Best rank k approximation for binary forms

TL;DR

This paper investigates the best rank k approximation problem for binary forms in tensor space, showing that critical points of the approximation lie in a hyperplane, extending known results from rank 1.

Contribution

It generalizes the understanding of critical points from rank 1 to rank k in the tensor space of binary forms, revealing their geometric structure.

Findings

Critical points of rank 1 approximation are eigenvectors.

Critical points of rank k approximation lie in a hyperplane.

Extends geometric understanding of approximation in tensor space.

Abstract

In the tensor space $\mathrm{Sym}^d {\mathbb R}^2$ of binary forms we study the best rank $k$ approximation problem. The critical points of the best rank $1$ approximation problem are the eigenvectors and it is known that they span a hyperplane. We prove that the critical points of the best rank $k$ approximation problem lie in the same hyperplane.