パーシステントホモロジー入門
青木利隆
神戸大学大学院人間発達環境学研究科
20/8/2024⽒名: ⻘⽊利隆 (Toshitaka Aoki)
専⾨: 多元環の表現論 ・傾理論(Tilting/Silting theory)bD(mod A).
加群圏 mod A. 導来圏<latexit sha1_base64="pSy/d3XmjVOio2Z+7JflZv6yJ7Y=">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</latexit><latexit sha1_base64="h4RjFJiLpla+WKQtrAEQWrvyibs=">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</latexit>

現在: 2023年2⽉より神⼾⼤学(エスカラ先⽣の研究室)に所属.
パーシステントホモロジー解析の表現論的な観点の研究に従事.
主にマルチパラメータ化に向けた理論構築データ記述科学
データの「かたち」と「うごき」に着⽬した記述⼦(=数学⾔語)の開発

位相的データ解析 (TDA=Topological Data Analysis)
パーシステントホモロジー解析 (2000年頃〜)
データの「かたち」=> ”⽳” の持続性(＝パーシステンス)に着⽬
情報をパーシステンス図として可視化
応⽤領域

数学領域

・材料科学
・進化⽣物学・画像解析 等々・代数/計算トポロジー
・多元環の表現論
・確率論
・機械学習 等々

Mapper解析

社会における様々な領域(⽣命科学・医療・産業・AI技術など)での取り組み

Contents
パーシステントホモロジー解析（１パラメータ）
数学的背景- ホモロジー群- ポセットの表現論・区間表現- パーシステンス図の安定性

マルチパラメータ化に向けた表現論

What’s this?

What’s this?

Let’s try Persistent Homology Analysis!

What’s this?
n 次元座標<latexit sha1_base64="12yjoZonWV5Q3m13aRfrsUOSgq4=">AAADuXichVJNLwNRFD01Pqq+2UhsRFOxal5FkNpIbCzbUppUIzPjYXS+Mu+1UY0/wJZYWJFYiJ9h4w9Y+AliSWJj4c50REqMN5l59517zn333rmaaxpCMvYU61A6u7p74r2Jvv6BwaHhkdEN4dQ8nRd1x3S8kqYKbho2L0pDmrzkely1NJNvatUV379Z554wHHtdNlxesdQ929g1dFUSlLe3h5MszYI1+dvIhEYS4co5I7E8trADBzpqsMBhQ5JtQoWgp4wMGFzCKmgS5pFlBH6OYyRIWyMWJ4ZKaJW+e3Qqh6hNZz+mCNQ63WLS65FyEin2yG7ZK3tgd+yZffwZqxnE8HNp0K61tNzdHjoZX3v/V2XRLrH/rYrMWWIXi0GuBuXuBohfhd7S148uXteyhVRzml2zF8r/ij2xe6rArr/pN3leuIyIvoM6RXXJ196Vw7a+RNUj6GwFaDTPr9uv2YrkfXVG0s3+/SLyf6rEMMN8G5Fx/fmo4iiSo/2TW3tHEjTVmZ8z/NvYmE1n5tPz+bnkcjac7zgmMIUZmuEFLGMVORQpNscpznCuLCmqsq8ctKgdsVAzhraliE+fJc32</latexit>

データ
画像データ
パーシステント
ホモロジー解析
パーシステンス図

Let’s try PH!

What’s this?
n 次元座標<latexit sha1_base64="12yjoZonWV5Q3m13aRfrsUOSgq4=">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</latexit>

データ
画像データ

逆解析

パーシステント
ホモロジー解析
パーシステンス図

Let’s try PH!

.
r

What are you doing?

What are you doing?

Let’s try PH!

点群 P ✓ RN に対し, 各点の周りで半径 r の球体を考える.
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半径 r をどんどん⼤きくしていく.
<latexit sha1_base64="rw02LOQTE5wGwGCvpONSwwhOBm8=">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</latexit><latexit sha1_base64="e3Wx7Nf/jUjAOvn7O9eX6Bt0nBg=">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</latexit><latexit sha1_base64="Qo76XvasFKXWq6ChOGZs3ayQCA8=">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</latexit>[

B(x, r)x2P“⽳”の発⽣・消滅時刻のペア(寿命)がデータの形を表わしている(と解釈).

<latexit sha1_base64="e3Wx7Nf/jUjAOvn7O9eX6Bt0nBg=">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</latexit>

半径 r = 0<latexit sha1_base64="YAlDzEx+pbYCXBzq1SbLtiuYNko=">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</latexit>r = 0.160<latexit sha1_base64="x3wTjIdCVVtwdjBO1wlIzqQ5zo8=">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</latexit>r = 0.166<latexit sha1_base64="GcAY1bBjDL20rcJhVLWOdDySQ+w=">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</latexit>r = 1.122<latexit sha1_base64="8ZWEI9fs0ZVLGGwjwIjlhYSFUTk=">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</latexit>r = 1.725<latexit sha1_base64="Z+6b2mObx/5iyh3xRxjWdN88w20=">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</latexit>

。

What are you doing?

Let’s try PH!

点群 P ✓ RN に対し, 各点の周りで半径 r の球体を考える.
<latexit sha1_base64="mY0lvKth7yUReUKrL5O9f10WEeQ=">AAAD0nichVJNSyNBEH1x3FWzH0a9CF6CwWVPoSOiiyfBiydJolHBuGF6bHXIfDndCZsEBfEm3j14UvQgon/Ci3/Agz9B9qjgZQ9bMxmRrOzYw0xXv3qvuqqmuGeZUjF2n+jSuj987OntS376/OVrf2pgcEm6Nd8QJcO1XH+F61JYpiNKylSWWPF8odvcEsu8Ohv4l+vCl6brLKqGJ9ZsfdMxN0xDVwRVUsP5dFnWuBRKbKfLtq62OG8Vd37OV1IZlmXhSr81cpGRQbTy7kCigDLW4cJADTYEHCiyLeiQ9KwiBwaPsDW0CPPJMkO/wA6SpK0RSxBDJ7RK3006rUaoQ+cgpgzVBt1i0euTMo0xdscu2CO7ZZfsgf35b6xWGCPIpUE7b2uFV+k/GF54fldl066w9aqKzVlhAz/CXE3K3QuRoAqjra83jx4XpotjrW/slP2m/E/YPbuhCpz6k3FeEMXjmOjrqFNUj3ydXfnV0Ze4eiSd7RCN5wV1BzXbsbyXzii6Obhfxv5PnRhWlG8jNm4wH1U0Yzn8ndw6O5Kkqc79O8NvjaXxbG4yO1mYyMxMR/PdixGM4jvN8BRmMIc8ShR7F2e4wrW2qDW1PW2/Te1KRJohdCzt8C8TAde3</latexit><latexit sha1_base64="rw02LOQTE5wGwGCvpONSwwhOBm8=">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</latexit>

半径 r をどんどん⼤きくしていく.
<latexit sha1_base64="rw02LOQTE5wGwGCvpONSwwhOBm8=">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</latexit><latexit sha1_base64="e3Wx7Nf/jUjAOvn7O9eX6Bt0nBg=">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</latexit><latexit sha1_base64="e3Wx7Nf/jUjAOvn7O9eX6Bt0nBg=">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</latexit>[

B(x, r)x2P“⽳”の発⽣・消滅時刻のペア(寿命)がデータの形を表わしている(と解釈).

