rank deficient least squares with minimum $\ell_1$ norm When a least square solution of a rank deficient least square problem is sought, there are multitude of solutions that give the same minimal residual vector $r=\mathbf{A}x-b$. I am familiar with Moore-Penrose pseudo-inverse that picks minimum $\ell_2$-norm least square solution among these solutions  i.e. $$ x_{LS} = \mathbf{A}^{\dagger}b$$
Computing this usually involves getting the SVD of A and screening all singular vectors associated with zero singular values. Is there a similar expression that gives the minimum $\ell_1$-norm? Even if no expression can be given, is there a algorithm to compute the minimum $\ell_1$-norm among all solutions that give the minimum residual vector? This seems to be a convex optimization problem but I could be mistaken.
If it helps I am interested in a problem with column-rank deficient $\mathbf{A}$ and it has more rows than columns. 
Update: A related problem where the residual norm is known can be solved via LASSO (using the homotopy approach to find the regularization parameter that matches the residual norm). LASSO indeed seems to give the minimum $\ell_1$ norm solution for the given residual norm. In this problem the residual vector is given and a minimum $\ell_1$ norm solution is sought.
Edit: After the comment below by @littleO I realized the mistake I was making that motivated the original question. I was planning to project the $b$ vector on to null space of $\mathbf{A}$ and call it the noise or residual vector but there could be a component of the noise vector in the range of $\mathbf{A}$ as well. So if I may change the question, how does one find out or estimate the norm of the noise?

First solve the unconstrained least squares problem, for example by using the Moore-Penrose pseudo-inverse (which picks the minimum $\ell_2$-norm least square solution), or any other method. Evaluating the two-norm of the residual at that optimal value of $x$ provides the $\epsilon$.  Then solve the $\ell_1$-norm minimization problem subject to the  $\ell_2$-norm constraint.
Using the Moore-Penrose pseudo-inverse least squares solution, that would be 
Minimize $\|x\|_1$ subject to $\|\mathbf{A}x - b\|_2 \le \|\mathbf{A}\mathbf{A}^{\dagger}b - b\|_2$
The CVX code for this convex optimization problem is:
cvx_begin
variable x(n)
minimize(norm(x,1))
norm(A*x-b) <= norm(A*pinv(A)*b-b)
cvx_end

