  We consider the problem of sparse signal recovery from noisy measurements.
Many of frequently used recovery methods rely on some sort of tuning depending
on either noise or signal parameters. If no estimates for either of them are
available, the noisy recovery problem is significantly harder. The square root
LASSO and the least absolute deviation LASSO are known to be noise-blind, in
the sense that the tuning parameter can be chosen independent on the noise and
the signal. We generalize those recovery methods to the \hrlone{} and give a
recovery guarantee once the tuning parameter is above a threshold. Moreover, we
analyze the effect of a bad chosen tuning parameter mistuning on a theoretic
level and prove the optimality of our recovery guarantee. Further, for Gaussian
matrices we give a refined analysis of the threshold of the tuning parameter
and proof a new relation of the tuning parameter on the dimensions. Indeed, for
a certain amount of measurements the tuning parameter becomes independent on
the sparsity. Finally, we verify that the least absolute deviation LASSO can be
used with random walk matrices of uniformly at random chosen left regular
biparitite graphs.
