On the cosine similarity and orthogonality between persistence diagrams
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912687087157248 |
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| author | Nordin, Azmeer Noorani, Mohd Salmi Md Masseran, Nurulkamal Ismail, Mohd Sabri Roslan, Nur Firyal |
| author_facet | Nordin, Azmeer Noorani, Mohd Salmi Md Masseran, Nurulkamal Ismail, Mohd Sabri Roslan, Nur Firyal |
| contents | Topological data analysis is an approach to study shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions. Multiple data sets can be compared by the similarity of their diagrams to understand their behaviors in relative to each other. The bottleneck and Wasserstein distances are often used as a tool to indicate the similarity. In this paper, we introduce cosine similarity as a new indicator for the similarity between persistence diagrams and investigate its properties. Furthermore, it leads to the new notion of orthogonality between persistence diagrams. It turns out that the orthogonality refers to perfect dissimilarity between persistence diagrams under the cosine similarity. Through data demonstration, the cosine similarity is shown to be more accurate than the standard distances to measure the similarity between persistence diagrams. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the cosine similarity and orthogonality between persistence diagrams Nordin, Azmeer Noorani, Mohd Salmi Md Masseran, Nurulkamal Ismail, Mohd Sabri Roslan, Nur Firyal Algebraic Topology Computational Geometry 55N31, 68T09, 46C05 Topological data analysis is an approach to study shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions. Multiple data sets can be compared by the similarity of their diagrams to understand their behaviors in relative to each other. The bottleneck and Wasserstein distances are often used as a tool to indicate the similarity. In this paper, we introduce cosine similarity as a new indicator for the similarity between persistence diagrams and investigate its properties. Furthermore, it leads to the new notion of orthogonality between persistence diagrams. It turns out that the orthogonality refers to perfect dissimilarity between persistence diagrams under the cosine similarity. Through data demonstration, the cosine similarity is shown to be more accurate than the standard distances to measure the similarity between persistence diagrams. |
| title | On the cosine similarity and orthogonality between persistence diagrams |
| topic | Algebraic Topology Computational Geometry 55N31, 68T09, 46C05 |
| url | https://arxiv.org/abs/2504.04361 |