Phaseless PCA: LowRank Matrix Recovery from Columnwise Phaseless Measurements
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:47624770, 2019.
Abstract
This work proposes the first set of simple, practically useful, and provable algorithms for two interrelated problems. (i) The first is lowrank matrix recovery from magnitudeonly (phaseless) linear projections of each of its columns. This finds important applications in phaseless dynamic imaging, e.g., Fourier Ptychographic imaging of live biological specimens. Our guarantee shows that, in the regime of small ranks, the sample complexity required is only a little larger than the orderoptimal one, and much smaller than what standard (unstructured) phase retrieval methods need. %Moreover our algorithm is fast and memoryefficient if only the minimum required number of measurements is used (ii) The second problem we study is a dynamic extension of the above: it allows the lowdimensional subspace from which each image/signal (each column of the lowrank matrix) is generated to change with time. We introduce a simple algorithm that is provably correct as long as the subspace changes are piecewise constant.
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