# On the Strong Restricted Isometry Property of Bernoulli Random Matrices

## TL;DR

This paper demonstrates that Bernoulli random matrices satisfy the strong restricted isometry property (SRIP) with high probability, ensuring robustness in compressed sensing even after row erasures, and extends the Johnson-Lindenstrauss lemma.

## Contribution

It proves Bernoulli matrices satisfy SRIP with high probability, enhancing understanding of their robustness in compressed sensing applications.

## Key findings

- Bernoulli matrices satisfy SRIP with overwhelming probability.
- The analysis extends the Johnson-Lindenstrauss lemma to Bernoulli matrices.
- Results confirm robustness of Bernoulli matrices under row erasures.

## Abstract

The study of the restricted isometry property (RIP) of corrupted random matrices is particularly important in the field of compressed sensing (CS) with corruptions. If a matrix still satisfies the RIP after that a certain portion of rows are erased, then we say that this matrix has the strong restricted isometry property (SRIP). In the field of compressed sensing, random matrices which satisfy certain moment conditions are of particular interest. Among these matrices, those with entries generated from i.i.d. Gaussian or symmetric Bernoulli random variables are often typically considered. Recent studies have shown that matrices with entries generated from i.i.d. Gaussian random variables satisfy the SRIP under arbitrary erasure of rows with high probability. In this paper, we study the erasure robustness property of Bernoulli random matrices. Our main result shows that with overwhelming probability, the SRIP holds for Bernoulli random matrices. Moreover, our analysis leads to a robust version of the famous Johnson-Lindenstrauss lemma for Bernoulli random matrices.