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Topology and data
TLDR
This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources. Expand
Computing persistent homology
TLDR
The homology of a filtered d-dimensional simplicial complex K is studied as a single algebraic entity and a correspondence is established that provides a simple description over fields that enables a natural algorithm for computing persistent homology over an arbitrary field in any dimension. Expand
Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition
TLDR
The method, called Mapper, is based on the idea of partial clustering of the data guided by a set of functions defined on the data, and is not dependent on any particular clustering algorithm, i.e. any clustering algorithms may be used with Mapper. Expand
The theory of multidimensional persistence
TLDR
This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension. Expand
Characterization, Stability and Convergence of Hierarchical Clustering Methods
TLDR
It is shown that within this framework, one can prove a theorem analogous to one of Kleinberg (2002), in which one obtains an existence and uniqueness theorem instead of a non-existence result. Expand
Topological estimation using witness complexes
TLDR
This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object, and produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology. Expand
On the Local Behavior of Spaces of Natural Images
TLDR
A theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle and a polynomial representation is used to give coordinatization to various subspaces ofℳ. Expand
Zigzag persistent homology and real-valued functions
TLDR
The algorithmic results provide a way to compute zigzag persistence for any sequence of homology groups, but combined with the structural results give a novel algorithm for computing extended persistence that is easily parallelizable and uses (asymptotically) less memory. Expand
Extracting insights from the shape of complex data using topology
TLDR
The method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets to find subgroups in data sets that traditional methodologies fail to find. Expand
Computing Persistent Homology
Abstract We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysisExpand
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