# Basis pursuit

**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 *L*_{1} 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.

## See also

## References & further reading

## External links

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

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