The importance of phase in complex compressive sensing
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We consider the question of estimating a real low-complexity signal (such as a sparse vector or a low-rank matrix) from the phase of complex random measurements, i.e., in a phase-only compressive sensing (PO-CS) scenario. We show that, with high probability and up to global unknown signal amplitude, we can perfectly recover such a signal if the sensing matrix is a complex, Gaussian random matrix and if the number of measurements is large compared to the complexity level of the signal space. Moreover, this recovery is still possible if each measurement is multiplied by an unknown sign. Our approach proceeds by recasting the (non-linear) PO-CS scheme as a linear compressive sensing model built from a signal normalization constraint, and a phase-consistency constraint imposing any signal estimate to match the observed phases in the measurement domain. Practically, stable and robust signal direction estimation is achieved from any instance optimal algorithm of the compressive sensing literature (e.g., basis pursuit denoising). This is ensured by proving that the matrix associated with this equivalent linear model satisfies with high probability the restricted isometry property under the above condition on the number of measurements. We finally observe experimentally that robust signal direction recovery is reached at about twice the number of measurements needed for signal recovery in compressive sensing.
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