Best Metric to Compare Sparsity of Vectors I solved the Basis Pursuit Denoising Problem looking for a sparse solution (I am in compressive sensing):
$$ {x}^{\ast} = \arg \min_{x} \left\{ \frac{1}{2} {\left\| A x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1} \right\} $$

for $ 100 $ different $ \lambda $ values with ADMM.  

Now I have $100$ different $x^*$ and want to find "the best one" out of them. Moreover, I solved several optimization problems for CS under noise and noisefree cases (e.g. also Lasso, etc.)

What is the best measure to determine sparsity? Found the paper Comparing Measures of Sparsity by Niall Hurley and Scott Rickard, saying it's the Gini index. Do you agree? In my opinion sparsity in this case is not all, isn't it? It's also about solving the equation as good as possible, e.g. having a small norm on:  

$$ {\left\| A {x}^{\ast} - y \right\|}_{1} $$

What do you recommend?
I am sorry I cannot comment your answer due to my low reputation. Gini and your suggested sparsity ratio ($l_1(x)/l_2(x)$) both give me the same value for $\lambda$. But 


The problem I still see is that I cannot take into account how well the vector is solving the equation $Ax-y$. I would like to combine the residuum $l_1(A\hat{x}-y)$ and the sparsity $l_1(\hat{x})/l_2(\hat{x})$ into a new metric value. Do you know something from the literature?
Gini only works with positive values, but my measurement vector has lots of negative values. In my eyes, this is a heavy disadvantage because many sparse vectors can have negative values.