16,532 research outputs found
Phase Retrieval via Smooth Amplitude Flow
Phase retrieval (PR) is an inverse problem about recovering a signal from
phaseless linear measurements. This problem can be effectively solved by
minimizing a nonconvex amplitude-based loss function. However, this loss
function is non-smooth. To address the non-smoothness, a series of methods have
been proposed by adding truncating, reweighting and smoothing operations to
adjust the gradient or the loss function and achieved better performance. But
these operations bring about extra rules and parameters that need to be
carefully designed. Unlike previous works, we present a smooth amplitude flow
method (SAF) which minimizes a novel loss function, without additionally
modifying the gradient or the loss function during gradient descending. Such a
new heuristic can be regarded as a smooth version of the original non-smooth
amplitude-based loss function. We prove that SAF can converge geometrically to
a global optimal point via the gradient algorithm with an elaborate
initialization stage with a high probability. Substantial numerical tests
empirically illustrate that the proposed heuristic is significantly superior to
the original amplitude-based loss function and SAF also outperforms other
state-of-the-art methods in terms of the recovery rate and the converging
speed. Specially, it is numerically shown that SAF can stably recover the
original signal when number of measurements is smaller than the
information-theoretic limit for both the real and the complex Gaussian models.Comment: 18 pages, 6 figures, two referrences adde
SPRSF: Sparse Phase Retrieval via Smoothing Function
Phase retrieval (PR) is an ill-conditioned inverse problem which can be found
in various science and engineering applications. Assuming sparse priority over
the signal of interest, recent algorithms have been developed to solve the
phase retrieval problem. Some examples include SparseAltMinPhase (SAMP), Sparse
Wirtinger flow (SWF) and Sparse Truncated Amplitude flow (SPARTA). However, the
optimization cost functions of the mentioned algorithms are non-convex and
non-smooth. In order to fix the non-smoothness of the cost function, the SPARTA
method uses truncation thresholds to calculate a truncated step update
direction. In practice, the truncation procedure requires calculating more
parameters to obtain a desired performance in the phase recovery. Therefore,
this paper proposes an algorithm called SPRSF (Sparse Phase retrieval via
Smoothing Function) to solve the sparse PR problem by introducing a smoothing
function. SPRSF is an iterative algorithm where the update step is obtained by
a hard thresholding over a gradient descent direction. Theoretical analyses
show that the smoothing function uniformly approximates the non-convex and
non-smooth sparse PR optimization problem. Moreover, SPRSF does not require the
truncation procedure used in SPARTA. Numerical tests demonstrate that SPRSF
performs better than state-of-the-art methods, especially when there is no
knowledge about the sparsity . In particular, SPRSF attains a higher mean
recovery rate in comparison with SPARTA, SAMP and SWF methods, when the
sparsity varies for the real and complex cases. Further, in terms of the
sampling complexity, the SPRSF method outperforms its competitive alternatives
Approximate Message Passing for Amplitude Based Optimization
We consider an -regularized non-convex optimization problem for
recovering signals from their noisy phaseless observations. We design and study
the performance of a message passing algorithm that aims to solve this
optimization problem. We consider the asymptotic setting , and obtain sharp performance bounds, where
is the number of measurements and is the signal dimension. We show that
for complex signals the algorithm can perform accurate recovery with only
measurements. Also, we
provide sharp analysis on the sensitivity of the algorithm to noise. We
highlight the following facts about our message passing algorithm: (i) Adding
regularization to the non-convex loss function can be beneficial even
in the noiseless setting; (ii) spectral initialization has marginal impact on
the performance of the algorithm.Comment: accepted by ICML; short version of arXiv:1801.01170 with more
simulations and other discussion
Sparse Signal Recovery from Phaseless Measurements via Hard Thresholding Pursuit
In this paper, we consider the sparse phase retrieval problem, recovering an
-sparse signal from phaseless samples
for . Existing
sparse phase retrieval algorithms are usually first-order and hence converge at
most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in
compressed sensing, we propose an efficient second-order algorithm for sparse
phase retrieval. Our proposed algorithm is theoretically guaranteed to give an
exact sparse signal recovery in finite (in particular, at most ) steps, when
are i.i.d. standard Gaussian random vector with and the initialization is in a neighborhood of the underlying
sparse signal. Together with a spectral initialization, our algorithm is
guaranteed to have an exact recovery from samples. Since the
computational cost per iteration of our proposed algorithm is the same order as
popular first-order algorithms, our algorithm is extremely efficient.
Experimental results show that our algorithm can be several times faster than
existing sparse phase retrieval algorithms
Frequency-Resolved Optical Gating Recovery via Smoothing Gradient
Frequency-resolved optical gating (FROG) is a popular technique for complete
characterization of ultrashort laser pulses. The acquired data in FROG, called
FROG trace, is the Fourier magnitude of the product of the unknown pulse with a
time-shifted version of itself, for several different shifts. To estimate the
pulse from the FROG trace, we propose an algorithm that minimizes a smoothed
non-convex least-squares objective function. The method consists of two steps.
