Note on sparse signal recovery

Definition 1 (Restricted isometry and orthogonality). The S-restricted isometry constant δS of a matrix F ∈ Rn×m is the smallest quantity such that (1− δS)‖x‖2 ≤ ‖FTx‖2 ≤ (1 + δS)‖x‖2 for all T ⊆ [m] with |T | ≤ S and all x ∈ R|T |. The (S, S′)-restricted orthogonality constant θS,S′ of F is the smallest quantity such that |FTx · FT ′x′| ≤ θS,S′‖c‖‖c‖ for all disjoint T, T ′ ⊆ [m] with |T | ≤ S and |T ′| ≤ S′, and all x ∈ R|T | and x ∈ R|T ′|. Definition 2 (Exact Reconstruction Principle). The matrix F ∈ Rn×m obeys the (deterministic) S-Exact Reconstruction Principle (S-ERP) if, for any h ∈ R with support on some J with |J | ≤ S, there exists w ∈ R satisfying w · vi = sgn(hi) ∀i ∈ J |w · vi| < 1 ∀i 6∈ J. Lemma 1. For all S, S′, we have θS,S′ ≤ δS+S′ ≤ θS,S′ +max(δS , δS′).