  The block-term tensor decomposition model with multilinear rank-$(L_r,L_r,1)$
terms (or, the "LL1 tensor decomposition" in short) offers a valuable
alternative for hyperspectral unmixing (HU) under the linear mixture model.
Particularly, the LL1 decomposition ensures the endmember/abundance
identifiability in scenarios where such guarantees are not supported by the
classic matrix factorization (MF) approaches. However, existing LL1-based HU
algorithms use a three-factor parameterization of the tensor (i.e., the
hyperspectral image cube), which leads to a number of challenges including high
per-iteration complexity, slow convergence, and difficulties in incorporating
structural prior information. This work puts forth an LL1 tensor
decomposition-based HU algorithm that uses a constrained two-factor
re-parameterization of the tensor data. As a consequence, a two-block
alternating gradient projection (GP)-based LL1 algorithm is proposed for HU.
With carefully designed projection solvers, the GP algorithm enjoys a
relatively low per-iteration complexity. Like in MF-based HU, the factors under
our parameterization correspond to the endmembers and abundances. Thus, the
proposed framework is natural to incorporate physics-motivated priors that
arise in HU. The proposed algorithm often attains orders-of-magnitude speedup
and substantial HU performance gains compared to the existing three-factor
parameterization-based HU algorithms.