⼊⼒

半径 r = 0<latexit sha1_base64="YAlDzEx+pbYCXBzq1SbLtiuYNko=">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</latexit><latexit sha1_base64="Qo76XvasFKXWq6ChOGZs3ayQCA8=">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</latexit>① 出⼒

r = 0.160<latexit sha1_base64="x3wTjIdCVVtwdjBO1wlIzqQ5zo8=">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</latexit>r = 0.166<latexit sha1_base64="GcAY1bBjDL20rcJhVLWOdDySQ+w=">AAADwXichVNNSyNBEH1x1I3x270IXoJB8RQqIlkRBMHLHo0aP1CRmbHVJvPFTCeYhP0De/XgYdeDggfxZ3jxD3jwJ4hHBS8erJmMSBTHHma6+nW96lfVNYZnyUAR3aU6tM6u7h/pnkxvX//A4NDwyFrgVn1TlE3Xcv0NQw+EJR1RVlJZYsPzhW4bllg3Kovh/npN+IF0nVVV98SOrR84cl+aumJo08/OZylfKBZ3h3KUp2hkPxuF2MghHkvucKqEbezBhYkqbAg4UGxb0BHws4UCCB5jO2gy5rMlo32BP8gwt8pegj10Riv8PeDVVow6vA5jBhHb5FMsfn1mZjFBt3RJj3RDV3RPL1/GakYxQi11no0WV3i7g39HV56/Zdk8Kxy+sxI1K+xjNtIqWbsXIWEWZotfa5w8rswtTzQn6ZweWP8Z3dE1Z+DUnsyLklj+lxB9DzWO6vFee1WO2uqSlE/AaztCk/3CvMOc7US/t8ooPjk8P0i8T509rFhvPTFu2B8VNBJ9jG+0tVckw11d+NjDn421af4D8sXSTG5hLu7vNMYwjinu4V9YwG8soRzV8Bj/caotalLzNL/l2pGKOT/RNrTmK8/rz8I=</latexit>×r = 1.122<latexit sha1_base64="8ZWEI9fs0ZVLGGwjwIjlhYSFUTk=">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</latexit>r = 1.725<latexit sha1_base64="Z+6b2mObx/5iyh3xRxjWdN88w20=">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</latexit>

⼤きい⽳(下部)の寿命<latexit sha1_base64="+HXoDId7bhdGBcwddybTJ0aIm3w=">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</latexit>

区間 [0.160, 1.725)

⼩さい⽳(上部)の寿命<latexit sha1_base64="CNoU1Jf+D+DVpyuYZYjOUR9tRvc=">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</latexit>

区間 [0.166, 1.122)

What are you doing?

Let’s try PH!

点群 P ✓ RN に対し, 各点の周りで半径 r の球体を考える.
<latexit sha1_base64="mY0lvKth7yUReUKrL5O9f10WEeQ=">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</latexit><latexit sha1_base64="rw02LOQTE5wGwGCvpONSwwhOBm8=">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</latexit>

半径 r をどんどん⼤きくしていく.
<latexit sha1_base64="rw02LOQTE5wGwGCvpONSwwhOBm8=">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</latexit><latexit sha1_base64="e3Wx7Nf/jUjAOvn7O9eX6Bt0nBg=">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</latexit><latexit sha1_base64="e3Wx7Nf/jUjAOvn7O9eX6Bt0nBg=">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</latexit>[

B(x, r)x2P“⽳”の発⽣・消滅時刻のペア(寿命)がデータの形を表わしている(と解釈).

⼊⼒

半径 r = 0<latexit sha1_base64="YAlDzEx+pbYCXBzq1SbLtiuYNko=">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</latexit><latexit sha1_base64="Qo76XvasFKXWq6ChOGZs3ayQCA8=">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</latexit>

出⼒

r = 0.160<latexit sha1_base64="x3wTjIdCVVtwdjBO1wlIzqQ5zo8=">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</latexit>r = 0.166
r = 0.205<latexit sha1_base64="GcAY1bBjDL20rcJhVLWOdDySQ+w=">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</latexit><latexit sha1_base64="In95saZAD1JfDIZEwDXAIXU7L2o=">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</latexit>r = 1.122
r = 0.881<latexit sha1_base64="8ZWEI9fs0ZVLGGwjwIjlhYSFUTk=">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</latexit><latexit sha1_base64="2OOiLyLZoWydCKTO16IA31XZOeg=">AAADwXichVJNT9tAEH3BQEP4DL0g9RIRgXqKxqiiERJSJC49EiCEChCyzSas4i/Zm6hJ1D/Qaw89tD0UiUPVn9ELf4ADP6HqMUhcOHTsGKGAatayd/bNvNk34zF9W4aK6Dozpo1PTL7ITuWmZ2bn5hfyi/uh1w4sUbM82wsOTCMUtnRFTUlliwM/EIZj2qJutrYif70jglB67p7q+uLYMZqubEjLUAy9DwqbBSqVy/rJQpFKFK/CU0NPjCKSte3lM1Uc4RQeLLThQMCFYtuGgZCfQ+gg+Iwdo89YwJaM/QIfkWNum6MERxiMtvjb5NNhgrp8jnKGMdviW2x+A2YWsEJX9JMGdEm/6A/d/TdXP84Raenybg65wj+Z/7S0e/ssy+Fd4eyBlapZoYFyrFWydj9GoiqsIb/T+zLY3dhZ6a/SOf1l/T/omn5zBW7nxrqoip2vKdlP0eGsPvtGu/JhpC9p9YR8dmI0PS6qO6rZSY2774zim6P7w9T/aXCEnejtpuaN5qOFXmqM+Yy20Y7keKr1xzP81NhfK+nrpfXqm2JlI5nvLF5hGa95ht+ignfYRi3u4Wd8w3dtS5OarwXD0LFMwnmJkaX1/wHdT8/G</latexit>r = 1.725<latexit sha1_base64="Z+6b2mObx/5iyh3xRxjWdN88w20=">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</latexit>材料科学における適⽤例

ヘモグロビンアモルファスシリカ(DEOXY HUMANHEMOGLOBIN)(AMORPHOUS SILICA)

出典: Protein Data
Bank Japan

出典: S. L. Roux,V.Petkov,
ISAACS ‒ Interactive
structure analysisof amorphous andcrystalline systems.
J. Appl. Cryst. 43,
181‒185 (2010).

