André L. F. de Almeida

Fazal-E Asim, André L. F. de Almeida, Bruno Sokal et al.

As beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) gain increasing attention in high-frequency wireless communications, accurate and scalable channel-estimation methods become essential. This paper develops a parametric channel-estimation and beamforming framework that deconstructs the composite BD-RIS channel into its generating directional factors, revealing the tensor structure induced jointly by propagation geometry and beyond-diagonal scattering. We propose two tensor-based estimators: Fourth-Order Tucker Channel Estimation (FORTE), which models the partially structured channel as a fourth-order Tucker tensor, and Fourth-Order PARAFAC Channel Estimation (FORPE), which captures the fully structured channel through a fourth-order PARAFAC model. By exploiting partial and full channel geometry, the proposed methods achieve higher estimation accuracy than Least Squares and Block Tucker Kronecker Factorization benchmarks. In particular, FORTE outperforms FORPE due to its more compact representation, attaining an NMSE of about 10^{-4} at 5 dB SNR. In contrast, FORPE provides essentially unique estimates of the composite-channel factor matrices, whereas FORTE identifies their subspaces. The proposed deconstruction also provides a structured representation useful for sensing-oriented parameter extraction and tensor-structured system optimization. Finally, the Tensor Optimization Framework for Beamforming, Combining, and Scattering (TenFormer) achieves spectral efficiency comparable to the benchmark design while significantly reducing computational complexity through parallel tensor-structured optimization.

Gérard Favier, André L. F. de Almeida

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product based approach is also formulated in terms of the index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for $N^{th}$ order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models.

Bruno Sokal, André L. F. de Almeida, Martin Haardt

We address joint active and passive beamforming for uplink RIS-assisted multi-user multi-stream MIMO systems with joint detection. The coupled design of the receive combiner, block-diagonal user precoders, and RIS phase vector is formulated through a third-order composite channel tensor. Exploiting this multilinear structure, we propose a multi-stream tensor alternating optimization method that updates the combiner, user precoders, and RIS coefficients via low-dimensional tensor projections. Simulations show that the proposed method approaches a multi-start alternating-optimization benchmark while reducing computational complexity and improving large-RIS scaling.