Simplicial complexes associated with cloud pointsSeminar algorithms

Jorge Cordero

Eindhoven University of Technology

8 May 2018

Outline

1 Motivation

2 Recap

3 Nerves

4 Čech complex

5 Vietoris-Rips complex

6 Delaunay complex

7 Alpha complexes

Topological data analysis

Data can be complex in terms of size or features.

Sometimes, data has shape.

Topological data analysis

Topological data analysis (TDA) help us to understand the structure(shape) of data.

Application:

Find coverage in networks of sensors

Understand protein interactions

Credit card fraud detection

Simplex

σ is an n-simplex spanned by n + 1 affinely independent points in P ∈ Rn.

A 0-simplex is a point

A 1-simplex is a line

A 2-simplex is a triangle

A 3-simplex is a tetrahedron

Simplicial complex

A simplicial complex K in Rn is a collection of simplices in Rn such that:

Every face of a simplex of K is in K

Every pair of distinct simplices of K has a disjoint interior

The intersection of two distinct simplices of K is a face of each ofthem

ASC and Homotopy

An abstract simplicial complex is a collection of finite non-empty subsetsof S.

If A ⊆ S , each non-empty subset B ⊆ A is also in S .

For our purposes, we can think of homotopy equivalent spaces (X ' Y) asspaces that can be deformed continuously one into another.

ASC and Homotopy

An abstract simplicial complex is a collection of finite non-empty subsetsof S.

If A ⊆ S , each non-empty subset B ⊆ A is also in S .

For our purposes, we can think of homotopy equivalent spaces (X ' Y) asspaces that can be deformed continuously one into another.

Simplicial representation

Our goal is to compute a simplicial representation of a set of points toapply topological tools for data analysis.

Nerves

We can use the nerve of a finite collection of sets S to create an abstractsimplicial complex.

The nerve of S consists of all non-empty subcollections whose sets have anon-empty common intersection,

NrvS = {X ⊆ S |⋂

X 6= ∅}.

Nerves

We can use the nerve of a finite collection of sets S to create an abstractsimplicial complex.

The nerve of S consists of all non-empty subcollections whose sets have anon-empty common intersection,

NrvS = {X ⊆ S |⋂

X 6= ∅}.

Nerves

From⋂X 6= ∅ and Y ⊆ X =⇒

⋂Y 6= ∅, it follows that NrvS is always

an abstract simplicial complex.

We represent the nerve as an abstract simplicial complex,

NrvS = {R,B,P,G , {R,B}, {B,P}, {P,G}, {G ,R}}.

Nerves

The topological space of S = {R,G ,B,Y } is a disk with three holes.

NrvS has the homotopy type of a sphere.

Hence, the homotopy types for NrvS and |S | are different.

Nerves

We want to compute a nerve that resembles the structure inherent to theset of points.

Nerve theorem

Let S be a finite collection of closed, convex sets in Euclidean space.Then, the nerve of S and the union of the sets in S have the samehomotopy type.

Nerves

We want to compute a nerve that resembles the structure inherent to theset of points.

Nerve theorem

Let S be a finite collection of closed, convex sets in Euclidean space.Then, the nerve of S and the union of the sets in S have the samehomotopy type.

Closed ball

A closed ball with center x and radius r is defined by,

Bx(r) = x + rBd = {y ∈ Rd | ||y − x || ≤ r}

Čech complex

Let’s construct a simplicial complex from a set of points.

Čech complex

Let’s construct a simplicial complex from a set of points.

Čech complex

Let’s construct a simplicial complex from a set of points.

Čech complex

Let’s construct a simplicial complex from a set of points.

Čech complex

Let S be a finite set of points in Rd .

The Čech complex of S and r is given by,

Č ech(r) = {σ ⊆ S |∩Bx(r) 6= ∅}

The Čech complex is equivalent to the nerve of the collection of balls.

Čech complex

Čech complexes for different values of r

Čech complex

Čech complexes for different values of r

Čech complex

Some properties of the Čech complex:

When ri ≤ rj , Č ech(ri ) ⊆ Č ech(rj).

Č ech(r) of a set S of points in Rd can always be represented as anabstract simplicial complex.

Computing the complex takes an exponential time in the size of S .

Čech complex

Some properties of the Čech complex:

When ri ≤ rj , Č ech(ri ) ⊆ Č ech(rj).Č ech(r) of a set S of points in Rd can always be represented as anabstract simplicial complex.

Computing the complex takes an exponential time in the size of S .

