The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911801902366720 |
|---|---|
| author | Connelly, Erin Agarwal, Sameer Ergur, Alperen Thomas, Rekha R. |
| author_facet | Connelly, Erin Agarwal, Sameer Ergur, Alperen Thomas, Rekha R. |
| contents | Given $k \leq 6$ points $(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2$, we characterize rank deficiency of the $k \times 9$ matrix $Z_k$ with rows $x_i^\top \otimes y_i^\top$ in terms of the geometry of the point configurations $\{x_i\}$ and $\{y_i\}$. While this question comes from computer vision the answer relies on tools from classical algebraic geometry: For $k \leq 5$, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case $k=6$ the rank-drop locus is captured by the classical theory of cubic surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09826 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products Connelly, Erin Agarwal, Sameer Ergur, Alperen Thomas, Rekha R. Algebraic Geometry Commutative Algebra Combinatorics Given $k \leq 6$ points $(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2$, we characterize rank deficiency of the $k \times 9$ matrix $Z_k$ with rows $x_i^\top \otimes y_i^\top$ in terms of the geometry of the point configurations $\{x_i\}$ and $\{y_i\}$. While this question comes from computer vision the answer relies on tools from classical algebraic geometry: For $k \leq 5$, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case $k=6$ the rank-drop locus is captured by the classical theory of cubic surfaces. |
| title | The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products |
| topic | Algebraic Geometry Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2301.09826 |