Gram Determinants of Real Binary Tensors

Abstract

A binary tensor consists of $2^n$ entries arranged into hypercube format $2\times 2\times \cdots \times 2$. There are $n$ ways to flatten such a tensor into a matrix of size $2\times 2^{n-1}$. For each flattening, $M$, we take the determinant of its Gram matrix, $\det (M{M}^T)$. We consider the map that sends a tensor to its $n$-tuple of Gram determinants. We propose a semi-algebraic characterization of the image of this map. This offers an answer to a question raised by Hackbusch and Uschmajew concerning the higher-order singular values of tensors.

arXiv comment18 pages