Čech complex

Some properties of the Čech complex:

When ri ≤ rj , Č ech(ri ) ⊆ Č ech(rj).Č ech(r) of a set S of points in Rd can always be represented as anabstract simplicial complex.

Computing the complex takes an exponential time in the size of S .

Vietoris-Rips complex

The Vietoris-Rips VR complex of S and r consists of all subsets ofdiameter at most 2r ,

VR(r) = {σ ⊆ S |diamσ ≤ 2r}.

The diameter of σ is the supremum of all pairwise distances betweenpoints in σ.

Comparing Čech and Vietoris-Rips complex

Čech complex

Vietoris-Rips complex

Vietoris-Rips

Notice that Č ech(r) ⊆ VR(r) but Č ech(r) 6' VR(r).

For appropriate values of ra and rb, we have VR(ra) ⊆ Č ech(rb).

Vietoris-Rips Lemma

Let S be a finite set of points in some Euclidean space and r ≥ 0. Itfollows that VR(r) ⊆ Č ech(

√2r).

Vietoris-Rips

Properties of VR complexes:

VR complexes avoid the intersection test used in Čech.

VR complexes also take exponential time to compute.

VR complexes might not have a geometric representation in theunderlying space of S.

Vietoris-Rips

Properties of VR complexes:

VR complexes avoid the intersection test used in Čech.

VR complexes also take exponential time to compute.

VR complexes might not have a geometric representation in theunderlying space of S.

Vietoris-Rips

Properties of VR complexes:

VR complexes avoid the intersection test used in Čech.

VR complexes also take exponential time to compute.

VR complexes might not have a geometric representation in theunderlying space of S.

Voronoi diagram

What if we want the simplicial complex to have a geometricrepresentation?.

Delaunay complexes can be used for this task.

To explore Delaunay complexes, we first introduce the Voronoi diagram.

Voronoi diagram

Consider a finite set of points S = {v1, v2, ..., vn} in Rd :

The Voronoi cell of vi ∈ S is the set of points in Rd closest to vi ,

Vvi = {x ∈ Rd | ||x − vi || ≤ ||x − vj ||, ∀vj ∈ S}.

Together, the Voronoi cells of all points vi cover the entire space.

Voronoi diagram

For a set of points in a plane:

Encode proximity information useful for answering point queries.

Can be computed in time O(nlogn) using Fortune’s algorithm.

It uses O(n) space.

Voronoi diagram

For a set of points in a plane:

Encode proximity information useful for answering point queries.

Can be computed in time O(nlogn) using Fortune’s algorithm.

It uses O(n) space.

Voronoi diagram

For a set of points in a plane:

Encode proximity information useful for answering point queries.

Can be computed in time O(nlogn) using Fortune’s algorithm.

It uses O(n) space.

Delaunay complex

The Delaunay complex of a finite set of points S ⊆ Rd is equivalent to thenerve of the Voronoi diagram,

Delaunay(S) = {σ ⊆ S |⋂ui∈σ

Vui 6= ∅}.

Delaunay complex

The Delaunay complex seems to always create triangles in R2.

Is this always the case?

No, unless the set of points is in general position.

Delaunay complex

The Delaunay complex seems to always create triangles in R2.

Is this always the case?

No, unless the set of points is in general position.

Delaunay complex

The Delaunay complex seems to always create triangles in R2.

Is this always the case?

No, unless the set of points is in general position.

Delaunay complex

A set of points S is in general position if not d + 2 points lie on a common(d − 1)-sphere.

For a finite set of points S ∈ Rd , assuming general position, the geometricrealization of a Delaunay complex fits in Rd .

Delaunay complex

For a set of points in the plane:

It can be computed from the Voronoi diagram.

It takes expected O(nlogn) time.

It uses O(n) space.

Alpha complexes

Let S be a finite set of points in Rd and r ≥ 0.

Let Ru(r) = Bu(r) ∩ Vu, with Vu being the Voronoi cell of u.

The alpha complex is defined as

Alpha(r) = {σ ⊆ S |⋂u∈σ

Ru(r) 6= ∅}.

Delaunay filtration

Different values of r to create different alpha complexes.

Delaunay filtration

Eventually, we obtain the Delaunay complex.

Delaunay filtration

The filtration of Km = Delaunay is represented as,

∅ = K0 ⊆ ... ⊆ Ki ⊆ ... ⊆ Km.

Ki corresponds to the i-th alpha complex from the sequence of differentalpha complexes obtained by varying r .

