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 $μ\vdash d$ and a $μ$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{μ,\mathbb{R}}$ of $μ$-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 $X_μ$ the complexification of $X_{μ,\mathbb{R}}$, the Euclidean Distance polynomial $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ of the dual variety of $X_μ$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ is the square of the $μ$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $μ$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ for tensors of format $2\times 2\times 2$.