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Phase retrieval of complex and vector-valued functions

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Pith's one-line read Phase retrieval reduces to a perpendicular-split test for complex and vector-valued functions.

desk verdict The vector-valued characterization is sound in finite dimensions, but Assumption 3.1 fails for natural infinite-dimensional measurements, so the infinite-dimensional claims overreach; the quaternion and graph results are genuinely worth keeping. read the letter →

arxiv 1909.02078 v1 pith:GIJAPTQT submitted 2019-09-04 math.FA cs.ITmath.IT

Yang Chen , Cheng Cheng , Qiyu Sun This is my paper · ORCID
classification math.FAcs.ITmath.IT MSC 42C1546C05
keywords phaseretrievalvector-valuedfunctionsquaternion-conjugateunitaryinvariancecomplementpropertyaffinevectorfieldsongraphsinfinite-dimensionalHilbertspaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a common geometric criterion for phase retrieval across three settings: complex functions in conjugation-invariant spaces, quaternion-valued functions, and vector-valued functions in unitary-invariant spaces with unitary-invariant measurements. In each case, a function is recoverable from the magnitudes of its linear measurements, up to the natural trivial ambiguity, if and only if it cannot be written as $f=u+v$ where every measurement of $u$ is orthogonal to the corresponding measurement of $v$, yet some pair of cross-measurements has a nonzero symmetric sum. The characterization makes no dimension restriction and, in the real scalar case, reduces to the classical complement property for measurement systems. The same machinery is then applied to reconstruct vector fields on graphs—up to an orthogonal matrix—from absolute magnitudes at vertices and relative magnitudes between neighboring vertices, and to affine phase retrieval of vector-valued functions. A sympathetic reader would care because the result supplies a single structural test that governs uniqueness in a wide family of phaseless-recovery problems.

What carries the argument

The load-bearing mechanism is the perpendicular-split identity. If a rival function $g$ has the same magnitude data as $f$ and is not a unitary image of $f$, then $u=(f+g)/2$ and $v=(f-g)/2$ produce exactly such a split: the measurement orthogonality comes from the equality of magnitudes, and the nonzero symmetric cross-term comes from the difference between the inner products of measurements of $g$ and of $f$. Conversely, any perpendicular split yields a rival $g=u-v$ with identical magnitudes and different pairwise measurement inner products. Assumption 3.1 is the bridge that lets magnitude data control those inner products: it states that equal pairwise inner products for all measurement pairs imply the two functions differ by a unitary, a condition that is automatic in finite dimensions but is an extra hypothesis in infinite dimensions.

What would settle it

Construct an infinite-dimensional real Hilbert space $H$, a unitary-invariant space $S$ of $H$-valued functions on a two-point domain, and a unitary-invariant measurement family $\Phi$ whose measurement map is injective but whose image does not determine all pairwise inner products—for instance, choose the two point evaluations so that the span of one function's measurements has an orthogonal complement of different dimension from the span of a competitor's measurements. If for such $f$ the pairwise-inner-product condition (3.8) holds for some $g$ that is not a unitary image of $f$, and no perpendicular split $u,v$ exists, then the 'if' direction of Theorem 3.2 fails and Assumption 3.1 is pinpointed as the obstruction.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 3.2: in a unitary-invariant space $S$ of $H$-valued functions with unitary-invariant measurements $\Phi$ satisfying Assumption 3.1, a function $f$ is phase retrieval—determined up to a unitary operator on $H$ by the magnitudes $\|\varphi(f)\|$—if and only if there are no $u,v\in S$ with $f=u+v$, $\langle\varphi(u),\varphi(v)\rangle=0$ for every $\varphi\in\Phi$, and $\langle\varphi_0(u),\varphi_1(v)\rangle+\langle\varphi_1(u),\varphi_0(v)\rangle\neq 0$ for some $\varphi_0,\varphi_1\in\Phi$. The same perpendicular-split condition, with the trivial ambiguity adjusted, characterizes complex conjugate phase retrieval of complex functions and quaternion conjugate phase retrieval of quaternion-valued functions. Direct corollaries are that a real scalar space is phase retrieval exactly when the measurement set has the complement property, that phase retrieval passes to projections onto subspaces, and that a vector field on a complete graph is determined up to an orthogonal matrix by its absolute and neighboring relative magnitudes, with a connected-simplex-graph condition covering non-complete graphs. Affine phase retrieval of vector-valued functions is characterized by injectivity of a family of linear maps built from the reference vectors.

