The Nullspace Property (NSP) is a concept in the field of convex optimization, particularly in relation to the formulation of certain convex problems, such as basis pursuit and sparse representation. It is closely associated with matrices and their structure in terms of representing linear systems.
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The null space property relative to is for every nonzero . It is equivalent to exact basis pursuit recovery of every vector supported in . Sufficiency follows from the triangle inequality; for necessity, compare and , which have the same image under . Recovery of a single fixed signed vector can hold without this uniform property.