Median-Truncated Nonconvex Approach for Phase Retrieval With Outliers

Zhang, Huishuai; Chi, Yuejie; Liang, Y. T.

doi:10.1109/tit.2018.2847695

Cited by 40 publications

(37 citation statements)

References 60 publications

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“…For a lifted forward map F satisfying RIP + with RIC-δ 1 , we first show that the initialization by spectral method yields an estimate that is in the set E( ). We then establish the regularity condition (40) for the objective function (3) in the -neighborhood defined by the initialization. These two results culminate into convergence to a global solution at a geometric rate as stated in Theorem 4.1.…”

Section: Proof Of Theorem 41mentioning

confidence: 99%

A Generalization of Wirtinger Flow for Exact Interferometric Inversion

Yonel¹,

Yazıcı²

2019

SIAM J. Imaging Sci.

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Interferometric inversion involves recovery of a signal from cross-correlations of its linear transformations. A close relative of interferometric inversion is the generalized phase retrieval problem, which consists of recovering a signal from the auto-correlations of its linear transformations. Recently, significant advancements have been made in phase retrieval methods despite the ill-posed, and non-convex nature of the problem. One such method is Wirtinger Flow (WF) [1], a non-convex optimization framework that provides high probability guarantees of exact recovery under certain measurement models, such as coded diffraction patterns, and Gaussian sampling vectors. In this paper, we develop a generalization of WF for interferometric inversion, which we refer to as Generalized Wirtinger Flow (GWF). GWF theory extends the probabilistic exact recovery results in [1] to arbitrary measurement models characterized in the equivalent lifted problem, hence covers a larger class of measurement models. GWF framework unifies the theory of low rank matrix recovery (LRMR) and the non-convex optimization approach of WF, thereby establishes theoretical advantages of the non-convex approach over LRMR. We show that the conditions for exact recovery via WF can be derived through a low rank matrix recovery formulation. We identify a new sufficient condition on the lifted forward model that directly implies exact recovery conditions of standard WF. This condition is less stringent than those of LRMR, which is the state of the art approach for exact interferometric inversion. We next establish our sufficient condition for the cross-correlations of linear measurements collected by complex Gaussian sampling vectors. In the particular case of the Gaussian model, we show that the exact recovery conditions of standard WF imply our sufficient condition, and that the regularity condition of WF is redundant for the interferometric inversion problem. Finally, we demonstrate the effectiveness of GWF numerically in a deterministic multi-static radar imaging scenario.

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Section: Proof Of Theorem 41mentioning

confidence: 99%

A Generalization of Wirtinger Flow for Exact Interferometric Inversion

Yonel¹,

Yazıcı²

2019

SIAM J. Imaging Sci.

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“…Lemma III.1. Assume that (15) holds with probability at least p for any x ∈ C N . Then, the Gram operator of the lifted forward model can be expressed as follows over the set of rank-1, PSD matrices:…”

Section: Resultsmentioning

confidence: 99%

A Deterministic Theory for Exact Non-Convex Phase Retrieval

Yonel,

Yazici

2020

Preprint

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In this work, we analyze the non-convex framework of Wirtinger Flow (WF) [1] for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we establish that the convergence of WF to a true solution is geometrically implied under a condition on the lifted forward model which relates to the concentration of the spectral matrix around its expectation given that the bound is sufficiently tight. As a result, a deterministic relationship between accuracy of spectral initialization and the validity of the regularity condition is derived, and a convergence rate that solely depends on the concentration bound is obtained. Notably, the developed framework addresses a theoretical gap in non-convex optimization literature on solving quadratic systems of equations with the convergence arguments that are deterministic. Finally, we quantify a lower bound on the signal-to-noise ratio such that theoretical guarantees are valid using the spectral initialization even in the absence of pre-processing or sample truncation.

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“…The sample median is well known to be more robust than the sample mean in statistics (Tyler 2008;Zhang, Chi, and Liang 2016). Hence, the sample median has been used in a variety of contexts to design robust algorithms in multi-armed bandit problems (Altschuler, Brunel, and Malek 2019), parameter recovery in phase retrieval (Zhang, Chi, and Liang 2018), and regression (Klivans, Kothari, and Meka 2018). In this paper, the analysis methods are novel and we provide high probability guarantees of O(log T ) regret (rather than just in average).…”

Section: Related Workmentioning

confidence: 99%

Robust Stochastic Bandit Algorithms under Probabilistic Unbounded Adversarial Attack

Guan

Bucci

et al. 2020

AAAI

Self Cite

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The multi-armed bandit formalism has been extensively studied under various attack models, in which an adversary can modify the reward revealed to the player. Previous studies focused on scenarios where the attack value either is bounded at each round or has a vanishing probability of occurrence. These models do not capture powerful adversaries that can catastrophically perturb the revealed reward. This paper investigates the attack model where an adversary attacks with a certain probability at each round, and its attack value can be arbitrary and unbounded if it attacks. Furthermore, the attack value does not necessarily follow a statistical distribution. We propose a novel sample median-based and exploration-aided UCB algorithm (called med-E-UCB) and a median-based ϵ-greedy algorithm (called med-ϵ-greedy). Both of these algorithms are provably robust to the aforementioned attack model. More specifically we show that both algorithms achieve O(log T) pseudo-regret (i.e., the optimal regret without attacks). We also provide a high probability guarantee of O(log T) regret with respect to random rewards and random occurrence of attacks. These bounds are achieved under arbitrary and unbounded reward perturbation as long as the attack probability does not exceed a certain constant threshold. We provide multiple synthetic simulations of the proposed algorithms to verify these claims and showcase the inability of existing techniques to achieve sublinear regret. We also provide experimental results of the algorithm operating in a cognitive radio setting using multiple software-defined radios.

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Median-Truncated Nonconvex Approach for Phase Retrieval With Outliers

Cited by 40 publications

References 60 publications

A Generalization of Wirtinger Flow for Exact Interferometric Inversion

A Generalization of Wirtinger Flow for Exact Interferometric Inversion

A Deterministic Theory for Exact Non-Convex Phase Retrieval

Robust Stochastic Bandit Algorithms under Probabilistic Unbounded Adversarial Attack

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