  In this paper we study the robustness properties of dimensionality reduction
with Gaussian random matrices having arbitrarily erased rows. We first study
the robustness property against erasure for the almost norm preservation
property of Gaussian random matrices by obtaining the optimal estimate of the
erasure ratio for a small given norm distortion rate. As a consequence, we
establish the robustness property of Johnson-Lindenstrauss lemma and the
robustness property of restricted isometry property with corruption for
Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal
lower and upper bounds of norm distortion rates of Gaussian random matrices
under a given erasure ratio. This allows us to establish the strong restricted
isometry property with the almost optimal RIP constants, which plays a central
role in the study of phaseless compressed sensing.
