16,813 research outputs found
Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study
We study the Kaczmarz methods for solving systems of quadratic equations,
i.e., the generalized phase retrieval problem. The methods extend the Kaczmarz
methods for solving systems of linear equations by integrating a phase
selection heuristic in each iteration and overall have the same per iteration
computational complexity. Extensive empirical performance comparisons establish
the computational advantages of the Kaczmarz methods over other
state-of-the-art phase retrieval algorithms both in terms of the number of
measurements needed for successful recovery and in terms of computation time.
Preliminary convergence analysis is presented for the randomized Kaczmarz
methods
PhasePack: A Phase Retrieval Library
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
Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns
This note shows that we can recover a complex vector x in C^n exactly from on
the order of n quadratic equations of the form ||^2 = b_i, i = 1, ...,
m, by using a semidefinite program known as PhaseLift. This improves upon
earlier bounds in [3], which required the number of equations to be at least on
the order of n log n. We also demonstrate optimal recovery results from noisy
quadratic measurements; these results are much sharper than previously known
results.Comment: 6 page
A Deterministic Theory for Exact Non-Convex Phase Retrieval
In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for
phase retrieval and identify a novel sufficient condition for universal exact
recovery through the lens of low rank matrix recovery theory. Via a perspective
in the lifted domain, we show that the convergence of the WF iterates to a true
solution is attained geometrically under a single condition on the lifted
forward model. As a result, a deterministic relationship between the accuracy
of spectral initialization and the validity of {the regularity condition} is
derived. In particular, we determine that a certain concentration property on
the spectral matrix must hold uniformly with a sufficiently tight constant.
This culminates into a sufficient condition that is equivalent to a restricted
isometry-type property over rank-1, positive semi-definite matrices, and
amounts to a less stringent requirement on the lifted forward model than those
of prominent low-rank-matrix-recovery methods in the literature. We
characterize the performance limits of our framework in terms of the tightness
of the concentration property via novel bounds on the convergence rate and on
the signal-to-noise ratio such that the theoretical guarantees are valid using
the spectral initialization at the proper sample complexity.Comment: In Revision for IEEE Transactions on Signal Processin
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