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October 2010 Sparse recovery under matrix uncertainty
Mathieu Rosenbaum, Alexandre B. Tsybakov
Ann. Statist. 38(5): 2620-2651 (October 2010). DOI: 10.1214/10-AOS793


We consider the model

y =  + ξ,

Z = X + Ξ,

where the random vector y ∈ ℝn and the random n × p matrix Z are observed, the n × p matrix X is unknown, Ξ is an n × p random noise matrix, ξ ∈ ℝn is a noise independent of Ξ, and θ is a vector of unknown parameters to be estimated. The matrix uncertainty is in the fact that X is observed with additive error. For dimensions p that can be much larger than the sample size n, we consider the estimation of sparse vectors θ. Under matrix uncertainty, the Lasso and Dantzig selector turn out to be extremely unstable in recovering the sparsity pattern (i.e., of the set of nonzero components of θ), even if the noise level is very small. We suggest new estimators called matrix uncertainty selectors (or, shortly, the MU-selectors) which are close to θ in different norms and in the prediction risk if the restricted eigenvalue assumption on X is satisfied. We also show that under somewhat stronger assumptions, these estimators recover correctly the sparsity pattern.


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Mathieu Rosenbaum. Alexandre B. Tsybakov. "Sparse recovery under matrix uncertainty." Ann. Statist. 38 (5) 2620 - 2651, October 2010.


Published: October 2010
First available in Project Euclid: 11 July 2010

zbMATH: 1373.62357
MathSciNet: MR2722451
Digital Object Identifier: 10.1214/10-AOS793

Primary: 62J05
Secondary: 62F12

Keywords: Errors-in-variables model , matrix uncertainty , measurement error , missing data , MU-selector , Oracle inequalities , portfolio replication , portfolio selection , restricted eigenvalue assumption , sign consistency , Sparsity

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 5 • October 2010
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