A kernel for multi-parameter persistent homology
•We propose the first kernel construction for multi-parameter persistent homology.•Our kernel is generic, stable and can be approximated in polynomial time.•Connect topological data analysis and machine learning for multivariate analysis.•Our technique is applicable to shape analysis, recognition an...
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| Vydané v: | Computers & graphics. X Ročník 2; s. 100005 |
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| Hlavní autori: | , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Elsevier Ltd
01.12.2019
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| ISSN: | 2590-1486, 2590-1486 |
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| Abstract | •We propose the first kernel construction for multi-parameter persistent homology.•Our kernel is generic, stable and can be approximated in polynomial time.•Connect topological data analysis and machine learning for multivariate analysis.•Our technique is applicable to shape analysis, recognition and classification.
[Display omitted]
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis. |
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| AbstractList | Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis. •We propose the first kernel construction for multi-parameter persistent homology.•Our kernel is generic, stable and can be approximated in polynomial time.•Connect topological data analysis and machine learning for multivariate analysis.•Our technique is applicable to shape analysis, recognition and classification. [Display omitted] Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis. Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis. |
| ArticleNumber | 100005 |
| Author | Wang, Bei Corbet, René Fugacci, Ulderico Kerber, Michael Landi, Claudia |
| AuthorAffiliation | b University of Modena and Reggio Emilia, Italy a Graz University of Technology, Austria c University of Utah, USA |
| AuthorAffiliation_xml | – name: a Graz University of Technology, Austria – name: b University of Modena and Reggio Emilia, Italy – name: c University of Utah, USA |
| Author_xml | – sequence: 1 givenname: René surname: Corbet fullname: Corbet, René email: maths@rene-corbet.de organization: Graz University of Technology, Austria – sequence: 2 givenname: Ulderico surname: Fugacci fullname: Fugacci, Ulderico organization: Graz University of Technology, Austria – sequence: 3 givenname: Michael surname: Kerber fullname: Kerber, Michael email: kerber@tugraz.at organization: Graz University of Technology, Austria – sequence: 4 givenname: Claudia surname: Landi fullname: Landi, Claudia email: clandi@unimore.it organization: University of Modena and Reggio Emilia, Italy – sequence: 5 givenname: Bei surname: Wang fullname: Wang, Bei email: beiwang@sci.utah.edu organization: University of Utah, USA |
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| Keywords | Persistent homology Multivariate analysis Topological data analysis Machine learning Persistent Homology Multivariate Analysis Machine Learning Topological Data Analysis |
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| Snippet | •We propose the first kernel construction for multi-parameter persistent homology.•Our kernel is generic, stable and can be approximated in polynomial... Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data... |
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| SubjectTerms | Machine learning Multivariate analysis Persistent homology Topological data analysis |
| Title | A kernel for multi-parameter persistent homology |
| URI | https://dx.doi.org/10.1016/j.cagx.2019.100005 https://www.ncbi.nlm.nih.gov/pubmed/33367228 https://www.proquest.com/docview/2473415090 https://pubmed.ncbi.nlm.nih.gov/PMC7755142 |
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