  This paper addresses the channel estimation problem for beyond diagonal
reconfigurable intelligent surface (BD-RIS) from a tensor decomposition
perspective. We first show that the received pilot signals can be arranged as a
three-way tensor, allowing us to recast the cascaded channel estimation problem
as a block Tucker decomposition problem that yields decoupled estimates for the
involved channel matrices while offering a substantial performance gain over
the conventional (matrix-based) least squares (LS) estimation method. More
specifically, we develop two solutions to solve the problem. The first one is a
closed-form solution that extracts the channel estimates via a block Tucker
Kronecker factorization (BTKF), which boils down to solving a set of parallel
rank-one matrix approximation problems. Exploiting such a low-rank property
yields a noise rejection gain compared to the standard LS estimation scheme
while allowing the two involved channels to be estimated separately. The second
solution is based on a block Tucker alternating least squares (BTALS) algorithm
that directly estimates the involved channel matrices using an iterative
estimation procedure. We discuss the uniqueness and identifiability issues and
their implications for training design. We also propose a tensor-based design
of the BD-RIS training tensor for each algorithm that ensures unique decoupled
channel estimates under trivial scaling ambiguities. Our numerical results shed
light on the tradeoffs offered by BTKF and BTALS methods. Specifically, while
the first enjoys fast and parallel extraction of the channel estimates in
closed form, the second has a more flexible training design, allowing for a
significantly reduced training overhead compared to the state-of-the-art LS
method.
