Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
Foundations of Computational Mathematics
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Modified-CS: modifying compressive sensing for problems with partially known support
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 1
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
On the Performance of Compressed Sensing with Partially Correct Support
Wireless Personal Communications: An International Journal
Performance analysis of partial segmented compressed sampling
Signal Processing
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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 iBPDN). Under the common @?"2-fidelity constraint made on the available measurements, this optimization promotes the (@?"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 by showing that iBPDN is @?"2-@?"1 instance optimal. The corresponding 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 also emphasize an interesting link between our method and the recent work of Davenport et al. on the @d-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.