  The many variants of the restricted isometry property (RIP) have proven to be
crucial theoretical tools in the fields of compressed sensing and matrix
completion. The study of extending compressed sensing to accommodate phaseless
measurements naturally motivates a strong notion of restricted isometry
property (SRIP), which we develop in this paper. We show that if $A \in
\mathbb{R}^{m\times n}$ satisfies SRIP and phaseless measurements $|Ax_0| = b$
are observed about a $k$-sparse signal $x_0 \in \mathbb{R}^n$, then minimizing
the $\ell_1$ norm subject to $ |Ax| = b $ recovers $x_0$ up to multiplication
by a global sign. Moreover, we establish that the SRIP holds for the random
Gaussian matrices typically used for standard compressed sensing, implying that
phaseless compressed sensing is possible from $O(k \log (n/k))$ measurements
with these matrices via $\ell_1$ minimization over $|Ax| = b$. Our analysis
also yields an erasure robust version of the Johnson-Lindenstrauss Lemma.
