2018/10/28 by Hiroshi Takeuchi, Takeuchi, Hiroshi
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #Algorithm #Biology #Cohomology #Combinatorics #Eigenvalues and eigenvectors #FOS: Mathematics #Filtration (mathematics) #Functor #Genetics #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Mayer–Vietoris sequence #Persistent homology #Pure mathematics #Quiver #Representation Theory (math.RT) #Singular homology #Topological and Geometric Data Analysis #Topology (electrical circuits) #math.AT #math.RT
paper · pdf · doi:10.48550/arxiv.1810.11774
published in arXiv (Cornell University) (Cornell University) · 33 pages
openalex publication_date 2018/10/28 · arxiv created 2019/12/18 · arxiv updated 2019/12/19 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
This paper aims to introduce a filtration analysis of sampled maps based on\npersistent homology, providing a new method for reconstructing the underlying\nmaps. The key idea is to extend the definition of homology induced maps of\ncorrespondences using the framework of quiver representations. Our definition\nof homology induced maps is given by most persistent direct summands of\nrepresentations, and the direct summands uniquely determine a persistent\nhomology. We provide stability theorems of this process and show that the\noutput persistent homology of the sampled map is the same as that of the\nunderlying map if the sample is dense enough. Compared to existing methods\nusing eigenspace functors, our filtration analysis has an advantage that no\nprior information on the eigenvalues of the underlying map is required. Some\nnumerical examples are given to illustrate the effectiveness of our method.\n