How hard is the tensor rank?

TL;DR

This paper establishes the computational complexity of tensor rank over various domains, showing it is polynomial-time equivalent to solving polynomial systems, with implications for undecidability and NP-hardness results.

Contribution

It provides a complete complexity characterization of tensor rank and symmetric rank, solving open problems and confirming conjectures in computational mathematics.

Findings

Tensor rank over integers is undecidable.

Computing symmetric rank over rationals is NP-hard.

The minimal rank matrix completion problem's complexity is characterized.

Abstract

We investigate the computational complexity of tensor rank, a concept that plays fundamental role in different topics of modern applied mathematics. For tensors over any integral domain, we prove that the rank problem is polynomial time equivalent to solving a system of polynomial equations over this integral domain. Our result gives a complete description of the algorithmic complexity of tensor rank and allows one to solve several known open problems. In particular, the tensor rank over $\mathbb{Z}$ turns out to be undecidable, which answers the question posed by Gonzalez and Ja'Ja' in 1980. We generalize our result and prove that the symmetric rank admits a similar description of computational complexity as the one we give for usual rank. In particular, computing the symmetric rank of a rational tensor is shown to be NP-hard, which proves a recent conjecture of Hillar and Lim. As a byproduct of our approach, we get a similar characterization of the algorithmic complexity of the minimal rank matrix completion problem, which gives a complete answer to the question discussed in 1999 by Buss, Frandsen, and Shallit.