Orthogonal Approximate Message Passing for Double Linear Transformation Model

The double linear transformation (DLT) model $\boldsymbol{Y}=\boldsymbol{A}\boldsymbol{X}\boldsymbol{B}+\boldsymbol{N}$ is widely used in signal processing and wireless communications, where $\boldsymbol{X}$ is estimated from the noisy observation $\boldsymbol{Y}$ with known matrices $\boldsymbol{A}$ and $\boldsymbol{B}$. Most existing algorithms reformulate the DLT model into a vectorized linear form with the equivalent transformation matrix $\boldsymbol{B}^{\rm T}\otimes\boldsymbol{A}$, upon which message passing principles are utilized to recover the signal. However, the Kronecker product causes numerous correlated submatrices in $\boldsymbol{B}^{\rm T}\otimes\boldsymbol{A}$, which significantly increases the matrix dimension and obscures the original statistical distribution properties of $\boldsymbol{A}$ and $\boldsymbol{B}$. To address this challenge, we propose a novel equivalent tri-decoupled double transformation (TDDT) framework that introduces an intermediate variable $\boldsymbol{Z}=\boldsymbol{X}\boldsymbol{B}$ to decouple the double linear constraint into two single linear constraints. This decoupling can preserve both the dimension and the statistical properties of $\boldsymbol{A}$ and $\boldsymbol{B}$ in the single linear problems. Building on the TDDT framework, a double linear orthogonal approximate message passing (DL-OAMP) algorithm is proposed, which incorporates three estimators for two decoupled single linear constraints and a more sophisticated nonlinear constraint on $\boldsymbol{X}$ beyond the independent and identically distributed assumption. Meanwhile, a state evolution analysis is established to accurately predict the asymptotic performance of DL-OAMP. Within the TDDT framework, we further apply the proposed DL-OAMP algorithm to compressed sensing reconstruction in signal processing and massive random access for near-field communications. Numerical results demonstrate that the proposed DL-OAMP algorithm achieves superior performance compared with state-of-the-art methods in practical applications.