The theory and applications ofpersistent homology
Ippei Obayashi
Center for Advanced Intelligence Project (AIP), RIKEN
Advanced Institute for Materials Research (AIMR), Tohoku University

Nov. 5, 2018

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Outline

123
4

Introduction
Homology and persistent homology
Applications of persistent homology
Software for persistent homology

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Introduction

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Topological Data Analysis (TDA)▶
▶

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

Connected components (islands), rings (holes), cavities

Persistent homology (PH) is one of the mostimportant tools for TDA▶
▶

Uses the concept of “homology”
Gives the good descriptor of the shape of data(persistence diagram)

Developed rapidly in 21st century
Mathematical theories and algorithms
Software▶ Applications to materials science, life science, etc.
▶
▶

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Probability - statistics and machine learning
Analysis - Fourier analysis and numerical analysis▶ Algebra - Symmetry analysis (for crystals)▶ Geometry and topology - TDA▶
▶

TDA is good for:heterogeneous data
disordered data
▶ data without complete randomness▶
▶

Mathematics and materials
Liquid and gas - random - probability theory andstatistical models▶ Crystals - ordered - group theory▶ Amorphous, polycrystalline, and porous media disordered - topology▶

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Example 1

Atomic configurations of amorphous silica and liquidsilica. Do you identify?
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From Y. Hiraoka, et al., PNAS 113(26):7035-40 (2016)

We can identify by using persistence diagram.
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Example 2

What is the characteristic diﬀerence between these twopointcloud ?
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We can distill the characteristic geometric patters by thecombination of PH and machine learning
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Homology andPersistent homology

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Homology
We can mathematically formalize “connectedcomponents”, “rings” “cavities” by homology.
Algebra is used for the formalization
We can identify the “type” of “holes” by a kind ofdimension (called degree)

dim 1: 1
dim 2: 0

dim 1: 0
dim 2: 1dim 1: 1
dim 2: 0

1 dim: You can see the inside from outside
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Theory and applications of PH

dim 1: 2
dim 2: 1

2 dim: You cannot see
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Count the rings
How many rings in this figure?

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4?

6?

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3?

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(4)
(1)
(2)

(3)

Linear algebra is the key to count the rings. Here wehave (1) + (2) + (3) = (4) since two arrows with opposite
directions are canceled. Therefore these four rings arelinearly dependent, and we can count the number oflinearly independent rings by using linear algebra.
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Persistent homology
Characterize the shape of data is diﬃcult problem▶

for 3D data or higher dimensional data.

Homology is used for that purpose, but we can only
count the number of holes
We need better way than homology
Computational homology is not robust to noise.
→ Use increasing sequences (filtrations)

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r-Ball model
largeholemediumholevery
smallhole

Input data is a set of point (a pointcloud)There is no holes in this pointcloud, but it looks likesome holes
Put discs of radii r on all points
Three holes▶

We can count the holes by homology

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Filtration
As the radius r become larger, some holes appear anddisappear. We can make pairs of appearance anddisappearance of a hole by using mathematical theory ofPH
radiusA holeappearbirth
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DividedOne hole
intodisappers
two holes
birth
death
Theory and applications of PH

Another holedisappears

death
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Persistence diagram
These pairs are called birth-death pairs. and the set of
all birth-death pairs are called persistence diagram (PD).
radiusA holeappearbirth

DividedOne hole
intodisappers
two holes
birth
death

Another holedisappearsdeath

1st persistence diagram

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PH is applicable to any dimensional data
But it is hard to intuitively understand higher
dimensional holes, 2D or 3D data is easy to analyze▶ Especially, PH is useful for 3D data▶

Various increasing sequence▶
▶

Image data
Especially 3D data, such as X-ray CT scan data

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The following two mathematical theorems are important:
Structural theorem for PH
Gives an algorithm of PDs▶ Uniqueness of a PD for a given input data▶

Stability theorem for PH▶

Ensures the robustness of a PD to noises

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Applications

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Back to Example 1
The atomic configuration of amorphous silica looks
like random▶

Similar to liquid silica

But amorphous silica has rigidity.
Some geometric structures are important for therigidity.
Y. Hiraoka, T. Nakamura, et al., Hierarchicalstructures of amorphous solids characterized bypersistent homology, PNAS 113 (26) 7035–7040,(2016)

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Back to example 2

Combination of Machine learning (ML) and PH
We have 200 pointclouds
100 pointclouds are labeled by 0, and other 100
pointclouds are labeled by 1
▶ Find characteristic geometric patterns by ML and PH▶

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Framework
Additional information

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

Characteristic
geometric
patterns
in data
Visualize

Persistence diagrams

Inverse analysis

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Each pointcloud is transformed into a PD
Vectorize PDs and apply a machine learning method
We can visualize the learned result in the form of aPD
We can identify important birth-death pairs bycomparing the learned result.
The important pairs are mapped on the original
input data by using the “inverse analysis of PDs”
Please see the demo

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Software

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Software

Software is important for practical data analysis by PH.
I introduce you HomCloud, data analysis software basedon PH.

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Various software
There are many software for PH.
Gudhi
dipha, phat, ripsereirine
RIVET
JavaPlex
Perseus
Dionysus..
.
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HomCloudFocus on applications, especially to materials
scienceMD simulation data
2D/3D image data
▶ Easy installation, user interface, machine learning,
inverse analysis▶
▶

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We can compute PDs from 2D/3D pointclouds and N
dimensional bitmap data.

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逆解析

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HomCloud Demo

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Summary
We can analyze the shape of data eﬀectively andquantitatively by using PH
Based on topology
PDs are good descriptors for the shape of data▶ Useful for 3D data▶
▶

Various applications▶
▶

Materials science
Life science, geology, etc.

The fusion of theoretical studies, softwaredevelopment, and practical data analysis isimportant.

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Appendix

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