A strong restricted isometry property, with an application to phaseless compressed sensing

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∈Rm×nA∈Rm×n satisfies SRIP and phaseless measurements |Ax0|=b|Ax0|=b are observed about a k  -sparse signal x0∈Rnx0∈Rn, then minimizing the ℓ1ℓ1 norm subject to |Ax|=b|Ax|=b recovers x0x0 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(klog⁡(en/k))O(klog⁡(en/k)) measurements with these matrices via ℓ1ℓ1 minimization over |Ax|=b|Ax|=b. Our analysis also yields an erasure robust version of the Johnson–Lindenstrauss Lemma.