A Precise Analysis of PhaseMax in Phase Retrieval

Salehi, Fariborz; Abbasi, Ehsan; Hassibi, Babak

doi:10.1109/isit.2018.8437494

Cited by 19 publications

(15 citation statements)

References 31 publications

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“…The PhaseLamp method has superior recovery performance over PhaseMax, but again it does not work when δ < 2 for real-valued models. A recent paper [38] extends the asymptotic analysis of [21] to the complex-valued setting, and it was shown that PhaseMax cannot work for δ < 4. On the other hand, AMP.A proposed in this paper achieves perfect recovery when δ > 1.5 and δ > 2.5, for the real and complex-valued models respectively.…”

Section: Existing Theoretical Workmentioning

confidence: 99%

Optimization-Based AMP for Phase Retrieval: The Impact of Initialization and $\ell_{2}$ Regularization

Maleki

2019

IEEE Trans. Inform. Theory

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We consider an 2-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting m, n → ∞, m/n → δ and obtain sharp performance bounds, where m is the number of measurements and n is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only m = 64 π 2 − 4 n ≈ 2.5n measurements. Also, we provide sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: (i) Adding 2 regularization to the nonconvex loss function can be beneficial. (ii) Spectral initialization has marginal impact on the performance of the algorithm. The sharp analyses in this paper, not only enable us to compare the performance of our method with other phase recovery schemes, but also shed light on designing better iterative algorithms for other non-convex optimization problems.

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Section: Existing Theoretical Workmentioning

confidence: 99%

Optimization-Based AMP for Phase Retrieval: The Impact of Initialization and $\ell_{2}$ Regularization

Maleki

2019

IEEE Trans. Inform. Theory

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“…For our analysis, we utilize the recently developed Convex Gaussian Min-max Theorem (CGMT) [31] which is a strengthened version of a classical Gaussian comparison inequality due to Gordon [10], and whose origins are in [26]. Previously, the CGMT has been successfully applied to derive the precise performance in a number of applications such as regularized M-estimators [30], analysis of the generalized lasso [19,31], data detection in massive MIMO [1,2,32], and PhaseMax in phase retrieval [6,24,23].…”

Section: Summary Of Contributionsmentioning

confidence: 99%

“…Theorem 3 (CGMT). [29] In (24), let S w , S u , be convex and compact sets, and assume ψ(•, •) is convexconcave on S w × S u . Also assume that G, g, and h all have entries i.i.d.…”

Section: A Convex Gaussian Min-max Theorem (Cgmt)mentioning

confidence: 99%

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The Impact of Regularization on High-dimensional Logistic Regression

Salehi¹,

Abbasi²,

Hassibi³

2019

Preprint

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Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes [27] have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-sparse, finite-alphabet, etc.) and so in this paper we study regularized logistic regression (RLR), where a convex regularizer that encourages the desired structure is added to the negative of the log-likelihood function. An advantage of RLR is that it allows parameter recovery even for instances where the (unconstrained) maximum likelihood estimate does not exist. We provide a precise analysis of the performance of RLR via the solution of a system of six nonlinear equations, through which any performance metric of interest (mean, mean-squared error, probability of support recovery, etc.) can be explicitly computed. Our results generalize those of Sur and Candes and we provide a detailed study for the cases of 2 2 -RLR and sparse ( 1-regularized) logistic regression. In both cases, we obtain explicit expressions for various performance metrics and can find the values of the regularizer parameter that optimizes the desired performance. The theory is validated by extensive numerical simulations across a range of parameter values and problem instances.

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“…Recently, [14], [15] proposed a convex optimization based algorithm for solving the phase retrieval problem that doesn't lift the underlying signal and hence does not square the number of variables. The precise analysis of it has been derived in [16], [17]. Typical non-convex approaches involve finding a good initialization, followed by iterative minimization of a loss function.…”

Section: Prior Workmentioning

confidence: 99%

Signal Reconstruction From Modulo Observations

Shah

Hegde

2019

2019 IEEE Global Conference on Signal and Information Processing (GlobalSIP)

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We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a (relatively) less wellknown imaging mechanism called modulo imaging, which can be used to extend the dynamic range of imaging systems; variations of this model have also been studied under the category of phase unwrapping. Signal reconstruction in the underdetermined regime with modulo observations is a challenging ill-posed problem, and existing reconstruction methods cannot be used directly. In this paper, we propose a novel approach to solving the inverse problem limited to two modulo periods, inspired by recent advances in algorithms for phase retrieval under sparsity constraints. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal and provides improved performance over other existing algorithms. We also provide experiments validating our approach on both synthetic and real data to depict its superior performance.

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A Precise Analysis of PhaseMax in Phase Retrieval

Cited by 19 publications

References 31 publications

Optimization-Based AMP for Phase Retrieval: The Impact of Initialization and $\ell_{2}$ Regularization

Optimization-Based AMP for Phase Retrieval: The Impact of Initialization and $\ell_{2}$ Regularization

The Impact of Regularization on High-dimensional Logistic Regression

Signal Reconstruction From Modulo Observations

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