On the structure of join tensors with applications to tensor eigenvalue problems
Provides theoretical and numerical insights into the structure of join tensors, which may benefit tensor eigenvalue problems, but the contribution is incremental and domain-specific.
The paper derives explicit polyadic and tensor-train decompositions for join tensors on general join semilattices, discusses optimality conditions, and numerically examines storage complexity for LCM tensors and sharpness of eigenvalue bounds.
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.