Typing Tensor Calculus in 2-Categories (I)
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866911084923846656 |
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| author | Ahmadi, Fatimah Rita |
| author_facet | Ahmadi, Fatimah Rita |
| contents | To formalize calculations in linear algebra for the development of efficient algorithms and a framework suitable for functional programming languages and faster parallelized computations, we adopt an approach that treats elements of linear algebra, such as matrices, as morphisms in the category of matrices, $\mathbf{Mat_{k}}$. This framework is further extended by generalizing the results to arbitrary monoidal semiadditive categories. To enrich this perspective and accommodate higher-rank matrices (tensors), we define semiadditive 2-categories, where matrices $T_{ij}$ are represented as 1-morphisms, and tensors with four indices $T_{ijkl}$ as 2-morphisms. This formalization provides an index-free, typed linear algebra framework that includes matrices and tensors with up to four indices. Furthermore, we extend the framework to monoidal semiadditive 2-categories and demonstrate detailed operations and vectorization within the 2-category of 2Vec introduced by Kapranov and Voevodsky. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_01212 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Typing Tensor Calculus in 2-Categories (I) Ahmadi, Fatimah Rita Category Theory Machine Learning Software Engineering To formalize calculations in linear algebra for the development of efficient algorithms and a framework suitable for functional programming languages and faster parallelized computations, we adopt an approach that treats elements of linear algebra, such as matrices, as morphisms in the category of matrices, $\mathbf{Mat_{k}}$. This framework is further extended by generalizing the results to arbitrary monoidal semiadditive categories. To enrich this perspective and accommodate higher-rank matrices (tensors), we define semiadditive 2-categories, where matrices $T_{ij}$ are represented as 1-morphisms, and tensors with four indices $T_{ijkl}$ as 2-morphisms. This formalization provides an index-free, typed linear algebra framework that includes matrices and tensors with up to four indices. Furthermore, we extend the framework to monoidal semiadditive 2-categories and demonstrate detailed operations and vectorization within the 2-category of 2Vec introduced by Kapranov and Voevodsky. |
| title | Typing Tensor Calculus in 2-Categories (I) |
| topic | Category Theory Machine Learning Software Engineering |
| url | https://arxiv.org/abs/1908.01212 |