Topological Data Analysis and Persistent Homology

Topological Data Analysis and
Persistent Homology
Supervisor: Prof. Francesco Vaccarino, Candidate: Carla Federica Melia
POLITECNICO DI TORINO
DIPARTIMENTO DI SCIENZE MATEMATICHE
Corso di Laurea Magistrale in Ingegneria Matematica
Graduation Session of December 2018
A.Y. 2017/18

 Topological Data Analysis (TDA) is a branch of applied mathematics that uses
notions and techniques of a miscellaneous set of scientific fields such as
 Its resulting tools allow to infer robust features about the “shape” of complex
datasets potentially corrupted by noise/incompleteness.
2
Introduction
Topological Data Analysis and Persistent Homology - Politecnico di Torino - 2018
StatisticsComputer ScienceAlgebraic Topology

TDA aims at inferring statistically significative information on the shape of the data.
3
The Shape of Data
[1]

[15] [16]
4
Finding Loops and Voids in Universe

Periodic behaviors Attractors
[7] [8]
5
Periodic Systems

6
Effective Brain Networks Analysis
[17][5]

 This thesis focuses on Persistent Homology (PH) technique.
 The purposes are:
7
Objectives 1/3
1) to provide a satisfying explanation of TDA and PH fundamentals
[2]

8
Objectives 2/3
2) to analyze the robustness and the reliability of the inferred features
[3]

9
Objectives 3/3
3) to implement some TDA techniques on some study cases

 The input is a finite set of elements coming with a notion of distance
between them.
 The elements are mapped into a point cloud (PCD).
 PCD is completed by building "continuous" shape, a complex, on it.
10
From PCD to Complex
[13]
A
B
C
B
A
C
A
B
C

 There are many ways to build simplicial complexes from a topological space.
 To be a useful, a simplicial complex has to have an homology that approximates the
one of the space we want to study.
11
[18]
Simplicial Complexes

12
Cech and Vietoris–Rips Complexes
 Reconstruction Theorem
 Nerve Theorem
 𝐶 𝛼(X) ⊂ 𝑉2𝛼(X) ⊂ 𝐶2𝛼(X)
Cech complex is difficult to calculate, but it is quite small
and accurate.
[19]
𝐶 𝛼 𝑋 ≔ 𝑝1, … , 𝑝k : 𝑝1, … , 𝑝k ⊂ 𝑋,∩𝑖 𝐵 𝛼 𝑝i ≠ ∅
𝑉𝛼 𝑋 ≔ 𝑝1, … , 𝑝k : 𝑝1, … , 𝑝k ⊂ 𝑋, 𝑚𝑎𝑥 𝑝i,𝑝j∈𝜎
(𝑑(𝑝i, 𝑝j)) ≤ 𝛼
Vietoris–Rips complex is easy to calculate, but it is usually very
big and less accurate.

 The homotopy equivalent items shares the same homology.
 Homology groups are a more computable alternative to homotopy ones.
13
Homology
Boundary Operator
𝜎 = [𝑣0, … , 𝑣 𝑘] 𝜕 𝑘 𝜎 = ෍
𝑖=0
𝑛
−1 𝑖
𝑣0,…, ො𝑣𝑖,…, 𝑣 𝑘
Simplex
𝜕 𝑣0, 𝑣1, 𝑣2, 𝑣3 = 𝑣1, 𝑣2, 𝑣3 - 𝑣0, 𝑣2, 𝑣3 + 𝑣0, 𝑣1, 𝑣3 - 𝑣0, 𝑣1, 𝑣2
[8]

14
Homology Groups
Cycles
𝑍 𝑘 = 𝐾𝑒𝑟(𝜕 𝑘)
Boundaries
𝐵 𝑘 = 𝐼𝑚(𝜕 𝑘+1)
k-th Homology Group
𝐻 𝑘 = 𝑍 𝑘/𝐵 𝑘
 The goal of homology is to discard cycles that are also boundaries, so we quotient
𝑍 𝑘 using the following equivalence relation
 The rank of 𝐻 𝑘 is 𝛽 𝑘.
𝛽0 = #Components 𝛽1 = #Loops 𝛽2 = #Voids 𝛽 𝑛 = # n-dim Holes
∀𝑧1, 𝑧2 ∈ 𝑍 𝑘, 𝑧1~𝑧2 ⇔ 𝑧1 − 𝑧2 ∈ 𝐵 𝑘

The sequence of simplicial complexes with its inclusion maps is a filtration.
[20]
15
Filtrations
Which 𝑑 should we choose?
The most persistent features are detected using PH. They are supposed to represent
true characteristics of the underlying space.

