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Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements.

dc.contributor.author Cheng, C
dc.contributor.author Daubechies, I
dc.contributor.author Dym, N
dc.contributor.author Lu, J
dc.date.accessioned 2020-12-14T19:38:51Z
dc.date.available 2020-12-14T19:38:51Z
dc.date.issued 2020
dc.identifier.uri https://hdl.handle.net/10161/21899
dc.description.abstract This paper is concerned with stable phase retrieval for a family of phase retrieval models we name "locally stable and conditionally connected" (LSCC) measurement schemes. For every signal $f$, we associate a corresponding weighted graph $G_f$, defined by the LSCC measurement scheme, and show that the phase retrievability of the signal $f$ is determined by the connectivity of $G_f$. We then characterize the phase retrieval stability of the signal $f$ by two measures that are commonly used in graph theory to quantify graph connectivity: the Cheeger constant of $G_f$ for real valued signals, and the algebraic connectivity of $G_f$ for complex valued signals. We use our results to study the stability of two phase retrieval models that can be cast as LSCC measurement schemes, and focus on understanding for which signals the "curse of dimensionality" can be avoided. The first model we discuss is a finite-dimensional model for locally supported measurements such as the windowed Fourier transform. For signals "without large holes", we show the stability constant exhibits only a mild polynomial growth in the dimension, in stark contrast with the exponential growth which uniform stability constants tend to suffer from; more precisely, in $R^d$ the constant grows proportionally to $d^{1/2}$, while in $C^d$ it grows proportionally to $d$. We also show the growth of the constant in the complex case cannot be reduced, suggesting that complex phase retrieval is substantially more difficult than real phase retrieval. The second model we consider is an infinite-dimensional phase retrieval problem in a principal shift invariant space. We show that despite the infinite dimensionality of this model, signals with monotone exponential decay will have a finite stability constant. In contrast, the stability bound provided by our results will be infinite if the signal's decay is polynomial.
dc.publisher Elsevier BV
dc.relation.ispartof CoRR
dc.subject cs.IT
dc.subject cs.IT
dc.subject cs.NA
dc.subject math.IT
dc.subject math.NA
dc.title Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements.
dc.type Journal article
duke.contributor.id Daubechies, I|0549786
duke.contributor.id Dym, N|0905973
duke.contributor.id Lu, J|0598771
dc.date.updated 2020-12-14T19:38:50Z
pubs.organisational-group Trinity College of Arts & Sciences
pubs.organisational-group Mathematics
pubs.organisational-group Duke
pubs.organisational-group Chemistry
pubs.organisational-group Physics
pubs.volume abs/2006.11709
duke.contributor.orcid Lu, J|0000-0001-6255-5165


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