Note on sparsity in signal recovery and in matrix identification

The standard approach to recover a signal with a sparse representation in D is to design a measurement matrix ΦD ∈ Cm×n which allows the recovery of any f ∈ Σk from the measurement vector ΦDf whenever k is sufficiently small. While a minimal requirement for the recovery of any f ∈ Σk is that the constructed map ΦD : Σ D k −→ C, f 7→ ΦDf is injective, the recent literature on sparse signal recovery focuses mainly on numerically robust recovery methods whose application require that ΦD is not only injective but well conditioned on its domain. Note that the Euclidean basis E = {ej} is commonly used as a dictionary. Then N = n and f ∈ Σk if f has at most k nonzero entries.