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The Complexity of Tensor Rank

Marcus Schaefer, Daniel Stefankovic

TL;DR

The paper establishes that the tensor rank decision problem over a field $$ is polynomial-time equivalent to the existential theory $ ext{ETh}()$, implying field-dependent complexity (e.g., $ ext{∃}C$-completeness over $$ and $ ext{∃}R$-completeness over $$ in general). It constructs a chain of reductions from $ ext{ETh}()$ to a quadratic-system minrank problem and then to tensor rank via a tensor $T_A$, showing that $ ext{minrank}_{}(A)=2m$ iff $ ext{rk}_{}(T_A)\le 2m+n$. The authors further prove an algebraic-universality result: for any algebraic set, there exists a tensor instance whose realization space mirrors the set, thereby encoding arbitrary algebraic geometry within tensor-rank problems. These results unify and sharpen prior hardness (notably Håstad’s NP-hardness over finite fields) and reveal that tensor-rank complexity ranges from near-NP to PSPACE regimes depending on the base field, while raising open questions about symmetry, fixed-rank cases, and undecidability over $b Z$.

Abstract

We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by Håstad~\cite{H90}. The hardness proof also implies an algebraic universality result.

The Complexity of Tensor Rank

TL;DR

The paper establishes that the tensor rank decision problem over a field is polynomial-time equivalent to the existential theory , implying field-dependent complexity (e.g., -completeness over and -completeness over in general). It constructs a chain of reductions from to a quadratic-system minrank problem and then to tensor rank via a tensor , showing that iff . The authors further prove an algebraic-universality result: for any algebraic set, there exists a tensor instance whose realization space mirrors the set, thereby encoding arbitrary algebraic geometry within tensor-rank problems. These results unify and sharpen prior hardness (notably Håstad’s NP-hardness over finite fields) and reveal that tensor-rank complexity ranges from near-NP to PSPACE regimes depending on the base field, while raising open questions about symmetry, fixed-rank cases, and undecidability over .

Abstract

We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by Håstad~\cite{H90}. The hardness proof also implies an algebraic universality result.

Paper Structure

This paper contains 10 sections, 10 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and Tools
  3. Tensors
  4. Logic and Complexity
  5. Algebraic Universality
  6. Hardness of Tensor Rank
  7. A Minimum Rank Problem
  8. A Tensor Rank Problem
  9. Universality
  10. Open Questions

Key Result

Theorem 1.1

Let $\mathbb{F}$ be a field. Given a statement $\varphi$ in $\mathop{\mathrm{\mathrm{ETh}}}\nolimits(\mathbb{F})$, the existential theory of $\mathbb{F}$, we can in polynomial time construct a tensor $T_\varphi$ and an integer $k$ so that $\varphi$ is true over $\mathbb{F}$ if and only if $T$ has te

Figures (1)

  • Figure 1: Complexity of the tensor rank problem over various rings. Previously all these problems were known to be $\mathbf{NP}$-hard using Håstad's argument H90HL13.

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Håstad H89
  • Lemma 2.2: Håstad H89
  • Lemma 2.3: Buss, Frandsen, Shallit BFS99
  • Lemma 2.4
  • Example 2.5: Hillar, Lim HL13
  • Definition 3.1
  • Lemma 3.2
  • Example 3.3
  • ...and 10 more