# On the product of the singular values of a binary tensor

## TL;DR

This paper generalizes the product of singular values for binary tensors to higher dimensions, linking it to algebraic geometry concepts like discriminants and dual varieties, and computes related polynomial coefficients for small tensor formats.

## Contribution

It introduces a generalized formula for the product of singular values of binary tensors in higher dimensions and explores their algebraic properties and polynomial representations.

## Key findings

- Derived a formula relating singular value products to discriminants and sum of squares polynomials.
- Analyzed the variety of tensors with fewer singular values than maximum.
- Computed explicit polynomial coefficients for 2x2x2 tensors.

## 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 $\mu\vdash d$ 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 $2\times 2\times 2$.