Depth-first search for tensor rank and border rank over finite fields

TL;DR

This paper introduces efficient algorithms for determining tensor rank and border rank over finite fields and rings, with polynomial space complexity, advancing computational methods in tensor analysis.

Contribution

The paper presents novel algorithms with specific time complexities for tensor rank and border rank over finite fields and rings, extending prior methods.

Findings

Algorithms operate in polynomial space.

Time complexity depends on tensor dimensions and rank.

Applicable to tensors over finite fields and rings.

Abstract

We present an $O^*\left(|\mathbb{F}|^{(R-n_*)\left(\sum_d n_d\right)+n_*}\right)$-time algorithm for determining whether a tensor of shape $n_0\times\dots\times n_{D-1}$ over a finite field $\mathbb{F}$ has rank $\le R$, where $n_*:=\max_d n_d$; we assume without loss of generality that $\forall d:n_d\le R$. We also extend this problem to its border rank analog, i.e., determining tensor rank over rings of the form $\mathbb{F}[x]/(x^H)$, and give an $O^*\left(|\mathbb{F}|^{H\sum_{1\le r\le R} \sum_d \min(r,n_d)}\right)$-time algorithm. Both of our algorithms use polynomial space.