Restricted Eigenvalue Properties for Correlated Gaussian Designs

Methods based on 1-relaxation, such as basis pursuit and the Lasso, are very popular for sparse regression in high dimensions. The conditions for success of these methods are now well-understood: (1) exact recovery in the noiseless setting is possible if and only if the design matrix X satisfies the restricted nullspace property, and (2) the squared 2-error of a Lasso estimate decays at the minimax optimal rate klog p n , where k is the sparsity of the p-dimensional regression problem with additive Gaussian noise, whenever the design satisfies a restricted eigenvalue condition. The key issue is thus to determine when the design matrix X satisfies these desirable properties. Thus far, there have been numerous results showing that the restricted isometry property, which implies both the restricted nullspace and eigenvalue conditions, is satisfied when all entries of X are independent and identically distributed (i.i.d.), or the rows are unitary. This paper proves directly that the restr...

• Type: Journal
• Year: 2010
• Where: JMLR
• Authors: Garvesh Raskutti, Martin J. Wainwright, Bin Yu
• Keywords: Design Matrix, Eigenvalue Conditions, JMLR 2010, Restricted Eigenvalue Condition