The problem of finding 
$$ \substack{{\rm min}\\x}\left( \|Ax-b\|^2\_2+\lambda \|x\|\_1\right),$$
where $\|\cdot\|\_2$ and $\|\cdot\|\_1$ are the $L\_2$ and $L\_1$ norms, respectively, is usually called the LASSO. $A$ is a matrix, $x$ and $b$ are vectors.

I believe this problem is robust against introducing some redundancy in the matrix $A$, by introducing for example in $A$ an extra column which is a linear combination of the original columns. Since the solution to LASSO is sparse, I expect this procedure does not change it much, i.e. the solution to
$$\substack{{\rm min}\\y}\left( \|\tilde{A}y-b\|^2\_2+\lambda \|y\|\_1\right),$$
should be very close to the original one.

Can this be proved? Can I find an upper bound to how much the solution changes?