Generalized phase retrieval: Measurement number, matrix recovery and beyond

Abstract

In this paper, we develop a framework of generalized phase retrieval in which one aims to reconstruct a vector x in R d or C d through quadratic samples x * A1x, . . . , x * AN x. The generalized phase retrieval includes as special cases the standard phase retrieval as well as the phase retrieval by orthogonal projections. We first explore the connections among generalized phase retrieval, low-rank matrix recovery and nonsingular bilinear form. Motivated by the connections, we present results on the minimal measurement number needed for recovering a matrix that lies in a set W ∈ C d×d . Applying the results to phase retrieval, we show that generic d × d matrices A1, . . . , AN have the phase retrieval property if N ≥ 2d − 1 in the real case and N ≥ 4d − 4 in the complex case for very general classes of A1, . . . , AN , e.g. matrices with prescribed ranks or orthogonal projections. Our method also leads to a novel proof for the classical Stiefel-Hopf condition on nonsingular bilinear form. We also give lower bounds on the minimal measurement number required for generalized phase retrieval. For several classes of dimensions d we obtain the precise values of the minimal measurement number. Our work unifies and enhances results from the standard phase retrieval, phase retrieval by projections and low-rank matrix recovery.2010 Mathematics Subject Classification. Primary 42C15, Secondary 94A12, 15A63, 15A83 .

Conjugate phase retrieval on CM by real vectors

“…On C 2 and C 3 , we fully solve the number of real measurement vectors needed for conjugate phase retrievability, and characterize all conjugate phase retrievable frame as real algebraic varieties. Building from the recent results by Wang and Xu [22], we prove that 4M − 6 is a sufficient number of generic measurements for conjugate phase retrieval in C M for M ≥ 4.…”

“…In this section, we will be proving the generic number required for conjugate phase retrieval using real vectors. To this end, we need some terminology from algebraic geometry and we will use a theorem in a recent paper by Wang and Xu [22].…”

Phase retrieval from the magnitudes of affine linear measurements

“…The result is sharp as we also construct an affine phase retrievable (A, b) for C d with m = 3d. This result shows that the nature of affine phase retrieval can be quite different from that of the standard phase retrieval in the complex setting, where it is known that 4d − O(log 2 d) measurements are needed for phase retrieval [16,21].…”

Data recovery on a manifold from linear samples: theory and computation

“…Again, we say a set of Hermitian matrices {A j } N j=1 in F d×d have the phase retrieval property if any x ∈ F d can be recovered up to a unimodular scalar from the quadratic measurements {x * A j x} N j=1 . This generalized version is studied in [11], and in special cases such as for orthogonal projection matrices {A j } N j=1 in other papers [8,10,12]. Let b j = x * A j x and X = xx * .…”

“…Here we will provide a framework for answering these questions. For phase retrieval and matrix completion we have developed techniques to make substantial progresses recently [21,11,18]. Our goal for this paper is more appropriately described as the combination of a survey and putting past work into a more unified framework.…”