Tutorial of topological data analysis (basis _ part 1)

1. Tutorial of
Topological Data Analysis
Tran Quoc Hoan
@k09hthaduonght.wordpress.com/
Hasegawa lab., Tokyo
The University of Tokyo
Part I - Basic Concepts

2. My TDA = Topology Data Analysis ’s road
TDA Road 2
Part I - Basic concepts &
applications
Part II - Advanced computation
Part III - Mapper Algorithm
Part V - Applications in…
Part VI - Applications in…
Part IV - Software Roadmap
He is following me

3. Outline
TDA - Basic Concepts 3
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology

4. Outline
TDA - Basic Concepts 4
1. Topology and holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
3. Definition of holes

5. Topology
I - Topology and Holes 5
The properties of space that are preserved under continuous
deformations, such as stretching and bending, but not tearing or
gluing
⇠= ⇠= ⇠=
⇠= ⇠= ⇠=
⇠=

6. Invariant
6
Question: what are invariant things in topology?
⇠= ⇠= ⇠=
⇠= ⇠=
⇠=
⇠=
Connected
Component Ring Cavity
1 0 0
2 0 0
1 1 0
1 10
Number of
I - Topology and Holes

7. Holes and dimension
7
Topology: consider the continuous deformation under the
same dimensional hole
✤ Concern to forming of shape: connected component, ring, cavity
• 0-dimensional “hole” = connected component
• 1-dimensional “hole” = ring
• 2-dimensional “hole” = cavity
How to define “hole”?
Use “algebraic” Homology group
I - Topology and Holes

8. Homology group
8
✤ For geometric object X, homology Hl satisfied:
k0 : number of connected components
k1 : number of rings
k2 : number of cavities
kq : number of q-dimensional holes
Betti-numbers
I - Topology and Holes
Image source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

9. Outline
TDA - Basic Concepts 9
1. Topology and holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
3. Definition of holes

10. Simplicial complexes
10
Simplicial complex:
A set of vertexes, edges, triangles, tetrahedrons, … that are closed
under taking faces and that have no improper intersections
vertex
(0-dimension)
edge
(1-dimension)
triangle
(2-dimension)
tetrahedron
(3-dimension)
simplicial
complex
not simplicial
complex
2 - Simplicial complexes
k-simplex

11. Simplicial
11
n-simplex:
The “smallest” convex hull of n+1 affinity independent points
vertex
(0-dimension)
edge
(1-dimension)
triangle
(2-dimension)
tetrahedron
(3-dimension)
n-simplex
= |v0v1...vn| = { 0v0 + 1v1 + ... + nvn| 0 + ... + n = 1, i 0}
A m-face of σ is the convex hull τ = |vi0…vim| of a non-empty subset
of {v0, v1, …, vn} (and it is proper if the subset is not the entire set)
⌧
2 - Simplicial complexes

12. Simplicial
12
Direction of simplicial:
The same direction with permutation <i0i1…in>
1-simplex
2-simplex
3-simplex
2 - Simplicial complexes

13. Simplicial complex
13
Definition:
A simplicial complex is a finite collection of simplifies K such that
(1) If 2 K and for all face ⌧ then ⌧ 2 K
(2) If , ⌧ 2 K and ⌧ 6= ? then ⌧ and ⌧ ⌧
The maximum dimension of simplex in K is the dimension of K
K2 = {|v0v1v2|, |v0v1|, |v0v2|, |v1v2|, |v0|, |v1|, |v2|}
K = K2 [ {|v3v4|, |v3|, |v4|}
NOT YES
2 - Simplicial complexes

14. Simplicial complexes
14
Hemoglobin
simplicial complex
Image source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
2 - Simplicial complexes

15. ✤ Let be a covering of
Nerve
15
= {Bi|i = 1, ..., m} X = [m
i=1Bi
✤ The nerve of is a simplicial complex N( ) = (V, ⌃)
2 - Simplicial complexes

16. Nerve theorem
16
✤ If is covered by a collection of convex closed
sets then X and are
homotopy equivalent
X ⊂ RN
= {Bi|i = 1, ..., m} N( )
2 - Simplicial complexes

