A Short Note on Compressed Sensing with Partially Known Signal Support

TL;DR

This paper explores a variation of compressed sensing where partial support knowledge is used to improve sparse signal recovery, extending previous results to noisy and compressible signals with practical iterative algorithms.

Contribution

It introduces an extension of the iBPDN method for partial support knowledge, including stability analysis for noisy and compressible signals, and proposes an iterative numerical solution.

Findings

Reconstruction stability depends on the sensing matrix satisfying RIP.

The method extends to noisy and compressible signals.

An iterative algorithm based on proximal methods is proposed.

Abstract

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.