Introduction to topological dataanalysis
Ippei Obayashi
Adavnced Institute for Materials Research, Tohoku University

Jan. 12, 2018

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 2018

1 / 32Persistent homology
Topological Data Analysis (TDA)▶
▶

Data analysis methods using topology from mathematics
Characterize the shape of data quantitatively⋆

By using connected components, rings, cavities, etc.

Persistent homology (PH) is a main tool of TDA▶
▶

The key idea is “Homology” from mathematics
Gives a good descriptor for the shape of data (called apersistence diagram)

Rapidly developed in 21st century
Mathematical theories
Software▶ Applications to materials science, sensor network,
phylogenetic network, etc.
▶
▶

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 20182 / 32

Example 1

These images are classified into two groups (left 4images and right 4 images). Do you find thecharacteristic shape to distinguish the two groups?
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 20183 / 32

Shapes around blue dots are “typical” for left images, and red dots
for right images
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 20184 / 32

Example 2

Atomic configurations of amorphous silica (SiO2 ) andliquid silica. Do you find the diﬀerence?
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 2018

5 / 32

From Y. Hiraoka, et al., PNAS 113(26):7035-40 (2016)

Persistence diagrams can capture the diﬀerence clearly
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 2018

6 / 32

Homology
Connected components, rings, and cavities aremathematically formalized by homology.
Algebra is used to formalize such geometricstructures
There are many types of holes and characterized by“dimension”

dim 1: 1
dim 2: 0

dim 1: 0
dim 2: 1

dim 1: 1
dim 2: 0

1 dim: You can see the inside from outside
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

dim 1: 2
dim 2: 1

2 dim: You cannot see
Jan. 12, 20187 / 32

How to count rings
How many rings/holes in the tetrahedron skelton?

Four?
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 20188 / 32

But if you see the tetrahedron from upside, the numberof rings is three.

What happened?
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 2018

9 / 32

(4)
(1)
(2)

(3)

We cosider the addition of rings. Then
(1) + (2) + (3) = (4) since two arrows with opposite
directions are vanished when added. This means that thefour rings are not linearly independent. We can formalize
the number of linearly independent rings by linear
algebra.
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201810 / 32

Persistent homology
Characterizing the shape of data is a diﬃcult
problem▶

Especially, for 3D data

Homology is one possible tool for that purpose, but
homology drops the details about the shape of data
too much▶

Homology can only count the number of holes

We want more information about the shape of datawith easy-to-use form
Computational homology is proposed in 20 century,
but it is sensitive to noise→ using increasing sequence (called filtration)
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201811 / 32

r-Ball model
large
hole

mediumholeverysmallhole

Input data is a set of points (called a point cloud)The points themselves have no “hole”, but there aresome hole-like structures
Put a disc whose radius is r onto each point
There are three holes▶

Homology can detect the number of holes

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201812 / 32

Filtration
By increasing the radii r gradually, many holes appear
and disappear. The theory of PH can makemathematically proper pairs of the radii of appearance
and disappearance.
radiusA holeappear

birthI. Obayashi (AIMR (Tohoku U.))

DividedOne hole
intodisappers
two holes
birth
death
Introduction to TDA

Another holedisappears

deathJan. 12, 201813 / 32

Persistence diagram
The pairs are called birth-death pairs. The pairs arevisualized by a scatter plot on (x, y)-plane.
radiusA holeappearbirth

DividedOne hole
intodisappers
two holes
birth
death

Another holedisappears

deathThis diagram visualizes 1-dimensional persistenthomology. This diagram is called persistence diagram.
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201814 / 32

We can apply PH to any dimensional data.
Practical for 2D and 3D
Because it is diﬃcult to understand high dimensional“holes”▶ Since it is hard to characterize the shape of 3D data, theapplication to 3D data is especially useful▶
▶

We can apply PH to various kinds of increasing
sequences
We can apply PH other than point clouds
Bitmap data
▶ PH is useful for 3D bitmap data such as X-ray CT data▶
▶

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201815 / 32

Mathematics of PH
PH relates various fields
Algebraic topology
Representation theory
Computational geometry
Combinatorics
Probability theory
Statistics
Various studies about fundamental theories are important

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201816 / 32

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201817 / 32

Amorphous Silica

What is glass?
Not liquid, not solid, but something in-between
Atomic configuration looks random
But it maintains rigidity
We require further geometric understandindgs ofatomic configurations

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201818 / 32

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201819 / 32

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201820 / 32

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201821 / 32

Combination of statistics/machine learning
Additional information

Machinelearning
・PCA・Regression
・Classiﬁcation:
Data (point clouds, images, etc.)

Characteristic
geometric
patterns
in data
Visualize

Persistence diagrams

Inverse analysis

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201822 / 32

Software

For the practical data analysis using PH, analysis
software is important.
I will introduce Homcloud.

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201823 / 32

Softwares for PH
Various analysis softwares are developed for their own
purpose and interest
Gudhi
dipha, phat, ripser
eirineRIVET
JavaPlex
Perseus
Dionysus..
.
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201824 / 32

HomcloudFocus on applications, especially to materials
science▶
▶

Data analysis for molecular dynamical simulations
Images from electric microscopy, 3D images from X-ray
CT

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201825 / 32

We can compute persistence diagrams from various
sources (point clouds, 2D/3D bitmap data)

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201826 / 32

Inverse analysis

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201827 / 32

Homcloud as a platform for thedevelopment of new methods
Getting an idea → Writing a code and trying it → Ifit works, we consider a background theory
We can quickly introduce such a new idea into dataanalysis▶

Collaborators also use the idea quickly

Try ideas found in papers by other researchers

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201828 / 32

I develop the software and analyze data together▶

Mainly data from materials science⋆▶
▶

Provided by collaborators

Dogfooding
Do not implement unused functionality▶

Collaborators also use Homcloud
Implemented mainly in python▶

Python is often used for data science

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201829 / 32

Homcloud Demo

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201830 / 32

Future plan of Homcloud
Better user interface
Performance improvement
Implement new methods▶

Parallel to theoretical researches

Publish in this winter▶

http://www.wpi-aimr.tohoku.ac.jp/hiraoka_labo/homcloud.html

If you want to use Homcloud, please contact with
us: ippei.obayashi.d8@tohoku.ac.jp

I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201831 / 32

Wrap up
Persistent homology enable us to analyze the shapeof data quantitatively and eﬀectively by using thepower of the mathematical theory of topology
A persistence diagram is a good descriptor for the shapeof data
▶ Applications to 3D data is most eﬀective, in my opinion▶

There are many applications
We mainly apply persistent homology to materials
science▶ Meteology▶ Brain science, life science, etc.
▶

Combination of theoretical researches, softwaredevelopment, and applications is important
I. Obayashi (AIMR (Tohoku U.))

Introduction to TDA

Jan. 12, 201832 / 32