Solving LASSO (Basis Pursuit Denoising Form) with LARS
There are 2 forms of the Basis Pursuit problem:

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

Real world model is:

$$ A x = y $$

Where $ x $ is a sparse vector.
Yet, in reality we don't have measurements of $ y $ but $ b = y + v $ where $ v $ is a vector with the properties of our measurement method.  

Hence we allow the model not to have strict equality which implies:

$$ {\left\| A x - b \right\|}_{2}^{2} \leq \epsilon = {\left\| v \right\|}_{2}^{2} $$

Now, the different models are equivalent as for any $ \epsilon $ there is a $ \lambda $ (Which depends on $ A $ and $ b $ unfortunately) which the models ( (1) and (2) ) are equivalent.

For instance I created simple simulation on for that simulation:



The full code is available on my StackExchange Cross Validated Q291962 GitHub Repository.

Since you linked to the paper you're reading - Sparse Geometric Representation Through Local Shape Probing the problem solved there is the $ \lambda $ form.
Since it is a strict Convex Problem there is a single solution. LARS isn't considered a very efficient or fast method to solve the problem, hence I recommend using a different solution.

I have a project with many solvers of the $ {L}_{1} $ Regularized Least Squares. I suggest you just take the Proximal Gradient Method - SolveLsL1Prox.m or the Accelerated Proximal Gradient Descent Method - SolveLsL1ProxAccel.m.

So you can just pick any of those MATLAB codes and solve the problem.