  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. The reconstruction method is based on a convex minimization program
coined "innovative Basis Pursuit DeNoise" (or iBPDN). Under the common
$\ell_2$-fidelity constraint made on the available measurements, this
optimization promotes the ($\ell_1$) 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. Our
proof relies on a small adaption of the results of Candes in 2008 for
characterizing the stability of the Basis Pursuit DeNoise (BPDN) program. We
emphasize also an interesting link between our method and the recent work of
Davenport et al. on the $\delta$-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. We conclude by
sketching an easy numerical method relying on monotone operator splitting and
proximal methods that iteratively solves iBPDN.
