Depth-first search for tensor rank and border rank over finite fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915030934487040 |
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| author | Yang, Jason |
| author_facet | Yang, Jason |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14676 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Depth-first search for tensor rank and border rank over finite fields Yang, Jason Computational Complexity 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. |
| title | Depth-first search for tensor rank and border rank over finite fields |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2411.14676 |