Stability properties of zigzag and image-zigzag persistent homology
- Topological data analysis consists in using methods in algebraic topology to extract qualitative and quantitative information from complex datasets. On a discrete, finite dataset embedded in a metric space, one can construct a filtration of simplicial complexes, thus providing the dataset with a topological structure and acting as a proxy for the space that gave rise to it. It is, then, possible to compute the homology groups of the simplicial complexes, thus yielding some topological information about the dataset. Two objects, following this approach, have been studied over the past 10 years: ordinary persistent homology, which looks at homological information of the entire filtration of simplicial complexes and provides some stability and robustness in the computations and zigzag persistent homology which analyzes a simplicial complex by looking at persistent homological features in subsamplings of the complex. This work sits at the intersection of these two approaches. We first demonstrate the limits of zigzag persistent homology by showing that, even in the most favorable case, topological bootstrapping may fail to provide accurate information regarding the simplicial complex of interest. Then, we develop a variation of this approach, image-zigzag, integrating ordinary persistent homology into topological bootstrapping and show that it exhibits the stability properties that are missing from ordinary zigzag. Finally, we provide two algorithms to compute image-zigzag persistent homology and compare the results from zigzag and image-zigzag applied to some simple examples.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Institute for Computational and Mathematical Engineering.
|Guibas, Leonidas J
|Guibas, Leonidas J
|Statement of responsibility
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis (Ph.D.)--Stanford University, 2013.
- © 2013 by Guillaume Axel Troianowski
- This work is licensed under a Creative Commons Attribution 3.0 Unported license (CC BY).
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