The goal of this short note is to present a refined analysis of the modified Basis Pursuit (-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 using an observation vector , where and in the highly underdetermined setting . Based on an initial and possibly erroneous guess of the signal's support , the Modified Basis Pursuit method of Vaswani and Lu consists of minimizing the norm of the estimate over the indices indexed by only. We prove exact recovery essentially under a Restricted Isometry Property assumption of order 2 times the cardinal of , i.e. the number of missed components.
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