Sparse recovery under matrix uncertainty

TL;DR

This paper addresses the challenge of sparse signal recovery when the design matrix is observed with additive noise, proposing new estimators called MU-selectors that improve stability and accuracy under matrix uncertainty.

Contribution

The paper introduces MU-selectors, a novel class of estimators that handle matrix uncertainty in high-dimensional sparse recovery, outperforming traditional methods like Lasso and Dantzig selector.

Findings

MU-selectors are close to the true parameter in various norms.

They achieve accurate prediction risk under restricted eigenvalue conditions.

Under stronger assumptions, they correctly recover the sparsity pattern.

Abstract

We consider the model {eqnarray*}y=X\theta^*+\xi, Z=X+\Xi,{eqnarray*} where the random vector $y\in\mathbb{R}^n$ and the random $n\times p$ matrix $Z$ are observed, the $n\times p$ matrix $X$ is unknown, $\Xi$ is an $n\times p$ random noise matrix, $\xi\in\mathbb{R}^n$ is a noise independent of $\Xi$, and $\theta^*$ is a vector of unknown parameters to be estimated. The matrix uncertainty is in the fact that $X$ is observed with additive error. For dimensions $p$ that can be much larger than the sample size $n$, we consider the estimation of sparse vectors $\theta^*$. Under matrix uncertainty, the Lasso and Dantzig selector turn out to be extremely unstable in recovering the sparsity pattern (i.e., of the set of nonzero components of $\theta^*$), even if the noise level is very small. We suggest new estimators called matrix uncertainty selectors (or, shortly, the MU-selectors) which are close to $\theta^*$ in different norms and in the prediction risk if the restricted eigenvalue assumption on $X$ is satisfied. We also show that under somewhat stronger assumptions, these estimators recover correctly the sparsity pattern.