  We study the complexity of isomorphism problems for d-way arrays, or tensors,
under natural actions by classical groups such as orthogonal, unitary, and
symplectic groups. Such problems arise naturally in statistical data analysis
and quantum information. We study two types of complexity-theoretic questions.
First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the
complexity of the isomorphism problem over a classical group to that over the
general linear group. Second, for a fixed group type (orthogonal, unitary, or
symplectic), we compare the complexity of the decision problems for different
actions.
  Our main results are as follows. First, for orthogonal and symplectic groups
acting on 3-way arrays, the isomorphism problems reduce to the corresponding
problem over the general linear group. Second, for orthogonal and unitary
groups, the isomorphism problems of five natural actions on 3-way arrays are
polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the
3-tensor isomorphism problem for any fixed d>3. For unitary groups, the
preceding result implies that LOCC classification of tripartite quantum states
is at least as difficult as LOCC classification of d-partite quantum states for
any d. Lastly, we also show that the graph isomorphism problem reduces to the
tensor isomorphism problem over orthogonal and unitary groups.
