This question has been bothering me for some time now. Please correct me if I've got anything wrong.
In compressed sensing, we try to find a unique k-sparse solution to an underdetermined system by using an RIP-satisfying measurement matrix $\Phi$ and $l_{1}$ minimization.
But what if the system doesn't have a unique k-sparse solution? I'm sure everybody familiar with this has seen the figure with the $l_{1}$ and $l_{2}$ ball showing why $l_{1}$ recovery works. Even in the $\mathbb{R}^{2}$ case, there are two 1-sparse solutions: at the point of intersection of $\Phi x=u$ with the axes. In general, $\Phi x=u$ will always be this hyperplane that intersects the coordinate axes at some point. So there are always multiple 1-sparse solutions. How is it that all these 1-sparse points give the same measurements? What difference would an RIP-satisfying $\Phi$ make in this picture? And why do we want the $l_{1}$ minimization to give us a unique k-sparse solution when the original problem does not have one?