PARAFAC-Based Channel Estimation for Beyond Diagonal Reconfigurable Surfaces

Channel estimation is a central bottleneck in BD-RIS-assisted MIMO systems. The richer inter-element coupling that enables large performance gains also makes training and hardware control substantially harder than in diagonal RIS architectures. Existing estimators either target only cascaded channels or require block-by-block reconfiguration of the BD-RIS interconnections, which is costly and difficult to implement in practice. To overcome this limitation, we propose a pilot-assisted tensor framework for group-connected BD-RIS under a two-timescale protocol, where the scattering structure is designed as a low-rank PARAFAC model with fixed factor matrices. This design keeps the interconnection topology constant across blocks and updates only phase shifts, enabling practical operation without sacrificing estimation quality. Building on this structure, we develop a PARAFAC-based alternating least-squares (PALS) receiver that recovers the individual channels. Numerical results confirm that PALS delivers markedly lower composite-channel NMSE than conventional LS, matches the accuracy of state-of-the-art tensor receivers, and sharply reduces BD-RIS design complexity