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**1 - 2**of**2**### FRAMES AND PHASELESS RECONSTRUCTION AMS SHORT COURSE: JOINT MATHEMATICS MEETINGS . . .

"... Frame design for phaseless reconstruction is now part of the broader problem of nonlinear recon- struction and is an emerging topic in harmonic analysis. The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redunda ..."

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Frame design for phaseless reconstruction is now part of the broader problem of nonlinear recon- struction and is an emerging topic in harmonic analysis. The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redundant system (frame), we want to reconstruct the unknown input. This problem has first occurred in X-ray crystallography starting from the early 20th century. The same nonlinear reconstruction problem shows up in speech processing, particularly in speech recognition. In this lecture we shall cover existing analysis results as well as algorithms for signal recovery including: necessary and sufficient conditions for injectivity, Lipschitz bounds of the nonlinear map and its left inverses, stochastic performance bounds, convex relaxation algorithms for inversion, least-squares inversion algorithms.

### Contemporary Mathematics On Lipschitz Inversion of Nonlinear Redundant Representations

"... Abstract. In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed us-ing Lipschitz continuous maps. Specifically we show that when the nonlin-ear analysis map α: H → Rm is injective, with (α(x))k = |〈x, fk〉|2, where {f ..."

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Abstract. In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed us-ing Lipschitz continuous maps. Specifically we show that when the nonlin-ear analysis map α: H → Rm is injective, with (α(x))k = |〈x, fk〉|2, where {f1, · · · , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω: Rm → H that is Lipschitz continuous. Additionally we obtain that the Lipschitz constant of this inverse map is at most 12 divided by the lower Lipschitz constant of α. 1.