Analysis Recovery With Coherent Frames and Correlated Measurements

Abstract

This paper introduces the restricted eigenvalue condition adapted to frame ${D}$ ( ${D}$ -RE), which is a natural extension to the standard restricted eigenvalue condition. The ${D}$ -RE condition is a relaxation of the ${D^\dagger }$ -RIP, where ${D^\dagger =(DD^{*})^{-1}D}$ is the canonical dual frame of ${D}$ . We establish the ${D}$ -RE condition for several classes of correlated measurement matrices, when the covariance matrix of row measurements satisfies the ${D}$ -RE condition. Furthermore, by the ${D}$ -RE condition, we get the error bounds in the analysis LASSO (ALASSO) and the analysis Dantzig Selector (ADS) under a sparsity scenario. In order to recover non-sparse signals, we consider the robust ${\ell _{2}}~{D}$ -nullspace property of correlated Gaussian matrices. Similarly, we get the error estimations in the ALASSO and the ADS in non-sparse case. The approximation equivalence between the ALASSO and the ADS is also established by calculating prediction loss difference.