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When does subtracting a rank-one approximation decrease tensor rank?

Emil Horobet, Ettore Teixeira Turatti

TL;DR

The article addresses when subtracting a critical rank-one approximation from a tensor can drive the tensor to zero through repeated steps. It develops the data-loci framework $DL_r$ via the Euclidean Distance Degree and ED duality, connecting distance optimization to the dual variety and its conormal, ultimately describing $DL_2$ and the higher layers recursively as $DL_r = DL_{X^* \cap DL_{r-1}}$. In the symmetric setting the authors prove chain stabilization at a finite step with an explicit bound, while for general tensors $DL_2$ is governed by bottleneck points and nodal singularities of the hyperdeterminant, and all $DL_r$ contain the weakly orthogonally decomposable tensors of border rank at most $r$. The work provides a structural bridge between low-border-rank data, algebraic geometry of rank-one varieties, and potential algorithmic pathways for higher-rank approximations, with illustrative matrix and $2\times2\times2$ tensor examples guiding intuition about stabilization and singular loci.

Abstract

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.

When does subtracting a rank-one approximation decrease tensor rank?

TL;DR

The article addresses when subtracting a critical rank-one approximation from a tensor can drive the tensor to zero through repeated steps. It develops the data-loci framework via the Euclidean Distance Degree and ED duality, connecting distance optimization to the dual variety and its conormal, ultimately describing and the higher layers recursively as . In the symmetric setting the authors prove chain stabilization at a finite step with an explicit bound, while for general tensors is governed by bottleneck points and nodal singularities of the hyperdeterminant, and all contain the weakly orthogonally decomposable tensors of border rank at most . The work provides a structural bridge between low-border-rank data, algebraic geometry of rank-one varieties, and potential algorithmic pathways for higher-rank approximations, with illustrative matrix and tensor examples guiding intuition about stabilization and singular loci.

Abstract

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.
Paper Structure (8 sections, 12 theorems, 46 equations, 1 table)

This paper contains 8 sections, 12 theorems, 46 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Joint ED correspondence and ED Duality
  4. Special data locus on the dual variety $X^*$
  5. The construction of the first layer of special tensor data locus $\operatorname{DL}_2$
  6. The sequence of higher border rank special tensor data loci
  7. Symmetric tensors
  8. Examples of $\operatorname{DL}_N$ and conclusion

Key Result

Theorem 2.2

Let $X\subseteq V$ be an irreducible affine cone, $X^*\subseteq V$ its dual variety, and $T\in V$ a general data point. The map $x\mapsto T-x$ gives a bijection from the critical points of $d_T$ on $X$ to the critical points of $d_T$ on $X^*$.

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.2: ED Duality
  • Remark 2.3
  • Remark 3.1
  • Proposition 3.2: Structure of $\operatorname{DL}_2$
  • proof
  • Remark 3.3
  • Proposition 3.4: Dimension of $\operatorname{DL}_A$ non-singular case
  • proof
  • Remark 3.5
  • ...and 23 more