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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| Published in: | IEEE transactions on information theory Vol. 62; no. 11; pp. 6535 - 6579 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
IEEE
01.11.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
| Online Access: | Get full text |
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| Abstract | 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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| AbstractList | 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. |
| Author | Zhi-Quan Luo Ruoyu Sun |
| Author_xml | – sequence: 1 givenname: Ruoyu orcidid: 0000-0003-2487-5322 surname: Sun fullname: Sun, Ruoyu – sequence: 2 givenname: Zhi-Quan surname: Luo fullname: Luo, Zhi-Quan |
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| SubjectTerms | 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 |
| Title | Guaranteed Matrix Completion via Non-Convex Factorization |
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