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Rank penalized estimators for high-dimensional matrices

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Date
2011
Link to item file
http://hal.archives-ouvertes.fr/hal-00583884/fr/
Dewey
Probabilités et mathématiques appliquées
Sujet
low rank matrix estimation; matrix completion; recovery of the rank; statistical learning
Journal issue
Electronic Journal of Statistics
Volume
5
Publication date
2011
Article pages
1161-1183
Publisher
Institute of Mathematical Statistics
DOI
http://dx.doi.org/10.1214/11-EJS637
URI
https://basepub.dauphine.fr/handle/123456789/5982
Collections
  • CEREMADE : Publications
Metadata
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Author
Klopp, Olga
Type
Article accepté pour publication ou publié
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.

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