Title: On the structure of join tensors with applications to tensor eigenvalue problems,
Abstract: We investigate the structure of join tensors, which may be regarded as the
multivariable extension of lattice-theoretic join matrices. Explicit formulae
for a polyadic decomposition (i.e., a linear combination of rank-1 tensors) and
a tensor-train decomposition of join tensors are derived on general join
semilattices. We discuss conditions under which the obtained decompositions are
optimal in rank, and examine numerically the storage complexity of the obtained
decompositions for a class of LCM tensors as a special case of join tensors. In
addition, we investigate numerically the sharpness of a theoretical upper bound
on the tensor eigenvalues of LCM tensors.