We study high-dimensional regression with missing entries in the covariates. A common strategy in practice is to emph{impute} the missing entries with an appropriate substitute and then implement a standard statistical procedure acting as if the covariates were fully observed. Recent literature on this subject proposes instead to design a specific, often complicated or non-convex, algorithm tailored to the case of missing covariates. We investigate a simpler approach where we fill-in the missing entries with their conditional mean given the observed covariates. We show that this imputation scheme coupled with standard off-the-shelf procedures such as the LASSO and square-root LASSO retains the minimax estimation rate in the random-design setting where the covariates are i.i.d. sub-Gaussian. We further show that the square-root LASSO remains emph{pivotal} in this setting. It is often the case that the conditional expectation cannot be computed exactly and must be approximated from data. We study two cases where the covariates either follow an autoregressive (AR) process, or are jointly Gaussian with sparse precision matrix. We propose tractable estimators for the conditional expectation and then perform linear regression via LASSO, and show similar estimation rates in both cases. We complement our theoretical results with simulations on synthetic and semi-synthetic examples, illustrating not only the sharpness of our bounds, but also the broader utility of this strategy beyond our theoretical assumptions.
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