Small coherence implies the weak Null Space Property
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In the Compressed Sensing community, it is well known that given a matrix X ∈ R^n× p with ℓ_2 normalized columns, the Restricted Isometry Property (RIP) implies the Null Space Property (NSP). It is also well known that a small Coherence μ implies a weak RIP, i.e. the singular values of X_T lie between 1-δ and 1+δ for "most" index subsets T ⊂{1,...,p} with size governed by μ and δ. In this short note, we show that a small Coherence implies a weak Null Space Property, i.e. h_T_2 < C h_T^c_1/√(s) for most T ⊂{1,...,p} with cardinality |T|< s. We moreover prove some singular value perturbation bounds that may also prove useful for other applications.
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