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Matrix completion from a few entries
Abstract
Let M be an n¿ × n matrix of rank r ¿ n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(r n) observed entries with relative root mean square error RMSE ¿ C(¿) (nr/|E|)1/2. Further, if r = O(1) and M is sufficiently unstructured, then it can be reconstructed exactly from |E| = O(n log n) entries. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|r log n), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices.
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Cited by
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Ma, S., Goldfarb, D., & Chen, L. (2011) · Mathematical Programming · doi:10.1007/s10107-009-0306-5 · Cited by 1,048
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Chatterjee, S. (2015) · The Annals of Statistics · doi:10.1214/14-aos1272 · Cited by 465
Statistical guarantees for the EM algorithm: From population to sample-based analysis
Balakrishnan, S., Wainwright, M. J., & Yu, B. (2017) · The Annals of Statistics · doi:10.1214/16-aos1435 · Cited by 384
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
Metadata
- Authors
- Keshavan, R. H., Oh, S., & Montanari, A.
- Venue
- 2009 IEEE International Symposium on Information Theory
- Publisher
- IEEE
- Published
- 2009-06
- Pages
- 324-328
- OpenAlex
- W2000157792
- Cited by
- 650
- References
- 0
- FWCI
- 81.70
- Type
- conference paper
- Language
- en
- Open access
- closed
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- Keshavan, R. H., Oh, S., & Montanari, A. (2009). Matrix completion from a few entries. 2009 IEEE International Symposium on Information Theory. https://doi.org/10.1109/isit.2009.5205567
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