Abstract
The double linear transformation model <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm Y = \bm {AXB}+ \bm W$</tex-math></inline-formula> plays an important role in a variety of science and engineering applications, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm X$</tex-math></inline-formula> is estimated through known transformation matrices <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm A$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm B$</tex-math></inline-formula> from the noisy measurement <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm Y$</tex-math></inline-formula> . Decoupling <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm X$</tex-math></inline-formula> from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm Y$</tex-math></inline-formula> is a formidable task due to the high complexity brought by the multiplication of the unknown matrix (vector) with the transformation matrix (M-UMTM). Unitary approximate message passing (UAMP) has been verified as a low complexity and strong robustness solution to the M-UMTM problems. However, it has only been used to tackle the problems with a single linear transformation matrix. In this work, we develop a generalized algorithm, namely, generalized double UAMP (GD-UAMP) for the target model, which not only inherits the low complexity of AMP, but also enhances robustness by employing double unitary transformation. As a generalized algorithm, GD-UAMP can be applied to address the generalized Bayesian inference problem, i.e., the arbitrary prior probability of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm X$</tex-math></inline-formula> and likelihood function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm Z$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm Z= \bm {AXB}$</tex-math></inline-formula> is the noiseless measurement. We verify the feasibility of the proposed algorithm in the channel estimation problem for various wireless communication systems. Numerical results demonstrate that the proposed algorithm can perfectly fit different scenarios and showcase superior performance compared with benchmarks.
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