On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem

Bălan, Radu; Zou, Dongmian

doi:10.1016/j.laa.2015.12.029

Cited by 30 publications

(25 citation statements)

References 23 publications

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“…In the finite-dimensional setting, this problem and its properties have been studied in e.g. [4,5,6,7,12]. More precisely, in these works it has been analyzed Date: November 8, 2018.…”

Section: Introductionmentioning

confidence: 99%

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Alaifari¹

2017

SIAM J. Math. Anal.

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Abstract. We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work [11], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame.We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over K = R or K = C) and verify that it is a necessary condition for uniqueness of the phase retrieval problem; when K = R the CP is also sufficient for uniqueness. In our general setting, we also prove a conjecture posed by Bandeira et al. [7], which was originally formulated for finite-dimensional spaces: for the case K = C the strong complement property (SCP) is a necessary condition for stability. To prove our main result, we show that the SCP can never hold for frames of infinite-dimensional Banach spaces.

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“…In the finite-dimensional setting, this problem and its properties have been studied in e.g. [4,5,6,7,12]. More precisely, in these works it has been analyzed Date: November 8, 2018.…”

Section: Introductionmentioning

confidence: 99%

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Alaifari¹

2017

SIAM J. Math. Anal.

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“…where σ k (xx * − x ′ x ′ * ) is the k-th singular value of the (at most rank-two) matrix xx * − x ′ x ′ * . In [3], Balan et al showed that if α and β are injective on C d / ∼, then β is bi-Lipschitz with respect to d 1 , and α is bi-Lipschitz with respect to D 2 , where in both cases R N is equipped with the Euclidean norm. Motivated by applications such as (Fourier) ptychography [18,22] and related numerical methods [13,14], we will study frames which are constructed as the shifts of a family of locally supported measurement vectors.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Lower Lipschitz bounds for phase retrieval from locally supported measurements

Iwen

Merhi

Perlmutter

2019

Applied and Computational Harmonic Analysis

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In this short note, we consider the worst case noise robustness of any phase retrieval algorithm which aims to reconstruct all nonvanishing vectors x ∈ C d (up to a single global phase multiple) from the magnitudes of an arbitrary collection of local correlation measurements. Examples of such measurements include both spectrogram measurements of x using locally supported windows and masked Fourier transform intensity measurements of x using bandlimited masks. As a result, the robustness results considered herein apply to a wide range of both ptychographic and Fourier ptychographic imaging scenarios. In particular, the main results imply that the accurate recovery of high-resolution images of extremely large samples using highly localized probes is likely to require an extremely large number of measurements in order to be robust to worst case measurement noise, independent of the recovery algorithm employed. Furthermore, recent pushes to achieve high-speed and high-resolution ptychographic imaging of integrated circuits for process verification and failure analysis will likely need to carefully balance probe design (e.g., their effective timefrequency support) against the total number of measurements acquired in order for their imaging techniques to be stable to measurement noise, no matter what reconstruction algorithms are applied.

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“…The first, as done in [4], is to modify the map to get a new map that is bi-Lipschitz. Another alternative which is explored in [2] is to replace the quotient metric with a different metric on C n /T with respect to which Φ is bi-Lipschitz. In this paper we will encounter a similar situation where we will have an initial G-invariant map which is injective but not Lipschitz.…”

Section: Then a Set S ⊂ C[x]mentioning

confidence: 99%

Complete set of translation invariant measurements with Lipschitz bounds

Cahill¹,

Contreras²,

Hip³

2019

Preprint

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In image and audio signal classification, a major problem is to build stable representations that are invariant under rigid motions and, more generally, to small diffeomorphisms. Translation invariant representations of signals in C n are of particular importance. The existence of such representations is intimately related to classical invariant theory, inverse problems in compressed sensing and deep learning. Despite an impressive body of litereature on the subject, most representations available are either: i) not stable due to the presence of high frequencies; ii) non discriminative; iii) non invariant when projected to finite dimensional subspaces. In the present paper, we construct low dimensional representations of signals in C n that are invariant under finite unitary group actions, as a special case we establish the existence of low-dimensional and complete Zm-invariant representations for any m ∈ N. Our construction yields a stable, discriminative transform with semi-explicit Lipschitz bounds on the dimension; this is particularly relevant for applications. Using some tools from Algebraic Geometry, we define a high dimensional homogeneous function that is injective. We then exploit the projective character of this embedding and see that the target space can be reduced significantly by using a generic linear transformation. Finally, we introduce the notion of non-parallel map, which is enjoyed by our function and employ this to construct a Lipschitz modification of it.

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On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem

Cited by 30 publications

References 23 publications

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Lower Lipschitz bounds for phase retrieval from locally supported measurements

Complete set of translation invariant measurements with Lipschitz bounds

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