Related papers: Solving Random Quadratic Systems of Equations Is N…
In this paper we consider a system of quadratic equations |<z_j, x>|^2 = b_j, j = 1, ..., m, where x in R^n is unknown while normal random vectors z_j in R_n and quadratic measurements b_j in R are known. The system is assumed to be…
A novel algorithm for the computation of the quadratic numerical range is presented and exemplified yielding much better results in less time compared to the random vector sampling method. Furthermore, a bound on the probability for the…
We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x of C^n about which we have phaseless samples of the form y_r = |< a_r,x >|^2, r = 1,2,...,m (knowledge…
A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y$. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the…
A novel approach termed \emph{stochastic truncated amplitude flow} (STAF) is developed to reconstruct an unknown $n$-dimensional real-/complex-valued signal $\bm{x}$ from $m$ `phaseless' quadratic equations of the form…
The problem of recovering a signal $\mathbf{x}\in \mathbb{R}^n$ from a set of magnitude measurements $y_i=|\langle \mathbf{a}_i, \mathbf{x} \rangle |, \; i=1,\ldots,m$ is referred as phase retrieval, which has many applications in fields of…
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian…
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time…
Solving Quadratic equation is one of the intrinsic interests as it is the simplest nonlinear equations. A novel approach for solving Quadratic Equation based on Genetic Algorithms (GAs) is presented. Genetic Algorithms (GAs) are a technique…
We present a novel algorithm attaining excessively fast, the sought solution of linear systems of equations. The algorithm is short in its basic formulation and, by definition, vectorized, while the memory allocation demands are trivial,…
Linear equations play a pivotal role in many areas of science and engineering, making efficient solutions to linear systems highly desirable. The development of quantum algorithms for solving linear systems has been a significant…
This article considers stochastic algorithms for efficiently solving a class of large scale non-linear least squares (NLS) problems which frequently arise in applications. We propose eight variants of a practical randomized algorithm where…
A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica and Matlab) do not offer a solution, is to find all the real solutions of a system of N nonlinear equations in a certain finite…
Solving linear systems of equations is essential for many problems in science and technology, including problems in machine learning. Existing quantum algorithms have demonstrated the potential for large speedups, but the required quantum…
An algorithm is proposed, analyzed, and tested experimentally for solving stochastic optimization problems in which the decision variables are constrained to satisfy equations defined by deterministic, smooth, and nonlinear functions. It is…
Discovering nonlinear differential equations that describe system dynamics from empirical data is a fundamental challenge in contemporary science. Here, we propose a methodology to identify dynamical laws by integrating denoising techniques…
The problem of recovering a signal $\boldsymbol x\in \mathbb{R}^n$ from a quadratic system $\{y_i=\boldsymbol x^\top\boldsymbol A_i\boldsymbol x,\ i=1,\ldots,m\}$ with full-rank matrices $\boldsymbol A_i$ frequently arises in applications…
The randomized row method is a popular representative of the iterative algorithm because of its efficiency in solving the overdetermined and consistent systems of linear equations. In this paper, we present an extended randomized multiple…
This paper investigates system identification problems with Gaussian inputs and quantized observations under fixed thresholds. By reinterpreting the nonlinear effects induced by quantization as the product of the unknown parameter and an…
In this work, we propose a new way to (non-interactively, verifiably) demonstrate Quantum Advantage by solving the average-case $\mathsf{NP}$ search problem of finding a solution to a system of (underdetermined) multivariate quadratic…