• Corpus ID: 195218652

Jaccard Filtration and Stable Paths in the Mapper

@article{Arendt2019JaccardFA,
  title={Jaccard Filtration and Stable Paths in the Mapper},
  author={Dustin L. Arendt and Matthew Broussard and Bala Krishnamoorthy and Nathaniel Saul},
  journal={ArXiv},
  year={2019},
  volume={abs/1906.08256}
}
The contributions of this paper are two-fold. We define a new filtration called the cover filtration built from a single cover based on a generalized Jaccard distance. We provide stability results for the cover filtration and show how the construction is equivalent to the Cech filtration under certain settings. We then develop a language and theory for stable paths within this filtration, inspired by ideas of persistent homology. We demonstrate how the filtration and paths can be applied to a… 

Figures from this paper

Interesting paths in the mapper complex

Interesting paths identified on several mapper graphs are used to explain how the genotype and environmental factors influence the growth rate, both in isolation as well as in combinations.

References

SHOWING 1-10 OF 39 REFERENCES

Mutiscale Mapper: A Framework for Topological Summarization of Data and Maps

A combinatorial version of the algorithm is presented that acts only ontex sets connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that is shown to hold.

Multiscale Mapper: Topological Summarization via Codomain Covers

A combinatorial version of the algorithm is presented that acts only on vertex sets connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that is shown to hold.

Interesting Paths in the Mapper

The problem of quantifying the interestingness of subpopulations in a Mapper, which appear as long paths, flares, or loops, is studied and polynomial time heuristics for IP and k-IP on DAGs are developed.

Persistence Images: A Stable Vector Representation of Persistent Homology

This work converts a PD to a finite-dimensional vector representation which it is called a persistence image, and proves the stability of this transformation with respect to small perturbations in the inputs.

Structure and Stability of the One-Dimensional Mapper

A theoretical framework is proposed that relates the structure of the Mappers to the one of the Reeb graphs, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability.

A quest to unravel the metric structure behind perturbed networks

The main result is that a simple filtering process based on the Jaccard index can recover the metric structure of G^* (which reflects that of the hidden space X as the authors will show) from the observed graph G under the network model.

Barcodes: The persistent topology of data

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The

Fast Evaluation of Union-Intersection Expressions

A novel combination of approximate set representations and word-level parallelism is used, which shows how to represent sets in a linear space data structure such that expressions involving unions and intersections of sets can be computed in a worst-case efficient way.

Understanding and Predicting Links in Graphs: A Persistent Homology Perspective

The use of persistent homology methods to capture structural and topological properties of graphs and use it to address the problem of link prediction is proposed.

A roadmap for the computation of persistent homology

A friendly introduction to PH is given, the pipeline for the computation of PH is navigated with an eye towards applications, and a range of synthetic and real-world data sets are used to evaluate currently available open-source implementations for the computations of PH.