On the modified Basis Pursuit reconstruction for Compressed Sensing with partially known support

TL;DR

This paper provides a refined theoretical analysis of the modified Basis Pursuit method for compressed sensing signal recovery when partial support information is available, establishing conditions for exact recovery under RIP assumptions.

Contribution

It offers a new analysis of the modified Basis Pursuit approach, clarifying the conditions for exact recovery with partially known support in compressed sensing.

Findings

Exact recovery guaranteed under RIP of order 2|T^c ∩ supp(x)|

Refined theoretical bounds for modified Basis Pursuit

Supports improved understanding of support estimation in compressed sensing

Abstract

The goal of this short note is to present a refined analysis of the modified Basis Pursuit ($\ell_1$-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 $x \in \mathbb R^p$ using an observation vector $y=Ax$, where $A \in \mathbb R^{n\times p}$ and in the highly underdetermined setting $n\ll p$. Based on an initial and possibly erroneous guess $T$ of the signal's support ${\rm supp}(x)$, the Modified Basis Pursuit method of Vaswani and Lu consists of minimizing the $\ell_1$ norm of the estimate over the indices indexed by $T^c$ only. We prove exact recovery essentially under a Restricted Isometry Property assumption of order 2 times the cardinal of $T^c \cap {\rm supp}(x)$, i.e. the number of missed components.