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Convolutional Phase Retrieval
Qing Qu · Yuqian Zhang · Yonina Eldar · John Wright

Tue Dec 05 06:30 PM -- 10:30 PM (PST) @ Pacific Ballroom #161 #None
We study the convolutional phase retrieval problem, which asks us to recover an unknown signal ${\mathbf x} $ of length $n$ from $m$ measurements consisting of the magnitude of its cyclic convolution with a known kernel $\mathbf a$ of length $m$. This model is motivated by applications to channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when $\mathbf a$ is random and $m \geq \Omega(\frac{ \| \mathbf C_{\mathbf x}\|^2}{ \|\mathbf x\|^2 } n \mathrm{poly} \log n)$, $\mathbf x$ can be efficiently recovered up to a global phase using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator; we overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis for alternating minimizing methods.

Author Information

Qing Qu (Columbia University)
Yuqian Zhang (Cornell University)
Yonina Eldar (Israel Institute of Technology)
John Wright (Columbia University)

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