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

Jason Yang

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.

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

Abstract

We present an -time algorithm for determining whether a tensor of shape over a finite field has rank , where ; we assume without loss of generality that . We also extend this problem to its border rank analog, i.e., determining tensor rank over rings of the form , and give an -time algorithm. Both of our algorithms use polynomial space.

Paper Structure

This paper contains 11 sections, 4 theorems, 1 equation, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Axis-reduction
  4. Tensor rank
  5. Algorithm
  6. Complexity
  7. Border rank
  8. Motivation and Definitions
  9. Border depth-first search
  10. Conclusion and Next Steps
  11. Acknowledgments

Key Result

Lemma 1

For any tensor $T\in\mathbb{F}^{n_0\times\dots\times n_{D-1}}$, $\mathrm{rk}\left(T\right)\le 1$ if and only if $\forall d: \mathrm{rk}\left(T_{\left(d\right)}\right)\le 1$. Furthermore, if this condition is satisfied, a minimum-rank CPD of $T$ can be found in polynomial time.

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof