Phase retrieval with outliers via median truncated amplitude flow

Zhang, Quanbing; Hu, Feihang; Liu, Dequn; Yuan, Yufan

doi:10.1145/3381271.3381287

Cited by 38 publications

(72 citation statements)

References 6 publications

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“…Theorem 28 (Median-truncated spectral method for robust phase retrieval [24]). Consider the robust phase retrieval problem in (89), and fix any ζ > 0.…”

Section: Truncated Spectral Methods For Removing Sparse Outliersmentioning

confidence: 99%

See 1 more Smart Citation

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Chi

Chen

2019

IEEE Trans. Signal Process.

320

284

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Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently.In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.

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“…Theorem 28 (Median-truncated spectral method for robust phase retrieval [24]). Consider the robust phase retrieval problem in (89), and fix any ζ > 0.…”

Section: Truncated Spectral Methods For Removing Sparse Outliersmentioning

confidence: 99%

“…Theorem 14 (Median-truncated GD for robust phase retrieval [24]). Consider the problem (89) with a fraction α of arbitrary outliers.…”

Section: Truncation For Removing Sparse Outliersmentioning

confidence: 99%

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Chi

Chen

2019

IEEE Trans. Signal Process.

320

284

View full text Add to dashboard Cite

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“…To illustrate these results, we present worked examples corresponding to two different sensing models. Numerical simulations demonstrate the performance improvements brought by the proposed optimal design over heuristic choices given in (4) and (5) as well as the functions T MM (·) constructed in [15]. To set the stage for proving our results, Section III recalls the asymptotic characterization of the spectral method obtained in previous work [14], [15].…”

Section: Introductionmentioning

confidence: 95%

“…In Figure 2, we compare the proposed optimal preprocessing function in (24) against the the trimming scheme in (4), the subset scheme in (5), as well as T MM (·) in (14). In our experiments, the signal dimension is set to n = 4096 and κ = 5.…”

Section: B Worked Examplesmentioning

confidence: 99%

Optimal Spectral Initialization for Signal Recovery With Applications to Phase Retrieval

Luo

Alghamdi

2019

IEEE Trans. Signal Process.

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We present the optimal design of a spectral method widely used to initialize nonconvex optimization algorithms for solving phase retrieval and other signal recovery problems. Our work leverages recent results that provide an exact characterization of the performance of the spectral method in the highdimensional limit. This characterization allows us to map the task of optimal design to a constrained optimization problem in a weighted L 2 function space. The latter has a closed-form solution.Interestingly, under a mild technical condition, our results show that there exists a fixed design that is uniformly optimal over all sampling ratios. Numerical simulations demonstrate the performance improvement brought by the proposed optimal design over existing constructions in the literature. In a recent work, Mondelli and Montanari have shown the existence of a weak reconstruction threshold below which the spectral method cannot provide useful estimates. Our results serve to complement that work by deriving the fundamental limit of the spectral method beyond the aforementioned threshold.

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“…If the unknown signal or image sequence is well modeled as S+LR, can this modeling be exploited to recover it from under-sampled phaseless measurements? Two precursors to this problem, low rank phase retrieval [79] and phase retrieval for a single outliercorrupted signal [80], have been recently studied. Finally, this article does not review the literature on deep learning based approaches to RPCA, e.g., [81], nor does it overview the recent work on robust or dynamic robust PCA for tensor data, e.g.…”

Section: Future Directionsmentioning

confidence: 99%

Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery

Vaswani¹,

Bouwmans²,

Javed³

et al. 2018

IEEE Signal Process. Mag.

292

114

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PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional subspace that can change over time, albeit gradually. The problem of tracking such data (and the subspaces) while being robust to outliers is called robust subspace tracking (RST). This article provides a magazine-style overview of the entire field of robust subspace learning and tracking. In particular solutions for three problems are discussed in detail: RPCA via sparse+lowrank matrix decomposition (S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an entire data vector is either an outlier or an inlier. The S+LR formulation instead assumes that outliers occur on only a few data vector indices and hence are well modeled as sparse corruptions.

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Phase retrieval with outliers via median truncated amplitude flow

Cited by 38 publications

References 6 publications

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Optimal Spectral Initialization for Signal Recovery With Applications to Phase Retrieval

Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery

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