17. Cech complex
17
P = {xi 2 RN
|i = 1, ..., m}
Br(xi) = {x 2 RN
| ||x xi||  r}
✤ The Cech complex C(P, r) is the nerve of
✤
= {Br(xi)| xi 2 P}
✤ From nerve theorem: C(P, r)
Xr = [m
i=1Br(xi) ' C(P, r)
✤ Filtration
ball with radius r
2 - Simplicial complexes

18. Cech complex
18
✤ The weighted Cech complex C(P, R) is the nerve of
✤ Computations to check the intersections of balls are not easy
ball with different radius= {Bri
(xi)| xi 2 P}
Alpha complex
2 - Simplicial complexes

19. Voronoi diagrams and Delaunay complex
19
✤ P = {xi 2 RN
|i = 1, ..., m}
Vi = {x 2 RN
| ||x xi||  ||x xj||, j 6= i}
RN
= [m
i=1Vi
Voronoi cell
✤ = {Vi|i = 1, ..., m}
D(P) = N( )
Voronoi decomposition
Delaunay complex
2 - Simplicial complexes

20. General position
20
✤ is in a general position, if there is no
✤ If all combination of N+2 points in P is in a general
position, then P is in a general position
x1, ..., xN+2 2 RN
x 2 RN
s.t.||x x1|| = ... = ||x xN+2||
✤ If P is in a general position then
The dimensions of Delaunay simplexes <= N
Geometric representation of D(P) can be
embedded in RN
2 - Simplicial complexes

21. Alpha complex
21
✤
✤
✤ The alpha complex is the nerve of
↵(P, r) = N( )
✤ From Nerve theorem:
Xr ' ↵(P, r)
2 - Simplicial complexes

22. Alpha complex
22
✤
✤
✤ The weighted alpha complex is defined
with different radius
if P is in a general position
filtration of alpha complexes
2 - Simplicial complexes

23. Alpha complex
23
✤ Computations are much easier than Cech complexes
✤ Software: CGAL
• Construct alpha complexes of points clouds data in RN with
N <= 3
Filtration of alpha complex
Image source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
2 - Simplicial complexes

24. Outline
TDA - Basic Concepts 24
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology

25. Definition of holes
25
Simplicial
complex
Chain
complex
Homology
group
Algebraic Holes
Geometrical
object
Algebraic
object
3 - Definition of Holes

26. What is hole?
26
✤ 1-dimensional hole: ring
not ring have ring
boundary
without
ring
without
boundary
Ring =
1-dimensional graph without boundary?
However, NOT
1-dimensional graph without
boundary but is 2-dimensional graph
’s boundary
Ring = 1-dimensional graph without boundary and is not boundary
of 2-dimensional graph
3 - Definition of Holes

27. What is hole?
27
✤ 2-dimensional hole: cavity
not cavity have cavity
boundary
without
cavity
without
boundary
However, NOT
2-dimensional graph without
boundary but is 3-dimensional graph
’s boundary
Cavity = 2-dimensional graph without boundary and is not boundary
of 3-dimensional graph
Cavity =
2-dimensional graph without boundary?
3 - Definition of Holes

28. Hole and boundary
28
q-dimensional hole
q-dimensional graph without boundary and
is not boundary of (q+1)-dimensional graph=
We try to make it clear by “Algebraic” language
3 - Definition of Holes

29. Chain complexes
29
Let K be a simplicial complex with dimension n. The group of q-
chains is defined as below:
The element of Cq(K) is called q chain.
Definition:
Cq(K) := {
X
↵i
⌦
vi0
...viq
↵
|↵i 2 R,
⌦
vi0
...viq
↵
: q simplicial in K}
0  q  nif
Cq(K) := 0, if q < 0 or q > n
3 - Definition of Holes

30. Boundary
30
Boundary of a q-simplex is the sum of its (q-1)-dimensional faces.
Definition:
vil is omitted
@|v0v1v2| := |v0v1| + |v1v2| + |v0v2|
3 - Definition of Holes

31. Boundary
31
Fundamental lemma
@q 1 @q = 0
@2 @1
For q = 2
In general
• For a q - simplex τ, the boundary ∂qτ, consists of all (q-1) faces of τ.
• Every (q-2)-face of τ belongs to exactly two (q-1)-faces, with different direction
@q 1@q⌧ = 0
3 - Definition of Holes

