Abstract
Matrix factorization is a popular approach for large-scale matrix completion. In this approach, the unknown low-rank matrix is expressed as the product of two much smaller matrices so that the low-rank property is automatically fulfilled. The resulting optimization problem, even with huge size, can be solved (to stationary points) very efficiently through standard optimization algorithms such as alternating minimization and stochastic gradient descent (SGD). 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 the factorization based formulation, and recover the true low-rank matrix. A major difference of our work 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. To the best of our knowledge, our result is the first one that provides exact recovery guarantee for many standard algorithms such as gradient descent, SGD and block coordinate gradient descent.
Original language | English (US) |
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Title of host publication | Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015 |
Publisher | IEEE Computer Society |
Pages | 270-289 |
Number of pages | 20 |
ISBN (Electronic) | 9781467381918 |
DOIs | |
State | Published - Dec 11 2015 |
Event | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States Duration: Oct 17 2015 → Oct 20 2015 |
Publication series
Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2015-December |
ISSN (Print) | 0272-5428 |
Conference
Conference | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 |
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Country/Territory | United States |
City | Berkeley |
Period | 10/17/15 → 10/20/15 |
Bibliographical note
Publisher Copyright:© 2015 IEEE.
Keywords
- matrix completion
- matrix factorization
- nonconvex optimization
- perturbation analysis