The Duality of Similarity and Metric Spaces
We introduce a new mathematical basis for similarity space. For the first time, we describe the relationship between distance and similarity from set theory. Then, we derive generally valid relations for the conversion between similarity and a metric and vice versa. We present a general solution for...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-02-01
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| Series: | Applied Sciences |
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| Online Access: | https://www.mdpi.com/2076-3417/11/4/1910 |
| _version_ | 1827588570799407104 |
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| author | Ondřej Rozinek Jan Mareš |
| author_facet | Ondřej Rozinek Jan Mareš |
| author_sort | Ondřej Rozinek |
| collection | DOAJ |
| description | We introduce a new mathematical basis for similarity space. For the first time, we describe the relationship between distance and similarity from set theory. Then, we derive generally valid relations for the conversion between similarity and a metric and vice versa. We present a general solution for the normalization of a given similarity space or metric space. The derived solutions lead to many already used similarity and distance functions, and combine them into a unified theory. The Jaccard coefficient, Tanimoto coefficient, Steinhaus distance, Ruzicka similarity, Gaussian similarity, edit distance and edit similarity satisfy this relationship, which verifies our fundamental theory. |
| first_indexed | 2024-03-09T00:39:12Z |
| format | Article |
| id | doaj.art-37736c0f13fb4fcfbadaf3d253442482 |
| institution | Directory Open Access Journal |
| issn | 2076-3417 |
| language | English |
| last_indexed | 2024-03-09T00:39:12Z |
| publishDate | 2021-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Applied Sciences |
| spelling | doaj.art-37736c0f13fb4fcfbadaf3d2534424822023-12-11T17:56:23ZengMDPI AGApplied Sciences2076-34172021-02-01114191010.3390/app11041910The Duality of Similarity and Metric SpacesOndřej Rozinek0Jan Mareš1Department of Process Control, Faculty of Electrical Engineering and Informatics, University of Pardubice, 530 02 Pardubice, Czech RepublicDepartment of Process Control, Faculty of Electrical Engineering and Informatics, University of Pardubice, 530 02 Pardubice, Czech RepublicWe introduce a new mathematical basis for similarity space. For the first time, we describe the relationship between distance and similarity from set theory. Then, we derive generally valid relations for the conversion between similarity and a metric and vice versa. We present a general solution for the normalization of a given similarity space or metric space. The derived solutions lead to many already used similarity and distance functions, and combine them into a unified theory. The Jaccard coefficient, Tanimoto coefficient, Steinhaus distance, Ruzicka similarity, Gaussian similarity, edit distance and edit similarity satisfy this relationship, which verifies our fundamental theory.https://www.mdpi.com/2076-3417/11/4/1910similarity metricsimilarity spacedistance metricmetric spacenormalized similarity metricnormalized distance metric |
| spellingShingle | Ondřej Rozinek Jan Mareš The Duality of Similarity and Metric Spaces Applied Sciences similarity metric similarity space distance metric metric space normalized similarity metric normalized distance metric |
| title | The Duality of Similarity and Metric Spaces |
| title_full | The Duality of Similarity and Metric Spaces |
| title_fullStr | The Duality of Similarity and Metric Spaces |
| title_full_unstemmed | The Duality of Similarity and Metric Spaces |
| title_short | The Duality of Similarity and Metric Spaces |
| title_sort | duality of similarity and metric spaces |
| topic | similarity metric similarity space distance metric metric space normalized similarity metric normalized distance metric |
| url | https://www.mdpi.com/2076-3417/11/4/1910 |
| work_keys_str_mv | AT ondrejrozinek thedualityofsimilarityandmetricspaces AT janmares thedualityofsimilarityandmetricspaces AT ondrejrozinek dualityofsimilarityandmetricspaces AT janmares dualityofsimilarityandmetricspaces |