Phase retrieval from the norms of affine transformations
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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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