Basis pursuit ($\min \{\Vert x{\Vert }_1:Ax=y\}$)

Basis pursuit minimizes an ${\ell }^1$ norm subject to exact linear measurements. Basis pursuit with a noise tolerance replaces the equality by $\Vert Ax-y{\Vert }_2\le \eta$. A unique minimizer has linearly independent active columns, and hence at most $\mathrm{rank}A$ nonzero coordinates: a null direction on its support would make the objective locally affine on a feasible line, contradicting minimality or uniqueness.

The null space property relative to $S$ is $\Vert h_S{\Vert }_1<\Vert h_{S^c}{\Vert }_1$ for every nonzero $h\in \ker A$. It is equivalent to exact basis pursuit recovery of every vector supported in $S$. Sufficiency follows from the triangle inequality; for necessity, compare $-h_S$ and $h_{S^c}$, which have the same image under $A$. Recovery of a single fixed signed vector can hold without this uniform property.

The displayed property holds for every $h$ and every $|S|\le s$, with $0\le \rho <1$ and $\tau >0$. It converts the cone inequality from basis pursuit minimality and the tube inequality from noisy feasibility into a two-constant bound $\Vert \hat{x}-x_0{\Vert }_2\le C_1\eta +C_2{\sigma }_s(x_0)/\sqrt{s}$. Both constants are needed in general; a fixed coefficient one on the approximation term does not follow.

For a real vector $x_0$ with support $S$, uniqueness in basis pursuit is equivalent to $|⟨\mathrm{sgn}(x_{0,S}),h_S⟩|<\Vert h_{S^c}{\Vert }_1$ for every nonzero $h\in \ker A$. The supporting-line inequality for the absolute value proves sufficiency. Taking small positive and negative multiples of $h$ before any active sign changes proves necessity.