Orthogonal tensor decompositions are nice, and if a given tensor has such a decomposition it can easily be found. But this notion is quite restrictive. If an x n x n tensor has an orthogonal decomposition, then its rank is at most n. In my talk I will discuss ways of weakening the notion of orthogonality while maintaining some of the nice properties. For example, I will introduce the notion of t-orthogonality. The usual orthogonal decomposition of a d-way tensor is d-orthogonal. But 2-orthogonal tensor decompositions (called Diagonal Singular Value Decompositions) may have rank up to the square root of the product of the dimensions of each of the modes. Moreover, the diagonal singular value decomposition is unique if the singular values are distinct, or if the decomposition is t-orthogonal for some real number t>2. An interesting challenge would be to develop an algorithm that finds the diagonal singular value decomposition if it exists.
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