First, we approximate the pulse by an iterative spectral algorithm. Then, the
attained initialization is refined based upon a sequence of block stochastic
gradient iterations. The algorithm is theoretically simple, numerically
scalable, and easy-to-implement. Empirically, our approach outperforms the
state-of-the-art when the FROG trace is incomplete, that is, when only few
shifts are recorded. Simulations also suggest that the proposed algorithm
exhibits similar computational cost compared to a state-of-the-art technique
for both complete and incomplete data. In addition, we prove that in the
vicinity of the true solution, the algorithm converges to a critical point. A
Matlab implementation is publicly available at
https://github.com/samuelpinilla/FROG.Comment: Simulations and comparisons are being adde
Accelerated Wirtinger Flow: A fast algorithm for ptychography
This paper presents a new algorithm, Accelerated Wirtinger Flow (AWF), for
ptychographic image reconstruction from phaseless diffraction pattern
measurements. AWF is based on combining Nesterov's acceleration approach with
Wirtinger gradient descent. Theoretical results enable prespecification of all
AWF algorithm parameters, with no need for computationally-expensive line
searches and no need for manual parameter tuning. AWF is evaluated in the
context of simulated X-ray ptychography, where we demonstrate fast convergence
and low per-iteration computational complexity. We also show examples where AWF
reaches higher image quality with less computation than classical algorithms.
AWF is also shown to have robustness to noise and probe misalignment
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
Substantial progress has been made recently on developing provably accurate
and efficient algorithms for low-rank matrix factorization via nonconvex
optimization. While conventional wisdom often takes a dim view of nonconvex
optimization algorithms due to their susceptibility to spurious local minima,
simple iterative methods such as gradient descent have been remarkably
successful in practice. The theoretical footings, however, had been largely
lacking until recently.
In this tutorial-style overview, we highlight the important role of
statistical models in enabling efficient nonconvex optimization with
performance guarantees. We review two contrasting approaches: (1) two-stage
algorithms, which consist of a tailored initialization step followed by
successive refinement; and (2) global landscape analysis and
initialization-free algorithms. Several canonical matrix factorization problems
are discussed, including but not limited to matrix sensing, phase retrieval,
matrix completion, blind deconvolution, robust principal component analysis,
phase synchronization, and joint alignment. Special care is taken to illustrate
the key technical insights underlying their analyses. This article serves as a
testament that the integrated consideration of optimization and statistics
leads to fruitful research findings.Comment: Invited overview articl
Reconstruction Methods in THz Single-pixel Imaging
The aim of this paper is to discuss some advanced aspects of image
reconstruction in single-pixel cameras, focusing in particular on detectors in
the THz regime. We discuss the reconstruction problem from a computational
imaging perspective and provide a comparison of the effects of several
state-of-the art regularization techniques.
Moreover, we focus on some advanced aspects arising in practice with THz
cameras, which lead to nonlinear reconstruction problems: the calibration of
the beam reminiscent of the Retinex problem in imaging and phase recovery
problems. Finally we provide an outlook to future challenges in the area
Phase Retrieval via Sparse Wirtinger Flow
Phase retrieval(PR) problem is a kind of ill-condition inverse problem which
can be found in various of applications. Utilizing the sparse priority, an
algorithm called SWF(Sparse Wirtinger Flow) is proposed in this paper to deal
with sparse PR problem based on the Wirtinger flow method. SWF firstly recovers
the support of the signal and then updates the evaluation by hard thresholding
method with an elaborate initialization. Theoretical analyses show that SWF has
a geometric convergence for any sparse length signal with the sampling
complexity . To get accuracy, the
computational complexity of SWF is
.
Numerical tests also demonstrate that SWF performs better than
state-of-the-art methods especially when we have no priori knowledge about
sparsity . Moreover, SWF is also robust to the nois
Quantitative, Comparable Coherent Anti-Stokes Raman Scattering (CARS) Spectroscopy: Correcting Errors in Phase Retrieval
Coherent anti-Stokes Raman scattering (CARS) microspectroscopy has
demonstrated significant potential for biological and materials imaging. To
date, however, the primary mechanism of disseminating CARS spectroscopic
information is through pseudocolor imagery, which explicitly neglects a vast
majority of the hyperspectral data. Furthermore, current paradigms in CARS
spectral processing do not lend themselves to quantitative sample-to-sample
comparability. The primary limitation stems from the need to accurately measure
the so-called nonresonant background (NRB) that is used to extract the
chemically-sensitive Raman information from the raw spectra. Measurement of the
NRB on a pixel-by-pixel basis is a nontrivial task; thus, reference NRB from
glass or water are typically utilized, resulting in error between the actual
and estimated amplitude and phase. In this manuscript, we present a new
methodology for extracting the Raman spectral features that significantly
suppresses these errors through phase detrending and scaling. Classic methods
of error-correction, such as baseline detrending, are demonstrated to be
inaccurate and to simply mask the underlying errors. The theoretical
justification is presented by re-developing the theory of phase retrieval via
the Kramers-Kronig relation, and we demonstrate that these results are also
applicable to maximum entropy method-based phase retrieval. This new
error-correction approach is experimentally applied to glycerol spectra and
tissue images, demonstrating marked consistency between spectra obtained using
different NRB estimates, and between spectra obtained on different instruments.
Additionally, in order to facilitate implementation of these approaches, we
have made many of the tools described herein available free for download.Comment: Body: 13 pages, 5 figures. Supplemental: 21 pages, 16 figure
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