CS: Weak Compressive Sensing

I have been asked by some of you if this or that maybe an instance of weak compressive sensing as discussed yesterday. Let's make it short, an example of weak compressive sensing measurement matrix is:

P( I + \epsilon A )

with P a projection operator (it takes only a random set of rows from its argument), I is the identity matrix, A a gaussian matrix. With \epsilon = 0, we have strict inpaiting whereas with very large \epsilon we asymptotically have some measurement matrix used in Compressive Sensing. Among the different things I'd ask somebody who would look into this type of construction:

• what is the effect of epsilon on the Donoho-Tanner phase transition ?
• what is the effect of additive and multiplicative noise on the Donoho-Tanner phase transition ?
• is this model relevant to the cross and bouquet approach of John Wright and Yi Ma [1]?
• are there any useful results out of work like [2] or [3] that bears on this problem ?

[1] Dense Error Correction via L1-Minimization by John Wright, Yi Ma

[2] Large deviations of the extreme eigenvalues of random deformations of matrices by Florent Benaych-Georges, Alice Guionnet, Mylène Maïda. 

[3] Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices by Florent Benaych-Georges, Alice Guionnet, Mylène Maïda.