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Subdivision Bifiltration
In topological data analysis, a subdivision bifiltration is a collection of filtered simplicial complexes, typically built upon a set of data points in a metric space, that captures shape and density information about the underlying data set. The subdivision bifiltration relies on a natural filtration of the barycentric subdivision of a simplicial complex by flags of minimum dimension, which encodes density information about the metric space upon which the complex is built. The subdivision bifiltration was first introduced by Donald Sheehy in 2011 as part of his doctoral thesis (later subsumed by a conference paper in 2012) as a discrete model of the multicover bifiltration, a continuous construction whose underlying framework dates back to the 1970s. In particular, Sheehy applied the construction to both the Vietoris-Rips and Čech filtrations, two common objects in the field of topological data analysis. Whereas single parameter filtrations are not robust with respect to outlie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Data Analysis
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools. The initial motivation is to study the shape of data. TDA has combined algebraic topology and other tools from pure mathematics to allow mathematically rigorous study of "shape". The main tool is persistent homology, an adaptation of homology to point cloud data. Persistent homology has been applied to many types of data across many fields. Moreover, its mathematical foundation is also of theoretica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