32. Hole and boundary
32
q-dimensional hole
q-dimensional graph without boundary and is
not boundary of (q+1)-dimensional graph
(1)
(2)
(1)
(2)
:= ker @q
:= im@q+1
(cycles group)
(boundary group)
Bq(K) ⇢ Zq(K) ⇢ Cq(K)
@q @q+1 = 0
3 - Definition of Holes

33. Hole and boundary
33
q-dimensional hole
q-dimensional graph without boundary and is
not boundary of (q+1)-dimensional graph
(1)
(2)
Elements in Zq(K) remain after make Bq(K) become zero
This operator is defined as Q
=
:= ker @q := im@q+1
Q(z0
) = Q(z) + Q(b) = Q(z)
(z and z’ are equivalent in
with respect to )
q-dimensional hole = an equivalence
class of vectors
ker @q
im @q+1
For z0
= z + b, z, z0
2 ker @q, b 2 im @q+1
3 - Definition of Holes

34. Homology group
34
Homology groups
The qth
Homology Group Hq is defined as Hq = Ker@q/Im@q+1
= {z + Im@q+1 | z 2 Ker@q } = {[z]|z 2 Ker@q}
Divided in groups with operator [z] + [z’] = [z + z’]
Betti Numbers
The qth
Betti Number is defined as the dimension of Hq
bq = dim(Hq)
H0(K): connected component H1(K): ring H2(K): cavity
3 - Definition of Holes

35. Computing Homology
35
v0
v1 v2
v3
All vectors in the column space of Ker@0 are equivalent with respect to Im@1
b0 = dim(H0) = 1
Im@2 has only the zero vector
b1 = dim(H1) = 1
H1 = { (|v0v1| + |v1v2| + |v2v3| + |v3v0|)}
3 - Definition of Holes

36. Computing Homology
36
v0
v1 v2
v3
H1 = { (hv0v1i + hv1v2i + hv2v3i hv0v3i)}
All vectors in the column space of Ker@0 are equivalent with respect to Im@1
b0 = dim(H0) = 1
Im@2 has only the zero vector
b1 = dim(H1) = 1
3 - Definition of Holes

37. Outline
TDA - Basic Concepts 37
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology

38. Persistent Homology
Persistent homology 38
✤ Consider filtration of finite type
K : K0
⇢ K1
⇢ ... ⇢ Kt
⇢ ...
9 ⇥ s.t. Kj
= K⇥
, 8j ⇥
✤ : total simplicial complexK = [t 0Kt
Kk
Kt
k
T( ) = t 2 Kt
Kt 1
: all k-simplexes in K
: all k-simplexes in K at time t
: birth time of the simplex
time
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

39. Persistent Homology
39
✤ Z2 - vector space
✤ Z2[x] - graded module
✤ Inclusion map
✤ is a free Z2[x] module with the baseCk(K)
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

40. Persistent Homology
40
✤ Boundary map
✤ From the graded structure
✤ Persistent homology
(graded homomorphism)
face of σ
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

41. Persistent Homology
41
✤ From the structure theorem of Z2[x] (PID)
✤ Persistent interval
✤ Persistent diagram
Ii(b): inf of Ii, Ii(d): sup of Ii
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

42. Persistent Homology
42
birth time
death time
✤ “Hole” appears close to the
diagonal may be the “noise”
✤ “Hole” appears far to the
diagonal may be the “noise”
✤ Detect the “structure hole”
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

43. Outline
TDA - Basic Concepts 43
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
see more at part2 of tutorial

44. Applications
5 - Some of applications 44
• Persistence to Protein compressibility
Marcio Gameiro et. al. (Japan J. Indust. Appl. Math (2015) 32:1-17)

45. Protein Structure
Persistence to protein compressibility 45
amino acid 1 amino acid 2
3-dim structure of hemoglobin
1-dim structure of protein
folding
peptide bond
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

46. Protein Structure
Persistence to protein compressibility 46
✤ Van der Waals radius of an atom
H: 1.2, C: 1.7, N: 1.55 (A0)
O: 1.52, S: 1.8, P: 1.8 (A0)
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
Van der Waals ball model of hemoglobin

47. Alpha Complex for Protein Modeling
Persistence to protein compressibility 47
✤
✤
✤
: position of atoms
: radius of i-th atom
: weighted Voronoi Decomposition
: power distance
: ball with radius ri
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

