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Persistence Module
A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of Topology, topological features of an object across a range of scale parameters. A persistence module often consists of a collection of Homology (mathematics), homology groups (or vector spaces if using Field (mathematics), field coefficients) corresponding to a Filtration (mathematics), filtration of topological spaces, and a collection of linear maps induced by the Inclusion map, inclusions of the filtration. The concept of a persistence module was first introduced in 2005 as an application of graded modules over polynomial rings, thus importing well-developed algebraic ideas from classical commutative algebra theory to the setting of persistent homology. Since then, persistence modules have been one of the primary algebraic structures studied in the field of applied topology. Definition Single Parameter Persistence Modules Let T be a t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Persistent Homology
:''See homology for an introduction to the notation.'' Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets. Definition Formally, consider a real-valued function on a simplicial complex f:K \rightarrow \mathbb that is non-decreasing on increasing sequences of faces, so f(\sigma) \leq f(\tau) whenever \sigma is a face of \tau in K. Then for every a \in \mathbb the sublevel set K_a=f^((-\infty, a]) is a subcomplex of ''K'', and the ordering of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
