Robust Instance-Optimal Recovery of Sparse Signals at Unknown Noise Levels

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.