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

07/17/2020

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by   Tae Hyung Kim, et al.

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We consider a setting in which it is desired to find an optimal complex vector 𝐱∈ℂ^N that satisfies 𝒜(𝐱) ≈𝐛 in a least-squares sense, where 𝐛∈ℂ^M is a data vector (possibly noise-corrupted), and 𝒜(·): ℂ^N →ℂ^M is a measurement operator. If 𝒜(·) 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 𝒜(·) 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 ℂ^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.