258 research outputs found
Parametrized Homology via Zigzag Persistence
This paper develops the idea of homology for 1-parameter families of
topological spaces. We express parametrized homology as a collection of real
intervals with each corresponding to a homological feature supported over that
interval or, equivalently, as a persistence diagram. By defining persistence in
terms of finite rectangle measures, we classify barcode intervals into four
classes. Each of these conveys how the homological features perish at both ends
of the interval over which they are defined
Applications of Zigzag Persistence to Topological Data Analysis
The theory of zigzag persistence is a substantial extension of persistent
homology, and its development has enabled the investigation of several
unexplored avenues in the area of topological data analysis. In this paper, we
discuss three applications of zigzag persistence: topological bootstrapping,
parameter thresholding, and the comparison of witness complexes
Fast Computation of Zigzag Persistence
Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. We propose an algorithm called FastZigzag which narrows this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell-wise non-zigzag filtration of a ?-complex with the same length, where the cells are copies of the input simplices. This conversion step in FastZigzag incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares
Persistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have
inclusions defined so that the corresponding directed graph between the spaces
is acyclic, as well as along any subgraph of this directed graph. This method
simultaneously generalizes standard persistent homology, zigzag persistence and
multidimensional persistence to arbitrary directed acyclic graphs, and it also
allows the study of more general families of topological spaces or point-cloud
data. We give an algorithm to compute the persistent homology groups
simultaneously for all subgraphs which contain a single source and a single
sink in arithmetic operations, where is the number of vertices in
the graph. We then demonstrate as an application of these tools a method to
overlay two distinct filtrations of the same underlying space, which allows us
to detect the most significant barcodes using considerably fewer points than
standard persistence.Comment: Revised versio
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