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Orthogonal Rank Decompositions for Tensors

The theory of orthogonal rank decompositions for matrices is well understood, but the same is not true for tensors. For tensors, even the notions of orthogonality and rank can be interpreted several diierent ways. Tensor decompositions are useful in applications such as principal component analysis for multiway data. We present two types of orthogonal rank decompositions and describe methods to compute them. Furthermore, we conjecture an extension of the Eckart-Young theorem for one of these decompositions and provide a counterexample to show that it does not hold in the other case.