A short note on compressed sensing with partially known signal support

This short note studies a variation of the compressed sensing paradigm introduced recently by Vaswani et al., i.e., the recovery of sparse signals from a certain number of linear measurements when the signal support is partially known. In this framework, we propose a reconstruction method based on a convex minimization program coined innovative Basis Pursuit DeNoise (or i  BPDN). Under the common ℓ2-fidelityℓ2-fidelity constraint made on the available measurements, this optimization promotes the (ℓ1)(ℓ1) sparsity of the candidate signal over the complement of this known part. In particular, this paper extends the results of Vaswani et al. to the cases of compressible signals and noisy measurements by showing that i  BPDN is ℓ2−ℓ1ℓ2−ℓ1 instance optimal. The corresponding proof relies on a small adaption of the results of Candès in 2008 for characterizing the stability of the Basis Pursuit DeNoise (BPDN) program.We also emphasize an interesting link between our method and the recent work of Davenport et al. on the δ-stableδ-stable embeddings and the cancel-then-recover strategy applied to our problem. For both approaches, reconstructions are indeed stabilized when the sensing matrix respects the Restricted Isometry Property for the same sparsity order.