PDBエントリーID
1A3N
標準アミノ酸を構成している
原⼦のx, y, z座標[Å]を図⽰Å: オングストローム(1Å = 0.1 ナノメートル)https://isaacs.sourceforge.io/ex.html

材料科学における適⽤例
平岡先⽣(京都⼤学)らによる
ヘモグロビン
ガラスの原⼦配置の解析(2016)(DEOXY HUMAN

PH解析の材料科学における最初期の成功例
HEMOGLOBIN)

アモルファスシリカ(AMORPHOUS SILICA)

パーシステントホモロジー解析により・中距離秩序構造の記述および出典: Protein Data
・液体とガラス状態の内部構造の違い
Bank JapanPDBエントリーIDの幾何学的特徴付けに成功.
1A3N
論⽂情報: Y. Hiraoka, T. Nakamura, A. Hirata, E. G. Escolar,
標準アミノ酸をK. Matsue, and Y. Nishiura, “Hierarchical structures
構成しているof amorphoussolids characterized by persistent homology”,
原⼦のx, y, of
z座標
Proceedings
the National Academy of Sciences.

[Å]を図⽰参考資料: ガラスの「形」を数学的に解明～トポロジーで読み解く無秩序の中の秩序～Å:オングストローム

https://www.jst.go.jp/pr/announce/20160614/index.html(1Å =0.1 ナノメートル)

出典: S. L. Roux,V.Petkov,
ISAACS ‒ Interactive
structure analysisof amorphous andcrystalline systems.
J. Appl. Cryst. 43,
181‒185 (2010).
https://isaacs.sourceforge.io/ex.html

Hmm… Their PD look quite different.

PH is not robust and unstable to outliers.

Hmm… Are they similar to each other?

Yes, they are similar by stability!

Oh… I can imagine.

Exercise.

計算ソフトウェア
GUDHI
HomCloud（逆解析の機能、国産） 今回の資料作成に使⽤
Ripser
RIVET（2パラメータの不変量計算）など他にも多数

。

ε 原等

パーシステントホモロジー解析の流れ

物体 の 配置特徴量

⼊⼒: データData<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

CT 画像
:

形を捉える
単体的複体のホモロジー
マルチスケール
で捉える

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>

F:<latexit sha1_base64="MaW40EE/LqHDhTICh6af03xZDp8=">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</latexit>

X0<latexit sha1_base64="rSySTNF50jYL6aquVPZ2aDh0uso=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">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</latexit>

X1<latexit sha1_base64="Xfqz7x57jhKAn6YUbflNt9lmENY=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">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</latexit>

X2<latexit sha1_base64="2oyBNIhdNnOZoWoHDYA/Eo8q4C0=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">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</latexit>

X3<latexit sha1_base64="zbiKDVxBn2/GiAY0ijj276Pri2A=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">AAADwXichVJLLwNRFP5qPOvNRmIjGmLV3IogVhIbS0U9gjQz47ZuOi9zbxvV+AO2FhZYkFiIn2HjD1j4CWJJYmPhzHREqjHuZOae+53znfudM8fwLCEVY8+JFq21rb2jsyvZ3dPb1z8wOLQh3bJv8pzpWq6/ZeiSW8LhOSWUxbc8n+u2YfFNo7QU+Dcr3JfCddZV1eN7tl50REGYuiJoe1eWDckVP8wPpFiahWus2chERgrRWnEHE1nsYh8uTJRhg8OBItuCDknPDjJg8AjbQ40wnywR+jlOkCRumaI4ReiEluhbpNNOhDp0DnLKkG3SLRa9PjHHMMGe2B17Y4/snr2wzz9z1cIcgZYq7Uady718/+nI2se/LJt2hYMfVqxmhQLmQ62CtHshElRh1vmV4/O3tYXVidoku2GvpP+aPbMHqsCpvJu3Wb56EZN9HxXK6pGvsStHDX2Jq0fS2Q7R+Lig7qBmOzbuuzOKbg7ul7H/U6cIK9Jbjc0bzEcJx7Exxj/aGjuSpKnO/J7hZmNjOp2ZTc9mZ1KL89F8d2IU45iiGZ7DIpaxglzYwzNc4kpb0oTmaX49tCURcYbRsLTaFw8F0Z4=</latexit>

X4<latexit sha1_base64="dh3JEfV/QhI5AWsVbCfx23FG6Tg=">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</latexit>

Persistence
Modules

フィルトレーション付き単体的複体<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="BBIaI+9fdWyliwHK33CwaCp118U=">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</latexit>

Hq (P ) ! Hq (X1 ) ! Hq (X2 ) ! Hq (X3 ) ! Hq (X4 )<latexit sha1_base64="WH5wjWSm0xE/riZ2dAGAQnGpWjQ=">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</latexit>

Hq (F ) ⇠
=

m
M
i=1

<latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

出⼒: PDV[ai ,bi ]Persistence
Diagrams

マルチスケールで形を捉える<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

パーシステンス
加群MATHEMATICAL FOUNDATIONS

単体的複体PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Definition（単体的複体）

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

V<latexit sha1_base64="eQOhTHymu1VmRN+niiRckFqX2tc=">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</latexit>

有限集合 V の部分集合の集まり K ✓ 2 が<latexit sha1_base64="B1fI7+/Ejz1WakKEvr6vfGPtQPQ=">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</latexit>• v 2 V =) {v} 2 K• ⌧ ✓ ✓ V, 2 K =) ⌧ 2 K<latexit sha1_base64="nGLkys6Sw/tjyaUf3MdQV3s9jkA=">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</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit><latexit sha1_base64="rlQUjrmHrNmK8y6hNM+ZqDV/0lQ=">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</latexit>を満たすとき, K を頂点に V を持つ（抽象的）単体的複体と呼ぶ.
<latexit sha1_base64="B1fI7+/Ejz1WakKEvr6vfGPtQPQ=">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</latexit><latexit sha1_base64="hZTdENr1RHzteGC4tI5jecmSJWU=">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</latexit>

ExamplePersistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit><latexit sha1_base64="kYG5TUjQWijIfdjEeyMqUn0/5fY=">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</latexit><latexit sha1_base64="bbHc1rE4ffGKZ/8NY4dOLTRySeA=">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</latexit>

V = {v1 , v2 , v4 , v5 },K := { 1 , . . . ,i := {vi }10 },2 = {v2 }where(vi 2 V ),3 := {v1 , v2 },10 := {v2 , v4 , v5 }, etc.

