Sparse estimation in high-dimensional linear errors-in-variables regression via a covariate relaxation method

Sparse signal recovery in high-dimensional settings via regularization techniques has been developed in the past two decades and produces fruitful results in various areas. Previous studies mainly focus on the idealized assumption where covariates are free of noise. However, in realistic scenarios, covariates are always corrupted by measurement errors, which may induce significant estimation bias when methods for clean data are naively applied. Recent studies begin to deal with the errors-in-variables models. Current method either depends on the distribution of covariate noise or does not depends on the distribution but is inconsistent in parameter estimation. A novel covariate relaxation method that does not depend on the distribution of covariate noise is proposed. Statistical consistency on parameter estimation is established. Numerical experiments are conducted and show that the covariate relaxation method achieves the same or even better estimation accuracy than that of the state of art nonconvex Lasso estimator. The advantage that the covariate relaxation method is independent of the distribution of covariate noise while produces a small estimation error suggests its prospect in practical applications.