Sparse recovery under matrix uncertainty

We consider the model y = Xθ* +ξ, Z = X +Ξ, where the random vector y ∈ ℝⁿ and the random n×p matrix Z are observed, the n × p matrix X is unknown, Ξ is an n × p random noise matrix, ξ ∈ ℝⁿ is a noise independent of Ξ, and θ* 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 θ*. 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 θ*), 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 θ* 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.