16
Persistent Homology
 With PH we study the homology of a filtration as a single algebraic entity.
 Its features can be then analyzed using its barcode representation and this is
formally justiﬁed by the Structure Theorem.
Definition
When 0 ≤ 𝑖 ≤ 𝑗 ≤ 𝑛, the inclusion 𝑥𝑖
𝑗
:𝑋𝑖 ↪ 𝑋𝑗 induces a homomorphism
𝐻(𝑥𝑖
𝑗
):𝐻(𝑋𝑖) → 𝐻(𝑋𝑗) whose images are the persistent homology groups.

17
Persistence Barcode
[21]

18
Persistence Diagram
 Persistence Diagram
 Given two persistent diagrams 𝐷1 and 𝐷2, the Bottleneck distance between them is
𝑑 𝐵 𝐷1, 𝐷2 = 𝑖𝑛𝑓𝛾 𝑠𝑢𝑝 𝑥∈𝐷1
| 𝑥 − 𝛾 𝑥 |∞ where 𝛾 ranges over all multi-bijections
𝐷1 → 𝐷2 .
[23]

19
Stability Results
Theorem
Let 𝑋 and 𝑌 be two compact metric spaces and let 𝐹𝑖𝑙𝑡(𝑋)
and 𝐹𝑖𝑙𝑡(𝑌) be the Vietoris–Rips filtrations built on top them.
Then
𝑑 𝐵(𝐷(𝐹𝑖𝑙𝑡 𝑋 ), 𝐷(𝐹𝑖𝑙𝑡 𝑌 )) ≤ 2𝑑 𝐺𝐻(𝑋, 𝑌)
Moreover, if 𝑋 and 𝑌 are embedded in the same space then
𝑑 𝐵(𝐷(𝐹𝑖𝑙𝑡 𝑋 ), 𝐷(𝐹𝑖𝑙𝑡 𝑌 )) ≤ 2𝑑 𝐻(𝑋, 𝑌)
[24]

20
Statistical Results
The most persistent features can be detected and separated from topological
noise using statistical methods as the Bootstrap.
Given a persistence diagram 𝑋 with an estimator ෠𝑋, with look for 𝛿 𝛼 such that
𝑃 𝑑 𝐵 𝑋, ෠𝑋 ≥ 𝛿 𝛼 ≤α, α∈(0,1)
The confidence set will be 𝑋: 𝑑 𝐵 𝑋, ෠𝑋 ≥ 𝛿 𝛼
[25]

21
Implementation
[27]
 To analyze the topological information of different datasets, a console application
was implemented using GUDHI in Python, TDA in R and QlikView.
 GUDHI proposed an efficient tree representation for simplicial complexes, the
simplex tree.
+ +

22
Data Upload
Data
Upload
Consistency
Checks
Dimension
Plot
Data …
pandas
matplotlib

23
TDA Application
Data Simplex Tree Persistence
Persistence
Diagram
Persistence
Barcode
Bootstrap
.pers
gudhi
TDA
Confidence
Band
…
Betti
Curves
𝛿 𝛼

24
Front End – QlikView

25
Results

BIBLIOGRAPHY
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[2] R. Ghrist, Three examples of applied and computational homology, 2008.
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Scientific Reports v. 3, Article number: 1236, 2013.
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Coastal and Shelf Science, Volume 67, Issue 4, Pages 605-617, 2006.
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[9] X. Feng, Y. Tong, G. W. Wei and K. Xia, Topological modeling of biomolecular data, Nanyang Technological University.
[10] B.Cottenceau, N.Delanoue ,L.Jaulin, Guaranteeing the homotopy type of a set defined by non-linear inequalities, DOI: 10.1007/s11155-007-9043-8, 2006.
[11] P. Lambrechts, The Poincaré conjecture and the shape of the universe slides, Wellesley College, 2009.
[12] E.A.Coutsias,S.Martin,A.ThompsonandJ.P.Watson,Topologyofcyclo-octaneenergylandscape, doi: 10.1063/1.3445267, 2010.
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[18] K. G. Wang, The Basic Theory of Presisten Homology, 2012.
[19] F. Chazal and B. Michel, An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists, arXiv:1710.04019v1 [math.ST], 2017.
[20] M. Wright, Introduction to Persistent Homology video on M. Wright channel https : //www.youtube.com/watch?v = 2PSqWBIrn90 consulted on 10/10/2018, 2016.
[21] R. Ghrist, Barcodes: The Persistent Topology Of Data, Bull. Amer. Math. Soc. 45 (2008), 61-75 , Doi: https://doi.org/10.1090/S0273-0979-07-01191-3, 2007.
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[23] The NIPS 2012 workshop on Algebraic Topology and Machine Learning.
[24] K.Fukumizu,Y.HiraokaandG.Kusano,PersistenceweightedGaussiankernelfortopologicaldata analysis, 2016.
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[27] J. Boissonnat, C. Maria, The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes, [Research Report] RR-7993, pp.20. <hal-00707901v1>, 2012.

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