Related papers: Convergence of the randomized Kaczmarz method for …
The Sparse Kaczmarz method is a famous and widely used iterative method for solving the regularized basis pursuit problem. A general scheme of the surrogate hyperplane sparse Kaczmarz method is proposed. In particular, a class of…
While characterization of coherent wavefields is essential to laser, x-ray and electron imaging, sensors measure the squared magnitude of the field, rather than the field itself. Holography or phase retrieval must be used to characterize…
In recent literature, a general two step procedure has been formulated for solving the problem of phase retrieval. First, a spectral technique is used to obtain a constant-error initial estimate, following which, the estimate is refined to…
We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of…
We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable…
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…
Consider a scenario in which an unknown signal is transformed by a known linear operator, and then the pointwise absolute value of the unknown output function is reported. This scenario appears in several applications, and the goal is to…
Companion paper [118] developed a powerful \emph{Random duality theory} (RDT) based analytical program to statistically characterize performance of \emph{descending} phase retrieval algorithms (dPR) (these include all variants of gradient…
The generalized phase retrieval problem over compact groups aims to recover a set of matrices -- representing an unknown signal -- from their associated Gram matrices. This framework generalizes the classical phase retrieval problem, which…
In many signal processing problems arising in practical applications, we wish to reconstruct an unknown signal from its phaseless measurements with respect to a frame. This inverse problem is known as the phase retrieval problem. For each…
Quantile-based randomized Kaczmarz (QRK) was recently introduced to efficiently solve sparsely corrupted linear systems $\mathbf{A} \mathbf{x}^*+\mathbf{\epsilon} = \mathbf{b}$ [SIAM J. Matrix Anal. Appl., 43(2), 605-637], where…
This paper investigates phase retrieval using the Reshaped Wirtinger Flow (RWF) algorithm, focusing on recovering target vector $\vx \in \R^n$ from magnitude measurements \(y_i = \left| \langle \va_i, \vx \rangle \right|, \; i = 1, \ldots,…
The distributed Kaczmarz algorithm is an adaptation of the standard Kaczmarz algorithm to the situation in which data is distributed throughout a network represented by a tree. We isolate substructures of the network and study convergence…
We present the first phase retrieval algorithm guaranteed to solve the multidimensional phase retrieval problem in polynomial arithmetic complexity without prior information. The method successfully terminates in O(N log(N)) operations for…
The randomized Kaczmarz ($\RK$) algorithm is a simple but powerful approach for solving consistent linear systems $Ax=b$. This paper proposes an accelerated randomized Kaczmarz ($\ARK$) algorithm with better convergence than the standard…
Due to the ever growing amounts of data leveraged for machine learning and scientific computing, it is increasingly important to develop algorithms that sample only a small portion of the data at a time. In the case of linear least-squares,…
Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination,…
We study nonconvex optimization for phase retrieval and the more general problem of semidefinite low-rank matrix sensing; in particular, we focus on the global nonconvex landscape of overparametrized versions of the nonsmooth amplitude…
The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have developed an acceleration technique for the Kaczmarz method that minimizes the distance…
Large-scale linear systems, $Ax=b$, frequently arise in practice and demand effective iterative solvers. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized…