The DOI matches its Crossref record; 63 of the 63 references it cites were found in a registry or an index.
Guaranteed Matrix Completion via Non-Convex Factorization
Open accessAbstract
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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Cites
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A Singular Value Thresholding Algorithm for Matrix Completion
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Exact Matrix Completion via Convex Optimization
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The Power of Convex Relaxation: Near-Optimal Matrix Completion
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A Unified Convergence Analysis of Block Successive Minimization Methods for Nonsmooth Optimization
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Self-citationEfficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems
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Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems
Chen, Y., & Candès, E. J. (2017) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.21638 · Cited by 477
Non-convex Optimization for Machine Learning
Jain, P., & Kar, P. (2017) · Foundations and Trends® in Machine Learning · doi:10.1561/2200000058 · Cited by 364
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
Chi, Y., Lu, Y. M., & Chen, Y. (2019) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2019.2937282 · Cited by 360
Complete Dictionary Recovery Over the Sphere I: Overview and the Geometric Picture
Sun, J., Qu, Q., & Wright, J. (2017) · IEEE Transactions on Information Theory · doi:10.1109/tit.2016.2632162 · Cited by 213
Subspace Estimation from Unbalanced and Incomplete Data Matrices: 𝓁2, ∞ Statistical Guarantees
Cai, C., Li, G., Chi, Y., Poor, H. V., & Chen, Y. (2021) · The Annals of Statistics · doi:10.1214/20-aos1986 · Cited by 37
Metadata
- Authors
- Sun, R., & Luo, Z.-Q.
- Venue
- IEEE Transactions on Information Theory
- Published
- 2016-11
- Volume
- 62
- Issue
- 11
- Pages
- 6535-6579
- ISSN
- 0018-9448 (ISSN-L)
- 1557-9654
- OpenAlex
- W2566608758
- Cited by
- 309
- References
- 63
- FWCI
- 37.71
- Type
- journal article
- Language
- en
- Open access
- green
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