Phaseless PCA and Subspace Tracking
This work introduces the first simple and provably correct solution for recovering a low-rank matrix from phaseless (magnitude-only) linear projections of each of its columns. This problem finds important applications in phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens. We demonstrate the practical advantage of our proposed approach, AltMinLowRaP, over existing work via extensive simulation, and some real-data, experiments. Under a right incoherence (denseness of right singular vectors) assumption, we can prove that the AltMinLowRaP sample complexity is only r^3 times the order-optimal value. In the regime of small ranks, this is close to optimal, and a significant improvement over standard (unstructured) Phase Retrieval solutions which necessarily need $n$ or more measurements per $n$-length signal (matrix column).
We also provide a solution for a dynamic extension of phaseless PCA. This allows the low-dimensional subspace from which each image/signal is generated to change with time in a piecewise constant fashion.