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

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Document pages: 19 pages

Abstract: We consider a setting in which it is desired to find an optimal complexvector $ mathbf{x} in mathbb{C}^N$ that satisfies $ mathcal{A}( mathbf{x}) approx mathbf{b}$ in a least-squares sense, where $ mathbf{b} in mathbb{C}^M$ is a data vector (possibly noise-corrupted), and$ mathcal{A}( cdot): mathbb{C}^N rightarrow mathbb{C}^M$ is a measurementoperator. If $ mathcal{A}( cdot)$ were linear, this reduces to the classicallinear least-squares problem, which has a well-known analytic solution as wellas powerful iterative solution algorithms. However, instead of linearleast-squares, this work considers the more complicated scenario where$ mathcal{A}( cdot)$ is nonlinear, but can be represented as the summationand or composition of some operators that are linear and some operators thatare antilinear. Some common nonlinear operations that have this structureinclude complex conjugation or taking the real-part or imaginary-part of acomplex vector. Previous literature has shown that this kind of mixedlinear antilinear least-squares problem can be mapped into a linearleast-squares problem by considering $ mathbf{x}$ as a vector in$ mathbb{R}^{2N}$ instead of $ mathbb{C}^N$. While this approach is valid, thereplacement of the original complex-valued optimization problem with areal-valued optimization problem can be complicated to implement, and can alsobe associated with increased computational complexity. In this work, wedescribe theory and computational methods that enable mixed linear antilinearleast-squares problems to be solved iteratively using standard linearleast-squares tools, while retaining all of the complex-valued structure of theoriginal inverse problem. An illustration is provided to demonstrate that thisapproach can simplify the implementation and reduce the computationalcomplexity of iterative solution algorithms.

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