Abstract
AbstractThis chapter exclusively considers the recovery of sparse vectors via ℓ 1-minimization, also known as basis pursuit. The idealized situation is investigated first, and the success of the recovery of all sparse vectors is shown to be equivalent to the null space property of the measurement matrix. In the realistic situation where sparse vectors are replaced by compressible vectors and where measurement errors occur, the analysis is extended by way of the stable and robust null space properties. Several conditions for the success of the recovery of an individual vector are given next. This is then interpreted geometrically as the survival of a face of the cross-polytope under projection. The chapter ends by pointing out the analogy with the recovery of low-rank matrices.Keywords ℓ 1-norm ℓ 1-minimizationnull space propertystabilityrobustnessnonconvex minimizationindividual recoverydual certificatetangent coneprojected cross-polytopenuclear norm minimization
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