Title: On the Erasure Robustness Property of Random Matrices,
Abstract: The study of the restricted isometry property (RIP) for corrupted random
matrices is particularly important in the field of compressed sensing (CS) with
corruptions. If a matrix still satisfy RIP after a certain portion of rows are
erased, then we say that the matrix has the strong restricted isometry property
(SRIP. In the field of compressed sensing, random matrices satisfies certain
moment conditions are of particular interest. Among these matrices, those with
entries generated from i.i.d Gaussian or i.i.d $\pm1$ random variables are
often typically considered. Recent studies have shown that a matrix generated
from i.i.d Gaussian random variables satisfies the strong restricted isometry
property under arbitrary erasure of rows. In the first part of this paper we
will work on $\pm 1$ random matrices. We study the erasure robustness of $\pm
1$ random matrices show that with overwhelming probability the SRIP will still
hold. Moreover the analysis will also lead to the robust version of the
Johnson-Lindenstrauss Lemma for $\pm 1$ matrices. Then in the second part of
this paper we work on finite frames. The study of the stability of finite
frames under corruptions shares a lot of similarity to CS with corruption. We
will focus on the Gaussian finite frames as a starter. We will improve existing
results and confirm that a Gaussian random frame is numerically stable under
arbitrary erasure of rows.