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The study of the restricted isometry property (RIP) of corrupted randommatrices is particularly important in the field of compressed sensing (CS) withcorruptions. If a matrix still satisfies the RIP after that a certain portionof rows are erased, then we say that this matrix has the strong restrictedisometry property (SRIP). In the field of compressed sensing, random matriceswhich satisfy certain moment conditions are of particular interest. Among thesematrices, those with entries generated from i.i.d. Gaussian or symmetricBernoulli random variables are often typically considered. Recent studies haveshown that matrices with entries generated from i.i.d. Gaussian randomvariables satisfy the SRIP under arbitrary erasure of rows with highprobability. In this paper, we study the erasure robustness property ofBernoulli random matrices. Our main result shows that with overwhelmingprobability, the SRIP holds for Bernoulli random matrices. Moreover, ouranalysis leads to a robust version of the famous Johnson-Lindenstrauss lemmafor Bernoulli random matrices.

On the Strong Restricted Isometry Property of Bernoulli Random Matrices