4 = {v4 }

7
10

3
9

0 -単体 頂点
1 = {v1 } 1 -単体
辺
2 -単体 ⾯<latexit sha1_base64="s5dolBA90cwoTWRTy6WyLFGGsvo=">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</latexit><latexit sha1_base64="xcu0NcqPDMrviXGbUq1frg+h26o=">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</latexit><latexit sha1_base64="j+yt33xFO8HNl7ZWJXjs0pnG4zE=">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</latexit>

68

5 = {v5 }・・・Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

ホモロジー群

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

K : 単体的複体. k : 体 (e.g., 複素数体 C , 有限体 Fp = Z/pZ )<latexit sha1_base64="mbMeoUdbA/O5lCzvUKxFmZf3KEQ=">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</latexit><latexit sha1_base64="XjEQkFIvksgaj/O3bYEHpmTecBY=">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</latexit><latexit sha1_base64="TaGylJVCuGo/tD9oNUV1YL/0628=">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</latexit><latexit sha1_base64="MFaBN9urEn/b5pF73SAr3+8HJXQ=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

鎖複体 C(K; k) = (Cq (K; k), @q )q2Z , where<latexit sha1_base64="R+Is/AnFgoKiLbnn9N+GF4yO3Go=">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</latexit>• Cq (K; k) := spank { |<latexit sha1_base64="DiBFNCAjhu1UstwEjHE00a0xsTI=">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</latexit>

2 K, # = q + 1}.

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>• @q : Cq (K; k) ! Cq 1 (K; k) s.t. @q 1 @q = 0,<latexit sha1_base64="ks5rZ7MJTMBBbNsQTqhqedK0VRA=">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</latexit><latexit sha1_base64="XLmd23ZmGbA2Sx4O+1G3qYIog8w=">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</latexit>= {v0 , . . . , vq } 7!

q
X
r=0

( 1)r {v0 , . . . , vr 1 , vr+1 , . . . , vq }.

Definition（ q 次ホモロジー群）
ExampleHq (K; k) := Ker @q /Im @q+1 .

Persistence
Modules

ホモロジーは関⼿的. Hq : Simp ! vect k.

Persistence
Diagrams<latexit sha1_base64="wCyESYZPLjQ/frmjJb1BnFRUMxw=">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</latexit><latexit sha1_base64="Fq5fHuS9VpqjFLb8980oxGEjVQo=">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</latexit><latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">AAADzXichVLLSitBED1xfEavRt0IbsINyl2FjohKVoIbdyZqVFAJM2PrHTIvpjshMepW8AdcuBAFQfEz3PgDLvwEcekFNy6snoxIlDv2MNPVp86prqopw7ctIRl7THRonV3dPb19yf6BX4NDqeGRNeFVA5OXTM/2gg1DF9y2XF6SlrT5hh9w3TFsvm5UFpR/vcYDYXnuqmz4fNvR91xr1zJ1SVA5NboleV02C4oiJHdNfpgupzIsy8KV/m7kIiODaBW84UQRW9iBBxNVOOBwIcm2oUPQs4kcGHzCttEkLCDLCv0ch0iStkosTgyd0Ap99+i0GaEunVVMEapNusWmNyBlGhPsgd2wF3bPbtkTe/tvrGYYQ+XSoN1oablfHjoZW3n9UeXQLvH3UxWbs8Qu5sJcLcrdDxFVhdnS1/ZPX1byyxPNSXbJnin/C/bI7qgCt/bPvCry5bOY6DuoUVSffO1dqbf1Ja4eQWcnRON5qm5VsxPL++iMpJvV/SL2f+rEsKN8G7Fx1XxUsB/LMX7Irb0jSZrq3NcZ/m6sTWVzM9mZ4nRmPh/Ndy/G8Rt/aIZnMY9FFFCi2A2c4xo32pJW1Q60oxa1IxFpRtG2tON3sLzWKw==</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit><latexit sha1_base64="2n6/2tPJwXalhn3aUFjbXOuacbk=">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</latexit>f : K ! L : 単体写像に対し, Hq (f ) : Hq (K) ! Hq (L) : 線型写像.
<latexit sha1_base64="DIqSR8aKlYpoE9nxcLjw0jhxJzw=">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</latexit><latexit sha1_base64="V4sa+kosCUpxGjVSJMRYWjbP7Ko=">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</latexit><latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">AAADzXichVLLSitBED1xfEavRt0IbsINyl2FjohKVoIbdyZqVFAJM2PrHTIvpjshMepW8AdcuBAFQfEz3PgDLvwEcekFNy6snoxIlDv2MNPVp86prqopw7ctIRl7THRonV3dPb19yf6BX4NDqeGRNeFVA5OXTM/2gg1DF9y2XF6SlrT5hh9w3TFsvm5UFpR/vcYDYXnuqmz4fNvR91xr1zJ1SVA5NboleV02C4oiJHdNfpgupzIsy8KV/m7kIiODaBW84UQRW9iBBxNVOOBwIcm2oUPQs4kcGHzCttEkLCDLCv0ch0iStkosTgyd0Ap99+i0GaEunVVMEapNusWmNyBlGhPsgd2wF3bPbtkTe/tvrGYYQ+XSoN1oablfHjoZW3n9UeXQLvH3UxWbs8Qu5sJcLcrdDxFVhdnS1/ZPX1byyxPNSXbJnin/C/bI7qgCt/bPvCry5bOY6DuoUVSffO1dqbf1Ja4eQWcnRON5qm5VsxPL++iMpJvV/SL2f+rEsKN8G7Fx1XxUsB/LMX7Irb0jSZrq3NcZ/m6sTWVzM9mZ4nRmPh/Ndy/G8Rt/aIZnMY9FFFCi2A2c4xo32pJW1Q60oxa1IxFpRtG2tON3sLzWKw==</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

ホモロジー群

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">AAADzXichVJNT9tAEH3BpYXQNqFcKvUSEVH1FG1QlVacqLhwawINQYIosp2FrrL+kHcTYQK9Vuof6KGHikqVivgZXPoHeuAnVD0GiQsHxo4RClGdteydffPe7Mx4LF8KpRm7yEwZD6YfPpqZzc49fvI0l59/tqW8bmDzuu1JL9i2TMWlcHldCy35th9w07Ekb1idtcjf6PFACc/9oEOfNx1z3xV7wjY1Qa38wq7mB7pfXS+8c00ZKqGOC618kZVYvArjRjkxikhW1ZvP1LCLNjzY6MIBhwtNtoQJRc8OymDwCWuiT1hAloj9HMfIkrZLLE4Mk9AOfffptJOgLp2jmCpW23SLpDcgZQFL7A87ZQP2m52xv+z6v7H6cYwol5B2a6jlfiv35fnm1USVQ7vGxztVas4ae3gb5yoodz9Goirsob53+HWwubKx1H/JfrB/lP8Ju2DnVIHbu7R/1vjGt5TobfQoqk++0a4cjPQlrR5FZydG03lR3VHNTirvtjOabo7uV6n/0ySGTPINU+NG89HBYSrHmpDbaEeyNNXl+zM8bmwtl8qVUqX2uri6ksz3DF5gEa9oht9gFeuook6xQ3zHL5wa742ucWR8GlKnMolmASPL+HwD4gzVog==</latexit>

