analysis
Machine learning
matrix factorization
nonconvex optimization
Optimization
Phase retrieval
Principal components analysis
Signal processing algorithms
Statistical models
statistics
ISSN:1053-587X, 1941-0476
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Shrnutí:Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, and robust principal component analysis. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.
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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2019.2937282