Load-bearing premise

For the vector-valued theorems, the argument assumes that any two functions whose measurements have the same pairwise dot products must be related by a single global rotation; this is automatic in finite dimensions but is a strictly stronger, unproved condition in infinite dimensions.

Editorial extensions

If this is right

  • In finite-dimensional Hilbert spaces, the vector-valued characterization is an 'if and only if' once the measurement map is injective, and in the real scalar case it reproduces the complement property as the exact condition for phase retrieval.
  • Phase retrieval is hereditary: if $f$ is phase retrieval in $S$, then its projection onto any subspace $\tilde H$ is phase retrieval in the projected space, so uniqueness on a subspace is forced by uniqueness on the whole space.
  • A velocity field on a complete graph is determined up to an orthogonal matrix by absolute speeds at vertices and relative speeds between every pair of vertices; for a general graph the same holds when the associated $d$-simplex graph is connected and covers all vertices.
  • With at least two reference vectors per measurement, affine phase retrieval of a vector-valued space is equivalent to injectivity of the measurement map whenever the reference differences span the Hilbert space.
  • For the shift-invariant space generated by the hat function, the paper identifies precisely which coefficient sequences are complex conjugate phase retrieval: all coefficients in the support interval must be nonzero and at most one adjacent pair may have a nonzero imaginary cross term.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Assumption 3.1 could be established for natural infinite-dimensional measurement systems—sampling at points or frames—the perpendicular-split test would become a practical certificate for uniqueness in those systems; the paper proves the assumption in finite dimensions and leaves the infinite-dimensional case open.
  • The perpendicular-split condition is algebraic: in the finite-dimensional range-space case it is equivalent to excluding a real matrix of rank at most two with zero diagonal traces, which suggests a low-rank matrix test for conjugate phase retrieval that could be checked by semidefinite relaxation.
  • The graph reconstruction theorem implies a distributed reconstruction algorithm: each $(d+1)$-clique fixes a local orthogonal frame, and connectivity of the $d$-simplex graph is exactly the condition for gluing these local frames into one global orthogonal transformation; where the graph disconnects, one should expect gauge ambiguities between components.
  • In affine phase retrieval, the paper's necessary-and-sufficient condition suggests that when the reference differences fail to span the space, the number of reference vectors required will grow with the codimension of their span—a quantitative question the paper does not address.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper develops phase-retrieval characterizations for complex-valued, quaternion-valued, and vector-valued functions from the magnitudes of linear measurements. Section 2 characterizes complex conjugate phase retrieval in a conjugate-invariant linear space by the non-existence of a split f=u+v satisfying the orthogonality condition (2.8) and the non-vanishing cross-correlation condition (2.9), with applications to matrix range spaces, shift-invariant spaces, and quaternion-valued functions. Section 3 introduces unitary-invariant spaces of vector-valued functions and, under Assumption 3.1, characterizes phase retrieval of a single function and of the whole space by the absence of a perpendicular split (Theorems 3.2 and 3.3), together with a projection theorem and a graph-vector-field reconstruction theorem. Section 4 treats affine phase retrieval through injectivity of the maps T_u. The paper states that all proofs are collected in Section 5.

Significance. If the results hold as stated, the paper provides a useful unifying framework: the scalar characterizations recover the classical complement-property equivalence, the quaternion characterization is new, and the graph-vector-field theorem gives a concrete connectivity condition for reconstruction up to an orthogonal matrix. The proofs that are actually supplied are generally clean, and the isomorphism argument in Theorem 2.10 is elegant. However, the infinite-dimensional vector-valued results rest on Assumption 3.1, for which the paper proves no infinite-dimensional instance and for which a natural point-evaluation family fails; moreover, Theorem 4.1 is missing from the proof section despite the paper's declared policy. The finite-dimensional and scalar parts appear sound, but the advertised infinite-dimensional vector-valued claims are not established.