K : 単体的複体. k : 体 (e.g., 複素数体 C , 有限体 Fp = Z/pZ )<latexit sha1_base64="mbMeoUdbA/O5lCzvUKxFmZf3KEQ=">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</latexit><latexit sha1_base64="XjEQkFIvksgaj/O3bYEHpmTecBY=">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</latexit><latexit sha1_base64="TaGylJVCuGo/tD9oNUV1YL/0628=">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</latexit><latexit sha1_base64="MFaBN9urEn/b5pF73SAr3+8HJXQ=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

鎖複体 C(K; k) = (Cq (K; k), @q )q2Z , where<latexit sha1_base64="R+Is/AnFgoKiLbnn9N+GF4yO3Go=">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</latexit>• Cq (K; k) := spank { |<latexit sha1_base64="DiBFNCAjhu1UstwEjHE00a0xsTI=">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</latexit>

2 K, # = q + 1}.

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">AAADy3ichVJNSxtRFD1xWmvTqkndFLqRhkhX4UWKLa4EQdwIRpsP0BBmxqc+8uaDeS/RJHXpxj/gQrpooRXpz+imf6CL/ITiMkI3LrwzGSlp6PiGmXffuefcd++da/lSKM1YPzVhPHo8+WTqafrZ8+mZ2Uz2RUV5rcDmZduTXlCzTMWlcHlZCy15zQ+46ViSV63mauivtnmghOd+0B2f1x3zwBX7wjY1QY1MdlfzY91bE1IHEXTSyORYgUVrftwoxkYO8dr0sqkSdrEHDzZacMDhQpMtYULRs4MiGHzC6ugRFpAlIj/HCdKkbRGLE8MktEnfAzrtxKhL5zCmitQ23SLpDUg5jzz7xa7YgP1k39lvdvvfWL0oRphLh3ZrqOV+Y/bs5fafB1UO7RqHf1WJOWvs432Uq6Dc/QgJq7CH+nb3fLC9vJXvLbAv7Jry/8z67AdV4LZv7K8lvnWREH0PbYrqk2+0K8cjfUmqR9HZidBkXlh3WLOTyLvvjKabw/tV4v80iSHjfDuJccP5aKKbyLEeyG20I2ma6uK/MzxuVBYLxaXCUultbmU5nu8pvMJrvKEZfocVrGMTZYp9hE/4hktjw1BG1/g4pE6kYs0cRpZxegeLBNWO</latexit>• @q : Cq (K; k) ! Cq 1 (K; k) s.t. @q 1 @q = 0,<latexit sha1_base64="ks5rZ7MJTMBBbNsQTqhqedK0VRA=">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</latexit><latexit sha1_base64="XLmd23ZmGbA2Sx4O+1G3qYIog8w=">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</latexit>= {v0 , . . . , vq } 7!

q
X
r=0

r

( 1) {v0 , . . . , vr 1 , vr+1 , . . . , vq }.

Definition（ q 次ホモロジー群）
ExampleHq (K; k) := Ker @q /Im @q+1 .
<latexit sha1_base64="wCyESYZPLjQ/frmjJb1BnFRUMxw=">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</latexit><latexit sha1_base64="Fq5fHuS9VpqjFLb8980oxGEjVQo=">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</latexit><latexit sha1_base64="z0YW6HqvhcpOrhlEjZEyJXCwJJc=">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</latexit>

2
7
4

10

9

1
8

5

H1
<latexit sha1_base64="ned1ab/0OnrRLbaIJJJRRc/EAaM=">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</latexit>⽳

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">AAADzXichVLLSitBED1xfEavRt0IbsINyl2FjohKVoIbdyZqVFAJM2PrHTIvpjshMepW8AdcuBAFQfEz3PgDLvwEcekFNy6snoxIlDv2MNPVp86prqopw7ctIRl7THRonV3dPb19yf6BX4NDqeGRNeFVA5OXTM/2gg1DF9y2XF6SlrT5hh9w3TFsvm5UFpR/vcYDYXnuqmz4fNvR91xr1zJ1SVA5NboleV02C4oiJHdNfpgupzIsy8KV/m7kIiODaBW84UQRW9iBBxNVOOBwIcm2oUPQs4kcGHzCttEkLCDLCv0ch0iStkosTgyd0Ap99+i0GaEunVVMEapNusWmNyBlGhPsgd2wF3bPbtkTe/tvrGYYQ+XSoN1oablfHjoZW3n9UeXQLvH3UxWbs8Qu5sJcLcrdDxFVhdnS1/ZPX1byyxPNSXbJnin/C/bI7qgCt/bPvCry5bOY6DuoUVSffO1dqbf1Ja4eQWcnRON5qm5VsxPL++iMpJvV/SL2f+rEsKN8G7Fx1XxUsB/LMX7Irb0jSZrq3NcZ/m6sTWVzM9mZ4nRmPh/Ndy/G8Rt/aIZnMY9FFFCi2A2c4xo32pJW1Q60oxa1IxFpRtG2tON3sLzWKw==</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

H2<latexit sha1_base64="0Ii33ntal/0zwf+H8UOqZPtzKR4=">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</latexit>

空洞<latexit sha1_base64="pV5a3qQBbCyZLdA5TwA/lrQ8rvk=">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</latexit>

3

6

8
>
<1
dimk Hq (K, k) = 1>:0

H0 連結成分<latexit sha1_base64="KlFU7KvFk1leVT4FlpNBpPQbUGM=">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</latexit>

q = 0,
q = 1,
q 2.

