Abstract
The task of matrix completion involves estimating the entries of a matrix, M ∈ ℝm×n, when a subset, Ω ⊂ {(i,j):1 ≤ i ≤ m,1 ≤ j ≤ n} of the entries are observed. A particular set of low rank models for this task approximate the matrix as a product of two low rank matrices, \(\widehat{M}=UV^T\), where U ∈ ℝm×k and V ∈ ℝn×k and k ≪ min {m,n}. A popular algorithm of choice in practice for recovering M from the partially observed matrix using the low rank assumption is alternating least square (ALS) minimization, which involves optimizing over U and V in an alternating manner to minimize the squared error over observed entries while keeping the other factor fixed. Despite being widely experimented in practice, only recently were theoretical guarantees established bounding the error of the matrix estimated from ALS to that of the original matrix M. In this work we extend the results for a noiseless setting and provide the first guarantees for recovery under noise for alternating minimization. We specifically show that for well conditioned matrices corrupted by random noise of bounded Frobenius norm, if the number of observed entries is \(\mathcal{O}\left(k^7n\log n\right)\), then the ALS algorithm recovers the original matrix within an error bound that depends on the norm of the noise matrix. The sample complexity is the same as derived in [7] for the noise–free matrix completion using ALS.
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References
Bertalmio, M., Vese, L., Sapiro, G., Osher, S.: Simultaneous structure and texture image inpainting. IEEE Transactions on Image Processing (2003)
Candès, E.J., Plan, Y.: Matrix completion with noise. CoRR (2009)
Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Foundations of Computational Mathematics (2009)
Candès, E.J., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Transactions on Information Theory (2010)
Golub, G.H., van Van Loan, C.F.: Matrix Computations (Johns Hopkins Studies in Mathematical Sciences), 3rd edn. The Johns Hopkins University Press (1996)
Jain, P., Netrapalli, P., Sanghavi, S.: Low-rank matrix completion using alternating minimization. ArXiv e-prints (December 2012)
Jain, P., Netrapalli, P., Sanghavi, S.: Low-rank matrix completion using alternating minimization. In: STOC (2013)
Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from a few entries. IEEE Transactions on Information Theory (2010)
Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from noisy entries. JMLR (2010)
Koren, Y., Bell, R., Volinsky, C.: Matrix factorization techniques for recommender systems. IEEE Computer (2009)
Mitra, K., Sheorey, S., Chellappa, R.: Large-scale matrix factorization with missing data under additional constraints. In: NIPS (2010)
So, A.M.C., Ye, Y.: Theory of semidefinite programming for sensor network localization. In: ACM-SIAM Symposium on Discrete Algorithms (2005)
Takács, G., Pilászy, I., Németh, B., Tikk, D.: Investigation of various matrix factorization methods for large recommender systems. In: KDD Workshop on Large-Scale Recommender Systems and the Netflix Prize Competition (2008)
Wang, Y., Xu, H.: Stability of matrix factorization for collaborative filtering. In: ICML (2012)
Yu, K., Tresp, V.: Learning to learn and collaborative filtering. In: NIPS Workshop on Inductive Transfer: 10 Years Later (2005)
Zhou, Y., Wilkinson, D., Schreiber, R., Pan, R.: Large-scale parallel collaborative filtering for the netflix prize. In: Fleischer, R., Xu, J. (eds.) AAIM 2008. LNCS, vol. 5034, pp. 337–348. Springer, Heidelberg (2008)
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Gunasekar, S., Acharya, A., Gaur, N., Ghosh, J. (2013). Noisy Matrix Completion Using Alternating Minimization. In: Blockeel, H., Kersting, K., Nijssen, S., Železný, F. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2013. Lecture Notes in Computer Science(), vol 8189. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40991-2_13
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DOI: https://doi.org/10.1007/978-3-642-40991-2_13
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