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Almost everywhere generalized phase retrieval
Applied and Computational Harmonic Analysis ( IF 2.5 ) Pub Date : 2020-08-26 , DOI: 10.1016/j.acha.2020.08.002
Meng Huang , Yi Rong , Yang Wang , Zhiqiang Xu

The aim of generalized phase retrieval is to recover xFd from the quadratic measurements xA1x,,xANx, where AjHd(F) and F=R or C. In this paper, we study the matrix set A=(Aj)j=1N which has the almost everywhere phase retrieval property. For the case F=R, we show that Nd+1 generic matrices with prescribed ranks have almost everywhere phase retrieval property. We also extend this result to the case where A1,,AN are orthogonal matrices and hence establish the almost everywhere phase retrieval property for the fusion frame phase retrieval. For the case where F=C, we obtain similar results under the assumption of N2d. We lower the measurement number d+1 (resp. 2d) with showing that there exist N=d (resp. 2d1) matrices A1,,ANHd(R) (resp. Hd(C)) which have the almost everywhere phase retrieval property. Our results are an extension of almost everywhere phase retrieval from the standard phase retrieval to the general setting and the proofs are often based on some new ideas about determinant variety.



中文翻译:

几乎所有地方的广义相位检索

广义相位检索的目的是恢复 XFd 从二次测量 X一种1个XX一种ñX,在哪里 一种ĴHdFF=[R 要么 C。在本文中,我们研究矩阵集一种=一种ĴĴ=1个ñ具有几乎所有相位检索特性。对于这种情况F=[R,我们证明 ñd+1个具有规定等级的通用矩阵几乎在任何地方都具有相检索特性。我们还将这个结果扩展到以下情况一种1个一种ñ是正交矩阵,因此为融合帧相位检索建立了几乎无处不在的相位检索特性。对于这种情况F=C,我们在 ñ2d。我们降低测量数d+1个(第2 d天)并显示存在ñ=d (分别 2d-1个)矩阵 一种1个一种ñHd[R (分别 HdC),几乎具有无处不在的相位检索特性。我们的结果是将几乎所有相位检索从标准相位检索扩展到常规设置,并且证明常常基于关于行列式变化的一些新思想。

更新日期:2020-08-26
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