Basis pursuit

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.

See also

• Compressed sensing
• Group testing
• Matching pursuit
• Sparse approximation