Phase retrieval from the norms of affine transformations

by   Meng Huang, et al.

In this paper, we consider the generalized phase retrieval from affine measurements. This problem aims to recover signals x∈ F^d from the affine measurements y_j=M_j^* + b_j^2, j=1,...,m, where M_j ∈ F^d× r, b_j∈ F^r, F∈{ R, C} and we call it as generalized affine phase retrieval. We develop a framework for generalized affine phase retrieval with presenting necessary and sufficient conditions for {(M_j, 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, b_j)}_j=1^m ⊂ F^d× r× F^r has generalized affine phase retrieval property, then m≥ d+d/r for F= R (m≥ 2d+d/r for F= 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 A={(M_j,b_j)}_j=1^m have the generalized phase retrieval property for F= R (resp. F= C).


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