Efficient iterative solutions to complex-valued nonlinear least-squares problems with mixed linear and antilinear operators

Abstract

AbstractWe consider a setting in which it is desired to find an optimal complex vector $${\mathbf {x}}\in {\mathbb {C}}^N$$ x ∈ C N that satisfies $${\mathcal {A}}({\mathbf {x}}) \approx {\mathbf {b}}$$ A ( x ) ≈ b in a least-squares sense, where $${\mathbf {b}} \in {\mathbb {C}}^M$$ b ∈ C M is a data vector (possibly noise-corrupted), and $${\mathcal {A}}(\cdot ): {\mathbb {C}}^N \rightarrow {\mathbb {C}}^M$$ A ( · ) : C N → C M is a measurement operator. If $${\mathcal {A}}(\cdot )$$ 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 $${\mathcal {A}}(\cdot )$$ 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 $${\mathbf {x}}$$ x as a vector in $${\mathbb {R}}^{2N}$$ R 2 N instead of $${\mathbb {C}}^N$$ 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.