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Accuracy guarantees for ℓ1-recovery

Anatoli Juditsky (juditsky***at***imag.fr)
Arkadi Nemirovski (nemirovs***at***isye.gatech.edu)

Abstract: We discuss two new methods of recovery of sparse signals from noisy observation based on ℓ1- minimization. They are closely related to the well-known techniques such as Lasso and Dantzig Selector. However, these estimators come with efficiently verifiable guaranties of performance. By optimizing these bounds with respect to the method parameters we are able to construct the estimators which possess better statistical properties than the commonly used ones. We also show how these techniques allow to provide efficiently computable accuracy bounds for Lasso and Dantzig Selector. We link our performance estimations to the well known results of Compressive Sensing and justify our proposed approach with an oracle inequality which links the properties of the recovery algorithms and the best estimation performance when the signal support is known. We also show how the estimates can be computed using the Non-Euclidean Basis Pursuit algorithm.

Keywords: sparse recovery, linear estimation, oracle inequalities, nonparametric estimation by convex optimization

Category 1: Applications -- Science and Engineering (Statistics )


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Entry Submitted: 10/28/2010
Entry Accepted: 10/28/2010
Entry Last Modified: 11/05/2010

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