Guaranteed Matrix Completion via Non-Convex Factorization

Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization, even with huge size, can be solved very efficiently through the standard optimization algorithms in practice. However, due to the non-convexity caused by the fact...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 62; no. 11; pp. 6535 - 6579
Main Authors: Sun, Ruoyu, Luo, Zhi-Quan
Format: Journal Article
Language:English
Published: New York IEEE 01.11.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithm design and analysis
Algorithms
alternating minimization
Complexity theory
Economic conditions
Factorization
Geometry
Information theory
Iterative methods
Mathematical models
Matrix
Matrix completion
matrix factorization
Minimization
nonconvex optimization
Optimization
Partitioning algorithms
perturbation analysis
SGD
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization, even with huge size, can be solved very efficiently through the standard optimization algorithms in practice. However, due to the non-convexity caused by the factorization model, there is a limited theoretical understanding of whether these algorithms will generate a good solution. In this paper, we establish a theoretical guarantee for the factorization-based formulation to correctly recover the underlying low-rank matrix. In particular, we show that under similar conditions to those in previous works, many standard optimization algorithms converge to the global optima of a factorization-based formulation and recover the true low-rank matrix. We study the local geometry of a properly regularized objective and prove that any stationary point in a certain local region is globally optimal. A major difference of this paper from the existing results is that we do not need resampling (i.e., using independent samples at each iteration) in either the algorithm or its analysis.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2016.2598574