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Rank penalized estimators for high-dimensional matrices

Klopp, Olga (2011), Rank penalized estimators for high-dimensional matrices, Electronic Journal of Statistics, 5, p. 1161-1183. http://dx.doi.org/10.1214/11-EJS637

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00583884/fr/
Date
2011
Journal name
Electronic Journal of Statistics
Volume
5
Publisher
Institute of Mathematical Statistics
Pages
1161-1183
Publication identifier
http://dx.doi.org/10.1214/11-EJS637
Metadata
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Author(s)
Klopp, Olga
Abstract (EN)
In this paper we consider the trace regression model. Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix $A_0$ corrupted by noise. We propose a new rank penalized estimator of $A_0$. For this estimator we establish general oracle inequality for the prediction error both in probability and in expectation. We also prove upper bounds for the rank of our estimator. Then we apply our general results to the problem of matrix completion when our estimator has a particularly simple form: it is obtained by hard thresholding of the singular values of a matrix constructed from the observations.
Subjects / Keywords
low rank matrix estimation; matrix completion; recovery of the rank; statistical learning

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