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

ホモロジー群<latexit sha1_base64="sciePH5h1LX0iiLLxhRa1TW6Xkw=">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</latexit>[

x2P

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

チェック複体B(x; r) ' |Č(P, r)| =) Hq[

x2P<latexit sha1_base64="aZJ1ttDW8mSYymqhxWznyNmn2A8=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

B(x; r) ' Hq (Č(P, r))<latexit sha1_base64="xC935B8AG1KK589Y++r4nUVhqIg=">AAAGw3icjVNLb9NAEJ4aCCU82sIFiYtFVdRKqbVOQotaVaqEkHrsgz6kpqrWm026imNb601KGuUP8Ac4cAKJA+KA+A1c+AM99AJnxLFIXDgwu3FA7sNkLa9nZ75v9ptZrxf5IlaEnIxYV65ey10fvZG/eev2nbHxibtbcdiSjG+y0A/ljkdj7ouAbyqhfL4TSU6bns+3vcZTHd9ucxmLMHiuOhHfa9J6IGqCUYWu/QlrpeLxugi6SjSOIsFUS/JevsLCUFZFQBW3pz13xqbKniYFMrN4JlTsh9wLQqUByz0XKg9YOjRVqQnf31UHgjUKqI4HihbCiDKhOvaSTZzyntEgOVM0qPtc8zFjVdLDPqsfn53VcvRcNnPJzO5fqN4F83k+HTCYkAzTEYfMP55ZtC/BFYfElYbElS/A8VpFdokz10OxphusQ4OCnXRhiTjFlOKKNLTLkcWhkaWhkeUzSB5UUz/N/vgkVmSGfd5wE2MSkrEaTox8ggpUIQQGLWgChwAU2j5QiPHZBRcIROjbgy76JFrCxDn0II/cFqI4Iih6GzjXcbWbeANc65yxYTPcxcdXItOGKXJM3pNT8oV8IN/J70tzdU0OraWDX6/P5dH+2Mv7G7/+y2riV8HBP1amZgU1eGK0CtQeGY+ugvX57aNXpxsL61PdR+Qt+YH635AT8hkrCNo/2bs1vv46I3sV2pg1wli6Ky9SfcmqJ8Z103izcbpuXXMzEzfojMKd9f5x5nlSRPiJ3k5mXv1/NOAohRn4fPx6WKf+jzqIjYwG3WOJXYmhgO8B+iLTnQL2jJvuSIMTaGmUk/B0hToWoaXPqz4EQxrFvtE84DBTHUNL65LIOczsRhdryO5t+kTzeCvds3fwvLFVdNw5Z26tPLm8kNzPUXgAD2Ea7+A8LMMKrMImMOujdWx9tb7lnuUaOZlTfag1knDuQWrken8AoMqr/Q==</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>r = 0.6

r = 0.8

dimk H0

1

1

dimk H1

10<latexit sha1_base64="mVNKurVIZ+OLB0RktkM8tX48M8I=">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</latexit><latexit sha1_base64="7rwksfqAHs6agMzkXPA1D7LNhaQ=">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</latexit><latexit sha1_base64="XSeRQ8zxybSv6R1x65xmvFVLpYg=">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</latexit><latexit sha1_base64="nls6m1NJ9VwOfcAjq0GIF/FNgCs=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

フィルトレーション

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit><latexit sha1_base64="anIa6Tg0xmXQKhBuNbX4JlAeONU=">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</latexit>def

F : R-filtration () F : R ! Top/Simp s.t. F (r) ✓ F (s) ( 8r  s)

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

Definition（sub-level set filtration）
ExampleX : 位相空間 or 単体的複体. 与えられた関数 f : X ! R に対し,X の部分空間の族 S " (f ) := (S " (f )r )r2R :<latexit sha1_base64="T5fpzHL9CFCj2o3D3h8I5njqPNU=">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</latexit><latexit sha1_base64="/Ywz48wlu4Bk1Ar28qf9huS5SW0=">AAACZHichVFNSwJBGH7cvswsLQmCICQxOskYYeFJ6NLRj/wAE9ndRltcd5fdVTDpD9S16NCpICL6GV36Ax38A0F0NOjSodd1IUqqd5iZZ555n3eemZEMVbFsxnoeYWx8YnLKO+2b8c/OBYLzCwVLb5kyz8u6qpslSbS4qmg8byu2ykuGycWmpPKi1NgZ7Bfb3LQUXduzOwavNMW6ptQUWbSJypSqwQiLMSfCoyDuggjcSOvBW+zjADpktNAEhwabsAoRFrUy4mAwiKugS5xJSHH2OY7hI22LsjhliMQ2aKzTquyyGq0HNS1HLdMpKnWTlGFE2RO7Y332yO7ZC/v4tVbXqTHw0qFZGmq5UQ2cLOXe/1U1abZx+KX607ONGrYdrwp5NxxmcAt5qG8fXfRzyWy0u8au2Sv5v2I99kA30Npv8k2GZy/how+I/3zuUVDYiMUTsURmM5JKul/hxTJWsU7vvYUUdpFGns7lOMUZzj3Pgl8ICYvDVMHjakL4FsLKJ7d2ids=</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit><latexit sha1_base64="1SgkooX8HzOkv1whgmJ5Bq8p8MM=">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</latexit><latexit sha1_base64="/Ywz48wlu4Bk1Ar28qf9huS5SW0=">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</latexit>

S " (f )r := f<latexit sha1_base64="ObBdvh295KgL042/rc1Px5fiufw=">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</latexit>1(( 1, r]) ✓ X

を f によるX のsub-level set filtrationと呼ぶ. 実際, 以下が成⽴.
<latexit sha1_base64="CwWRdiDpyteXyJtkliZPJFAgXKE=">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</latexit><latexit sha1_base64="/Ywz48wlu4Bk1Ar28qf9huS5SW0=">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</latexit><latexit sha1_base64="VgNzTvJNpvgPnmJ1f0hiA1uYtCM=">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</latexit>r  s =) S " (f )r ✓ S " (f )s .

ただし, X が単体的複体の場合は以下の条件を課すものとする.

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit><latexit sha1_base64="/Ywz48wlu4Bk1Ar28qf9huS5SW0=">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</latexit>⌧✓<latexit sha1_base64="bFClZ6/km2JCI0Jz0CHV1ozJvp0=">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</latexit>=) f (⌧ )  f ( ).

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

フィルトレーション

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Example (点を膨らませる構成はsub-level set filtrationである.)
N

N

Data
<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

有限集合 P ⇢ R に対し, X := R 上の関数 fP を次で定める.
<latexit sha1_base64="K4TbTJ9//WU/e2jzPyHYda8F7q8=">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</latexit><latexit sha1_base64="A3qD3aHENspOTjCybwl1FcPXdYs=">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</latexit><latexit sha1_base64="zSqy4wKvgDmbdbrL9NQ5Y6nS1yc=">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</latexit>f P : RN

/R

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>

22<latexit sha1_base64="gJELRCxDtFfHzzKflz1oEYTVxNo=">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</latexit>/ inf kx

y

x2P
"

yk.

