Basis pursuit

Basis pursuit is the mathematical optimization problem of the form

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

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.

Basis pursuit is equivalent to linear programming.[1]

See also

• Compressed sensing
• Group testing
• Lasso (statistics)
• Matching pursuit
• Sparse approximation
• Basis pursuit denoising

Notes

1. A. M. Tillmann Equivalence of Linear Programming and Basis Pursuit, PAMM (Proceedings in Applied Mathematics and Mechanics) Volume 15, 2015, pp. 735-738, DOI: 10.1002/PAMM.201510351

References & further reading

• Stephen Boyd, Lieven Vandenbergh: Convex Optimization, Cambridge University Press, 2004, ISBN 9780521833783, pp. 337–337
• Simon Foucart, Holger Rauhut: A Mathematical Introduction to Compressive Sensing. Springer, 2013, ISBN 9780817649487, pp. 77–110

External links

• Shaobing Chen, David Donoho: Basis Pursuit
• Terence Tao: Compressed Sensing. Mahler Lecture Series (slides)