Conjugate phase retrieval on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msup></mml:math> by real vectors

Evans, Luke; Lai, Chun–Kit

doi:10.1016/j.laa.2019.11.001

Cited by 14 publications

(10 citation statements)

References 22 publications

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“…Then the set of all Ψ that perform phase or sign retrieval is Zariski open in K d×m , by [3]. The analogous observation has been made for conjugate phase retrieval in [11].…”

Section: Phase Retrieval For Group Framesmentioning

confidence: 74%

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Phase retrieval for affine groups over prime fields

Bartusel¹,

Führ²,

Oussa³

2021

Preprint

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We study phase retrieval for group frames arising from permutation representations, focusing on the action of the affine group of a finite field. We investigate various versions of the phase retrieval problem, including conjugate phase retrieval, sign retrieval, and matrix recovery. Our main result establishes that the canonical irreducible representation of the affine group Zp ⋊ Z * p (with p prime), acting on the vectors in C p with zero-sum, has the strongest retrieval property, allowing to reconstruct matrices from scalar products with a group orbit consisting of rank-one projections. We explicitly characterize the generating vectors that ensure this property, provide a linear matrix recovery algorithm and explicit examples of vectors that allow matrix recovery. We also comment on more general permutation representations.

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“…Then the set of all Ψ that perform phase or sign retrieval is Zariski open in K d×m , by [3]. The analogous observation has been made for conjugate phase retrieval in [11].…”

Section: Phase Retrieval For Group Framesmentioning

confidence: 74%

“…[3]. The intermediate notion of conjugate phase retrieval is due to Evans and Lai [11]. Matrix recovery and associated linear reconstruction methods were introduced to this discussion in [2]; see also [6].…”

Section: Phase Retrieval For Group Framesmentioning

confidence: 99%

Phase retrieval for affine groups over prime fields

Bartusel¹,

Führ²,

Oussa³

2021

Preprint

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“…In [9] m = 4n−4 measurements are shown to be sufficient in the complex setting. In [17] the conjugate phase retrieval is discussed: here the vectors w i are real measurements of complex signals in C n . These measurements are invariant with respect to global phase multiplication and also complex conjugation, and it is shown that 4n − 6 generic measurements are separating with respect to this group action.…”

Section: Related Workmentioning

confidence: 99%

Low Dimensional Invariant Embeddings for Universal Geometric Learning

Dym¹,

Gortler²

2022

Preprint

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This paper studies separating invariants: mappings on d-dimensional semi-algebraic subsets of D dimensional Euclidean domains which are invariant to semi-algebraic group actions and separate orbits. The motivation for this study comes from the usefulness of separating invariants in proving universality of equivariant neural network architectures.We observe that in several cases the cardinality of separating invariants proposed in the machine learning literature is much larger than the ambient dimension D. As a result, the theoretical universal constructions based on these separating invariants is unrealistically large. Our goal in this paper is to resolve this issue.We show that when a continuous family of semi-algebraic separating invariants is available, separation can be obtained by randomly selecting 2d+1 of these invariants. We apply this methodology to obtain an efficient scheme for computing separating invariants for several classical group actions which have been studied in the invariant learning literature. Examples include matrix multiplication actions on point clouds by permutations, rotations, and various other linear groups.

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“…To demonstrate that Theorem 3 is not vacuous, we present an example here. For this purpose, we require a result from [13] concerning conjugate phase retrieval in C 2 and C 3 :…”

Section: 2mentioning

confidence: 99%

“…Modern phase retrieval is often considered in the case of frames [6,5,7]. Conjugate phase retrieval for frames was introduced in [13] (see also [11,33]). Phase retrieval in the context of wavelets and other systems appear in [26,9,1].…”

Section: Introductionmentioning

confidence: 99%

Conjugate Phase Retrieval in Paley-Wiener Space

Lai

Littmann

Weber

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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We consider the problem of conjugate phase retrieval in Paley-Wiener space P W π . The goal of conjugate phase retrieval is to recover a signal f from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning f (t) and f (t) are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in P W π by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in P W π by sampling both f and f only on the real line. Moreover, we demonstrate experimentally that the Gerchberg-Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate.

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Conjugate phase retrieval on CM by real vectors

Cited by 14 publications

References 22 publications

Phase retrieval for affine groups over prime fields

Phase retrieval for affine groups over prime fields

Low Dimensional Invariant Embeddings for Universal Geometric Learning

Conjugate Phase Retrieval in Paley-Wiener Space

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