TOPOLOGICAL DATA ANALYSIS
3.3 Simplicial complexes
It is desirable to work with simplicial complexes because the homology of a simplicial complex is easily computable [70, 71].
Here, we give a brief description of simplicial complexes. We refer the reader to [72, 73] for a more in-depth discussion.
Abstract simplicial complexes
Definition 3.1 (abstract simplicial complex). An abstract simplicial complex is a collection Σ of finite non-empty sets, such that if σ ∈ Σ, and τ is a non-empty subset ofσ, thenτ∈ Σ.
An elementσofΣis asimplexofΣ. Each non-empty subsetτof a simplexσ is aface ofσ. Thevertex set VofΣis the collection of the one-point elements of Σ. We do not distinguish between the vertex v ∈ V and the 0-simplex {v} ∈Σ.
Thedimension of a simplexis one less than the number of its elements. The dimension of a simplicial complexis the greatest dimension of its simplices or is infinite if there is no such greatest dimension.
A subcollection ofΣthat is itself a simplicial complex is called asubcomplex ofΣ.
Nerve and Nerve Lemma
Given a topological spaceX, there are many different simplicial complexes that can be constructed from it. One particularly useful construction that we will devote our attention to is thenerve.
Definition 3.2(nerve). Let Abe a collection of subsets of the spaceX. The nerveof A, denoted N(A), is the abstract simplicial complex with vertices consisting of the elements ofAand simplices given by the finite subcollec- tionsn
Ai0,· · · ,Ain
oofAsuch that
Ai0∩ · · · ∩Ain ,∅.
WhenAis a covering ofX, the nerveN(A)can be thought of as a combina- torial representation ofX.
The nerve ofX is a useful construction because the Nerve Lemma [74, 75]
provides criteria that guarantee that the nerve of a covering ofXis homotopy equivalent toX.
Theorem 3.3 (Nerve Lemma [73]). Let X be a paracompact space. If U is an open cover of X such that every non-empty intersection of finitely many sets inU is contractible, then X is homotopy-equivalent to the nerve N(U).
Throughout this thesis, we assume all covers are good open covers so that the Nerve Lemma applies.
Cech complexesˇ
When X is a metric space, we can consider the covering ofX byε-balls to form aCech complex.ˇ
Definition 3.4( ˇCech complex). Let (X,d) be a metric space. Forε > 0, let A⊆Xsuch thatX =Sa∈ABε(a), whereBε(a):={x∈X: d(x,a) < ε}is a ball of radiusεcentered ata∈A. TheCech complexˇ of(A,ε), denoted ˘C(A,ε), is the nerve of the covering
Bε(a) a∈A.
The ˇCech complex can be used to approximate the homology of a manifold.
Theorem 3.5 ([76]). Let M be a compact Riemannian manifold. There exists a positive number e so thatC˘(M,ε) is homotopy equivalent to M wheneverε ≤ e.
Moreover, for everyε ≤ e, there is a finite subset V ⊆ M so that the subcomplex C˘(V,ε)⊆C˘(M,ε)is also homotopy equivalent to M.
Vietoris-Rips complexes
Unfortunately, the ˇCech complex can be computationally intensive to com- pute. One can consider the Vietoris-Rips complex as an alternative, which allows us to consider only pairs instead of all subcollections.
Definition 3.6 (Vietoris-Rips complex). Let (X,d) be a metric space. The Vietoris-Rips complexVR(X,ε)is the simplicial complex with vertex setX, such that a set{x0,· · · ,xk}spans ak-simplex if and only ifd
xi,xj
≤εfor all 0≤i,j≤k.
We can relate the ˇCech complex and Vietoris-Rips complex through the following inclusions:
Proposition 3.7([76]). Forε >0, we have the inclusions C˘(X,ε)⊆VR(X,ε)⊆C˘(X, 2ε). 3.4 Persistent homology
Persistent homology is motivated by the desire to distinguish noise from features in data and the need to examine data over a range of scales. We use persistent homology to measure the scale, or resolution, of a topological feature.
We follow the expositions of Edelsbrunner and Harer [77, 78] as we introduce background material for persistent homology. Although persistence can be defined for any sequence of vector spaces connected by homomorphisms, in our applications, we are only concerned with persistent homology for sim- plicial complexes. We refer the reader to [72, 73] for a thorough discussion of homology.
Throughout this chapter, we consider only homology with Z/2Z coeffi- cients. We write Hr(X)forHr(X;Z/2Z), and we call its rank the rth Betti number of X, denotedβr(X).
Filtrations
In order to define persistent homology for simplicial complexes, we intro- duce the notion of afiltration of a simplicial complex given by amonotonic map.
Definition 3.8 (monotonic). Let K be a simplicial complex. We say that a function f : K→R is monotonic if it is non-decreasing along increasing chains of faces; that is, f (τ) ≤ f (σ)wheneverτis a face ofσ.
If f :K →Ris monotonic, then the sublevel set K(a) = f−1(−∞,a] is a subcomplex ofKfor everya∈R.
In particular, ifKis a simplicial complex withmsimplices, then we can find an increasing sequence ofn+1≤m+1 different subcomplexes:
∅=K0⊆K1 ⊆ · · · ⊆Kn =K.
Lettinga1 < · · · < an be the images of the simplices inK under f : K → R, anda0=−∞, we haveKi =K(ai)for eachi.
We call this nested sequence of subcomplexes afiltrationof(K, f).
The inclusion mapsKi−1 →Kibetween subcomplexes induce the following homomorphisms by functorality:
0=H∗(K0) →H∗(K1) → · · · →H∗(Km) = H∗(K).
As simplices get added toKi−1to formKi, new homology classes may come into existence, and existing homology classes may become trivial or merge with one another.
Persistent homology groups
Definition 3.9(pth persistent homology groups). Let fpi,j: Hp(Ki)→Hp
Kj be the homomorphism induced by the inclusionKi → Kj, fori≤ j. Thepth persistent homology groupsare the images
Hi,jp =imfpi,j,
for 0≤i≤ j≤n.
The ranks of these groups are called thepth persistent Betti numbers βi,pj=rankHi,pj.
Note that the persistent homology groupsHi,pj consist of homology classes ofKithat still exist inKj, andHi,ip =Hp(Ki).
A homology classα∈Hp(Ki)isbornatKiif it is not in the image of the map induced by the inclusion Ki−1 ⊂ Ki. We sayα dies entering Kj if the image of the map induced byKi−1 ⊂ Kj−1 does not contain the image ofαbut the image of the map induced byKi−1 ⊂ Kj does. Thepersistence ofαisaj−ai or j−i, depending on the application. A class isessentialif it does not die within the filtration.
Persistence diagrams
A persistence diagram is a multiset of points in the extended planeR2∞. Letµi,pjbe the number ofp-dimensional classes born atKiand dying enter- ing Kj. Then the pth persistence diagram of the filtration of (K, f), denoted dgmp(f), consists of points(ai,aj), each with multiplicityµi,pj.
3.5 Reeb graphs and Reeb spaces