Michael Kech
We explicitly give a frame of cardinality $5n-6$ such that every signal in $\mathbb{C}^n$ can be recovered up to a phase from its associated intensity measurements via the PhaseLift approach. Furthermore, we give explicit linear measurements with $4r(n-r)+n-2r$ outcomes that enable the recovery of every positive semidefinite $n\times n$ matrix of rank at most $r$.
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The aim of sparse phase retrieval is to recover a $k$-sparse signal $\mathbf{x}_0\in \mathbb{C}^{d}$ from quadratic measurements $|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2$ where $\mathbf{a}_i\in \mathbb{C}^d, i=1,\ldots,m$. Noting $|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2={\text{Tr}}(A_iX_0)$ with $A_i=\mathbf{a}_i\mathbf{a}_i^*\in \mathbb{C}^{d\times d}, X_0=\mathbf{x}_0\mathbf{x}_0^*\in \mathbb{C}^{d\times d}$, one can recast sparse phase retrieval as a problem of recovering a rank-one sparse matrix from linear measurements. Yin and Xin introduced PhaseLiftOff which presents a proxy …
We consider a variant of the phase retrieval problem, where vectors are replaced by unitary matrices, i.e., the unknown signal is a unitary matrix U, and the measurements consist of squared inner products |Tr(C*U)|^2 with unitary matrices C that are chosen by the observer. This problem has applications to quantum process tomography, when the unknown process is a unitary operation. We show that PhaseLift, a convex programming algorithm for phase retrieval, can be adapted to this matrix setting, using measurements that are sampled from unitary 4- and 2-designs. In the case of unitary 4-design me…
The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally efficient, numerically stable, and comes with rigorous performance guarantees. PhaseLift is optimal in the sense that the number of amplitude measurements required for phase reconstruction scales linearly with the dimension of the signal. However, it specifically demands Gaussian random measurement vectors - a limitation that restricts practical utility and obscure…
Recovery Performance of PhaseLift for Phase Retrieval from Coded Diffraction Patterns
本文研究了含噪编码衍射图案下 PhaseLift 相位恢复的稳定性，证明了对抗性噪声下的恢复误差界限，并给出了次高斯噪声下的误差上下界，推进了 Soltanolkotabi 猜想。
PhaseLift for Coded Diffraction Patterns: Optimal Sampling Rate
本文针对编码衍射图案的结构化相位恢复问题，证明了PhaseLift可行性规划在标准随机掩模模型下达到最优掩模复杂度与总采样率，证明采用结合自适应掩模分配和维度无关截断阈值的改进高尔夫方案构造近似对偶证书。
In this work we analyze the problem of phase retrieval from Fourier measurements with random diffraction patterns. To this end, we consider the recently introduced PhaseLift algorithm, which expresses the problem in the language of convex optimization. We provide recovery guarantees which require O(log^2 d) different diffraction patterns, thus improving on recent results by Candes et al. [arXiv:1310.3240], which require O(log^4 d) different patterns.
