The Experts below are selected from a list of 29388 Experts worldwide ranked by ideXlab platform
Christoph Studer - One of the best experts on this subject based on the ideXlab platform.
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Linear Spectral Estimators and an Application to Phase Retrieval
arXiv: Information Theory, 2018Co-Authors: Ramina Ghods, Tom Goldstein, Andrew S. Lan, Christoph StuderAbstract:Phase Retrieval refers to the problem of recovering real- or complex-valued vectors from magnitude measurements. The best-known algorithms for this problem are iterative in nature and rely on so-called spectral initializers that provide accurate initialization vectors. We propose a novel class of estimators suitable for general nonlinear measurement systems, called linear spectral estimators (LSPEs), which can be used to compute accurate initialization vectors for Phase Retrieval problems. The proposed LSPEs not only provide accurate initialization vectors for noisy Phase Retrieval systems with structured or random measurement matrices, but also enable the derivation of sharp and nonasymptotic mean-squared error bounds. We demonstrate the efficacy of LSPEs on synthetic and real-world Phase Retrieval problems, and show that our estimators significantly outperform existing methods for structured measurement systems that arise in practice.
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PhaseLin: Linear Phase Retrieval
arXiv: Information Theory, 2018Co-Authors: Ramina Ghods, Tom Goldstein, Andrew S. Lan, Christoph StuderAbstract:Phase Retrieval deals with the recovery of complex- or real-valued signals from magnitude measurements. As shown recently, the method PhaseMax enables Phase Retrieval via convex optimization and without lifting the problem to a higher dimension. To succeed, PhaseMax requires an initial guess of the solution, which can be calculated via spectral initializers. In this paper, we show that with the availability of an initial guess, Phase Retrieval can be carried out with an ever simpler, linear procedure. Our algorithm, called PhaseLin, is the linear estimator that minimizes the mean squared error (MSE) when applied to the magnitude measurements. The linear nature of PhaseLin enables an exact and nonasymptotic MSE analysis for arbitrary measurement matrices. We furthermore demonstrate that by iteratively using PhaseLin, one arrives at an efficient Phase Retrieval algorithm that performs on par with existing convex and nonconvex methods on synthetic and real-world data.
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PhaseMax: Convex Phase Retrieval via Basis Pursuit
IEEE Transactions on Information Theory, 2018Co-Authors: Tom Goldstein, Christoph StuderAbstract:We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as Phase Retrieval. We formulate Phase Retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the Phase Retrieval problem to a higher dimension, PhaseMax is a “non-lifting” relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is basis pursuit, which implies that the Phase Retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.
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CISS - PhaseLin: Linear Phase Retrieval
2018 52nd Annual Conference on Information Sciences and Systems (CISS), 2018Co-Authors: Ramina Ghods, Tom Goldstein, Andrew S. Lan, Christoph StuderAbstract:Phase Retrieval deals with the recovery of complex-or real-valued signals from magnitude measurements. As shown recently, the method PhaseMax enables Phase Retrieval via convex optimization and without lifting the problem to a higher dimension. To succeed, PhaseMax requires an initial guess of the solution, which can be calculated via spectral initializers. In this paper, we show that with the availability of an initial guess, Phase Retrieval can be carried out with an ever simpler, linear procedure. Our algorithm, called PhaseLin, is the linear estimator that minimizes the mean squared error (MSE) when applied to the magnitude measurements. The linear nature of PhaseLin enables an exact and nonasymptotic MSE analysis for arbitrary measurement matrices. We furthermore demonstrate that by iteratively using PhaseLin, one arrives at an efficient Phase Retrieval algorithm that performs on par with existing convex and nonconvex methods on synthetic and real-world data.