major comments (2)
  1. [Section 3.1 (Assumption 3.1; Theorems 3.2–3.3)] The central vector-valued characterization is conditional on Assumption 3.1, but the paper only proves this assumption in the finite-dimensional case (Remark 3.4) and supplies no infinite-dimensional example satisfying it. In infinite dimensions the assumption is strictly stronger than injectivity of T_Φ, and the natural point-evaluation family on the unitary-invariant space S=ℓ²(N;ℓ²(N)) violates it: let f(n)=2^{-n}e_{n+1} and g(n)=2^{-n}e_{n+2}. Then ⟨g(n),g(m)⟩=4^{-n}δ_{nm}=⟨f(n),f(m)⟩ for all n,m, so (3.8) holds for Φ={δ_n}, but any U with g=Uf would satisfy Ue_{n+1}=e_{n+2} for all n≥1; such a U is not onto H, contradicting unitarity. Thus the abstract's promise of an infinite-dimensional vector-valued characterization is not established; the theorems hold only under a hypothesis whose verified instances are finite-dimensional. The authors should either prove Assumption 3.1 for a nontrivial infinite-dimensional measurement family or explicitly restrict the vector-valued theorems to the finite-dimensional case.
  2. [Section 4 (Theorem 4.1)] Theorem 4.1 is stated as a characterization of affine phase retrieval of a linear space, yet Section 5—which the paper says collects all proofs—does not prove it. The paragraph preceding the theorem only derives the identity ⟨φ(f−g),φ(f+g)+2b_{φ,i}⟩=0 from equal affine magnitudes; it does not establish the asserted equivalence, in particular the necessity direction that injectivity of every T_u follows from affine phase retrieval. Please provide a complete proof of both directions, or explicitly include the missing argument in Section 5.
minor comments (4)
  1. [Section 5.6 (proof of Proposition 2.8)] In the displayed expansion after (5.37), the final term '2µ_iµ_j a_1 a_1^T' should presumably be '2µ_iµ_j a_2 a_2^T'.
  2. [Section 2.3] The notation Q_8 for the quaternions is nonstandard and is easily confused with the quaternion group of order 8; consider using H or a different symbol.
  3. [Figure 1 caption] The caption is hard to parse, especially the clause beginning 'where we assume that the hyperplane containing two vectors ...'; please rephrase it for clarity.
  4. [Section 1 and Section 5] The statement 'All proofs are collected in Section 5' is inaccurate also for Theorem 3.3, whose proof is not listed there, although it is a direct consequence of Theorem 3.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central vector-valued characterization is a direct equivalence under an explicit assumption, and the only self-citation is a non-load-bearing prior lemma.

full rationale

I find no circular step. The main claims are derived from definitions and explicit hypotheses rather than from fitted quantities or self-referential renaming. Theorem 3.2 is proved in Section 5.1 by a genuine equivalence: given u, v satisfying (3.11)-(3.13), the function g = u - v has the same magnitude measurements but a different pairwise-geometry, contradicting phase retrieval; conversely, given g in M_{f,Phi} that is not a unitary image, Assumption 3.1 supplies a pairwise inner-product mismatch, and setting u = (f+g)/2, v = (f-g)/2 produces exactly the forbidden split. Assumption 3.1 is explicitly stated, not derived from the conclusion, and the finite-dimensional sufficiency in Remark 3.4 is a proof, not a definitional shortcut. The scalar corollaries are consequences, with the known complement-property equivalence credited to [4,8,9,12]. The only self-citation used as a proof input is [17, Theorem 3.2] (Lemma 5.2) in the hat-function application Proposition 2.9; this is a parameter-free real-scalar lemma with stated assumptions and is not the target vector-valued result, so it is external support rather than load-bearing circularity. The fragility of Assumption 3.1 in infinite dimensions, where injectivity of T_Phi is necessary but not sufficient and a point-evaluation counterexample can violate it, is a limitation of applicability rather than an equivalence-by-construction: the characterization is conditional but not circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

