Basis Pursuit is an optimization technique used in the field of signal processing and compressed sensing, primarily for recovering sparse signals from limited or incomplete measurements. The fundamental idea behind Basis Pursuit is to express a signal as a linear combination of basis functions and to find the representation that uses the fewest non-zero coefficients, thereby focusing on the sparsest solution.
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Basis pursuit minimizes an norm subject to exact linear measurements. Basis pursuit with a noise tolerance replaces the equality by . A unique minimizer has linearly independent active columns, and hence at most nonzero coordinates: a null direction on its support would make the objective locally affine on a feasible line, contradicting minimality or uniqueness.