Simplicial complexes associated with cloud points Seminar algorithms Jorge Cordero Eindhoven University of Technology 8 May 2018
Outline Motivation 1 Recap 2 Nerves 3 ˇ Cech complex 4 Vietoris-Rips complex 5 Delaunay complex 6 Alpha complexes 7
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 ∈ R n . 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 R n is a collection of simplices in R n 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
ASC and Homotopy An abstract simplicial complex is a collection of finite non-empty subsets of S. If A ⊆ S , each non-empty subset B ⊆ A is also in S .
ASC and Homotopy An abstract simplicial complex is a collection of finite non-empty subsets of 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.
Simplicial representation Our goal is to compute a simplicial representation of a set of points to apply topological tools for data analysis.
Nerves We can use the nerve of a finite collection of sets S to create an abstract simplicial complex.
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 � = ∅} .
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 }} .
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 the set of points.
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.
Closed ball A closed ball with center x and radius r is defined by, B x ( r ) = x + r B d = { y ∈ R d | || y − x || ≤ r }
ˇ Cech complex Let’s construct a simplicial complex from a set of points.
ˇ Cech complex Let’s construct a simplicial complex from a set of points.
ˇ Cech complex Let’s construct a simplicial complex from a set of points.
ˇ Cech complex Let’s construct a simplicial complex from a set of points.
ˇ Cech complex Let S be a finite set of points in R d . The ˇ Cech complex of S and r is given by, ˇ Cech ( r ) = { σ ⊆ S |∩ B x ( r ) � = ∅} The ˇ Cech complex is equivalent to the nerve of the collection of balls.
ˇ Cech complex ˇ Cech complexes for different values of r
ˇ Cech complex ˇ Cech complexes for different values of r
ˇ Cech complex Some properties of the ˇ Cech complex: When r i ≤ r j , ˇ Cech ( r i ) ⊆ ˇ Cech ( r j ).
ˇ Cech complex Some properties of the ˇ Cech complex: When r i ≤ r j , ˇ Cech ( r i ) ⊆ ˇ Cech ( r j ). Cech ( r ) of a set S of points in R d can always be represented as an ˇ abstract simplicial complex.
ˇ Cech complex Some properties of the ˇ Cech complex: When r i ≤ r j , ˇ Cech ( r i ) ⊆ ˇ Cech ( r j ). Cech ( r ) of a set S of points in R d can always be represented as an ˇ abstract 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 of diameter at most 2 r , VR ( r ) = { σ ⊆ S | diam σ ≤ 2 r } . The diameter of σ is the supremum of all pairwise distances between points in σ .
Comparing ˇ Cech and Vietoris-Rips complex ˇ Cech complex Vietoris-Rips complex
Vietoris-Rips Notice that ˇ Cech ( r ) ⊆ VR ( r ) but ˇ Cech ( r ) �≃ VR ( r ). For appropriate values of r a and r b , we have VR ( r a ) ⊆ ˇ Cech ( r b ). 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 ( 2 r ).
Vietoris-Rips Properties of VR complexes: VR complexes avoid the intersection test used in ˇ Cech.
Vietoris-Rips Properties of VR complexes: VR complexes avoid the intersection test used in ˇ Cech. VR complexes also take exponential time to compute.
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.
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.
Voronoi diagram Consider a finite set of points S = { v 1 , v 2 , ..., v n } in R d : The Voronoi cell of v i ∈ S is the set of points in R d closest to v i , Vv i = { x ∈ R d | || x − v i || ≤ || x − v j || , ∀ v j ∈ S } . Together, the Voronoi cells of all points v i cover the entire space.
Voronoi diagram For a set of points in a plane: Encode proximity information useful for answering point queries.
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.
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 ⊆ R d is equivalent to the nerve of the Voronoi diagram, � Delaunay ( S ) = { σ ⊆ S | V u i � = ∅} . u i ∈ σ
Delaunay complex The Delaunay complex seems to always create triangles in R 2 .
Delaunay complex The Delaunay complex seems to always create triangles in R 2 . Is this always the case?
Delaunay complex The Delaunay complex seems to always create triangles in R 2 . 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 ∈ R d , assuming general position, the geometric realization of a Delaunay complex fits in R d .
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 R d and r ≥ 0. Let R u ( r ) = B u ( r ) ∩ V u , with V u being the Voronoi cell of u . The alpha complex is defined as � Alpha ( r ) = { σ ⊆ S | R u ( r ) � = ∅} . u ∈ σ
Delaunay filtration Different values of r to create different alpha complexes.
Delaunay filtration Eventually, we obtain the Delaunay complex.
Delaunay filtration The filtration of K m = Delaunay is represented as, ∅ = K 0 ⊆ ... ⊆ K i ⊆ ... ⊆ K m . K i corresponds to the i -th alpha complex from the sequence of different alpha complexes obtained by varying r .
Summary The nerve of a set of points S allows us to compute simplicial complexes.
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 .
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.
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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