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PhasePack: A Phase Retrieval Library
arXiv: Optimization and Control, 2017Co-Authors: Rohan Chandra, Christoph Studer, Ziyuan Zhong, Justin Hontz, Val Mcculloch, Tom GoldsteinAbstract:Phase Retrieval deals with the estimation of complex-valued signals solely from the magnitudes of linear measurements. While there has been a recent explosion in the development of Phase Retrieval algorithms, the lack of a common interface has made it difficult to compare new methods against the state-of-the-art. The purpose of PhasePack is to create a common software interface for a wide range of Phase Retrieval algorithms and to provide a common testbed using both synthetic data and empirical imaging datasets. PhasePack is able to benchmark a large number of recent Phase Retrieval methods against one another to generate comparisons using a range of different performance metrics. The software package handles single method testing as well as multiple method comparisons. The algorithm implementations in PhasePack differ slightly from their original descriptions in the literature in order to achieve faster speed and improved robustness. In particular, PhasePack uses adaptive stepsizes, line-search methods, and fast eigensolvers to speed up and automate convergence.
Dustin G. Mixon - One of the best experts on this subject based on the ideXlab platform.
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Recent Advances in Phase Retrieval [Lecture Notes]
IEEE Signal Processing Magazine, 2016Co-Authors: Yonina C. Eldar, Nethaniel Hammen, Dustin G. MixonAbstract:In many applications in science and engineering, one is given the modulus squared of the Fourier transform of an unknown signal and then tasked with solving the corresponding inverse problem, known as Phase Retrieval. Solutions to this problem have led to some noteworthy accomplishments, such as identifying the double helix structure of DNA from diffraction patterns, as well as characterizing aberrations in the Hubble Space Telescope from point spread functions. Recently, Phase Retrieval has found interesting connections with algebraic geometry, low-rank matrix recovery, and compressed sensing. These connections, together with various new imaging techniques developed in optics, have spurred a surge of research into the theory, algorithms, and applications of Phase Retrieval. In this lecture note, we outline these recent connections and highlight some of the main results in contemporary Phase Retrieval.
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Phase Transitions in Phase Retrieval
Excursions in Harmonic Analysis Volume 4, 2015Co-Authors: Dustin G. MixonAbstract:Consider a scenario in which an unknown signal is transformed by a known linear operator, and then the pointwise absolute value of the unknown output function is reported. This scenario appears in several applications, and the goal is to recover the unknown signal – this is called Phase Retrieval. Phase Retrieval has been a popular subject of research in the last few years, both in determining whether complete information is available with a given linear operator and in finding efficient and stable Phase Retrieval algorithms in the cases where complete information is available. Interestingly, there are a few ways to measure information completeness, and each way appears to be governed by a Phase transition of sorts. This chapter will survey the state of the art with some of these Phase transitions, and identify a few open problems for further research.
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Phase Retrieval from very few measurements
Linear Algebra and its Applications, 2014Co-Authors: Matthew Fickus, Dustin G. Mixon, Aaron A. Nelson, Yang WangAbstract:Abstract In many applications, signals are measured according to a linear process, but the Phases of these measurements are often unreliable or not available. To reconstruct the signal, one must perform a process known as Phase Retrieval. This paper focuses on completely determining signals with as few intensity measurements as possible, and on efficient Phase Retrieval algorithms from such measurements. For the case of complex M-dimensional signals, we construct a measurement ensemble of size 4 M − 4 which yields injective intensity measurements; this is conjectured to be the smallest such ensemble. For the case of real signals, we devise a theory of “almost” injective intensity measurements, and we characterize such ensembles. Later, we show that Phase Retrieval from M + 1 almost injective intensity measurements is NP -hard, indicating that computationally efficient Phase Retrieval must come at the price of measurement redundancy.
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Phase Retrieval with Polarization
SIAM Journal on Imaging Sciences, 2014Co-Authors: Boris Alexeev, Afonso S. Bandeira, Matthew Fickus, Dustin G. MixonAbstract:In many areas of imaging science, it is difficult to measure the Phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform Phase Retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient Phase Retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [Candes, Strohmer, and Voroninski, PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming, preprint, arXiv:1109.4499, 2011]. We use numerical simulations to illustrate the performance of our Phase Retrieval procedure, and we compare reconstruction error and runtime with a common alternating-projections-type procedure.
