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.
['Yu Xia', 'Song Li']
Analysis Recovery With Coherent Frames and Correlated Measurements