Linear and Conic Programming Estimators in High-Dimensional Errors-in-variables Models

TL;DR

This paper studies high-dimensional errors-in-variables linear regression, proposing new estimators that are nearly optimal and computationally feasible via linear and conic programming, outperforming traditional methods like Lasso.

Contribution

It introduces new estimators based on linear and second order cone programming that are nearly minimax optimal and efficiently computable in high-dimensional errors-in-variables models.

Findings

The proposed estimator is nearly minimax optimal.

It can be computed via a single linear or conic programming problem.

The estimator attains the minimax efficiency bound.

Abstract

We consider the linear regression model with observation error in the design. In this setting, we allow the number of covariates to be much larger than the sample size. Several new estimation methods have been recently introduced for this model. Indeed, the standard Lasso estimator or Dantzig selector turn out to become unreliable when only noisy regressors are available, which is quite common in practice. We show in this work that under suitable sparsity assumptions, the procedure introduced in Rosenbaum and Tsybakov (2013) is almost optimal in a minimax sense and, despite non-convexities, can be efficiently computed by a single linear programming problem. Furthermore, we provide an estimator attaining the minimax efficiency bound. This estimator is written as a second order cone programming minimisation problem which can be solved numerically in polynomial time.