Summary

The nerve of a set of points S allows us to compute simplicialcomplexes.

Čech complexes and Vietoris-Rips complexes allows us to constructabstract simplicial complexes from S .

The Delaunay triangulation allows to create simplicial complexes withgeometric realizations.

The alpha complex can be seen as a filtered version of the Delaunaycomplex.

Summary

The nerve of a set of points S allows us to compute simplicialcomplexes.

Čech complexes and Vietoris-Rips complexes allows us to constructabstract simplicial complexes from S .

The Delaunay triangulation allows to create simplicial complexes withgeometric realizations.

The alpha complex can be seen as a filtered version of the Delaunaycomplex.

Summary

The nerve of a set of points S allows us to compute simplicialcomplexes.

Čech complexes and Vietoris-Rips complexes allows us to constructabstract simplicial complexes from S .

The Delaunay triangulation allows to create simplicial complexes withgeometric realizations.

The alpha complex can be seen as a filtered version of the Delaunaycomplex.

Summary

The nerve of a set of points S allows us to compute simplicialcomplexes.

Čech complexes and Vietoris-Rips complexes allows us to constructabstract simplicial complexes from S .

The Delaunay triangulation allows to create simplicial complexes withgeometric realizations.

The alpha complex can be seen as a filtered version of the Delaunaycomplex.

Sources

H. Edelsbrunner, J. L. Harer, Computational topology. Anintroduction. American Mathematical Society, Providence, RI, 2010.

Mark de Berg, Otfried Cheong, Marc van Kreveld, and MarkOvermars. 2008. Computational Geometry: Algorithms andApplications (3rd ed.).

Lecture notes and videos on Computational Topology. Lectures 11,12, and 13.

Topological data analysis of genes

Čech and Vietoris-Rips complex

Persistent homology

http://www.math.wsu.edu/faculty/bkrishna/FilesMath574/S16/LecNotes/welcome.htmlhttps://www.ayasdi.com/blog/topology/shape-genome-going-beyond-double-helix-topological-data-analysis/https://jeremykun.com/2015/08/06/cech-vietoris-rips-complex/https://www.youtube.com/watch?annotation_id=annotation_405791327&feature=iv&src_vid=h0bnG1Wavag&v=2PSqWBIrn90

Images

https://mathematica.stackexchange.com/questions/61021/

triangulated-mesh-from-voronoi-diagram

https:

//www.researchgate.net/figure/Delaunay-Triangulation_fig3_44897706

http://www.pnas.org/content/pnas/113/26/7035/F10.large.jpg

http://www.ugc.edu.hk/minisite/rgc_newsletter/rgcnews18/eng/05.htm

http://www.pnas.org/content/pnas/113/26/7035/F10.large.jpg

http://www.dei.unipd.it/~schenato/pics/SensorNetwork.jpg

https://www.johndcook.com/persistent_homology.png

https:

//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_

complex_example.svg/532px-Simplicial_complex_example.svg.png

https://s3.amazonaws.com/cdn.ayasdi.com/wp-content/uploads/2017/05/

09112616/srep43845-f2.jpg

https://mathematica.stackexchange.com/questions/61021/triangulated-mesh-from-voronoi-diagramhttps://mathematica.stackexchange.com/questions/61021/triangulated-mesh-from-voronoi-diagramhttps://www.researchgate.net/figure/Delaunay-Triangulation_fig3_44897706https://www.researchgate.net/figure/Delaunay-Triangulation_fig3_44897706http://www.pnas.org/content/pnas/113/26/7035/F10.large.jpg http://www.ugc.edu.hk/minisite/rgc_newsletter/rgcnews18/eng/05.htmhttp://www.pnas.org/content/pnas/113/26/7035/F10.large.jpghttp://www.dei.unipd.it/~schenato/pics/SensorNetwork.jpghttps://www.johndcook.com/persistent_homology.pnghttps://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_complex_example.svg/532px-Simplicial_complex_example.svg.pnghttps://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_complex_example.svg/532px-Simplicial_complex_example.svg.pnghttps://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_complex_example.svg/532px-Simplicial_complex_example.svg.pnghttps://s3.amazonaws.com/cdn.ayasdi.com/wp-content/uploads/2017/05/09112616/srep43845-f2.jpghttps://s3.amazonaws.com/cdn.ayasdi.com/wp-content/uploads/2017/05/09112616/srep43845-f2.jpg

MotivationRecapNervesCech complexVietoris-Rips complexDelaunay complexAlpha complexes