2016/08/22 by Gao, Bing, Sun, Qiyu, Wang, Yang +1 · 1 citation
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1608.06117
In this paper, we consider the phase retrieval problem in which one aims to recover a signal from the magnitudes of affine measurements. Let \\mathbf aj\j=1m ⊂ \mathbb Hd and \mathbf b=(b1, …, bm)^\top∈\mathbb Hm, where \mathbb H=\mathbb R or \mathbb C. We say \\mathbf aj\j=1m and \mathbf b are affine phase retrievable for \mathbb Hd if any \mathbf x∈\mathbb Hd can be recovered from the magnitudes of the affine measurements \|+bj|, 1≤ j≤ m\. We develop general framework for affine phase retrieval and prove necessary and sufficient conditions for \\mathbf aj\j=1m and \mathbf b to be affine phase retrievable. We establish results on minimal measurements and generic measurements for affine phase retrieval as well as on sparse affine phase retrieval. In particular, we also highlight some notable differences between affine phase retrieval and the standard phase retrieval in which one aims to recover a signal \mathbf x from the magnitudes of its linear measurements. In standard phase retrieval, one can only recover \mathbf x up to a unimodular constant, while affine phase retrieval removes this ambiguity. We prove that unlike standard phase retrieval, the affine phase retrievable measurements \\mathbf aj\j=1m and \mathbf b do not form an open set in \mathbb Hm× d× \mathbb Hm. Also in the complex setting, the standard phase retrieval requires 4d-O(log2d) measurements, while the affine phase retrieval only needs m=3d measurements.