1. Persistent homology I Peter Kálnai Autumn school Department of Algebra
Ústupky, 24th – 27th November 2016

2. Algebraic topology
“Don’t be afraid of these ideas – you see them for the first time. When you see them for the tenth time, you won’t be afraid any more. They will have been safely stored on the list of things that you simply don’t understand.” Martin Markl

3. Outline
Introducing the idea
Mathematical formalism
Software

4. Introducing the idea

5. Introduction What is Persistent homology?
A method of Topological data analysis
An algebraic tool for discerning topological features of data
“Extracting knowledge of data”
Topological features
Topology is a “study of shape”
Clusters, holes, graphs, components…
Data sets:
Long vectors
Viewed as discrete points in a space with a metric
A few relevant coordinates
Do not know which ones are they
it is now often the case
that we are given data in the form of very long vectors, where all but a few of the
coordinates turn out to be irrelevant to the questions

6. Topological data analysis (TDA)
“Fundamental advance in machine learning” G.Carlsson
Coordinate invariance
- No change with rotating or changing coordinates
Deformation invariance
- No change with squeezing, inflating, twisting…
- A donut = a cup
Compressed representation
A sphere: infinitely many points
An icosahedron: 12 points, 30 edges, 20 faces =
it is now often the case that we are given data in the form of very long vectors, where all but a few of the
coordinates turn out to be irrelevant to the questions

7. Deformation invariance
Yes:
No:

8. Motivation
Problem:
Given a large, n-dimensional data set, how can one determine its shape and structure?
Input: a discrete sampling X0 of an unknown m-dimensional manifold X in an Euclidean space Rn, where m << n
Goal: to compute the homological properties of X
Conclusion: the model of X0 has the homology of X
Complications: assume X0 can not be refined; X0 with noise
Strategy: connect points in X0 into spaces with graspable homology
Point cloud  Coverings  Simplicial complexes (Nerves)
Covering created with a proximity parameter ε
Programme: (Grothendieck) The topology of a given space is framed in the mappings to or from that space.

9. Mathematical formalism

10. Homotopy
We recall that two continuous maps f, g between topological spaces X, Y are said to be homotopic if there is a continuous map
H: X × [0, 1] → Y
so that H(x, 0) = f(x) and H(x, 1) = g(x), for all in X. Denoted f ≈ g and H is called a homotopy.
A map f : X → Y is a homotopy equivalence if there is a map g : Y → X so that both hold:
f ◦ g ≈ 1Y
g ◦ f ≈ 1X
The map g is called a homotopic inverse of f.

11. Homotopy
Two topological spaces X,Y are homotopy equivalent if there exist a homotopy equivalence between them. Denoted X ≈ Y
A space that is homotopy equivalent to the one point space
is said to be contractible.
Equivalence classes under ≈ are called homotopy types.

12. Singular homology
For any topological space X, any PID D, and any integer k ≥ 0 imagine a D-module Hk(X,D) with:
Functoriality: f : X → Y continuous, then there is an homomorphism of D-modules Hk(f ,D): Hk(X,D) → Hk(Y,D) satisfying Hk(g,D) ◦ Hk(f ,D) = Hk(g ◦ f) and Hk(1X,D) = 1
Homotopy invariance: if f ≈ g then Hk(f ,D) = Hk(g ,D), if X ≈ Y then Hk(X,D), Hk(Y,D) are isomorphic
Normalisation: Hk({x},D) is isomorphic to DD
Betti numbers: define k-th Betti number βk(X,D) of X with coefficients in D as the rank of the free submodule of Hk(X,D)

13. Simplicial complexes
Let K be a set , let S be collection of subsets of K.
The singleton {v} is called a vertex of K for all v ϵ K.
A subset of a set in S is called a face.
A pair (K; S) is a simplicial complex if the collection S contains all vertices and all faces.
An element σ of the collection S is called a k-simplex if the cardinality of the set is k + 1, e.g.: k=0,1,2,3
An orientation of σ is an equivalence class of orderings of the vertices of σ under the sign of their permutation  [σ]
Zomorodian A., Carlsson G.: “Computing Persistent Homology”, Disc. Comp. Geom 33:(2005)

14. Chain complexes
The k-th chain group Ck of (K;S) is the free abelian group on its set of oriented k-simplices, where
[σ] = −[τ] if σ = τ and σ and τ are oriented differently.
An element of the chain group:
c = Σi=0,1,…,k ni [σi], σi ∈ K with coefficients ni ∈ Z.
A boundary operator ∂k : (∂k ◦ ∂k+1) = 0
∂kσ = Σi=0,1,..,k (−1)i [v0, v1, , vi , , vk ],
A chain complex C∗:
…  Ck+1  Ck  Ck-1  …

15. Chain complexes σ = [v1,v2] δ(σ) = [v2] – [v1] σ = [v1,v2,v3]

16. Chain complexes for PIDs
Abelian group = Mod-Z (left Z-modules)
Z is a principal ideal domain (PID)
F field  F[x] the polynomial ring is PID
 Chain complexes with coefficients in any PID D
Structural theorem for PIDs:

17. Simplicial Homology the cycle group Zk := ker δk
the boundary group Bk := δk+1(Ck+1)
the kth homology group Hk := Zk/Bk with elements that are classes of homologous cycles
This definition satisfies functoriality, homotopy invariance and normalisation with A = Z
The k-th Betti number corresponds to an informal notion of the number of independent k-dimensional surfaces.

