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Barcodes -or how to Discover Shapes in Complex Data
David Meintrup
University of Applied Sciences IngolstadtSeptember 15, 2015
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Motivation
How can we discover the shapes behind theses point clouds?
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Contents
1 A short history of algebraic topology
2 From point clouds to barcodes
3 Applications
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Geometry versus Topology I
Leonhard Euler9, 1707 - 1783Swiss Mathematician andPhysicist866 publications
Seven Bridges of Königsberg:
Is there a walk that crosses each bridge exactly once7?
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Geometry versus Topology II
The London underground map
(A) geometrically (B) topologically
Geometry deals with distances and measuresTopology deals with shapes and relations
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Allowed Operations
allowed: all continuous smooth deformations
not allowed: cutting, tearing, joining
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Topologically identical objects I
(A) Some versions of theUnknot6
(A) Square = Disk
(B) Sphere = Cube
Topology is the “rubber band geometry“
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Topologically identical objects II
The oldest joke about a topologist
A topologist is someone who who can’t tell the difference between acoffee mug and a doughnut2.
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Creating Shapes I
What is associated to the word gluing ?
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Creating Shapes II
Take a rectangleMark arrows on opposite sites as shownGlue edges togetherWhat kind of shapes do you get?
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Creating Shapes III(A) Do nothing:
Disk
(B) Glue A on A:
Tube
(C) Glue A on A, and B on B:
Torus
(D) Glue all 4 edges together:
Sphere
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Creating Shapes IV(E) Glue A on A,but turn the orientationaround!
Moebius band
(F) Glue A on A, and B on B,but for B turn the orientationaround!
Klein bottle
Felix Klein9, 1849 - 1925
German Mathematicianlearned General Relativity Theorywith 70
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Algebraic Invariants
ALGEBRAIC topology
Sir Michael Atiyah9, 1929*
British Mathematician“Algebra is the offer made by thedevil to the mathematician.“
We distinguish shapes by assigning numbers(so called algebraic invariants4)Idea: invariants different ⇒ shapes differentExample of invariants: Betti numbers10
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Betti Numbers I
Betti Numbers: βi, i = 0, 1, 2, . . .
β0: counts pieces (how many separate parts?)β1: counts independent circles (how many holes?)β2: counts cavities/voids (how many empty volumes?)
Examples:
β0 = 1, β1 = 0, β2 = 0. β0 = 1, β1 = 1, β2 = 0.
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Betti Numbers IIExamples: (β0 pieces, β1 holes, β2 voids)
Two loops
β0 = 2, β1 = 2, β2 = 0.
Double loop
β0 = 1, β1 = 2, β2 = 0.
Sphere
β0 = 1, β1 = 0, β2 = 1.
Torus
β0 = 1, β1 = 2, β2 = 1.David Meintrup Barcodes JMP Discovery 2015 16 / 27
Contents
1 A short history of algebraic topology
2 From point clouds to barcodes
3 Applications
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TDA I
Topological data analysis in a nutshell:
1 start with a point cloud2 inflate the points to get a topological shape3 compute the Betti numbers4 classify the shape
G. Seurat9 (1859-1891)David Meintrup Barcodes JMP Discovery 2015 19 / 27
TDA II - Example
(1) Point Cloud
(3) Betti numbers:β0 = 1, β1 = 1, β2 = 0.
(2) Topological shape:
(4) Shape: Circle
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TDA IIIObvious question: How big should one inflate the points?
too small right size too big
Answer: move from small to big and record the Betti numbers!
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TDA IV - ExamplePoint cloud with 5 points:
Conclusion about underlying space: Circle
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Contents
1 A short history of algebraic topology
2 From point clouds to barcodes
3 Applications
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JMP demo
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Final Thoughts
Aleksandr Solzhenitsyn, In the First Circle“Topology! The stratosphere of human thought! In the twenty-fourthcentury it might possibly be of use to someone . . . “
“The existence of the Klein bottle isnot just a mathematical artifact.Instead, its presence is intimately tiedto the geometry of cyclo-octaneconformation, and . . . can be used to. . . explain the molecular motion ofcyclo-octane.“Topology of cyclo-octane energylandscape, Martin et al. (2010) [5]
Thank you!
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References
References
[1] G. Carlsson, Topology and data, Bulletin of the American Mathematical Society (NewSeries) 46 (2009), no. 2.
[2] M. Coelho and J. Zigelbaum, Shape-changing interfaces, Personal Ubiquitous Comput.15, 2 (2011).
[3] R. Ghrist, Barcodes: the persistent topology of data, Bulletin of the American Mathemati-cal Society (New Series) 45 (2008), no. 1.
[4] A. Hatcher, Algebraic topology, Cambridge University Press, New York, NY, 2002.
[5] S. Martin, A. Thompson, E. A. Coutsias, and J.-P. Watson, Topology of cyclo-octane en-ergy landscape, Journal of Chemical Physics 132 (2010).
[6] popmath.org.uk, Centre for the Popularisation of Mathematics, University of Wales, Ban-gor, visited August 12, 2015.
[7] storyofmathematics.com/18th_euler.html, visited August 12, 2015.
[8] S. Weinberger, What is . . . Persistent Homology?, Notices Amer. Math. Soc. (2011).
[9] Wikipedia.org, Articles and Pictures: Leonhard Euler, Felix Klein, Michael Atiyah, KleinBottle, Georges Seurat, visited August 12, 2015.
[10] Afra Zomorodian, Topological data analysis, in: Advances in Applied and ComputationalTopology, Proc. Symp. Applied Math., Vol. 70.
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