When does subtracting a rank-one approximation decrease tensor rank?
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916576173752320 |
|---|---|
| author | Horobet, Emil Turatti, Ettore Teixeira |
| author_facet | Horobet, Emil Turatti, Ettore Teixeira |
| contents | Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14985 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | When does subtracting a rank-one approximation decrease tensor rank? Horobet, Emil Turatti, Ettore Teixeira Algebraic Geometry Metric Geometry Optimization and Control 14Q99, 15A69, 58K05, 14M12, 13P99, 41A65 Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors. |
| title | When does subtracting a rank-one approximation decrease tensor rank? |
| topic | Algebraic Geometry Metric Geometry Optimization and Control 14Q99, 15A69, 58K05, 14M12, 13P99, 41A65 |
| url | https://arxiv.org/abs/2303.14985 |