Robust Outlier Bound Condition to Phase Retrieval with Adversarial Sparse Outliers
研究从含对抗性稀疏离群值的无相位测量中恢复信号的问题，证明 Robust-PhaseLift 模型在离群值比例 $s<0.1185$ 时能精确恢复信号，并在 $s>0.1185$ 时以高概率失败。
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
针对相位缺失的强度测量问题，提出通过迹范数最小化的半定规划实现信号精确恢复，证明在测量数 m 约为 n log n 时以高概率成功，且对加性噪声具有鲁棒性。
Phase retrieval with polarization
在成像科学的许多领域中，难以测量线性测量的相位。因此，人们通常希望从强度测量重构信号，即，执行相位恢复。在本文中，我们提供了一种新的测量设计，它的灵感来自干涉测量，并利用扩展图的某些属性。我们还给出了一个有效的相位恢复程序，并使用谱图理论中的最新结果来产生一个稳定的性能保证，该保证可与[Candes et al. 2011]中的PhaseLift保证相媲美。我们使用数值模拟来说明我们的相位恢复程序的性能，我们比较重建误差和运行时间与一个共同的交替投影型程序。
In the context of the phase retrieval problem, it is known that certain natural classes of measurements, such as Fourier measurements and random Bernoulli measurements, do not lead to the unique reconstruction of all possible signals, even in combination with certain practically feasible random masks. To avoid this difficulty, the analysis is often restricted to measurement ensembles (or masks) that satisfy a small-ball probability condition, in order to ensure that the reconstruction is unique. This paper shows a complementary result: for random Bernoulli measurements, there is still a large …
Phase retrieval from power spectra of masked signals
在衍射成像中，人们的任务是从信号的功率谱重建信号。为了解决这个逆问题中的模糊性，人们可能会调用有关信号的先验知识，但在这种情况下，相位恢复算法已经取得了有限的成功。一种替代方案是通过多次照射信号来在测量过程中产生冗余，每次使用不同的掩模使信号失真。尽管最近在相位恢复方面取得了一些进展，但该社区尚未构建一个唯一确定所有信号并承认有效重建算法的掩模集合。在本文中，我们利用最近提出的偏振方法来构建这样的合奏。我们还提出了数值模拟，以说明在这种情况下的偏振方法的稳定性。在一个国家的最先进的相位恢复算法称为PhaseLift相比，我们发现，偏振速度快得多，具有可比的稳定性。
The phase retrieval problem has garnered significant attention since the development of the PhaseLift algorithm, which is a convex program that operates in a lifted space of matrices. Because of the substantial computational cost due to lifting, many approaches to phase retrieval have been developed, including non-convex optimization algorithms which operate in the natural parameter space, such as Wirtinger Flow. Very recently, a convex formulation called PhaseMax has been discovered, and it has been proven to achieve phase retrieval via linear programming in the natural parameter space under …
Phase Retrieval from Coded Diffraction Patterns
本文研究了从仅强度测量中恢复物体相位的问题，提出通过随机调制产生编码衍射图案，并证明PhaseLift算法能以对数次数的调制精确恢复相位信息。
We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with "lifting" and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric "slab". Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that i…
Phase retrieval refers to the problem of reconstructing an unknown vector $x_0 \in \mathbb{C}^n$ or $x_0 \in \mathbb{R}^n $ from $m$ measurements of the form $y_i = \big\vert \langle ξ^{\left(i\right)}, x_0 \rangle \big\vert^2 $, where $ \left\{ ξ^{\left(i\right)} \right\}^m_{i=1} \subset \mathbb{C}^m $ are known measurement vectors. While Gaussian measurements allow for recovery of arbitrary signals provided the number of measurements scales at least linearly in the number of dimensions, it has been shown that ambiguities may arise for certain other classes of measurements $ \left\{ ξ^{\left(…
The paper considers the phase retrieval problem in N-dimensional complex vector spaces. It provides two sets of deterministic measurement vectors which guarantee signal recovery for all signals, excluding only a specific subspace and a union of subspaces, respectively. A stable analytic reconstruction procedure of low complexity is given. Additionally it is proven that signal recovery from these measurements can be solved exactly via a semidefinite program. A practical implementation with 4 deterministic diffraction patterns is provided and some numerical experiments with noisy measurements co…
Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns
针对编码衍射图样相位恢复，证明使用对数数量随机掩模时，PhaseLift型凸程序达到高斯型稳定恢复界，且非凸分解损失在过参数化宽度下具有良性景观，所有临界点对应真实信号。
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 con…
In this paper, we introduce two symmetric directed graphs depending on supports of signals and windows, and we show that the connectivity of those graphs provides either necessary and sufficient conditions to phase retrieval of a signal from magnitude measurements of its multiple-window short-time Fourier transform. Also we propose an algebraic reconstruction algorithm, and provide an error estimate to our algorithm when magnitude measurements are corrupted by deterministic/random noises.
