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Basis pursuit is the mathematical optimization problem of the form:

\( \min_x \|x\|_1 \quad \mbox{subject to} \quad y = Ax. \)

where x is a N × 1 solution vector (signal), y is a M × 1 vector of observations (measurements), A is a M × N transform matrix (usually measurement matrix) and M < N.

It is usually applied in cases where there is an underdetermined system of linear equations y = Ax that must be exactly satisfied, and the sparsest solution in the L1 sense is desired.

When it is desirable to trade off exact congruence of Ax and y in exchange for a sparser x, basis pursuit denoising is preferred.