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Matrix Completion From a Few Entries
Abstract
Let M be an n¿ × n matrix of rank r, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm, which we call OptSpace, that reconstructs M from |E| = O(rn) observed entries with relative root mean square error 1/2 RMSE ¿ C(¿) (nr/|E|)1/2with probability larger than 1 - 1/n3. Further, if r = O(1) and M is sufficiently unstructured, then OptSpace reconstructs it exactly from |E| = O(n log n) entries with probability larger than 1 - 1/n3. 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
Matrix Completion Methods for Causal Panel Data Models
Athey, S., Bayati, M., Doudchenko, N., Imbens, G., & Khosravi, K. (2018) · doi:10.3386/w25132 · Cited by 497
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
- Keshavan, R. H., Montanari, A., & Oh, S.
- Venue
- IEEE Transactions on Information Theory
- Published
- 2010-06
- Volume
- 56
- Issue
- 6
- Pages
- 2980-2998
- ISSN
- 0018-9448 (ISSN-L)
- OpenAlex
- W2144730813
- Cited by
- 974
- References
- 0
- FWCI
- 38.85
- Type
- journal article
- Language
- en
- Open access
- closed
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