There are no fitted constants or numerical data. The theorems rely on explicit structural assumptions (unitary invariance, conjugate invariance, Assumption 3.1) and on standard linear algebra facts. Lemma 5.2 from the authors' prior work [17] is used for the hat-function example. No physical entities are postulated; the d-simplex graph is a definitional construction, not an entity with independent empirical content.

assumptions (7)
  • domain assumption Unitary invariant space S and unitary invariant measurement family Phi (Section 3, equations (3.1)-(3.3)).
    The vector-valued phase retrieval definition and Theorems 3.2 and 3.3 require S and Phi to commute with all unitaries on H; without this, the orbit Uf is not guaranteed to be contained in the set of functions with the same magnitude observations.
  • domain assumption Assumption 3.1: equal pairwise measurement Gram matrices imply unitary equivalence.
    This is the bridge in the sufficiency direction of Theorem 3.2. In finite dimensions it is equivalent to injectivity of the measurement map (Remark 3.4), but in infinite dimensions it is a genuine extra hypothesis.
  • domain assumption The linear space C is complex conjugate invariant (Section 2.1).
    Conjugate phase retrieval is meaningful only when the conjugate of every function lies in C; the R^2 embedding proof of Theorem 2.1 uses this invariance.
  • domain assumption The linear space W is quaternion conjugate invariant (Section 2.3, equation (2.17)).
    The quaternion analogue needs qf and (qf)^* to remain in W so that the four real components f1,...,f4 and their quaternion combinations stay in W.
  • domain assumption Lemma 5.2 from [17] characterizing phase retrieval in the real hat-function shift-invariant space.
    Proposition 2.9 uses this external characterization, from the authors' prior work, to decide which coefficient patterns are conjugate phase retrieval; it is imported rather than proved here.
  • standard math Complement property equivalence for scalar frames (Corollary 3.6).
    The paper re-derives the known Balan-Casazza-Edidin equivalence; it is used as a check on the scalar case rather than as an input to the vector-valued theorems.
  • standard math Gram-Schmidt orthonormalization and unitary operators on Hilbert space.
    Remark 3.4 and Lemma 5.1 rely on the standard fact that equal Gram matrices imply an isometry between spans, which is standard linear algebra.
invented entities (1)
  • d-simplex graph G_f associated to a vector field f
    purpose: Encodes which (d+1)-tuples of vertices have affinely independent vector values, so local reconstructions on d-simplices can be stitched together.
    This is a defined combinatorial object, not an empirical entity; it is included because the graph theorem depends on it, and its connectedness is a checkable condition from the measured magnitudes (Remark 3.11).

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Pith. "Pith review of Phase retrieval of complex and vector-valued functions." pith.science (2026). https://pith.science/paper/GIJAPTQT

@misc{pith2026190902078,
  author       = {Pith},
  title        = {Pith review of: Phase retrieval of complex and vector-valued functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIJAPTQT}},
  note         = {Machine review of arXiv:1909.02078}
}
abstract

The phase retrieval problem in the classical setting is to reconstruct real/complex functions from the magnitudes of their Fourier/frame measurements. In this paper, we consider a new phase retrieval paradigm in the complex/quaternion/vector-valued setting, and we provide several characterizations to determine complex/quaternion/vector-valued functions $f$ in a linear space $S$ of (in)finite dimensions, up to a trivial ambiguity, from the magnitudes $\|\phi(f)\|$ of their linear measurements $\phi(f), \phi\in \Phi$. Our characterization in the scalar setting implies the well-known equivalence between the complement property for linear measurements $\Phi$ and the phase retrieval of linear space $S$. In this paper, we also discuss the affine phase retrieval of vector-valued functions in a linear space and the reconstruction of vector fields on a graph, up to an orthogonal matrix, from their absolute magnitudes at vertices and relative magnitudes between neighboring vertices.

Figures

Figures reproduced from arXiv: 1909.02078 by the authors.

