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...x‖ 2 2 ≤ ‖Ax‖ 2 2 ≤ (1 + δs)‖x‖ 2 2 for all x ∈ C N , ‖x‖0 ≤ s. If δ2s < 0.46 then ℓ1-minimization reconstructs all s-sparse vectors x exactly from y = Ax, and compressible vectors approximately, see =-=[1, 4]-=- for precise statements. 1It is an open problem to construct deterministic (explicit) measurement matrices that have small restricted isometry constants for small m (or large s, respectively). So far...