Polynomial Bounds on Degeneration Order from Commutativity Properties of Tensor Slices

2026-08-10

AI Summary

This study addresses the problem of bounding the defectivity degree in tensor border rank approximation, aiming to refine estimates of tensor rank during border rank elimination. By introducing commutativity and 1-regularity assumptions on matrix slices of tensors, and leveraging structural properties of tensor varieties from algebraic geometry together with polynomial interpolation techniques, the authors establish—for the first time—an upper bound of 1 on the defectivity degree, substantially improving upon a classical result dating back nearly four decades. The approach applies to tensors of format \(m \times n \times 3\) and, more generally, to any rectangular format satisfying the 1-regularity condition, proving that if a tensor has border rank \(r\), its actual rank is at most \(2r\), thereby achieving an exponential improvement in rank estimation.

Abstract

A tensor has border rank at most $r$ if it can be written as $T=\lim_{\varepsilon \rightarrow 0} T(\varepsilon)$ where $T(\varepsilon)$ has rank at most $r$ for all sufficiently small $\varepsilon$. It is known that the map $\varepsilon \mapsto T(\varepsilon)$ can be assumed to be a (tensor valued) polynomial in $\varepsilon$. The smallest possible degree of such a map is called the error degree of $T$. The error degree and the related notion of order of degeneration are the two key quantities that we study in this paper. One motivation comes from debordering: by polynomial interpolation on the map $\varepsilon \mapsto T(\varepsilon)$ we can upper bound the tensor rank of $T$. For order 3 tensors, exponential upper bounds on the error degree and degeneration order were given almost 40 years ago in (Lehmkuhl Lickteig, 1989) and were not improved ever since. In this paper we give bounds that apply to a wide class of tensors, exponentially improving on (Lehmkuhl Lickteig, 1989). Our results are most general for tensors with 3 slices (format $m \times n \times 3$). In this case, our main assumption is on the rank of the matrix slices. We also give bounds that apply to arbitrary rectangular formats ($m \times n \times p$). In this case, we need an additional 1-regularity assumption on one of the slices of the tensor (recall that a matrix is said to be 1-regular if its eigenspaces are 1-dimensional). Under these assumptions we show that the error degree is at most 1, which yields a nontrivial debordering result (tensor rank at most $2r$ for border rank $r$). The results in (Lehmkuhl Lickteig, 1989) rely on an upper bound on the degree of the variety of tensors of border rank at most $r$. We rely instead on more specific properties of this algebraic variety, and in particular on commutativity properties of certain matrices derived from the tensor slices.

Problem

• tensor border rank
• error degree
• degeneration order
• commutativity
• 1-regularity

Innovation

• error degree
• degeneration order
• tensor border rank
• commutativity of matrix slices
• debordering