Gram Determinants of Real Binary Tensors

TL;DR

This paper studies the algebraic properties of binary tensors by analyzing the determinants of Gram matrices from different flattenings, providing a semi-algebraic description of their possible values and addressing questions about higher-order singular values.

Contribution

It introduces a semi-algebraic characterization of the Gram determinants' image for binary tensors, advancing understanding of their higher-order singular values.

Findings

Provides a semi-algebraic description of Gram determinant images

Answers a question on higher-order singular values of tensors

Characterizes the algebraic structure of binary tensor flattenings

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, ${\rm det }(M M^T)$. 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.