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

Abstract

The goal of this short note is to present a refined analysis of the modified Basis Pursuit (1\ell_1-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 xRpx \in \mathbb R^p using an observation vector y=Axy=Ax, where ARn×pA \in \mathbb R^{n\times p} and in the highly underdetermined setting npn\ll p. Based on an initial and possibly erroneous guess TT of the signal's support supp(x){\rm supp}(x), the Modified Basis Pursuit method of Vaswani and Lu consists of minimizing the 1\ell_1 norm of the estimate over the indices indexed by TcT^c only. We prove exact recovery essentially under a Restricted Isometry Property assumption of order 2 times the cardinal of Tcsupp(x)T^c \cap {\rm supp}(x), i.e. the number of missed components.

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