In this paper, we address the challenging problem of selecting tuning parameters for high-dimensional sparse regression. We propose a simple and computationally efficient method, called path thresholding (PaTh), that transforms any tuning parameter-dependent sparse regression algorithm into an asymptotically tuning-free sparse regression algorithm. More specifically, we prove that, as the problem size becomes large (in the number of variables and in the number of observations), PaTh performs accurate sparse regression, under appropriate conditions, without specifying a tuning parameter. In finite-dimensional settings, we demonstrate that PaTh can alleviate the computational burden of model selection algorithms by significantly reducing the search space of tuning parameters.

We consider the high-dimensional sparse linear regression problem of accurately estimating a sparse vector using a small number of linear measurements that are contaminated by noise. It is well known that the standard cadre of computationally tractable sparse regression algorithms---such as the Lasso, Orthogonal Matching Pursuit (OMP), and their extensions---perform poorly when the measurement matrix contains highly correlated columns. To address this shortcoming, we develop a simple greedy algorithm, called SWAP, that iteratively swaps variables until convergence. SWAP is surprisingly effective in handling measurement matrices with high correlations. In fact, we prove that SWAP outputs the true support, the locations of the non-zero entries in the sparse vector, under a relatively mild condition on the measurement matrix. Furthermore, we show that SWAP can be used to boost the performance of any sparse regression algorithm. We empirically demonstrate the advantages of SWAP by comparing it with several state-of-the-art sparse regression algorithms.