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On the modified Basis Pursuit reconstruction for Compressed Sensing with partially known support

2 June 2009
Stéphane Chrétien
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Abstract

The goal of this short note is to present a refined analysis of the modified Basis Pursuit (ℓ1\ell_1ℓ1​-minimization) approach to signal recovery in Compressed Sensing with partially known support, as introduced by Vaswani and Lu. The problem is to recover a signal x∈Rpx \in \mathbb R^px∈Rp using an observation vector y=Axy=Axy=Ax, where A∈Rn×pA \in \mathbb R^{n\times p}A∈Rn×p and in the highly underdetermined setting n≪pn\ll pn≪p. Based on an initial and possibly erroneous guess TTT of the signal's support supp(x){\rm supp}(x)supp(x), the Modified Basis Pursuit method of Vaswani and Lu consists of minimizing the ℓ1\ell_1ℓ1​ norm of the estimate over the indices indexed by TcT^cTc only. We prove exact recovery essentially under a Restricted Isometry Property assumption of order 2 times the cardinal of Tc∩supp(x)T^c \cap {\rm supp}(x)Tc∩supp(x), i.e. the number of missed components.

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