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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
Samuel Micka 08 August 2018 2 / 18
Motivation
Problem
Motivation
Topological descriptors can be used as a statistic for shapecomparison and classification.
New methods of storing and representing geometric objects.
Samuel Micka 08 August 2018 3 / 18
Background
Background
Graphs
Graph G = 〈V ,E 〉 where V are the vertices...
Samuel Micka 08 August 2018 4 / 18
Background
Background
Graphs
Graph G = 〈V ,E 〉 where V are the vertices and E are the edges
Samuel Micka 08 August 2018 4 / 18
Background
Background
Graphs
Assumption: G has with straight-line embedding inR2 and is planar.
Samuel Micka 08 August 2018 4 / 18
Background
Background
Filtration Intuition
Filtration on graph G : sequence of sugraphs G0 . . .Gn such thatif j ≤ i then Gj ⊂ Gi .
Samuel Micka 08 August 2018 5 / 18
Background
Background
Filtration Intuition
Filtration on graph G : sequence of sugraphs G0 . . .Gn such thatif j ≤ i then Gj ⊂ Gi .
Samuel Micka 08 August 2018 5 / 18
Background
Background
Filtration Intuition
Filtration on graph G : sequence of sugraphs G0 . . .Gn such thatif j ≤ i then Gj ⊂ Gi .
Samuel Micka 08 August 2018 5 / 18
Background
Background
Example with height filtration
As subgraphs are discovered, new features(connected components) are born and killed.
Samuel Micka 08 August 2018 6 / 18
Background
Background
Example with height filtration
As subgraphs are discovered, new features(connected components) are born and killed.
Samuel Micka 08 August 2018 6 / 18
Background
Background
Example with height filtration
As subgraphs are discovered, new features(connected components) are born and killed.
Samuel Micka 08 August 2018 6 / 18
Background
Background
Example with height filtration
As subgraphs are discovered, new features(connected components) are born and killed.
Samuel Micka 08 August 2018 6 / 18
Background
Background
Example with height filtration
As subgraphs are discovered, new features(connected components) are born and killed.
Samuel Micka 08 August 2018 6 / 18
Background
Background
Example with height filtration
As subgraphs are discovered, new features(connected components) are born and killed.
Samuel Micka 08 August 2018 6 / 18
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.
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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
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Problem Statement
Ultimate Goal
Learn Simplicial Complexes from Persistence Diagrams
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Methods
Vertex Locations
Vertex Locations
General Idea
Height index filtration onsimplicial complexes
Reconstruction done with threepersistence diagrams
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Methods
Vertex Locations
Vertex Locations
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Methods
Vertex Locations
Vertex Locations
Samuel Micka 08 August 2018 10 / 18
Methods
Vertex Locations
Vertex Locations
Samuel Micka 08 August 2018 10 / 18
Methods
Vertex Locations
Vertex Locations
Samuel Micka 08 August 2018 10 / 18
Methods
Vertex Locations
Vertex Locations
Samuel Micka 08 August 2018 10 / 18
Methods
Vertex Locations
Vertex Locations
Theorem 5 (Vertex Reconstruction)
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Methods
Vertex Locations
Vertex Locations
Theorem 5 (Vertex Reconstruction)
Samuel Micka 08 August 2018 11 / 18
Methods
Vertex Locations
Vertex Locations
Theorem 5 (Vertex Reconstruction)
Samuel Micka 08 August 2018 11 / 18
Methods
Vertex Locations
Vertex Locations
Theorem 5 (Vertex Reconstruction)
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Methods
Edge Locations
Determining Edge Locations
We have vertex locations...
?
How do we test if there exists an edge between two vertices?
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Methods
Indegree of Vertex
Indegree of vertex
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Lemma 7 (Indegree from Diagram)
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Methods
Indegree of Vertex
Indegree of vertex
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Lemma 7 (Indegree from Diagram)
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Methods
Indegree of Vertex
Indegree of vertex
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Lemma 7 (Indegree from Diagram)
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Methods
Indegree of Vertex
Indegree of vertex
4
3
Lemma 7 (Indegree from Diagram)
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Methods
Edge Locations
Determining Edge Locations
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Methods
Edge Locations
Determining Edge Locations
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Lemma 9 (Edge Existence)
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Methods
Edge Locations
Determining Edge Locations
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Lemma 9 (Edge Existence)
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Methods
Edge Locations
Determining Edge Locations
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Theorem 10 (Edge Reconstruction)
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Conclusion
Overall Idea
Summary
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Theorem 11 (Plane Graph Reconstruction)
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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.
Samuel Micka 08 August 2018 17 / 18
Conclusion
Questions?
Thanks!
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samuel.a.micka@gmail.com
Samuel Micka 08 August 2018 18 / 18
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