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Simplicial complexes associated with cloud points Seminar algorithms - - PowerPoint PPT Presentation
Simplicial complexes associated with cloud points Seminar algorithms - - PowerPoint PPT Presentation
Simplicial complexes associated with cloud points Seminar algorithms Jorge Cordero Eindhoven University of Technology 8 May 2018 Outline Motivation 1 Recap 2 Nerves 3 Cech complex 4 Vietoris-Rips complex 5 Delaunay complex 6
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Topological data analysis
Data can be complex in terms of size or features. Sometimes, data has shape.
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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
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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
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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 of them
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ASC and Homotopy
An abstract simplicial complex is a collection of finite non-empty subsets
- f S.
If A ⊆ S, each non-empty subset B ⊆ A is also in S.
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ASC and Homotopy
An abstract simplicial complex is a collection of finite non-empty subsets
- f 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) as spaces that can be deformed continuously one into another.
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Simplicial representation
Our goal is to compute a simplicial representation of a set of points to apply topological tools for data analysis.
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Nerves
We can use the nerve of a finite collection of sets S to create an abstract simplicial complex.
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Nerves
We can use the nerve of a finite collection of sets S to create an abstract simplicial complex. The nerve of S consists of all non-empty subcollections whose sets have a non-empty common intersection, NrvS = {X ⊆ S|
- X = ∅}.
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Nerves
From X = ∅ and Y ⊆ X = ⇒ Y = ∅, 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}}.
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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.
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Nerves
We want to compute a nerve that resembles the structure inherent to the set of points.
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Nerves
We want to compute a nerve that resembles the structure inherent to the set 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 same homotopy type.
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Closed ball
A closed ball with center x and radius r is defined by, Bx(r) = x + rBd = {y ∈ Rd| ||y − x|| ≤ r}
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ˇ Cech complex
Let’s construct a simplicial complex from a set of points.
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ˇ Cech complex
Let’s construct a simplicial complex from a set of points.
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ˇ Cech complex
Let’s construct a simplicial complex from a set of points.
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ˇ Cech complex
Let’s construct a simplicial complex from a set of points.
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ˇ Cech complex
Let S be a finite set of points in Rd. The ˇ Cech complex of S and r is given by, ˇ Cech(r) = {σ ⊆ S|∩Bx(r) = ∅} The ˇ Cech complex is equivalent to the nerve of the collection of balls.
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ˇ Cech complex
ˇ Cech complexes for different values of r
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ˇ Cech complex
ˇ Cech complexes for different values of r
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ˇ Cech complex
Some properties of the ˇ Cech complex: When ri ≤ rj, ˇ Cech(ri) ⊆ ˇ Cech(rj).
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ˇ Cech complex
Some properties of the ˇ Cech complex: When ri ≤ rj, ˇ Cech(ri) ⊆ ˇ Cech(rj). ˇ Cech(r) of a set S of points in Rd can always be represented as an abstract simplicial complex.
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ˇ Cech complex
Some properties of the ˇ Cech complex: When ri ≤ rj, ˇ Cech(ri) ⊆ ˇ Cech(rj). ˇ Cech(r) of a set S of points in Rd can always be represented as an abstract simplicial complex. Computing the complex takes an exponential time in the size of S.
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Vietoris-Rips complex
The Vietoris-Rips VR complex of S and r consists of all subsets of diameter at most 2r, VR(r) = {σ ⊆ S|diamσ ≤ 2r}. The diameter of σ is the supremum of all pairwise distances between points in σ.
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Comparing ˇ Cech and Vietoris-Rips complex
ˇ Cech complex Vietoris-Rips complex
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Vietoris-Rips
Notice that ˇ Cech(r) ⊆ VR(r) but ˇ Cech(r) ≃ VR(r). For appropriate values of ra and rb, we have VR(ra) ⊆ ˇ Cech(rb).
Vietoris-Rips Lemma
Let S be a finite set of points in some Euclidean space and r ≥ 0. It follows that VR(r) ⊆ ˇ Cech( √ 2r).
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Vietoris-Rips
Properties of VR complexes: VR complexes avoid the intersection test used in ˇ Cech.
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Vietoris-Rips
Properties of VR complexes: VR complexes avoid the intersection test used in ˇ Cech. VR complexes also take exponential time to compute.
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Vietoris-Rips
Properties of VR complexes: VR complexes avoid the intersection test used in ˇ Cech. VR complexes also take exponential time to compute. VR complexes might not have a geometric representation in the underlying space of S.
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Voronoi diagram
What if we want the simplicial complex to have a geometric representation?. Delaunay complexes can be used for this task. To explore Delaunay complexes, we first introduce the Voronoi diagram.
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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.
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Voronoi diagram
For a set of points in a plane: Encode proximity information useful for answering point queries.
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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.
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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.
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Delaunay complex
The Delaunay complex of a finite set of points S ⊆ Rd is equivalent to the nerve of the Voronoi diagram, Delaunay(S) = {σ ⊆ S|
- ui∈σ
Vui = ∅}.
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Delaunay complex
The Delaunay complex seems to always create triangles in R2.
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Delaunay complex
The Delaunay complex seems to always create triangles in R2. Is this always the case?
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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.
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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 geometric realization of a Delaunay complex fits in Rd.
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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.
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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) = ∅}.
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Delaunay filtration
Different values of r to create different alpha complexes.
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Delaunay filtration
Eventually, we obtain the Delaunay complex.
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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 different alpha complexes obtained by varying r.
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Summary
The nerve of a set of points S allows us to compute simplicial complexes.
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Summary
The nerve of a set of points S allows us to compute simplicial complexes. ˇ Cech complexes and Vietoris-Rips complexes allows us to construct abstract simplicial complexes from S.
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Summary
The nerve of a set of points S allows us to compute simplicial complexes. ˇ Cech complexes and Vietoris-Rips complexes allows us to construct abstract simplicial complexes from S. The Delaunay triangulation allows to create simplicial complexes with geometric realizations.
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Summary
The nerve of a set of points S allows us to compute simplicial complexes. ˇ Cech complexes and Vietoris-Rips complexes allows us to construct abstract simplicial complexes from S. The Delaunay triangulation allows to create simplicial complexes with geometric realizations. The alpha complex can be seen as a filtered version of the Delaunay complex.
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Sources
- H. Edelsbrunner, J. L. Harer, Computational topology. An
- introduction. American Mathematical Society, Providence, RI, 2010.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark
- Overmars. 2008. Computational Geometry: Algorithms and
Applications (3rd ed.). Lecture notes and videos on Computational Topology. Lectures 11, 12, and 13. Topological data analysis of genes ˇ Cech and Vietoris-Rips complex Persistent homology
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