Improved Matrix Uncertainty Selector

TL;DR

This paper introduces a modified matrix uncertainty selector for high-dimensional sparse regression models with random noise in the design matrix, improving estimation accuracy over the original MU selector.

Contribution

The paper proposes a new Compensated MU selector that accounts for random noise in the design matrix, enhancing estimation stability and accuracy in high-dimensional settings.

Findings

The Compensated MU selector outperforms the original MU selector in simulations.

Theoretical analysis confirms improved estimation bounds.

Numerical experiments demonstrate better accuracy in missing data scenarios.

Abstract

We consider the regression model with observation error in the design: y=X\theta* + e, Z=X+N. Here the random vector y in R^n and the random n*p matrix Z are observed, the n*p matrix X is unknown, N is an n*p random noise matrix, e in R^n is a random noise vector, and \theta* is a vector of unknown parameters to be estimated. We consider the setting where the dimension p can be much larger than the sample size n and \theta* is sparse. Because of the presence of the noise matrix N, the commonly used Lasso and Dantzig selector are unstable. An alternative procedure called the Matrix Uncertainty (MU) selector has been proposed in Rosenbaum and Tsybakov (2010) in order to account for the noise. The properties of the MU selector have been studied in Rosenbaum and Tsybakov (2010) for sparse \theta* under the assumption that the noise matrix N is deterministic and its values are small. In this paper, we propose a modification of the MU selector when N is a random matrix with zero-mean entries having the variances that can be estimated. This is, for example, the case in the model where the entries of X are missing at random. We show both theoretically and numerically that, under these conditions, the new estimator called the Compensated MU selector achieves better accuracy of estimation than the original MU selector.