# On the structure of join tensors with applications to tensor eigenvalue problems

## TL;DR

This paper explores the structure of join tensors, providing explicit decompositions and analyzing their properties, with applications to tensor eigenvalue problems and specific cases like LCM tensors.

## Contribution

It introduces explicit polyadic and tensor-train decompositions for join tensors on general join semilattices, including conditions for optimal rank and numerical analysis.

## Key findings

- Derived explicit decompositions for join tensors.
- Analyzed storage complexity for LCM tensors.
- Numerically examined eigenvalue bounds for LCM tensors.

## 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.