Square-root lasso under correlated regressors: Tight statistical analysis with a wireless communications application

This paper provided a comprehensive analysis of sparse signal estimation from noisy and possibly underdetermined linear observations in the high-dimensional asymptotic regime. The focus was on the square-root lasso (sqrt-lasso), a popular convex optimization method used for sparse signal recovery. We analyzed its performance using several metrics, such as root-mean-squared error (r.m.s.e.), mean absolute error (m.a.e.), cosine similarity, and true/false recovery rates. The analysis assumed a normally distributed design matrix with left-sided correlation and Gaussian noise. In addition to theoretical contributions, we applied these results to a real-world wireless communications problem by examining the error performance of sqrt-lasso in generalized space shift keying (GSSK) modulation for multiple-input multiple-output (MIMO) systems. This application was particularly relevant, as the GSSK modulation generates sparse data symbols, making it an ideal scenario for sparse recovery techniques. Our study offered tight asymptotic approximations for the performance of sqrt-lasso in such systems. Beyond the wireless communications application, the results had broader implications for other high-dimensional applications, including compressed sensing, machine learning, and statistical inference. The analysis presented in this paper, supported by numerical simulations, provided practical insights into how sqrt-lasso behaved under correlated designs, offering useful guidelines for optimizing its use in real-world scenarios. The expressions and insights obtained from this study can be used to optimally choose the penalization parameter of the sqrt-lasso. By applying these results, one can make informed decisions about performance and fine-tuning the sqrt-lasso, considering the presence of correlated regressors in a high-dimensional context.