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Categorified Reeb Graphs
A natural construction for smoothing a Reeb graph to reduce its topological complexity is obtained and an ‘interleaving’ distance is defined which is stable under the perturbation of a function.
A User’s Guide to Topological Data Analysis
- E. Munch
- Computer Science
- 5 July 2017
This article introduces two of the most commonly used topological signatures, the persistence diagram and the mapper graph, which represent loops and holes in the space by considering connectivity of the data points for a continuum of values rather than a single fixed value.
Persistent Homology of Complex Networks for Dynamic State Detection
It is shown how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior.
Chatter detection in turning using persistent homology
Topological and statistical behavior classifiers for tracking applications
- Paul Bendich, S. Chin, Adam Watkins
- Computer ScienceIEEE Transactions on Aerospace and Electronic…
- 1 June 2014
A periodic track appraisal based on behavior is introduced, and it adjusts the traditional kinematic data association likelihood using an established formulation for feature-aided data association.
Convergence between Categorical Representations of Reeb Space and Mapper
Using tools from category theory, it is formally proved that the convergence between the Reeb space and mapper is proved in terms of an interleaving distance between their categorical representations.
Intrinsic Interleaving Distance for Merge Trees
- Ellen Gasparovic, E. Munch, S. Oudot, Katharine Turner, Bei Wang, Yusu Wang
- Computer ScienceArXiv
- 31 July 2019
It is shown that the interleaving distance is intrinsic on the space of labeled merge trees and provided an algorithm to construct metric 1-centers for collections of labeling merge trees, and it is proved that the intrinsic property of the Interleaving Distance also holds for thespace of unlabeled merge trees.
Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
This paper shows that the two metrics are strongly equivalent on the space of Reeb graphs, and gives an immediate proof of bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
Approximating Continuous Functions on Persistence Diagrams Using Template Functions
This paper describes a mathematical framework for featurizing the persistence diagram space using template functions, and discusses two example realizations of these functions: tent functions and Chybeyshev interpolating polynomials.
Probabilistic Fréchet means for time varying persistence diagrams
- E. Munch, Katharine Turner, Paul Bendich, S. Mukherjee, J. Mattingly, J. Harer
- 24 July 2013
This work alters the original definition of Fr\'echet mean so that it now becomes a probability measure on the set of persistence diagrams and shows that this map is Holder continuous on finite diagrams and thus can be used to build a useful statistic on time-varying persistence diagrams, better known as vineyards.