Significance of $ \lambda $ in Basis Pursuit
There are 2 forms of the Basis Pursuit problem:

$$\begin{align*}
\text{The $ \lambda $ Form:} &amp; \quad &amp;&amp; \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1} \\
\text{The $ \epsilon$ Form:} &amp;&amp; \quad &amp; \arg \min_{x} {\left\| x \right\|}_{1} \\
&amp; \text{subject to} &amp;&amp;  \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} \leq \epsilon
\end{align*}$$

Sometimes it is more intuitive to have a look on the $ \epsilon $ form as it tell us exactly the trade off.
The lower $ \epsilon $ it means we require teh solution to hold the Linear Equation and we pay for that with the $ {L}_{1} $ Norm of the solution.

The nice things is that for any value $ \epsilon $ we can find $ \lambda $ for the other form in such way the generate the same solution:

$$ \forall \epsilon, \; \exists \lambda : x \left( \epsilon \right) = x \left
( \lambda \right) $$

Where $ x \left( \epsilon \right) $ and $ x \left ( \lambda \right) $ are the solution of the $ \epsilon $ form and the $ \lambda $ form respectively.

You may see an example in my answer at Cross Validated Question 291962.