Efficient Iterative Solutions to Complex-Valued Nonlinear Least-Squares Problems with Mixed Linear and Antilinear Operators

07/17/2020
βˆ™
by   Tae Hyung Kim, et al.
βˆ™
0
βˆ™

We consider a setting in which it is desired to find an optimal complex vector π±βˆˆβ„‚^N that satisfies π’œ(𝐱) β‰ˆπ› in a least-squares sense, where π›βˆˆβ„‚^M is a data vector (possibly noise-corrupted), and π’œ(Β·): β„‚^N β†’β„‚^M is a measurement operator. If π’œ(Β·) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where π’œ(Β·) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering 𝐱 as a vector in ℝ^2N instead of β„‚^N. While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
βˆ™ 01/21/2022

Extended Randomized Kaczmarz Method for Sparse Least Squares and Impulsive Noise Problems

The Extended Randomized Kaczmarz method is a well known iterative scheme...
research
βˆ™ 07/06/2011

Integrating Generic Sensor Fusion Algorithms with Sound State Representations through Encapsulation of Manifolds

Common estimation algorithms, such as least squares estimation or the Ka...
research
βˆ™ 12/28/2019

Scaled Relative Graph of Normal Matrices

The Scaled Relative Graph (SRG) by Ryu, Hannah, and Yin (arXiv:1902.0978...
research
βˆ™ 07/29/2019

Multivariate approximation of functions on irregular domains by weighted least-squares methods

We propose and analyse numerical algorithms based on weighted least squa...
research
βˆ™ 06/29/2020

A Nonmonotone Matrix-Free Algorithm for Nonlinear Equality-Constrained Inverse Problems

Least squares form one of the most prominent classes of optimization pro...
research
βˆ™ 08/10/2019

Separable nonlinear least-squares parameter estimation for complex dynamic systems

Nonlinear dynamic models are widely used for characterizing functional f...
research
βˆ™ 04/19/2021

On the Complexity of Inverse Mixed Integer Linear Optimization

Inverse optimization is the problem of determining the values of missing...

Please sign up or login with your details

Forgot password? Click here to reset