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Phase Retrieval with polarization
arXiv: Information Theory, 2012Co-Authors: Boris Alexeev, Afonso S. Bandeira, Matthew Fickus, Dustin G. MixonAbstract:In many areas of imaging science, it is difficult to measure the Phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform Phase Retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient Phase Retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [Candes et al. 2011]. We use numerical simulations to illustrate the performance of our Phase Retrieval procedure, and we compare reconstruction error and runtime with a common alternating-projections-type procedure.
Tom Goldstein - One of the best experts on this subject based on the ideXlab platform.
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Linear Spectral Estimators and an Application to Phase Retrieval
arXiv: Information Theory, 2018Co-Authors: Ramina Ghods, Tom Goldstein, Andrew S. Lan, Christoph StuderAbstract:Phase Retrieval refers to the problem of recovering real- or complex-valued vectors from magnitude measurements. The best-known algorithms for this problem are iterative in nature and rely on so-called spectral initializers that provide accurate initialization vectors. We propose a novel class of estimators suitable for general nonlinear measurement systems, called linear spectral estimators (LSPEs), which can be used to compute accurate initialization vectors for Phase Retrieval problems. The proposed LSPEs not only provide accurate initialization vectors for noisy Phase Retrieval systems with structured or random measurement matrices, but also enable the derivation of sharp and nonasymptotic mean-squared error bounds. We demonstrate the efficacy of LSPEs on synthetic and real-world Phase Retrieval problems, and show that our estimators significantly outperform existing methods for structured measurement systems that arise in practice.
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PhaseLin: Linear Phase Retrieval
arXiv: Information Theory, 2018Co-Authors: Ramina Ghods, Tom Goldstein, Andrew S. Lan, Christoph StuderAbstract:Phase Retrieval deals with the recovery of complex- or real-valued signals from magnitude measurements. As shown recently, the method PhaseMax enables Phase Retrieval via convex optimization and without lifting the problem to a higher dimension. To succeed, PhaseMax requires an initial guess of the solution, which can be calculated via spectral initializers. In this paper, we show that with the availability of an initial guess, Phase Retrieval can be carried out with an ever simpler, linear procedure. Our algorithm, called PhaseLin, is the linear estimator that minimizes the mean squared error (MSE) when applied to the magnitude measurements. The linear nature of PhaseLin enables an exact and nonasymptotic MSE analysis for arbitrary measurement matrices. We furthermore demonstrate that by iteratively using PhaseLin, one arrives at an efficient Phase Retrieval algorithm that performs on par with existing convex and nonconvex methods on synthetic and real-world data.
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PhaseMax: Convex Phase Retrieval via Basis Pursuit
IEEE Transactions on Information Theory, 2018Co-Authors: Tom Goldstein, Christoph StuderAbstract:We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as Phase Retrieval. We formulate Phase Retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the Phase Retrieval problem to a higher dimension, PhaseMax is a “non-lifting” relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is basis pursuit, which implies that the Phase Retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.
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CISS - PhaseLin: Linear Phase Retrieval
2018 52nd Annual Conference on Information Sciences and Systems (CISS), 2018Co-Authors: Ramina Ghods, Tom Goldstein, Andrew S. Lan, Christoph StuderAbstract:Phase Retrieval deals with the recovery of complex-or real-valued signals from magnitude measurements. As shown recently, the method PhaseMax enables Phase Retrieval via convex optimization and without lifting the problem to a higher dimension. To succeed, PhaseMax requires an initial guess of the solution, which can be calculated via spectral initializers. In this paper, we show that with the availability of an initial guess, Phase Retrieval can be carried out with an ever simpler, linear procedure. Our algorithm, called PhaseLin, is the linear estimator that minimizes the mean squared error (MSE) when applied to the magnitude measurements. The linear nature of PhaseLin enables an exact and nonasymptotic MSE analysis for arbitrary measurement matrices. We furthermore demonstrate that by iteratively using PhaseLin, one arrives at an efficient Phase Retrieval algorithm that performs on par with existing convex and nonconvex methods on synthetic and real-world data.
