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

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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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