In phase retrieval problems, a signal of interest (SOI) is reconstructed based on the magnitude of a linear transformation of the SOI observed with additive noise. The linear transform is typically referred to as a measurement matrix. Many works on phase retrieval assume that the measurement matrix is a random Gaussian matrix, which, in the noiseless scenario with sufficiently many measurements, guarantees invertability of the transformation between the SOI and the observations, up to an inherent phase ambiguity. However, in many practical applications, the measurement matrix corresponds to an…
In phase retrieval, the goal is to recover a complex signal from the magnitude of its linear measurements. While many well-known algorithms guarantee deterministic recovery of the unknown signal using i.i.d. random measurement matrices, they suffer serious convergence issues some ill-conditioned matrices. As an example, this happens in optical imagers using binary intensity-only spatial light modulators to shape the input wavefront. The problem of ill-conditioned measurement matrices has also been a topic of interest for compressed sensing researchers during the past decade. In this paper, usi…
Fast Phase Retrieval from Local Correlation Measurements
本文提出了一种快速相位恢复方法，利用多种局部相位无关相关测量在近线性时间内恢复信号，同时证明了在特定测量条件下能保证恢复非稀疏信号，并展示了随机化测量在任意信号上的非均匀概率恢复保证。
In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functionals (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization sub-problems. For each subproblem, the Nesterov's accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm is used. In the setting of sparse phase retrieval, a standard $\ell_1$ norm…
In the phase retrieval problem one seeks to recover an unknown $n$ dimensional signal vector $\mathbf{x}$ from $m$ measurements of the form $y_i = |(\mathbf{A} \mathbf{x})_i|$, where $\mathbf{A}$ denotes the sensing matrix. Many algorithms for this problem are based on approximate message passing. For these algorithms, it is known that if the sensing matrix $\mathbf{A}$ is generated by sub-sampling $n$ columns of a uniformly random (i.e., Haar distributed) orthogonal matrix, in the high dimensional asymptotic regime ($m,n \rightarrow \infty, n/m \rightarrow κ$), the dynamics of the algorithm a…
An Algorithm for Exact Super-resolution and Phase Retrieval
本文提出一种两阶段算法，能从少量低通幅度测量中实现精确超分辨率，适用于连续盘内任意信号位置，且不增加采样量。
In this work we characterize all ambiguities of the linear (aperiodic) one-dimensional convolution on two fixed finite-dimensional complex vector spaces. It will be shown that the convolution ambiguities can be mapped one-to-one to factorization ambiguities in the $z-$domain, which are generated by swapping the zeros of the input signals. We use this polynomial description to show a deterministic version of a recently introduced masked Fourier phase retrieval design. In the noise-free case a (convex) semi-definite program can be used to recover exactly the input signals if they share no common…
Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces [7] and possibly severely ill-conditioned in finite-dimensional Hilbert spaces [7]. Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll f…
A Local Analysis of Block Coordinate Descent for Gaussian Phase Retrieval
本文研究了非凸问题中块坐标下降法的收敛性，通过高斯相位恢复问题的特殊结构，证明了该方法在确定性初始化下以线性速率收敛到全局极小值。
Stable Phase Retrieval with Mirror Descent
本文将无噪声镜像下降相位恢复框架拓展至含噪场景，证明了算法对小加性噪声的稳定性及高斯测量下的全局收敛性，结合谱初始化可降低样本复杂度，数值实验验证了其计算与统计效率。
The phase retrieval problem asks to recover a natural signal $y_0 \in \mathbb{R}^n$ from $m$ quadratic observations, where $m$ is to be minimized. As is common in many imaging problems, natural signals are considered sparse with respect to a known basis, and the generic sparsity prior is enforced via $\ell_1$ regularization. While successful in the realm of linear inverse problems, such $\ell_1$ methods have encountered possibly fundamental limitations, as no computationally efficient algorithm for phase retrieval of a $k$-sparse signal has been proven to succeed with fewer than $O(k^2\log n)$…
3D Tomographic Phase Retrieval and Unwrapping
本文建立三维相位恢复的唯一性理论，提出编码/非编码测量方案、三维相位解缠条件和投影层析唯一性证明，将三维相位恢复约化为相位投影层析问题。
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