Basis pursuit is the mathematical optimization problem of the form where x is a N-dimensional solution vector (signal), y is a M-dimensional vector of observations (measurements), A is a M × N transform matrix (usually measurement matrix) and M < N. Basis pursuit is NP-hard.
[1] 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 problems can be converted to linear programming problems in polynomial time and vice versa, making the two types of problems polynomially equivalent.
[2] A basis pursuit problem can be converted to a linear programming problem by first noting that where
,
x
i
This construction is derived from the constraint
, where the value of
is intended to be stored in
depending on whether
is greater or less than zero, respectively.
Although a range of
values can potentially satisfy this constraint, solvers using the simplex algorithm will find solutions where one or both of
is zero, resulting in the relation
From this expansion, the problem can be recast in canonical form as:[2]