Persistent Topological Laplacians—A Survey
Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to c...
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MDPI AG
2025-01-01
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| author | Xiaoqi Wei Guo-Wei Wei |
| author_facet | Xiaoqi Wei Guo-Wei Wei |
| author_sort | Xiaoqi Wei |
| collection | DOAJ |
| description | Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and <i>N</i>-chain complexes. |
| format | Article |
| id | doaj-art-d3720113602b4b3b85383e0c7d4af6ab |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-d3720113602b4b3b85383e0c7d4af6ab2025-01-24T13:39:44ZengMDPI AGMathematics2227-73902025-01-0113220810.3390/math13020208Persistent Topological Laplacians—A SurveyXiaoqi Wei0Guo-Wei Wei1Department of Mathematics, Michigan State University, East Lansing, MI 48824, USADepartment of Mathematics, Michigan State University, East Lansing, MI 48824, USAPersistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and <i>N</i>-chain complexes.https://www.mdpi.com/2227-7390/13/2/208topological data analysistopological Laplacianspersistent spectral theory |
| spellingShingle | Xiaoqi Wei Guo-Wei Wei Persistent Topological Laplacians—A Survey Mathematics topological data analysis topological Laplacians persistent spectral theory |
| title | Persistent Topological Laplacians—A Survey |
| title_full | Persistent Topological Laplacians—A Survey |
| title_fullStr | Persistent Topological Laplacians—A Survey |
| title_full_unstemmed | Persistent Topological Laplacians—A Survey |
| title_short | Persistent Topological Laplacians—A Survey |
| title_sort | persistent topological laplacians a survey |
| topic | topological data analysis topological Laplacians persistent spectral theory |
| url | https://www.mdpi.com/2227-7390/13/2/208 |
| work_keys_str_mv | AT xiaoqiwei persistenttopologicallaplaciansasurvey AT guoweiwei persistenttopologicallaplaciansasurvey |