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PhasePack: A Phase Retrieval Library
arXiv: Optimization and Control, 2017Co-Authors: Rohan Chandra, Christoph Studer, Ziyuan Zhong, Justin Hontz, Val Mcculloch, Tom GoldsteinAbstract:Phase Retrieval deals with the estimation of complex-valued signals solely from the magnitudes of linear measurements. While there has been a recent explosion in the development of Phase Retrieval algorithms, the lack of a common interface has made it difficult to compare new methods against the state-of-the-art. The purpose of PhasePack is to create a common software interface for a wide range of Phase Retrieval algorithms and to provide a common testbed using both synthetic data and empirical imaging datasets. PhasePack is able to benchmark a large number of recent Phase Retrieval methods against one another to generate comparisons using a range of different performance metrics. The software package handles single method testing as well as multiple method comparisons. The algorithm implementations in PhasePack differ slightly from their original descriptions in the literature in order to achieve faster speed and improved robustness. In particular, PhasePack uses adaptive stepsizes, line-search methods, and fast eigensolvers to speed up and automate convergence.
Yang Wang - One of the best experts on this subject based on the ideXlab platform.
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Phase Retrieval for sub-Gaussian measurements.
arXiv: Optimization and Control, 2019Co-Authors: Bing Gao, Haixia Liu, Yang WangAbstract:Generally, Phase Retrieval problem can be viewed as the reconstruction of a function/signal from only the magnitude of the linear measurements. These measurements can be, for example, the Fourier transform of the density function. Computationally the Phase Retrieval problem is very challenging. Many algorithms for Phase Retrieval are based on i.i.d. Gaussian random measurements. However, Gaussian random measurements remain one of the very few classes of measurements. In this paper, we develop an efficient Phase Retrieval algorithm for sub-gaussian random frames. We provide a general condition for measurements and develop a modified spectral initialization. In the algorithm, we first obtain a good approximation of the solution through the initialization, and from there we useWirtinger Flow to solve for the solution. We prove that the algorithm converges to the global minimizer linearly.
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Almost Everywhere Generalized Phase Retrieval
arXiv: Functional Analysis, 2019Co-Authors: Meng Huang, Yi Rong, Yang WangAbstract:The aim of generalized Phase Retrieval is to recover $\mathbf{x}\in \mathbb{F}^d$ from the quadratic measurements $\mathbf{x}^*A_1\mathbf{x},\ldots,\mathbf{x}^*A_N\mathbf{x}$, where $A_j\in \mathbf{H}_d(\mathbb{F})$ and $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. In this paper, we study the matrix set $\mathcal{A}=(A_j)_{j=1}^N$ which has the almost everywhere Phase Retrieval property. For the case $\mathbb{F}=\mathbb{R}$, we show that $N\geq d+1$ generic matrices with prescribed ranks have almost everywhere Phase Retrieval property. We also extend this result to the case where $A_1,\ldots,A_N$ are orthogonal matrices and hence establish the almost everywhere Phase Retrieval property for the fusion frame Phase Retrieval. For the case where $\mathbb{F}=\mathbb{C}$, we obtain similar results under the assumption of $N\geq 2d$. We lower the measurement number $d+1$ (resp. $2d$) with showing that there exist $N=d$ (resp. $2d-1$) matrices $A_1,\ldots, A_N\in \mathbf{H}_d(\mathbb{R})$ (resp. $\mathbf{H}_d(\mathbb{C})$) which have the almost everywhere Phase Retrieval property. Our results are an extension of almost everywhere Phase Retrieval from the standard Phase Retrieval to the general setting and the proofs are often based on some new ideas about determinant variety.
