High dimensional structured estimation with noisy designs

Structured estimation methods, such as LASSO, have received considerable attention in recent years and substantial progress has been made in extending such methods to general norms and non-Gaussian design matrices. In real world problems, however, covariates are usually corrupted with noise and there have been efforts to generalize structured estimation method for noisy covariate setting. In this paper we first show that without any information about the noise in covariates, currently established techniques of bounding statistical error of estimation fail to provide consistency guarantees. However, when information about noise covariance is available or can be estimated, then we prove consistency guarantees for any norm regularizer, which is a more general result than the state of the art. Next, we investigate empirical performance of structured estimation, specifically LASSO, when covariates are noisy and empirically show that LASSO is not consistent or stable in the presence of additive noise. However, prediction performance improves quite substantially when the noise covariance is available for incorporating in the estimator.