パーシステントホモロジーと機械学習

平岡 裕章
Yasuaki Hiraoka
東北大学
原子分子材料科学高等研究機構
WPI-AIMR
JST CREST
SIP革新的構造材料
JST イノベーションハブMI^2I

位相的データ解析
位相的データ解析 (Topological Data Analysis, TDA)
今世紀に数学者が開発したデータ解析手法

Data Has Shape, Shape Has Meaning, Meaning Drives Value
Gunnar Carlsson’s Gr. (math. Stanford, AYASDI)- （ビッグ）データ解析, ソーシャルネットワーク, 医療, 金融 etc
Robert Ghrist’s Gr. (math. UPenn)- 情報ネットワーク, センサーネットワーク
Konstantin Mischaikow’s Gr. (math. Rutgers)- 流体解析, データ時系列解析
東北大AIMR, CREST, SIP, MI^2I- 材料科学（ガラス, 粉体, 高分子, 金属, 蓄電池 etc）

位相的データ解析の材料科学への応用高分子アモルファス
D1 (Aamo )CPBOCOCT

expansion

AFM
imagesilica

metallic glass

粉体

elsewhere ? ? . Our experi
mography (XCT) and threto accurately determine g(< 10 3 µm) and grain’s ddense pack⇡ 10 2 µm ? .
Figure ??(a) shows a 3D
tallized structure. The brig
deformation of octa.
ordered aggregates of bead
between different crystal
random and crystalline ph??(b) shows the disordere
Joint
with ANU of a partially crystallised packing
Helical XCT is utilised
Fig. 1 left)
3D visualisation
the packings with a spatia(M.Saadatfar
containing200,000 beads.etc)
right) MegaTwo
analyses have been carrie

無秩序系の記述言語を提案

Shape of Data
Data

point cloud, atomic configuration, digital image,sensor net work

Geometric Model

Polyhedron (Alpha, Čech, cell complex)

Algebraic Description Persistent homology, sheaf
cohomology, category4

Edelsbrunner & Mücke ‘94

Alpha shape
X = {xiRm | i = 1, . . . , n} : point cloud

Rm = [i Vi : Voronoi decomp.
[i Bi (r) = [i (Bi (r) \ Vi )

X3

X4

X2

X1

X5
X6

Alpha shape A(X, r) : dual of {Bi (r)(simplicial complex)

Nerve theorem: [i Bi (r) ' A(X, r)easier to analyze by computers

A(X, r)A(X, s) for r < s

Vi | i = 1, . . . , n}filtration:changing resolution

(ref. Edelsbrunner, Mucke)パーシステントホモロジー
データ

入力

出力

PH

次元の穴のサイズや形状の情報
パーシステント図(PD)ヘモグロビン20

パーシステントホモロジー_ (death)

1510=1

500

PD構成法

発生

1520- 発生軸は穴の発生パラメータdb10_ (birth)- PDの各点はデータ内の穴消滅

球を膨らませる

5PDd- 消滅軸は穴の消滅パラメータ- 対角線付近の点はノイズ消滅b発生- 対角線から離れた点はロバスト- 数学者Edelsbrunner, Carlsson等によって開発 (2003, 2005年)- 穴のサイズ, 形状, 階層構造などを扱える- 数学的進化と同時に，現在急速に諸科学への応用が進められている

Edelsbrunner, Letscher, Zomorodian, Carlsson, de Silva

Persistent Homology
X1X2

X4

X3

filtration X : X1

X2

persistent homology···

PH1 (X )I[3, 4]

X5

Xn

representations on An

H` (X ) : H` (X1 ) ! H` (X2 ) ! · · · ! H` (Xn )1n2interval decomposition (Gabriel’s Theorem)sMH` (X ) 'I[bi , di ]I[b, d] : 0···0i=1Kat Xb

generator···

K0at Xd···

0

Persistence Diagram
Interval decomp: Hk (X )pi=1 I[bi , di ]I[b, d] represents appearance and disappearance ofbirth time

death timea topological feature at t = b, d in X = {X(t)}tb0

Dk (X ) = {(bi , di )=dd

R2 0 : i = 1, . . . , p} : persistence diagram

deathglass

dbb : lifetime

birthliquidt

TDA on Materials Science

What is glass?
Not yet fully answered to “what is glass?”
Not liquid, not solid, but something in-between
Atomic configuration looks random, but
sufficient cohesion to maintain rigidity
Further geometric understandings of atomic
configurations are required

supercooled
liquidliquidvolumeSolar Energy Glass, DVD, BD, etc.

glass
crystaltemperature

TDA on Materials Science

Atomic configurations of silica (SiO2)
30

obtained bymoleculardynamics(MD)simulation

25
20
15
10
5025
20

25
15

20
15

10

glass

10

5500

30

1025

520

0

15

−510−105−15
10025
20

25
15

15

51020
15

1010

5500

liquid500−5−5−10crystal

Y.H., et al. PNAS (2016)

TDA on Materials Science

1 dim persistence diagrams of silica
32
1.8

2.5
1.6
1.4
2

1.5

10.8

PD1 of alphafiltrations

Multiplicity

1.2[ Å 2 ]

