Basis pursuit is the mathematical optimization problem of the form:
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.
References & further reading
- Shaobing Chen, David Donoho: Basis Pursuit
- Terence Tao: Compressed Sensing. Mahler Lecture Series (slides)