Phase retrieval from the norms of affine transformations

2018/05/21 by Huang, Meng, Xu, Zhiqiang
#Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.1805.07899

Abstract

In this paper, we consider the generalized phase retrieval from affine measurements. This problem aims to recover signals \mathbf x ∈ \mathbb F^d from the affine measurements y_j=\normM_j^*\vx +\mathbb b_j², j=1,…,m, where M_j ∈ \mathbb F^{d× r}, \mathbf b_j∈ \mathbb F^r, \mathbb F∈ \\mathbb R,\mathbb C\ and we call it as \em generalized affine phase retrieval. We develop a framework for generalized affine phase retrieval with presenting necessary and sufficient conditions for \(M_j,\mathbf b_j)\_j=1^m having generalized affine phase retrieval property. We also establish results on minimal measurement number for generalized affine phase retrieval. Particularly, we show if \(M_j,\mathbf b_j)\_j=1^m ⊂ \mathbb F^{d× r}× \mathbb F^r has generalized affine phase retrieval property, then m≥ d+\floord/r for \mathbb F=\mathbb R (m≥ 2d+\floord/r for \mathbb F=\mathbb C ). We also show that the bound is tight provided r| d. These results imply that one can reduce the measurement number by raising r, i.e. the rank of M_j. This highlights a notable difference between generalized affine phase retrieval and generalized phase retrieval. Furthermore, using tools of algebraic geometry, we show that m≥ 2d (resp. m≥ 4d-1) generic measurements \mathcal A=\(M_j,b_j)\_j=1^m have the generalized phase retrieval property for \mathbb F=\mathbb R (resp. \mathbb F=\mathbb C).

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