Saving phase: Injectivity and stability for phase retrieval

By Afonso S. Bandeira, Jameson Cahill, Dustin G. Mixon, Aaron A. Nelson. 2013

343 citations recorded by the source corpus.

Abstract Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles yield stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4 M − 4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M = 2 , 3 . Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer–Rao lower bounds to identify stability with stronger versions of the injectivity characterizations.

Subfields: Computer Vision and Pattern Recognition, Electrical and Electronic Engineering, Radiation

View Paper: https://www.semanticscholar.org/paper/c61a5df979a0806381dc5d831ddfff420be6f802

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Citations in the corpus

Phase Retrieval via Wirtinger Flow: Theory and Algorithms (2014, background): “If the loss function is non-negative and vanishes only when x = y, then a solution to the generalized phase retrieval problem (1.1) is any solution to minimize f(z) ∶= 1 2m m ∑ r=1 `(yr, ∣a ∗ rz∣ 2 ), z ∈ Cn. (2.1) Although one could study many loss functions, we shall focus in this paper on the…”
Phase retrieval in infinite-dimensional Hilbert spaces (2016, background): “This was observed by the authors of [5] where they presented a much more complicated proof for this case.”
Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow (2016, background): “It has been established that m ≥ 2n − 1 or m ≥ 4n − 4 generic measurements {(ai; yi)}mi=1 as in (1) suffice for uniquely determining an n-dimensional real-valued or complex-valued vector x [1], [2], respectively, while the former with m = 2n− 1 has also been shown to be necessary [1], [3].”
Phase Retrieval from Coded Diffraction Patterns (2013, methodology): “Having said this, there are similarities in the proof techniques, especially in the role played by the robust injectivity property, compare our Lemma 3.7 from Section 3.3 with [32, Section 3.3].”
Phase Retrieval with Application to Optical Imaging: A contemporary overview (2015, background): “Complex signAl C ConjeCture: 4N−4 generiC meAsurements Are suFFiCient [43]”
Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems (2015, neutral): “[5, 18]).”