Figure 1
Figure 1. Plotted are the graph G with vertices marked by blue dots from A to K and edges in black dashed lines, and the 2-simplex graph Gf with vertices {4ACD, 4BCE, 4CDE, 4CDF, 4CEF, 4DEF, 4DF H, 4EF G, 4F GH, 4GHI, 4HIK, 4IJK} marked by the red filled triangle located around the centroid and edges between 2-simplices in red solid lines, where we assume that the hyperplane containing two vectors of the vector field f locate… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Random phaseless sampling for causal signals in shift-invariant spaces: a zero distribution perspective

    cs.IT 2019-08 conditional novelty 7.0 of 10

    If a signal generator satisfies a new zero-set condition, causal signals in complex-generated shift-invariant spaces can be recovered from three random magnitude samples per unit interval, with probability one.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aldroubi and K

    A. Aldroubi and K. Gr¨ ochenig, Non-uniform sampling and reconstruction in shift- invariant spaces, SIAM Rev., 43(2001), 585–620

    work page 2001

  2. [2]

    Aldroubi, Q

    A. Aldroubi, Q. Sun and W.-S. Tang, Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces, J. Fourier Anal. Appl. , 11(2005), 215–244

    work page 2005

  3. [3]

    Alaifari, I

    R. Alaifari, I. Daubechies, P. Grohs and R. Yin, Stable phase retrieval in infinite dimensions, Found. Comput. Math. , to appear

    work page

  4. [4]

    Alaifari and P

    R. Alaifari and P. Grohs, Phase retrieval in the general setting of continuous frames for Banach spaces, SIAM J. Math. Anal. , 49(2017), 1895–1911. PHASE RETRIEVAL OF COMPLEX AND VECTOR-VALUED FUNCTIONS 27

    work page 2017

  5. [5]

    Alexeev, A

    B. Alexeev, A. S. Bandeira, M. Fickus and D. G. Mixon, Phase retrieval with polar- ization, SIAM J. Imaging Sci. , 7(2014), 35–66

    work page 2014

  6. [6]

    S. L. Altmann, Rotations, Quaternions, and Double Groups , Oxford, England: Clarendon Press, 1986

    work page 1986

  7. [7]

    M. A. Alvarez, L. Rosasco and N. D. Lawrence, Kernels for vector-valued functions: A review, Found. Trends Mach. Learn. , 4(2012), 195–266

    work page 2012

  8. [8]

    Balan, B

    R. Balan, B. G. Bodmann, P. G. Casazza and D. Edidin, Painless reconstruction from magnitudes of frame coefficients, J. Fourier Anal. Appl. , 15(2009), 488–501

    work page 2009

Show all 56 references
  1. [9]

    Balan, P

    R. Balan, P. G. Casazza and D. Edidin, On signal reconstruction without phase, Appl. Comput. Harmon. Anal. , 20(2006), 345–356

    2006

  2. [10]

    Bamieh, F

    B. Bamieh, F. Paganini and M.A. Dahleh, Distributed control of spatially-invariant systems, IEEE Trans. Autom. Control , 47(2002), 1091–1107

    2002

  3. [11]

    A. S. Bandeira, J. Cahill, D. G. Mixon and A. A. Nelson, Saving phase: injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal. , 37(2014), 106–125

    2014

  4. [12]

    Cahill, P

    J. Cahill, P. Casazza and I. Daubechies, Phase retrieval in infinite-dimensional Hilbert spaces, Trans. Amer. Math. Soc. Ser. B , 3(2016), 63–76

    2016

  5. [13]

    Cahill, P

    J. Cahill, P. Casazza, J. Peterson and L. Woodland, Phase retrieval by projections, Houston J. Math. , 42(2016), 537–558

    2016

  6. [14]

    E. J. Candes, T. Strohmer and V. Voroninski, PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming, Commun. Pure Appl. Math. , 66(2013), 1241–1274

    2013

  7. [15]

    Carmeli, E

    C. Carmeli, E. de Vito, A. Toigo and V. Umanita, Vector valued reproducing kernel Hilbert spaces and universality, Anal. Appl. (Singap.) , 8(2010), 19–61

    2010

  8. [16]

    A. L. Casanovas, G. Monaci, P. Vandergheynst and R. Gribonval, Blind audiovisual source separation based on sparse redundant representations, IEEE Trans. Multime- dia, 12(2010), 358–371

    2010

  9. [17]

    Y. Chen, C. Cheng, Q. Sun and H. Wang, Phase retrieval of real-valued signals in a shift-invariant space, Appl. Comput. Harmon. Anal. , to appear

