Abstract
Phase retrieval refers to the problem of reconstructing an unknown vector x_0 in {mathbb {C}}^n or x_0 in {mathbb {R}}^n from m measurements of the form y_i = big vert langle xi ^{left( iright) }, x_0 rangle big vert ^2 , where left{ xi ^{left( iright) } right} ^m_{i=1} subset {mathbb {C}}^m 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 been shown that ambiguities may arise for certain other classes of measurements left{ xi ^{left( iright) } right} ^{m}_{i=1} such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector xi ^{left( iright) } in {mathbb {C}}^m does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals x_0 in {mathbb {R}}^n 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 x_0 in {mathbb {C}}^n up to an additional assumption which, as we show, is necessary.
Highlights
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, (1.1)Communicated by Holger Rauhut. 89 Page 2 of 27Journal of Fourier Analysis and Applications (2020) 26:89 where the ξ (i) ∈ Cn are known measurement vectors and wi ∈ R represents additive noise
We extend our results x0 ∈ Cn up to an additional assumption which, as we show, is necessary
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]
Summary
Journal of Fourier Analysis and Applications (2020) 26:89 where the ξ (i) ∈ Cn are known measurement vectors and wi ∈ R represents additive noise. This paper provides an analysis both for real-valued and complex-valued signals We believe that this understanding will be of importance for the subsequent study of structured scenarios such as coded diffraction patterns, which are intrinsically complex in nature. We control one of these cones using a restricted isometry property and one via the small-ball method We think that this novel viewpoint and some of the techniques developed in this paper will be useful for the analysis of other interesting measurement scenarios, such as the case of heavy-tailed measurement vectors ξ (i) or the case that ξ (i) has only entries 0 and 1
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