18. Simplicial Homology
Simplicial homology is canonically isomorphic to the singular homology of the corresponding topological space
the homology of simplicial complexes is algorithmically computable! (the Smith normal form of matrices with columns parametrised by the bases of chain groups)
Aim: To compute the homology of a space X
Options:
Find a homotopy equivalence f:X → (K;S) for some simplicial complex (K;S)
a) Find a space Y homotopy equivalent to X (X ≈ Y)
b) Find a homotopy equivalence g:Y → (K’;S’) for some simplicial complex (K’;S’)

19. Čech complexes
Construction: creating a simplicial complex from a covering Ų = (Ui | i ϵ X0) of X
We say that the simplicial complex (X0; S) is the nerve of Ų if S contains exactly families of those points of X0 whose covers give a non-empty intersection. Denoted nerve(Ų)
Does not depend on any distance!
If X is an Euclidean space like Rn, then we can use the Euclidean distance and form coverings parameterised with a diameter ε

20. Čech complexes Nerve theorem:
Let X be a topological space and let Ų be a covering of X indexed by X0 that consists of open sets and is numerable. (e.g. in a metric space, every point-finite open covering is numerable)
Suppose that the intersection of any family index by a non-empty subset of X0 is either empty or contractible.
Then nerve(Ų) ≈ X.

21. Čech complexes
Let X be a metric space, let ε > 0 and let X0 be such that the balls with diameter ε with centers in X0 covers X.
Definition: Čech complex C(X0, ε) is the nerve of the covering consisting of ε-balls centered by points in X0
Corollary: Let M be a compact Riemannian manifold. Then there is e > 0 so that C(M, ε) ≈ M for all ε ≤ e.

22. Čech complexes 0 < ε0 ε0 < ε1 ε1 < ε2 ε2 < ε3 ε3 < ε4
ε4 < ε5
ε5 < ε6
Bubenik P: “Stat. Top. Data Analysis using Persistence Landscapes”, J. M.L.R. 16, (2015)

23. Čech complexes “The shape of the data lies not in a single
ε too small  C(X0, ε) is a discrete set;
ε too large  C(X0, ε) is a single simplex
The golden mean
may not exist!
Consider the ordered sequence of spaces (C(X0, ε) |ε > 0), with inclusion maps ιε,ε’ : C(X0, ε’) → C(X0, ε) for ε < ε′
The homology of this sequence captures
which homological features in the data persist over the range of parameters [ε, ε′].
“The shape of the data lies not in a single
space, but in a diagram of spaces”, R.Ghrist

24. Vietoris-Ribs complexes
Let (X,d) denote a metric space. Then the Vietoris-Rips complex for X, attached to the parameter ε, denoted by VR(X, ε), is the simplicial complex whose vertex set is X, and where {x0, x1, , xk} spans a
k-simplex if and only if d(xi, xj) ≤ ε for all 0 ≤ i, j ≤ k.
“a clique complex of ε -nearest neighbour graph”
Let (X,d) be a convex subset of n-dimensional Euclidean space. Then for any point cloud X0 in X:
C(X0, ε) contained in VR(X, 2ε) contained in C(X0, 2ε)

25. Vietoris-Ribs complexes
Ghrist R. „Barcodes: The persistent topology of data“, Bull. AMS 45/1, 2008

26. Čech VS Vietoris-Ribs complexes
S1 V S1 V S1
S1 V S1
Ghrist R. „Barcodes: The persistent topology of data“, Bull. AMS 45/1, 2008

27. Barcodes
Ghrist R. „Barcodes: The persistent topology of data“, Bull. AMS 45/1, 2008

28. Persistent diagram (3, 4) (1, 2) (2, 5) (0, 1) (0, ∞)  (0, 5) [ ) [ )
[ )
[ )
[ ) [
(0, ∞) = (0, 5) 3rd argument in
ripsDiag(XX, maxdimension,maxscale, …)
(3, 4)
(1, 2)
(2, 5)
(0, 1)
(0, ∞)  (0, 5)

29. Persistent diagram
(3, 4)
(2, 5)
(1, 2)
(0, 1)
(0, 5)

30. Software

31. Software for PH Fast C++ Projects: Dionysus (2007-2012)
Persistent Homology Algorithm Toolbox (PHAT, )
GUDHI (2014-)
Interfaces:
Python binding for Dionysus
TDA package for R over the C++ projects

32. Dionysus By Dmitriy Morozov (http://mrzv.org/)
Tested on Windows (+-)
Dependencies: Python, Cmake, Boost, CGAL, PyQt4, PyOpenGL, Numpy

33. Dionysus – rips.py

34. Dionysus – rips.py ε = 22.7 ε = 12.7 Input: points of chain linkages
Parameters: 2-skeleton; ε
Output: #simplex = 285; 823
ε = 22.7
ε = 12.7

35. Dionysus – rips.py ε = 1.6 ε = 1.7 Input: points of a trefoil knot
Parameters: 2-skeleton; ε
Output: #simplex =
ε = 1.6
ε = 1.7

36. TDA for R
400 points
from a circle

37. TDA for R
60 points from
two circles 

38. Questions
If anything in the first half made sense then it was your fault. (Dylan Moran)