learning simplicial complexes from persistence diagramssam.micka/files/cccg18slides.pdf ·...

Learning Simplicial Complexes from PersistenceDiagrams

Robin Lynne Belton1 Brittany Terese Fasy1 2 Rostik Mertz2

Samuel Micka2 David L. Millman2 Daniel Salinas2

Anna Schenfisch1 Jordan Schupbach1 Lucia Williams2

08 August 2018, WinnipegCCCG 2018

1Depart. of Mathematical Sciences, Montana State U.2School of Computing, Montana State U.

Samuel Micka 08 August 2018 1 / 18

Collaborators

Collaborator Recognition


Motivation

Problem

Motivation

Topological descriptors can be used as a statistic for shapecomparison and classification.

New methods of storing and representing geometric objects.


Background

Background

Graphs

Graph G = 〈V ,E 〉 where V are the vertices...


Background

Background

Graphs

Graph G = 〈V ,E 〉 where V are the vertices and E are the edges


Background

Background

Graphs

Assumption: G has with straight-line embedding inR2 and is planar.


Background

Background

Filtration Intuition

Filtration on graph G : sequence of sugraphs G0 . . .Gn such thatif j ≤ i then Gj ⊂ Gi .


Background

Background

Example with height filtration

As subgraphs are discovered, new features(connected components) are born and killed.


Background

Previous Work

Previous Work

Turner et al. Show that persistence diagrams are representative ofunderlying three-dimensional shape a.

Require infinite number of directions to offer exact representation.

We extend theoretically with algorithm for determining the number ofdirections necessary and how to choose them.

1

aTurner, K., Mukherjee, S., and Boyer, D. M. Persistent homologytransform for modeling shapes and surfaces.Information and Inference: A Journal of the IMA 3, 4 (2014), 310–344


Problem Statement

Ultimate Goal

Learn Simplicial Complexes from Persistence Diagrams


Methods

Vertex Locations

Vertex Locations

General Idea

Height index filtration onsimplicial complexes

Reconstruction done with threepersistence diagrams


Methods

Vertex Locations

Vertex Locations


Methods

Vertex Locations

Vertex Locations

Theorem 5 (Vertex Reconstruction)


Methods

Edge Locations

Determining Edge Locations

We have vertex locations...

?

How do we test if there exists an edge between two vertices?


Methods

Indegree of Vertex

Indegree of vertex

4

Lemma 7 (Indegree from Diagram)


Methods

Indegree of Vertex

Indegree of vertex

4

3

Lemma 7 (Indegree from Diagram)


Methods

Edge Locations



Methods

Edge Locations


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Lemma 9 (Edge Existence)


Methods

Edge Locations


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Lemma 9 (Edge Existence)


Methods

Edge Locations


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Theorem 10 (Edge Reconstruction)


Conclusion

Overall Idea

Summary

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Theorem 11 (Plane Graph Reconstruction)


Conclusion

Future Work

In Progress

Remove reliance on diagonal.

Utilize Euler Characteristic Curves.

Implement to test effectiveness of classification.

Extend to higher dimensions and different types of complexes.

Explore new representations and storage approaches for simplicialcomplexes.


Conclusion

Questions?

Thanks!

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[email protected]


[email protected]

learning simplicial complexes from persistence diagramssam.micka/files/cccg18slides.pdf ·...

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