Orthogonal Approximate Message Passing for Double Linear Transformation Model

The double linear transformation (DLT) modelY = AXB+Nis widely used in signal processing and wireless communications, whereXis estimated from the noisy observation Y with known matricesAandB. Most existing algorithms reformulate the DLT model into a vectorized linear form with the equivalent transformation matrixBT⊗A, upon which message passing principles are utilized to recover the signal. However, the Kronecker product causes numerous correlated submatrices inBT⊗A, which significantly increases the matrix dimension and obscures the original statistical distribution properties ofAandB. To address this challenge, we propose a novel equivalent tri-decoupled double transformation (TDDT) framework that introduces an intermediate variableZ = XBto decouple the double linear constraint into two single linear constraints. This decoupling can preserve both the dimension and the statistical properties ofAandBin 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 onXbeyond 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.