# Basis pursuit

Basis pursuit is the mathematical optimization problem of the form:

${\displaystyle \min _{x}\|x\|_{1}\quad {\mbox{subject to}}\quad y=Ax.}$

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 equality of Ax and y in exchange for a sparser x, basis pursuit denoising is preferred.