  10. [18]

    Cheng, Y

    C. Cheng, Y. Jiang and Q. Sun, Spatially distributed sampling and reconstruction, Appl. Comput. Harmon. Anal. , 47(2019), 109–148

    2019

  11. [19]

    Cheng, J

    C. Cheng, J. Jiang and Q. Sun, Phaseless sampling and reconstruction of real-valued signals in shift-invariant spaces, J. Fourier Anal. Appl. , 25(2019), 1361–1394

    2019

  12. [20]

    Cheng and Q

    C. Cheng and Q. Sun, Stable phaseless sampling and reconstruction of real-valued signals with finite rate of innovations, arXiv:1801.05538

    arXiv

  13. [21]

    de Boor, R

    C. de Boor, R. A. DeVore and A. Ron, The structure of finitely generated shift- invariant spaces in L2(Rd), J. Funct. Anal. , 119(1994), 37–78

    1994

  14. [22]

    D. L. Donoho and G. Kutyniok, Microlocal analysis of the geometric separation prob- lem, Commun. Pure Appl. Math. , 66(2013), 1–47

    2013

  15. [23]

    D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decompositions, IEEE Trans. Inform. Theory , 47(2001), 2845–2862

    2001

  16. [24]

    Edidin, Projections and phase retrieval, Appl

    D. Edidin, Projections and phase retrieval, Appl. Comput. Harmon. Anal. , 42(2017), 350–359

    2017

  17. [25]

    Evans and C

    L. Evans and C. K. Lai, Conjugate phase retrieval on CM by real vectors, arXiv: 1709.08836

    arXiv

  18. [26]

    J. R. Fienup, Reconstruction of an object from the modulus of its Fourier transform, Opt. Lett., 3(1978), 27–29

    1978

  19. [27]

    B. Gao, Q. Sun, Y. Wang and Z. Xu, Phase retrieval from the magitudes of affine linear measurements, Adv. Appl. Math. , 93(2018), 121–141

    2018

  20. [28]

    Grohs and M

    P. Grohs and M. Rathmair, Stable Gabor phase retrieval and spectral clustering, Commun. Pure Appl. Math. , 72(2019), 981–1043

    2019

  21. [29]

    M. H. Hayes, J. S. Lim and A. V. Oppenheim, Signal reconstruction from phase or magnitude, IEEE Trans. Acoust. Speech Signal Process. , 28(1980), 672–680. 28 YANG CHEN, CHENG CHENG AND QIYU SUN

    1980

  22. [30]

    Huang and Z

    M. Huang and Z. Xu, Phase retrieval from the norms of affine transformations, arXiv: 1805.07899

    arXiv

  23. [31]

    N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction, Springer, 2001

    2001

  24. [32]

    Jaganathan, Y

    K. Jaganathan, Y. C. Eldar and B. Hassibi, Phase retrieval: an overview of recent developments, In Optical Compressive Imaging, edited by A. Stern, Taylor & Francis, 2016

    2016

  25. [33]

    Jiang, C

    J. Jiang, C. Cheng and Q. Sun, Nonsubsampled graph filter banks: theory and dis- tributed algorithms, IEEE Trans. Signal Process. , 67(2019), 3938–3953

    2019

  26. [34]

    M. V. Klibanov, P. E. Sacks and A. V. Tikhonravov, The phase retrieval problem, Inverse problems, 11(1995), 1–28

    1995

  27. [35]

    C. K. Lai, F. Littmann and E. S. Weber, Conjugate phase retrieval in Paley-Wiener space, in Proc. SampTA, 2019

    2019

  28. [36]

    L. Li, T. Juste, J. Brennan, C. Cheng and D. Han, Phase retrievable projective representation frames for finite abelian groups, J. Fourier Anal. Appl. , 25(2019), 86–100

    2019

  29. [37]

    J. K. Kuipers, Quaternions and Rotation Sequences: A Primer with Applications to Oribits, Aerospace, and Virtual Reality , Princeton University Press, 1998

    1998

  30. [38]

