André L. F. de Almeida

Beyond Diagonal (BD) RIS has recently been proposed and proven theoretically to enhance channel gain and coverage. The performance of BD-RIS architectures and modes depends heavily on the accuracy of the channel state information. However, the conventional least squares (LS) estimator requires significant training and disregards the built-in block Kronecker structure of the composite channel. In this talk, the presenter links the channel estimation problem for BD-RIS to a tensor decomposition problem. Specifically, he will demonstrate that the received pilot signals can be organized as a three-way (3D) array or a third-order tensor that follows a block Tucker decomposition model. First, he discusses how the specific BD-RIS architecture affects the resulting tensor decomposition structure. Next, he demonstrates how interpreting the received pilot signals as a three-way tensor allows us to reformulate the cascaded channel estimation problem as a block-Tucker tensor decomposition problem. This yields decoupled estimates for the involved channel matrices, substantially improving performance over the matrix-based LS method. He discusses two algorithms to solve this problem. The first is a closed-form solution that extracts channel estimates via block-Tucker Kronecker factorization (BTKF). This boils down to solving a set of parallel rank-one matrix approximation problems. The second algorithm is based on a block-Tucker alternating least squares (BTALS) method that directly estimates the channel matrices using an iterative procedure. He highlights the trade-offs of the BTKF and BTALS methods. BTKF offers fast, parallel extraction of channel estimates in closed form, while BTALS provides a more flexible training design that reduces training overhead significantly compared to state-of-the-art methods