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Simplicial Homology and Topological Data Analysis Patrick O’Neil George Mason University October 15, 2013 Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 1 / 23
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Page 1: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Simplicial Homologyand Topological Data Analysis

Patrick O’Neil

George Mason University

October 15, 2013

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 1 / 23

Page 2: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Overview

1 Motivation

2 Constructing Simplicial ComplexesSimplicial ComplexsDelaunay and Alpha Complexes

3 Simplicial HomologyChain Groups and Boundary MapsInvariants

4 Persistence Homology

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 2 / 23

Page 3: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Data Analysis

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Page 4: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Point Clouds

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Page 5: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

What is a Simplex?

A simplex is a generalization of a triangle to arbitrary dimensions.

Definition

A k-simplex is a k-dimensional polytope which is the convex hull of k + 1vertices.

For lower dimensions, these have familiar names: vertex (0), edge (1),triangle (2), tetrahedron (3).

Figure: 3-simplex

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 5 / 23

Page 6: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Simplex: Abstract Representation

Given k + 1 vertices {v0, v1, ..., vk}, we can denote the k-simplex formedby these vertices as σ = (v0, v1, ..., vk). A face of σ is any subset of σ.

Example: the 2-simplex (v0, v1, v3) is a face of the 4-simplex(v0, v1, v2, v3, v4)

Note that given an abstract k-simplex σ, it is possible to form a geometricrealization of the complex. The result will be the convex hull of thevertices in σ, embedded in Rd , d ≥ k .

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 6 / 23

Page 7: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

What is a Simplicial Complex?

One can form a Simplicial Complex by joining simplicies in a certain way.

Definition

A Simplicial Complex K is a set of simplicies such that

a) K is closed under the subset relation: If τ ⊆ σ ∈ K, then τ ∈ K.

b) (∗) If σ1, σ2 ∈ K, then σ1 ∩ σ2 is a face of both σ1 and σ2.

We only need (∗) when discussing a geometric realization of a simplicialcomplex.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 7 / 23

Page 8: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Simplicial Complex Example

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Page 9: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Simplicial Complex Non-Example

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Page 10: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Cover and Nerve: DefinitionWe now define an open cover and the associated nerve of the open cover:

Definition

Given a topological space X, an open cover O of a set S ⊆ X is

O = {Ui}i∈I

where Ui ⊆ X and I is an indexing set, such that S ⊆⋃

i∈I Ui

Definition

Given an open cover O = {Ui}i∈I , the nerve N of O is a subset of P(I )such that

a) ∅ ∈ N

b) If ∩j∈JUj 6= ∅ for J ⊆ I , then J ∈ N

From these definitions, it is clear that the nerve of an open cover is asimplicial complex!

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 10 / 23

Page 11: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Cover and Nerve: Example

Pretty simple simplicial complex.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 11 / 23

Page 12: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Open Cover Dependence

The simplicial complex given by the nerve is completely dependent uponthe open cover that is chosen.

There are many ways to choose an open cover. We will present theVoronoi Diagram which produces the Delaunay Complex.

We will then present the Alpha complex which allows us to analyze thedata at different levels of detail.

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Page 13: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Delaunay Complex: Definition

Given a subset S of a metric space X = (X , d), we define the VoronoiRegion R(x) of a point x ∈ S to be

R(x) = {y ∈ X |d(x , y) ≤ d(s, y),∀s ∈ S\{x}}

and the Voronoi Diagram of S , V (S), to be the set

V (S) = {R(x)|x ∈ S}

The Voronoi Diagram forms a closed cover of S and so we can considerthe nerve of V (S), which is called the Delaunay Complex.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 13 / 23

Page 14: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Delaunay Complex: Example

(a) Voronoi Region (b) Delaunay Complex

This is a very “filled” simplicial complex. Is there a way to capture moreof the intricacies of the data?

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 14 / 23

Page 15: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Alpha Complex: Example

The Alpha complex is obtained by taking ε-balls around each point in S ,intersecting these balls with the Voronoi regions, and taking the nerve ofthis cover.

(c) Alpha Balls and Nerve (d) Alpha Complex

Notice that the Alpha complex is a subcomplex of the Delaunay complex.If ε =∞, then the alpha complex is the Delaunay complex.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 15 / 23

Page 16: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

1 Motivation

2 Constructing Simplicial ComplexesSimplicial ComplexsDelaunay and Alpha Complexes

3 Simplicial HomologyChain Groups and Boundary MapsInvariants

4 Persistence Homology

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 16 / 23

Page 17: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

nth Chain Group

We now turn to simplicial homology to analyze the shape of a givensimplicial complex. Here, we do not want to differentiate between differentorientations of simplicies and so we will denote an oriented simplex as [σ]for simplex σ.

Definition

Given a simplicial complex K, the nth Chain Group Cn(K) is the freeAbelian group on K’s oriented n-simplicies.

Thus, elements in Cn(K) can be expressed as finite sums...

c =∑i

ci [σi ]

where σi ∈ K are n-simplicies and ci ∈ Z for all i .

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 17 / 23

Page 18: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Boundary Homomorphism

Given the Chain Groups, we can define the n-boundary homomorphism∂n : Cn → Cn−1 to be,

∂n[v0, ..., vn] =n∑

i=0

(−1)i [v0, ..., vi , ..., vn]

where vi denotes removing vi from the vertex sequence.

Example: Triangle (on board)

∂1([a, c]− [b, c] + [b, a]) = ([c]− [a])− ([c]− [b]) + ([a]− [b]) = 0

Thus, [a, c]− [b, c] + [b, a] ∈ ker ∂1.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 18 / 23

Page 19: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Simplicial Homology

Note that ∂n ◦ ∂n+1 ≡ 0 (exercise) and therefore, the boundary operatorconnects the chain groups into a chain complex:

...→ Cn+1∂n+1−−−→ Cn

∂n−→ Cn−1 → ...

on which we can analyze the nth homology group Hn,

Hn = ker ∂n/im∂n+1

Elements in ker ∂n are cycles and elements in im∂n+1 are boundaries.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 19 / 23

Page 20: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Betti Numbers

We can consider the Betti Number of the nth homology group,

βn = rank(Hn) = min{|X | : X ⊆ Hn, 〈X 〉Hn}

This gives the number of n-dimensional holes in our space. Note that βnwill be non-zero for only finitely many n. Additionally, since K is finite, Hn

is finitely generated and therefore βn <∞ for all n.

A nice feature due to the Euler-Poincare formula states that

χ(X) =∞∑n=1

(−1)nβn

where χ is the Euler characteristic.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23

Page 21: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Single-Scale to Multi-Scale

Recall that we derived the alpha complex from a single parameter ε. Thatis, we constructed balls of radius ε around our data points and intersectedthese balls with the Voronoi regions at each point. Taking the nerve ofthis cover yielded the alpha complex.

Therefore, the invariants we calculated were dependent on our choice of ε.Persistence Homology allows one to vary the value of ε and look forsimplicies and invariants which do not change with varying ε. That is,features which persist through changes in scale.

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 21 / 23

Page 22: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

Discussion of persistence homology will be left for another talk.

Thanks!

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 22 / 23

Page 23: Simplicial Homology and Topological Data Analysismason.gmu.edu/~poneil/TopologicalDataAnalysis.pdf · Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 20 / 23. Single-Scale

References

Afra Zomorodian (2011)

Topological Data Analysis

Proceedings of Symposia in Applied Mathematics

Patrick O’Neil (GMU) Simplicial Homology & TDA October 15, 2013 23 / 23


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