<latexit sha1_base64="x1gEw1IQpzxT0YJTCji0K9x25Io=">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</latexit>

このとき, sub-level set filtration S (fP ) について以下が成⽴.
<latexit sha1_base64="7Q4RopcU+v4iiM6J5gzgT+huXo4=">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</latexit>

S " (fP )r =[x2P

B(x, r) ( ' |Č(P, r)|).

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">AAADyXichVI9axtBEH3SJbGsfEiOm4AbEeGQSqxCsI0rQRqDCUh2JBlsI+5Oa+XQ3odvV0IfqHLnP+BCVYJVGP8MN/kDKfQTQkoF3LjI3OlCUEzOe9zt7Jv3ZmfmxvCEJRVj00RSe/T4yVJqOf302fMXmezKy5p0O77Jq6YrXP/A0CUXlsOrylKCH3g+121D8LrR/hD4613uS8t1Pqm+x49tveVYJ5apK4Ia2cyR4j01/Og2O4LLUa6RzbMCC1fuvlGMjDyiVXZXEhUcoQkXJjqwweFAkS2gQ9JziCIYPMKOMSTMJ8sK/RwjpEnbIRYnhk5om74tOh1GqEPnIKYM1SbdIuj1SZnDOvvOrtiMfWPX7Ae7+2+sYRgjyKVPuzHXcq+ROX+1f/ugyqZd4fNfVWzOCifYCnO1KHcvRIIqzLm+O7iY7W/vrQ/fsK/sJ+X/hU3ZDVXgdH+ZkwrfG8dEb6JLUT3yLXalt9CXuHokne0QjecFdQc127G8P51RdHNwv4z9nzoxRJRvPzZuMB9tDGI5xgO5LXYkTVNd/HeG7xu1d4XiRmGj8j5f2o7mO4U1vMZbmuFNlLCDMqrhnI9xiYm2q51qPW0wpyYTkWYVC0s7+w1mgtRX</latexit>

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>

他にもVietoris-Rips filtrationや, super-level bi-filtrationなども
sub-level set filtrationとして構成できる.

<latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

フィルトレーション

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">AAADzXichVJNT9tAEH3BpYXQNqFcKvUSEVH1FG1QlVacqLhwawINQYIosp2FrrL+kHcTYQK9Vuof6KGHikqVivgZXPoHeuAnVD0GiQsHxo4RClGdteydffPe7Mx4LF8KpRm7yEwZD6YfPpqZzc49fvI0l59/tqW8bmDzuu1JL9i2TMWlcHldCy35th9w07Ekb1idtcjf6PFACc/9oEOfNx1z3xV7wjY1Qa38wq7mB7pfXS+8c00ZKqGOC618kZVYvArjRjkxikhW1ZvP1LCLNjzY6MIBhwtNtoQJRc8OymDwCWuiT1hAloj9HMfIkrZLLE4Mk9AOfffptJOgLp2jmCpW23SLpDcgZQFL7A87ZQP2m52xv+z6v7H6cYwol5B2a6jlfiv35fnm1USVQ7vGxztVas4ae3gb5yoodz9Goirsob53+HWwubKx1H/JfrB/lP8Ju2DnVIHbu7R/1vjGt5TobfQoqk++0a4cjPQlrR5FZydG03lR3VHNTirvtjOabo7uV6n/0ySGTPINU+NG89HBYSrHmpDbaEeyNNXl+zM8bmwtl8qVUqX2uri6ksz3DF5gEa9oht9gFeuook6xQ3zHL5wa742ucWR8GlKnMolmASPL+HwD4gzVog==</latexit>

Observation

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>f : X ! R に対し(e.g. f = fP ), sub-level set filtration<latexit sha1_base64="T5fpzHL9CFCj2o3D3h8I5njqPNU=">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</latexit><latexit sha1_base64="gQttjmpbIhZGzm0caTZczk0auI4=">AAADvXichVJNLwRBEH1rfK5vLhIXsSFOm14RRCIkLo67WCTIZmb00tmej8z0LmvjD7g4Ik4kDuJnuPgDDn6COJK4OKiZHZElRk9muvrVe9VVNWW4UviKsadEk9bc0trW3pHs7Oru6e3rH1j3nbJn8rzpSMfbNHSfS2HzvBJK8k3X47plSL5hlJYC/0aFe75w7DVVdfmOpe/ZoihMXRGUL84XC9lCX4qlWbhGfhuZyEghWlmnP5HDNnbhwEQZFjhsKLIldPj0bCEDBpewHdQI88gSoZ/jGEnSlonFiaETWqLvHp22ItSmcxDTD9Um3SLp9Ug5gjH2yG7ZK3tgd+yZffwZqxbGCHKp0m7Utdwt9J4Mrb7/q7JoV9j/VsXmrFDEbJiroNzdEAmqMOv6ytHZ6+rcylhtnF2zF8r/ij2xe6rArryZNzm+chkTfRcViuqSr7Erhw19iavHp7MVovG8oO6gZiuW99UZRTcH9/ux/1MnhozyrcbGDeajhKNYjvFPbo0dSdJUZ37O8G9jfTKdmU5P56ZSi3PRfLdjGKOYoBmewSKWkUWeYguc4hwX2oLGNanZdWpTItIMomFpB5+JXM9o</latexit>

S " (f ) = (S " (f )r )r2R where S " (f )r = f<latexit sha1_base64="w90XWfrNPbaMMyJA7Y2Z/Vp6lIk=">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</latexit>1(( 1, r])

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">AAADy3ichVJNSxtRFD1xWmvTqkndFLqRhkhX4UWKLa4EQdwIRpsP0BBmxqc+8uaDeS/RJHXpxj/gQrpooRXpz+imf6CL/ITiMkI3LrwzGSlp6PiGmXffuefcd++da/lSKM1YPzVhPHo8+WTqafrZ8+mZ2Uz2RUV5rcDmZduTXlCzTMWlcHlZCy15zQ+46ViSV63mauivtnmghOd+0B2f1x3zwBX7wjY1QY1MdlfzY91bE1IHEXTSyORYgUVrftwoxkYO8dr0sqkSdrEHDzZacMDhQpMtYULRs4MiGHzC6ugRFpAlIj/HCdKkbRGLE8MktEnfAzrtxKhL5zCmitQ23SLpDUg5jzz7xa7YgP1k39lvdvvfWL0oRphLh3ZrqOV+Y/bs5fafB1UO7RqHf1WJOWvs432Uq6Dc/QgJq7CH+nb3fLC9vJXvLbAv7Jry/8z67AdV4LZv7K8lvnWREH0PbYrqk2+0K8cjfUmqR9HZidBkXlh3WLOTyLvvjKabw/tV4v80iSHjfDuJccP5aKKbyLEeyG20I2ma6uK/MzxuVBYLxaXCUultbmU5nu8pvMJrvKEZfocVrGMTZYp9hE/4hktjw1BG1/g4pE6kYs0cRpZxegeLBNWO</latexit>を得た. 特に, 任意の増⼤列 r0 < r1 < r2 < · · · に対し<latexit sha1_base64="y42yqh2YR5t81Nh6OGVeQ/zTe60=">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</latexit><latexit sha1_base64="o7uBiDHC1L1R2dxROXMJGKVtFLw=">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</latexit>

"

◆0

◆1

"

◆2

"

S (f )r0 , ! S (f )r1 , ! S (f )r2 , ! · · · .

