Phase Retrieval Using Feasible Point Pursuit: Algorithms and Cramér–Rao Bound

Qian, Cheng; Sidiropoulos, Nicholas D.; Huang, Kejun; Huang, Lei; So, Hing Cheung

doi:10.1109/tsp.2016.2593688

Cited by 38 publications

(30 citation statements)

References 33 publications

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“…Experience shows that the rank deficiency of FIM is usually related to trivial ambiguities of the problem, but not the critical ones. Take phase retrieval as an example, it has been shown that the FIM corresponding to this problem is always rank one deficient [150], [151], which, by identifying its null space, seems to be highly related to the global phase ambiguity inherent to this problem. For certain measurement systems, e.g., 1D Fourier measurement, the problem is not identifiable besides the trivial phase ambiguity, but the rank deficiency is still one [152], [153], which means the critical non-uniqueness issue is not revealed by the singularity of FIM.…”

Section: F Crb For Matrix and Cp Tensor Factorizationmentioning

confidence: 99%

Tensor Decomposition for Signal Processing and Machine Learning

Sidiropoulos

Lathauwer

et al. 2017

IEEE Trans. Signal Process.

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1,299

989

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Tensors or {\em multi-way arrays} are functions of three or more indices $(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing, statistics, data mining and machine learning. This overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors. As such, it focuses on fundamentals and motivation (using various application examples), aiming to strike an appropriate balance of breadth {\em and depth} that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and/or developing tensor algorithms and software. Some background in applied optimization is useful but not strictly required. The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties (including fairly good coverage of identifiability); broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.Comment: revised version, overview articl

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Section: F Crb For Matrix and Cp Tensor Factorizationmentioning

confidence: 99%

Tensor Decomposition for Signal Processing and Machine Learning

Sidiropoulos

Lathauwer

et al. 2017

IEEE Trans. Signal Process.

Self Cite

1,299

989

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“…Gaussian {a i } satisfies σ 1 := A ≤ (1 + δ ) √ m for some δ > 0 with probability exceeding 1 − 2e −c0m as soon as m ≥ c 1 n for sufficiently large c 1 > 0, where c 1 > 0 is a universal constant depending on δ [59, Remark 5.25]. Combining (42), (43), and (44) yields…”

Section: B Exact Recovery From Noiseless Datamentioning

confidence: 99%

Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

Wang

Giannakis

Eldar

2018

IEEE Trans. Inform. Theory

321

418

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This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form yi = | ai, x | 2 , where ai's are given random measurement vectors. This problem is known to be NP-hard in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adopts the amplitude-based empirical loss function, and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and ai's are real-valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.

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“…Having elaborated on the properties of RIP matrices, we are ready to derive bounds for the three terms on the right hand side of (29). Regarding the first term, it is easy to check that…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…A popular class of nonconvex approaches is based on alternating projections including the seminal works by Gerchberg-Saxton [12] and Fienup [4], [13], [14], alternating minimization with re-sampling (AltMinPhase) [5], (stochastic) truncated amplitude flow (TAF) [15,9,16,17,18] and the Wirtinger flow (WF) variants [7,8,19,20], trust-region [21], (stochastic) proximal linear algorithms [22,23]. See also related discussion in [10,1,24,25,26,27,28,29]. Specifically, the WF variants and the trust-region methods minimize the intensity (modulus squared) based empirical risk, while AltMinPhase and TAF cope with the amplitude-based empirical risk.…”

mentioning

confidence: 99%

Sparse Phase Retrieval via Truncated Amplitude Flow

Wang

Zhang

Giannakis

et al. 2018

IEEE Trans. Signal Process.

129

170

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This paper develops a novel algorithm, termed SPARse Truncated Amplitude flow (SPARTA), to reconstruct a sparse signal from a small number of magnitude-only measurements. It deals with what is also known as sparse phase retrieval (PR), which is NP-hard in general and emerges in many science and engineering applications. Upon formulating sparse PR as an amplitudebased nonconvex optimization task, SPARTA works iteratively in two stages: In stage one, the support of the underlying sparse signal is recovered using an analytically well-justified rule, and subsequently a sparse orthogonality-promoting initialization is obtained via power iterations restricted on the support; and, in the second stage, the initialization is successively refined by means of hard thresholding based gradient-type iterations. SPARTA is a simple yet effective, scalable, and fast sparse PR solver. On the theoretical side, for any n-dimensional k-sparse (k n) signal x with minimum (in modulus) nonzero entries on the order of (1/ √ k) x 2, SPARTA recovers the signal exactly (up to a global unimodular constant) from about k 2 log n random Gaussian measurements with high probability. Furthermore, SPARTA incurs computational complexity on the order of k 2 n log n with total runtime proportional to the time required to read the data, which improves upon the state-of-the-art by at least a factor of k. Finally, SPARTA is robust against additive noise of bounded support. Extensive numerical tests corroborate markedly improved recovery performance and speedups of SPARTA relative to existing alternatives. and ptychography, astronomy, optics, as well as array and coherent diffraction imaging. In these settings, optical sensors and detectors such as charge-coupled device cameras, photosensitive films, and human eyes record only the intensity (squared magnitude) of a light wave, but not the phase. In particular, solution to PR has led to significant accomplishments, including the discovery in 1953 of DNA double helical structure from diffraction patterns, and the characterization of aberrations in the Hubble Space Telescope from measured point spread functions [1]. Due to the absence of Fourier phase information, the one-dimensional (1D) Fourier PR problem is generally ill-posed. It can be shown that there are in fact exponentially many non-equivalent solutions beyond trivial ambiguities in the 1D PR case [2]. A common approach to overcome this ill-posedness is exploiting additional information on the unknown signal such as non-negativity, sparsity, or bounded magnitude [3,4,5]. Other viable solutions consist of introducing redundancy into the measurement transforming system to obtain over-sampled and short-time Fourier transform (STFT) measurements [6], random Gaussian measurements [7,8,9], and coded diffraction patterns using structured illumination and random masks [10,11,7], just to name a few; see [10] for contemporary reviews on the theory and practice of PR.Past PR approaches can be mainly categorized as convex and nonconvex ones. A popular class of ...

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Phase Retrieval Using Feasible Point Pursuit: Algorithms and Cramér–Rao Bound

Cited by 38 publications

References 33 publications

Tensor Decomposition for Signal Processing and Machine Learning

Tensor Decomposition for Signal Processing and Machine Learning

Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

Sparse Phase Retrieval via Truncated Amplitude Flow

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