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Phase Retrieval from the magnitudes of affine linear measurements
Advances in Applied Mathematics, 2018Co-Authors: Bing Gao, Qiyu Sun, Yang WangAbstract:Abstract In this paper, we consider the affine Phase Retrieval problem in which one aims to recover a signal from the magnitudes of affine measurements. Let { a j } j = 1 m ⊂ H d and b = ( b 1 , … , b m ) ⊤ ∈ H m , where H = R or C . We say { a j } j = 1 m and b are affine Phase retrievable for H d if any x ∈ H d can be recovered from the magnitudes of the affine measurements { | 〈 a j , x 〉 + b j | , 1 ≤ j ≤ m } . We develop general framework for affine Phase Retrieval and prove necessary and sufficient conditions for { a j } j = 1 m and b to be affine Phase retrievable. We establish results on minimal measurements and generic measurements for affine Phase Retrieval as well as on sparse affine Phase Retrieval. In particular, we also highlight some notable differences between affine Phase Retrieval and the standard Phase Retrieval in which one aims to recover a signal x from the magnitudes of its linear measurements. In standard Phase Retrieval, one can only recover x up to a unimodular constant, while affine Phase Retrieval removes this ambiguity. We prove that unlike standard Phase Retrieval, the affine Phase retrievable measurements { a j } j = 1 m and b do not form an open set in H m × d × H m . Also in the complex setting, the standard Phase Retrieval requires 4 d − O ( log 2 d ) measurements, while the affine Phase Retrieval only needs m = 3 d measurements.
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generalized Phase Retrieval measurement number matrix recovery and beyond
Applied and Computational Harmonic Analysis, 2017Co-Authors: Yang Wang, Zhiqiang XuAbstract:Abstract In this paper, we develop a framework of generalized Phase Retrieval in which one aims to reconstruct a vector x in R d or C d through quadratic samples x ⁎ A 1 x , … , x ⁎ A N x . The generalized Phase Retrieval includes as special cases the standard Phase Retrieval as well as the Phase Retrieval by orthogonal projections. We first explore the connections among generalized Phase Retrieval, low-rank matrix recovery and nonsingular bilinear form. Motivated by the connections, we present results on the minimal measurement number needed for recovering a matrix that lies in a set W ∈ C d × d . Applying the results to Phase Retrieval, we show that generic d × d matrices A 1 , … , A N have the Phase Retrieval property if N ≥ 2 d − 1 in the real case and N ≥ 4 d − 4 in the complex case for very general classes of A 1 , … , A N , e.g. matrices with prescribed ranks or orthogonal projections. We also give lower bounds on the minimal measurement number required for generalized Phase Retrieval. For several classes of dimensions d we obtain the precise values of the minimal measurement number. Our work unifies and enhances results from the standard Phase Retrieval, Phase Retrieval by projections and low-rank matrix recovery.
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Phase Retrieval From the Magnitudes of Affine Linear Measurements
arXiv: Information Theory, 2016Co-Authors: Bing Gao, Qiyu Sun, Yang WangAbstract:In this paper, we consider the Phase Retrieval problem in which one aims to recover a signal from the magnitudes of affine measurements. Let $\{{\mathbf a}_j\}_{j=1}^m \subset {\mathbb H}^d$ and ${\mathbf b}=(b_1, \ldots, b_m)^\top\in{\mathbb H}^m$, where ${\mathbb H}={\mathbb R}$ or ${\mathbb C}$. We say $\{{\mathbf a}_j\}_{j=1}^m$ and $\mathbf b$ are affine Phase retrievable for ${\mathbb H}^d$ if any ${\mathbf x}\in{\mathbb H}^d$ can be recovered from the magnitudes of the affine measurements $\{| +b_j|,\, 1\leq j\leq m\}$. We develop general framework for affine Phase Retrieval and prove necessary and sufficient conditions for $\{{\mathbf a}_j\}_{j=1}^m$ and $\mathbf b$ to be affine Phase retrievable. We establish results on minimal measurements and generic measurements for affine Phase Retrieval as well as on sparse affine Phase Retrieval. In particular, we also highlight some notable differences between affine Phase Retrieval and the standard Phase Retrieval in which one aims to recover a signal $\mathbf x$ from the magnitudes of its linear measurements. In standard Phase Retrieval, one can only recover $\mathbf x$ up to a unimodular constant, while affine Phase Retrieval removes this ambiguity. We prove that unlike standard Phase Retrieval, the affine Phase retrievable measurements $\{{\mathbf a}_j\}_{j=1}^m$ and $\mathbf b$ do not form an open set in ${\mathbb H}^{m\times d}\times {\mathbb H}^m$. Also in the complex setting, the standard Phase Retrieval requires $4d-O(\log_2d)$ measurements, while the affine Phase Retrieval only needs $m=3d$ measurements.
