Gram Determinants of Real Binary Tensors

A binary tensor consists of 2 entries arranged into hypercube format 2 × 2 × ⋯ × 2. There are n ways to flatten such a tensor into a matrix of size 2 × 2 . For each flattening, M, we take the determinant of its Gram matrix, det(MM ). 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.