Linear and Conic Programming Approaches to High-Dimensional Errors-in-variables Models

We consider the regression model with observation error in the design when the dimension can be much larger than the sample size and the true parameter is sparse. We propose two new estimators, based on linear and conic programming, and we prove that they satisfy oracle inequalities similar to those for the model with exactly known covariates. The only difference is that they contain additional scaling with the l1 or l2 norm of the true parameter. The scaling with the l2 norm is minimax optimal and it is achieved on conic programming, while the scaling with the l1 norm is achieved on the linear programming estimator, which is easier to implement.
This is a joint work with Mathieu Rosenbaum.