48. Alpha Complex for Protein Modeling
Persistence to protein compressibility 48
✤
✤
✤
Alpha complex nerve
k - simplex
Nerve lemma
Changing radius
to form a filtration (by w)
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

49. Topology of Ovalbumin
Persistence to protein compressibility 49
birth time
deathtime
birth time
deathtime
1st betti
plot
2nd betti
plot
PD1 PD2
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

50. Compressibility
Persistence to protein compressibility 50
3-dim structureFunctionality
Softness
Compressibility
Experiments Quantification
Persistence diagrams
(Difficult)
…..…..
Select generators and fitting parameters
with experimental compressibility
holes

51. Denoising
Persistence to protein compressibility 51
birth time
deathtime
✤ Topological noise
✤ Non-robust topological features depend on a status of
fluctuations
✤ The quantification should not be dependent on a
status of fluctuations
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

52. Holes with Sparse or Dense Boundary
Persistence to protein compressibility 52
✤ A sparse hole structure is deformable to a much larger
extent than the dense hole → greater compressibility
✤ Effective sparse holes
: van der Waals ball
: enlarged ball
birth time
deathtime
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

53. # of generators v.s. compressibility
Persistence to protein compressibility 53
# of generators v.s. compressibility
Topological Measurement Cp
Compressibility
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

54. Applications
5 - Some of applications 54
• Persistence to Phylogenetic Trees

55. Protein Phylogenetic Tree
Persistence to Phylogenetic Trees 55
✤ Phylogenetic tree is defined by a distance matrix for a
set of species (human, dog, frog, fish,…)
✤ The distance matrix is calculated by a score function
based on similarity of amino acid sequences
amino acid sequences
fish hemoglobin
frog hemoglobin
human hemoglobin
distance matrix of
hemoglobin
fish
frog
human
dog
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

56. Persistence Distance and Classification of Proteins
Persistence to Phylogenetic Trees 56
✤ The score function based on amnio acid sequences does not
contain information of 3-dim structure of proteins
✤ Wasserstein distance (of degree p)
Cohen-Steiner, Edelsbrunner, Harer, and Mileyko, FCM, 2010
on persistence diagrams reflects similarity of persistence
diagram (3-dim structures) of proteins
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

57. Persistence Distance and Classification of Proteins
Persistence to Phylogenetic Trees 57
birth time
deathtime
birth time
birth time
deathtime
deathtimeWasserstein distance
Bijection
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

58. Distance between persistence diagrams
Persistence to Phylogenetic Trees 58
Persistence of sub level sets
Stability Theorem (Cohen-Steiner et al., 2010)
birth time
deathtime
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

59. Phylogenetic Tree by Persistence
Persistence to Phylogenetic Trees 59
✤ Apply the distance on persistence diagrams to classify
proteins
Persistence diagram used the noise band same as
in the computations of compressibility
3DHT
3D1A
1QPW
3LQD
1FAW
1C40
2FZB
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

60. Future work
TDA - Basic Concepts 60
✤ Principle to de-noise fluctuations in persistence diagrams (NMR
experiments)
✤ Finding minimum generators to identify specific regions in a
protein (e.g., a region inducing high compressibility, hereditarily
important regions)
✤ Zigzag persistence for robust topological features among a
specific group of proteins (quiver representation)
✤ Multi-dimensional persistence (PID → Grobner basic)
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf

61. Applications more in part … of tutorials
5 - Some of applications 61
✤ Robotics
✤ Computer Visions
✤ Sensor network
✤ Concurrency & database
✤ Visualization
Prof. Robert Ghrist
Department of Mathematics
University of Pennsylvania
One of pioneers in applications
Michael Farber Edelsbrunner
Mischaikow Gaucher Bubenik
Zomorodian
Carlsson

62. Software
TDA - Basic Concepts 62
• Alpha complex by CGAL
http://www.cgal.org/
• Persistence diagrams by Perseus (coded by Vidit Nanda)
http://www.sas.upenn.edu/~vnanda/perseus/index.html
http://chomp.rutgers.edu/Project.html
• CHomP project

63. Reference links
TDA - Basic Concepts 63
• Yasuaki Hiraoka associate professor homepage
http://www2.math.kyushu-u.ac.jp/~hiraoka/site/About_Me.html
http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
www.msys.sys.i.kyoto-u.ac.jp/~kazunori/paper/nist20081219.pdf
• Applications in sensor network