Complex Phase Retrieval from Subgaussian Measurements
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Complex Phase Retrieval from Subgaussian Measurements Felix Krahmer1 · Dominik Stöger1,2 Received: 3 July 2019 / Revised: 17 July 2020 / Accepted: 19 August 2020 / Published online: 20 November 2020 © The Author(s) 2020
Abstract Phase retrieval refers to the problem of reconstructing an unknown vector x0 ∈ Cn or 2 m x0 ∈ Rn from m measurements of the form yi = ξ (i) , x0 , where ξ (i) i=1 ⊂ Cm are known measurement vectors. While Gaussian measurements allow for recovery of arbitrary signals provided the number of measurements scales at least linearly in the number of dimensions, it has m shown that ambiguities may arise for certain been other classes of measurements ξ (i) i=1 such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector ξ (i) ∈ Cm does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals x0 ∈ Rn from the measurements. This extends recent work by Krahmer and Liu from the real-valued to the complex-valued case. However, our proof strategy is quite different and we expect some of the new proof ideas to be useful in several other measurement scenarios as well. We then extend our results x0 ∈ Cn up to an additional assumption which, as we show, is necessary. Keywords Phase retrieval · Small-ball method · Convex optimization · Descent cone analysis Mathematics Subject Classification 94A17
1 Introduction Phase retrieval refers to the problem of reconstructing an unknown vector x0 ∈ Cn from m measurements of the form yi = |ξ (i) , x0 |2 + wi
(where i ∈ [m]) ,
Communicated by Holger Rauhut.
B
Dominik Stöger [email protected]
1
Technische Universität München, Munich, Germany
2
Present Address: University of Southern California, Los Angeles, USA
(1.1)
89
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Journal of Fourier Analysis and Applications (2020) 26:89
where the ξ (i) ∈ Cn are known measurement vectors and wi ∈ R represents additive noise. Such problems are ubiquituous in many areas of science and engineering such as X-ray crystallography [23,32], astronomical imaging [18], ptychography [35], and quantum tomography [28]. The foundational papers [4,7,13] proposed to reconstruct x0 via the PhaseLift method, a convex relaxation of the original problem. These papers have triggered many followup works since they were the first to establish rigorous recovery guarantees under the assumption that the measurement vectors ξ (i) are sampled uniformly at random from the sphere. Since then several papers have analyzed scenarios where the measurement vectors possess a significantly reduced amount of randomness, in particular spherical designs [21] and coded diffraction patterns [5,22]. However, the theoretical results for coded diffraction patterns rely on the assumption that the modulus of the illumination patterns is varying. Indeed, it was shown in [17] that for certain illumination patterns with constant modulus ambiguities can arise, i.e., it is not possible to determine x0 uniquely from th
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