10.6

0
−0.5

0.4

glass

0.5

00.5

1

1.5

2

2.5

0.2
3

[ Å 2 ]
32
1.8

2.5
1.6
1.4
1.2
1.5

10.8

1
0.6
0.4

liquid0.5

0−0.5

crystal
00.5

11.5[ Å 2 ]

22.5

0.2
3

Multiplicity

2[ Å 2 ]All PD computations are
performed by HomCloudY.H., et al. PNAS (2016)

TDA on Materials Science

Support dim and order parameter
32
1.8

2.5
1.6
1.4

[ Å 2 ]

1.2
1.5

10.8

Multiplicity

20-dim support results from periodic
atomic locations of crystals

10.6

0
−0.5

0.4

crystal

0.5

00.5

11.5

22.5

transition

0.2
3[ Å 2 ]32
1.8

1-dim support (curves!) appears

2.5
1.6
1.4

[ Å 2 ]

1.2
1.5

10.8

10.6

glass

0.5

0−0.5

00.5

11.5

22.5

0.4
0.2

Multiplicity

2transition

3[ Å 2 ]2-dim support results from random
atomic locations of liquids
liquidRemark : This suggests a topological approach for glass transition.

Y.H., et al. PNAS (2016)

TDA on Materials Science

Geometric origins of curves: inverse problem
what characterizes glassstructures?

what is the geometric originof the curves?
glass PD

inverse problem!
-optimal cycle
Dey, et.al, 2011, Escolar and H, 2015

-continuation
Gameiro, Obayashi, H. Physica D, 2015

hierarchical ring structure
CP: primary rings generating the others
CO: three oxygen rings
CT: triangles on tetrahedra

BO: oxygen rings (four)Kusano, Fukumizu, Y.H. ICML (2016)

topological statistics

PD and kernel method
vectorization of PDs for statistics (persistence landscape, etc)
Reininghaus et al. proposes the persistence scale-space kernel (PSSK) on
PDs based on a heat diffusion
Our method: persistence weighted Gaussian kernel (PWGK)PWGK can explicitly control the effect of persistenceLet ⌦ be a LCH and k : ⌦ ⇥ ⌦ ! R be a positive def. kernel
Hk : reproducing kernel Hilbert space (RKHS) s.t. f (x) = hf, k(·, x)iHk
8x 2 ⌦, 8f 2 HkMb (⌦) : space of all finite signed Radon measures on ⌦k is called C0 -universal ifk(x, x) is a C0 -function (cont. & vanishing at 1 )Hk is dense in C0 (⌦)
Z
Thm: Kernel embedding of distributions(Fukumizu et al, 2011)k(·, x)dµ(x) is injective.
k : Mb (⌦) ! Hk , µ 7!

dk (µ, ⌫) = kk (µ)

k (⌫)kHk

defines a distance on Mb (⌦).

Kusano, Fukumizu, Y.H. ICML (2016)

topological statistics

Persistence weighted Gaussian kernel (PWGK)p)Birthex) warc (x) = tan 1 (Cpers(x)hD e atx2D

D e atD e athhweighted persistence diagramDXµww(x) x
D =BirthBirth

C and p explicitly control the effect of persistence in statistics.

PWGK
The Gaussian kernel kG (x, y) = ek G : Mb () ! H kG , µ w
D 7!

dw
kG (D, E) = ||wkG (µD )||x y||22 2

Z

is C0-universalkG (·, x)dµw
D (x) becomes injectivew
kG (µE )||HkG

defines a distanceStability Theorem
Let X, Y ⇢ Rd be finite subsets and p > d + 1 . Then,arcdw
kG (D(X), D(Y ))  LdH (X, Y )Kusano, Fukumizu, Y.H. ICML (2016)

topological statistics

PD detects glass transition and its geometry
PWGK
0.6
glass transition!

Glass: KPCA, PWGK

red: > 3100K
0.4

blue: < 3100K

0.2

0-0.2

from liquid to glass

3100K

1020
20-0.4

40
40

Index

60
60

80
80

-0.6-0.5

kernel Fisher discriminant

00.5

kernel PCA✦ PWGK successfully detects GT✦ What is the geometric trigger to generate the phase transition?

Inverse problem!

1

Kusano, Fukumizu, Y.H. ICML (2016)

topological statistics

PD detects glass transition and its geometry
PWGK
0.6
glass transition!

glass

Glass: KPCA, PWGK

red: > 3100K

Co starts to appear!

0.4

blue: < 3100K

0.20liquid

-0.2

3100K

1020
20-0.44040Index

60
60

80
80

-0.6-0.5

kernel Fisher discriminant

00.5

kernel PCA✦ PWGK successfully detects GT✦ What is the geometric trigger to generate the phase transition?

Inverse problem!

1

Y.H., et al. PNAS (2016)

TDA on Materials Science

Curves and constrains32

configuration space

1.8

Tx M

2.5
1.6
1.4

[ Å 2 ]

1.2
1.5

1

inadmissible
00.5

11.5

2

2.5

rigiditye0−0.5

ibl
iss

0.4
0.5

M

xm

0.6

ina

ledmissib

adadmissible

10.8

Multiplicity

2

not allowed to move in thenormal direction

constrains of the rings

0.2
32[ Å ]

O-O-O structures (MRO)
O-O-O ring constrains are
discoverednecessary to study both distanceand angle distributions
simultaneously (conventionalmethods cannot detect)