    P. Li, W. Chen and Q. Sun, Inertial proximal ADMM for separable multi-block convex optimizations and compressive affine phase retrieval, In preparation

  31. [39]

    Li and D

    Y. Li and D. Han, Two-step PR-scheme for recovering signals in detectable union of cones by magnitude measurements, arXiv: 1809.08136

    arXiv

  32. [40]

    Liebling, T

    M. Liebling, T. Blu, E. Cuche, P. Marquet, C. D. Depeursinge and M. Unser, Local amplitude and phase retrieval method for digital holography applied to microscopy, In Proc. SPIE, 5143(2003), 210–214

    2003

  33. [41]

    J. Lin, S. Li and Y. Shen, Compressed data separation with redundant dictionaries, IEEE Trans. Inform. Theory , 59(2013), 4309–4315

    2013

  34. [42]

    R. Lin, G. Song and H. Zhang, Multi-task learning in vector-valued reproducing kernel Banach spaces with the 𝓁1 norm, arXiv:1901.01036

    1901 arXiv

  35. [43]

    Mallat and I

    S. Mallat and I. Waldspurger, Phase retrieval for the Cauchy wavelet transform, J. Fourier Anal. Appl. , 21(2015), 1251–1309

    2015

  36. [44]

    McDonald, Phase retrieval and magnitude retrieval of entire functions, J

    J. McDonald, Phase retrieval and magnitude retrieval of entire functions, J. Fourier Anal. Appl. , 10(2004), 259–267

    2004

  37. [45]

    C. A. Micchelli and M. Pontil, On learning vector-valued functions, Neural Comput., 17(2005), 177–204

    2005

  38. [46]

    R. P. Millane, Phase retrieval in crystallography and optics, J. Opt. Soc. Am. A. , 7(1990), 394–411

    1990

  39. [47]

    Motee and A

    N. Motee and A. Jadbabaie, Optimal control of spatially distributed systems, IEEE Trans. Autom. Control, 53(2008), 1616–1629

    2008

  40. [48]

    Motee and Q

    N. Motee and Q. Sun, Sparsity and spatial localization measures for spatially dis- tributed systems, SIAM J. Control Optim. , 55(2017), 200–235

    2017

  41. [49]

    V. Pohl, F. Yang and H. Boche, Phaseless signal recovery in infinite dimensional spaces using structured modulations, J. Fourier Anal. Appl. , 20(2014), 1212–1233

    2014

  42. [50]

    Shechtman, Y

    Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao and M. Segev, Phase retrieval with application to optical imaging: a contemporary overview, IEEE Signal Process. Mag., 32(2015), 87–109

    2015

  43. [51]

    B. A. Shenoy, S. Mulleti and C. S. Seelamantula, Exact phase retrieval in principal shift-invariant spaces, IEEE Trans. Signal Process. , 64(2015), 406–416

    2015

  44. [52]

    Sun, Local and global phaseless sampling in real spline spaces, arXiv: 1705.00836

    W. Sun, Local and global phaseless sampling in real spline spaces, arXiv: 1705.00836

    arXiv

  45. [53]

    Thakur, Reconstruction of bandlimited functions from unsigned samples, J

    G. Thakur, Reconstruction of bandlimited functions from unsigned samples, J. Fourier Anal. Appl. , 17(2011), 720–732

    2011

  46. [54]

    Waldspurger, A

    I. Waldspurger, A. d’Aspremont and S. Mallat, Phase recovery, MaxCut and complex semidefinite programming, Math. Program., 149(2015), 47–81. PHASE RETRIEVAL OF COMPLEX AND VECTOR-VALUED FUNCTIONS 29

    2015

  47. [55]

    Wang and Z

    Y. Wang and Z. Xu, Phase retrieval for sparse signals, Appl. Comput. Harmon. Anal. , 37(2014), 531–544

    2014

  48. [56]

    Wang and Z

    Y. Wang and Z. Xu, Generalized phase retrieval: Measurement number, matrix re- covery and beyond, Appl. Comput. Harmon. Anal. , 45(2019), 423–446. Chen: Key Laboratory of Computing and Stochastic Mathematics (Min- istry of Education), School of Mathematics and Statistics, Huna...

    2019

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