  We present a robust framework to perform linear regression with missing
entries in the features. By considering an elliptical data distribution, and
specifically a multivariate normal model, we are able to conditionally
formulate a distribution for the missing entries and present a robust
framework, which minimizes the worst case error caused by the uncertainty about
the missing data. We show that the proposed formulation, which naturally takes
into account the dependency between different variables, ultimately reduces to
a convex program, for which a customized and scalable solver can be delivered.
In addition to a detailed analysis to deliver such solver, we also asymptoticly
analyze the behavior of the proposed framework, and present technical
discussions to estimate the required input parameters. We complement our
analysis with experiments performed on synthetic, semi-synthetic, and real
data, and show how the proposed formulation improves the prediction accuracy
and robustness, and outperforms the competing techniques.
  Missing data is a common problem associated with many datasets in machine
learning. With the significant increase in using robust optimization techniques
to train machine learning models, this paper presents a novel robust regression
framework that operates by minimizing the uncertainty associated with missing
data. The proposed approach allows training models with incomplete data, while
minimizing the impact of uncertainty associated with the unavailable data. The
ideas developed in this paper can be generalized beyond linear models and
elliptical data distributions.
