if $\mathbf x$ is $N\times 1$ sparse vector, and $\mathbf A$ is an $M\times N$ matrix with $M<<N$, and we measure $\mathbf y=\mathbf{Ax}$, then compressed sensing theory tells us that we can exactly recover $\mathbf x$ from $\mathbf y$ under some conditions for $\mathbf A$ like RIP.
But what if $\mathbf A$ is an $M\times N$ matrix with $M>>N$ or $M=N$. Now the equation is not under-determined. But the equation can not be solved directly because $\mathbf A$ is now ill-conditioned matrix. To approximate to solution we can apply regularization. Since $\mathbf x$ still sparse we apply $l_1$ reguarization scheme as $$ ||y-Ax||_2^2 + \lambda ||x||_1$$ and find a sparse solution for x where sparsity level is dependent on $\lambda$.
The questions are
1) Do we still need to consider RIP for $\mathbf A$? What effects the acuuracy of the solution of $l_1$ regularization scheme except $\lambda$?
2) Why do we need RIP in compressive sensing and does it have any relation with the values of $M$ and$N$?
3) If $x$ is not sparse does this regularization fail for sure? Or it can still be work like $l_2$ regularization scheme. Maybe better?