Sparse recovery guarantees in compressive sensing and related optimization problems often assume incoherence between the 'sensing' and 'sparsity' domains. In practice, incoherence is rarely satisfied due to physical constraints and limitations. Here we discuss the notion of local coherence, and show that by matching the sampling strategy to the local coherence at hand, sparse recovery guarantees extend to a rich new class of sensing problems beyond incoherent systems. We discuss particular applications to compressive MRI imaging and polynomial interpolation.
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