We demonstrate that the phase of complex linear measurements of signals preserves significant information about the angles between those signals. We provide stable angle embedding guarantees, akin to the restricted isometry property in classical compressive sensing, that characterize how well the angle information is preserved. They also suggest that a number of measurements linear in the sparsity and logarithmic in the dimensionality of the signal contains sufficient information to acquire and reconstruct a sparse signal within a positive scalar factor. We further show that the reconstruction can be formulated and solved using standard convex and greedy algorithms taken directly from the CS literature. Even though the theoretical results only provide approximate reconstruction guarantees, our experiments suggest that exact reconstruction is possible.
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