Complex Phase Retrieval from Subgaussian Measurements

Abstract

Phase retrieval refers to the problem of reconstructing an unknown vector $$x_0 \in {\mathbb {C}}^n$$ x 0 ∈ C n or $$x_0 \in {\mathbb {R}}^n $$ x 0 ∈ R n from m measurements of the form $$y_i = \big \vert \langle \xi ^{\left( i\right) }, x_0 \rangle \big \vert ^2 $$ y i = | ⟨ ξ i , x 0 ⟩ | 2 , where $$ \left\{ \xi ^{\left( i\right) } \right\} ^m_{i=1} \subset {\mathbb {C}}^m $$ ξ i i = 1 m ⊂ 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 $$ \left\{ \xi ^{\left( i\right) } \right\} ^{m}_{i=1}$$ ξ i i = 1 m such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector $$ \xi ^{\left( i\right) } \in {\mathbb {C}}^m $$ ξ i ∈ C m does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals $$x_0 \in {\mathbb {R}}^n$$ x 0 ∈ 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 $$x_0 \in {\mathbb {C}}^n $$ x 0 ∈ C n up to an additional assumption which, as we show, is necessary.

Error Bound of Empirical $\ell_2$ Risk Minimization for Noisy Standard and Generalized Phase Retrieval Problems

“…This involves an exhausting calculation if the fourth power of quadratic chaos, but the calculation can be avoided by applying Lemma 3 instead. Similar argument was also used in [33] for extension of [7] to sub-Gaussian measurement.…”

“…Under Gaussian measurement, a stable reconstruction result || x x * − x 0 x * 0 || F = O ||η|| was given in the first study of phaselift [9], and the bound was improved to O ||η|| 1 n under a different lifting formulation [7]. A more recent work [33] extended these results to sub-Gaussian measurement. Nevertheless, phaselift cannot handle large-scale problem due to the prohibitive storage and computational load, hence in many cases the reconstruction can only be carried out in the original vector field.…”

Proof methods for robust low-rank matrix recovery

“…The problem here is that all the mass of 𝑍 is concentrated on its diagonal. The proof in [KS20a] shows that this cannot be the case, if 𝑥 0 is incoherent.…”

“…Such ambiguities can be addressed by an additional constraints on the measurements; then the proof techniques sketched above carries over. We refer the interested reader to [KS20a] for details.…”

The numerics of phase retrieval

“…The biggest impact PhaseLift has had on phase retrieval is that on the one hand it triggered a broad and systematic study of numerical algorithms for phase retrieval, and on the other hand it ignited a sophisticated design of initializations for non-convex solvers. Beyond phase retrieval, it ignited research in related areas, such as in bilinear compressive sensing (Ling and Strohmer 2015), including blind deconvolution (Ahmed, Recht and Romberg 2013, Li, Lee and Bresler 2016, Krahmer and Stöger 2019) and blind demixing (Ling and Strohmer 2017). Moreover, the techniques behind PhaseLift and sparse recovery have influenced other areas directly related to phase retrieval, namely low-rank phase retrieval problems as they appear for instance in quantum tomography, as well as utilizing sparsity in phase retrieval.…”