On the product of the singular values of a binary tensor

Abstract

A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge 2$. Given a partition $\mu ⊢d$ and a $\mu$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{\mu ,\mathbb{R}}$ of $\mu$-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of $t$. Denoting with $X_{\mu }$ the complexification of $X_{\mu ,\mathbb{R}}$, the Euclidean Distance polynomial ${\mathrm{EDpoly}}_{X_{\mu }^{\vee },t}({\epsilon }^2)$ of the dual variety of $X_{\mu }$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of ${\mathrm{EDpoly}}_{X_{\mu }^{\vee },t}({\epsilon }^2)$ is the square of the $\mu$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $\mu$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of ${\mathrm{EDpoly}}_{X_{\mu }^{\vee },t}({\epsilon }^2)$ for tensors of format $2\times 2\times 2$.

21 pages, 5 figures