CIF: Small: Structured High-dimensional Data Recovery from Phaseless Measurements

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Phase retrieval (PR), or 'signal recovery from phaseless measurements', is a problem that occurs in numerous signal/image acquisition domains, such as Fourier ptychography and sub-diffraction imaging, in which only the magnitude (intensity) of certain linear projections of the signal or image can be measured. While PR is a classical problem, in recent years there has been renewed interest in PR with the goal of developing provably correct and fast algorithms. Much of this work, however, does not assume any structure on the signal, and as a result necessarily requires more measurements than the unknown signal's length. This can be a challenge when moving to very high resolution imaging because it implies a proportionally higher cost of data acquisition (in terms of time, number of sensors, or power consumption). Dynamic imaging of time-varying scenes, e.g., live biological samples, poses an even greater challenge. We address this limitation by exploiting two common classes of structural assumptions - sparsity and low-rank -- to enable fast and low cost high-resolution imaging. A diverse group of graduate and undergraduate students is involved in the research.This project develops the first set of provably correct, fast, and low-sample-complexity algorithms for phaseless low rank matrix recovery in two settings. The first involves recovery from phaseless linear projections of each column of the matrix. This finds applications in phaseless dynamic imaging when the (vectorized) image sequence is well approximated by a low rank matrix, e.g., slow changing dynamic scenes. The second setting involves recovery from phaseless linear projections of the entire matrix. This is useful when the image itself can be modeled as being low rank. This project also develops provably fast and statistically efficient sparse PR algorithms and explores extensions to learning generalized linear models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.