Phase Transitions in Phase-Only Compressed Sensing

The goal of phase-only compressed sensing is to recover a structured signal $\mathbf{x}$ from the phases $\mathbf{z} = {\rm sign}(\mathbf{\Phi}\mathbf{x})$ under some complex-valued sensing matrix $\mathbf{\Phi}$. Exact reconstruction of the signal's direction is possible: we can reformulate it as a linear compressed sensing problem and use basis pursuit (i.e., constrained norm minimization). For $\mathbf{\Phi}$ with i.i.d. complex-valued Gaussian entries, this paper shows that the phase transition is approximately located at the statistical dimension of the descent cone of a signal-dependent norm. Leveraging this insight, we derive asymptotically precise formulas for the phase transition locations in phase-only sensing of both sparse signals and low-rank matrices. Our results prove that the minimum number of measurements required for exact recovery is smaller for phase-only measurements than for traditional linear compressed sensing. For instance, in recovering a 1-sparse signal with sufficiently large dimension, phase-only compressed sensing requires approximately 68% of the measurements needed for linear compressed sensing. This result disproves earlier conjecture suggesting that the two phase transitions coincide. Our proof hinges on the Gaussian min-max theorem and the key observation that, up to a signal-dependent orthogonal transformation, the sensing matrix in the reformulated problem behaves as a nearly Gaussian matrix.