Restricted eigenvalue property for corrupted Gaussian designs

Motivated by the construction of robust estimators via convex relaxations, known to be computationally efficient, we present conditions on the sample size which guarantee an augmented notion of Restricted Eigenvalue-type condition for Gaussian designs. Such notion is suitable for the construction of robust estimators of a multivariate Gaussian model whose samples are corrupted by outliers. Our proof technique relies on simultaneous lower and upper bounds of two different random bilinear forms so to balance the interaction between the parameter vector and the estimated corruption vector. This argument has the advantage of not relying on known bounds of the extreme singular values of the associated Gaussian ensemble nor on the use of mutual incoherence arguments. An important theoretical and practical consequence of such approach of analysis is that a sharper restricted eigenvalue constant can be obtained and the sparsity levels of the unknown parameter and corruption vectors are allowed to be completely independent of each other.