Rank and symmetries of signature tensors
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910546048057344 |
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| author | Galuppi, Francesco Santarsiero, Pierpaola |
| author_facet | Galuppi, Francesco Santarsiero, Pierpaola |
| contents | The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. In this paper we propose a systematic study of basic properties of signature tensors, starting from their rank, symmetries and conciseness. We prove a sharp upper bound on the rank of signature tensors of piecewise linear paths. We show that there are no skew-symmetric signature tensors of order three or more, and we also prove that specific instances of partial symmetry can only happen for tensors of order three. Finally, we give a simple geometric characterization of paths whose signature tensors are not concise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20405 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rank and symmetries of signature tensors Galuppi, Francesco Santarsiero, Pierpaola Algebraic Geometry 14N07, 15A69, 15A72, 60L10 The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. In this paper we propose a systematic study of basic properties of signature tensors, starting from their rank, symmetries and conciseness. We prove a sharp upper bound on the rank of signature tensors of piecewise linear paths. We show that there are no skew-symmetric signature tensors of order three or more, and we also prove that specific instances of partial symmetry can only happen for tensors of order three. Finally, we give a simple geometric characterization of paths whose signature tensors are not concise. |
| title | Rank and symmetries of signature tensors |
| topic | Algebraic Geometry 14N07, 15A69, 15A72, 60L10 |
| url | https://arxiv.org/abs/2407.20405 |