Topological data analysis can extract effective information from higher-dimensional data.Its mathematical basis is persistent homology.The persistent homology can calculate topological features at different spatiotemp...Topological data analysis can extract effective information from higher-dimensional data.Its mathematical basis is persistent homology.The persistent homology can calculate topological features at different spatiotemporal scales of the dataset,that is,establishing the integrated taxonomic relation among points,lines,and simplices.Here,the simplicial network composed of all-order simplices in a simplicial complex is essential.Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values,a method based on the concept of critical simplices can be developed by searching all-order spanning trees.Employing this new method,not only the Morse function values with the theoretical minimum number of critical simplices can be obtained,but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly.Finally,this method is used to analyze some examples and compared with other methods,showing its effectiveness and feasibility.展开更多
基金support of the National Natural Science Foundation of China under Grant Nos.62173095 and 12005001the Hong Kong Shun Hing Education and Charity Fund Chair Professor in Engineering.
文摘Topological data analysis can extract effective information from higher-dimensional data.Its mathematical basis is persistent homology.The persistent homology can calculate topological features at different spatiotemporal scales of the dataset,that is,establishing the integrated taxonomic relation among points,lines,and simplices.Here,the simplicial network composed of all-order simplices in a simplicial complex is essential.Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values,a method based on the concept of critical simplices can be developed by searching all-order spanning trees.Employing this new method,not only the Morse function values with the theoretical minimum number of critical simplices can be obtained,but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly.Finally,this method is used to analyze some examples and compared with other methods,showing its effectiveness and feasibility.