Rujie Yin - One of the best experts on this subject based on the ideXlab platform.
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stable Phase Retrieval in infinite dimensions
Foundations of Computational Mathematics, 2019Co-Authors: Rima Alaifari, Philipp Grohs, Ingrid Daubechies, Rujie YinAbstract:The problem of Phase Retrieval is to determine a signal \(f\in \mathcal {H}\), with \( \mathcal {H}\) a Hilbert space, from intensity measurements \(|F(\omega )|\), where \(F(\omega ):=\langle f, \varphi _\omega \rangle \) are measurements of f with respect to a measurement system \((\varphi _\omega )_{\omega \in \Omega }\subset \mathcal {H}\). Although Phase Retrieval is always stable in the finite-dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if \(\mathcal {H}\) is infinite-dimensional: in that case Phase Retrieval is never uniformly stable (Alaifari and Grohs in SIAM J Math Anal 49(3):1895–1911, 2017; Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016); moreover, the stability deteriorates severely in the dimension of the problem (Cahill et al. 2016). On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function |F| of intensity measurements is concentrated on disjoint sets \(D_j\subset \Omega \), i.e. when \(F= \sum _{j=1}^k F_j\) where each \(F_j\) is concentrated on \(D_j\) (and \(k \ge 2\)). Motivated by these considerations, we propose a new paradigm for stable Phase Retrieval by considering the problem of reconstructing F up to a Phase factor that is not global, but that can be different for each of the subsets \(D_j\), i.e. recovering F up to the equivalence $$\begin{aligned} F \sim \sum _{j=1}^k e^{\mathrm {i}\alpha _j} F_j. \end{aligned}$$ We present concrete applications (for example in audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable Phase Retrieval can actually be achieved, for instance, if the measurement system is a Gabor frame or a frame of Cauchy wavelets.
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Stable Phase Retrieval in Infinite Dimensions
arXiv: Functional Analysis, 2016Co-Authors: Rima Alaifari, Philipp Grohs, Ingrid Daubechies, Rujie YinAbstract:The problem of Phase Retrieval is to determine a signal $f\in \mathcal{H}$, with $\mathcal{H}$ a Hilbert space, from intensity measurements $|F(\omega)|$, where $F(\omega):=\langle f , \varphi_\omega\rangle$ are measurements of $f$ with respect to a measurement system $(\varphi_\omega)_{\omega\in \Omega}\subset \mathcal{H}$. Although Phase Retrieval is always stable in the finite dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if $\mathcal{H}$ is infinite-dimensional: in that case Phase Retrieval is never uniformly stable [8, 4]; moreover the stability deteriorates severely in the dimension of the problem [8]. On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function $|F|$ of intensity measurements is concentrated on disjoint sets $D_j\subset \Omega$, i.e., when $F= \sum_{j=1}^k F_j$ where each $F_j$ is concentrated on $D_j$ (and $k \geq 2$). Motivated by these considerations we propose a new paradigm for stable Phase Retrieval by considering the problem of reconstructing $F$ up to a Phase factor that is not global, but that can be different for each of the subsets $D_j$, i.e., recovering $F$ up to the equivalence $$ F \sim \sum_{j=1}^k e^{i \alpha_j} F_j.$$ We present concrete applications (for example in audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable Phase Retrieval can actually be achieved, for instance if the measurement system is a Gabor frame or a frame of Cauchy wavelets.