<latexit sha1_base64="/ZbzoyfxcULBT3HPAlrNAds1hgc=">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</latexit>"

Hq (S (f )r0 )Hq (◆0 )"! Hq (S (f )r1 )Hq (◆1 )"! Hq (S (f )r2 )Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>

Hq の関⼿性より, 次のベクトル空間と線型写像の列を得る.
<latexit sha1_base64="2ukarTXSiPvE+MSfEqpZ8pHWsQA=">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</latexit>

Hq (◆2 )<latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>! ···Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">AAADynichVI9TxtBEH3mSADnw3bSINFYsYhSWWuEDHKFlBQpUtgYAxIga+9YnJXvS7frE+bkLlX+QApoggQS4mfQ5A+k8E+IUhIpTYrMnS+KHJRjT3c7++a92Zm5MX1bKs3YODdjzD54ODe/kH/0+MnTQrH0bFt5g8ASHcuzvWDX5ErY0hUdLbUtdv1AcMe0xY7Zfx37d0IRKOm5W3roiwOH91x5JC2uCeoWi/taHOvojeS9gDtqVO4WK6zKklW+a9RSo4J0Nb1SroV9HMKDhQEcCLjQZNvgUPTsoQYGn7ADRIQFZMnELzBCnrQDYglicEL79O3RaS9FXTrHMVWitugWm96AlGUss6/sit2yL+yafWO//hsrSmLEuQxpNyda4XcLHxfbP+9VObRrvP+rysxZ4wjrSa6ScvcTJK7CmujDk0+37cbmcvSSnbPvlP9nNmY3VIEb/rAuWmLzNCP6IUKK6pNvuivHU33JqkfR2UnQbF5cd1yzk8n70xlNN8f3q8z/yYlhp/kOM+PG89HHSSbHvCe36Y7kaapr/87wXWN7pVqrV+ut1cpGI53veSzhBV7RDK9hA2/RRIdihzjDBS6Nd0ZgDI1oQp3JpZrnmFrGh9+aKdSw</latexit>

フィルトレーション

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Observation

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>f : X ! R に対し(e.g. f = fP ), sub-level set filtration<latexit sha1_base64="T5fpzHL9CFCj2o3D3h8I5njqPNU=">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</latexit><latexit sha1_base64="gQttjmpbIhZGzm0caTZczk0auI4=">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</latexit>

S " (f ) = (S " (f )r )r2R where S " (f )r = f<latexit sha1_base64="w90XWfrNPbaMMyJA7Y2Z/Vp6lIk=">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</latexit>1(( 1, r])

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>を得た. 特に, 任意の増⼤列 r0 < r1 < r2 < · · · に対し<latexit sha1_base64="y42yqh2YR5t81Nh6OGVeQ/zTe60=">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</latexit>◆0<latexit sha1_base64="o7uBiDHC1L1R2dxROXMJGKVtFLw=">AAAI1nic1VXPTxNBFH4F1IoooBejl40Egolppj2gQUlI9ODFBKhAExab7XbYbrq/3N1SSrN6IibGgxfjwZMmmhj/Ai9evJh49oBHL8YYTxi9ePCb2ZaK2BWObNOdN++97833vpl2Sp5lBiFjm6me3r5Dh4+kj/YfGzh+YnBo+ORC4NZ8nc/rruX6hZIWcMt0+HxohhYveD7X7JLFF0vVqyK+uMr9wHSdm2HD48u2ZjjmiqlrIVzFoRf5W0215mm+79aj8ZXzxaZfZJGirlVct+qbRiWUoaaqqKYbakWmKpHyD0w2UrqDsl1AuYSFcgKjqHrZDYOMUhwaYRkmH2W3kW0ZI9NnNkZ+3L3xZsYd7n1AKpXJJZ1qZBMnh0LYFmkU4LNEWWLkwbdMTfh8WKaMc4qoH9gasjgyNHireBuYLbW8DuaiZiDROlax8PWBVGiUfWAv2RZ7x16xL+xX11pNWUNwaWAsxVjuFQfvn87//C/KxhhSpYNK5BzSCl2SXE1w96RHdKHH+NX1x1v5ybnR5hh7xr6C/1O2yd6iA2f1u/58ls89SeQTYG5Lb3fl2rwFZzsxr91ZCEWFrkHifmjIsGRna9Axqa7Y3yqtJ+b8uZOdrDbSwlhCTJyWhswu0wXoV0aOLlGazHWhbEAZWT3uV8Q8WEJ9o1XbAaoucbbs2UGdJvyWxBvIFv1XgTTlLJTr+vDVUUFkllDBkDU73QlOEXTRJRsbq3LJMu52Su6VqChOfoBxHf4L25k+fDETBbZAWFIhvv0LmsK6Y1LvCFaH0RJGoUWb9bLMi7N4q7edHOXJI+VRnV3LF9S+PWkSIupiDY5Y+YApE3NXttkH+9NotrJHjTqqBHQbJ1jMDoI+HbYGsI39qcPes8/sW/yuf8p7e1KqLP+7KgfsHLVZi1X3pZH3OvootMFdmv375txtLOQy2YnMxCwu1UmKnzSdpXM0jpvzIk3TdZqhedJTA6lc6nLqSrqQvpPeSN+LU3tSLcwp2vGkH/4GbiT9Yg==</latexit>

"

◆1

"

◆2

"

S (f )r0 , ! S (f )r1 , ! S (f )r2 , ! · · · .

<latexit sha1_base64="/ZbzoyfxcULBT3HPAlrNAds1hgc=">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</latexit>"

Hq (S (f )r0 )<latexit sha1_base64="1Y6Tg+4TOSyyuYb3+nb/DtZ82lI=">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</latexit>R"

Hq (◆0 )"! Hq (S (f )r1 )Hq (◆1 )"! Hq (S (f )r2 )Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>

Hq の関⼿性より, 次のベクトル空間と線型写像の列を得る.
<latexit sha1_base64="2ukarTXSiPvE+MSfEqpZ8pHWsQA=">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</latexit>

Hq (◆2 )<latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>! ···Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>

S (f )! Top/Simp

Hq ( ;k)! vect k : functor.

<latexit sha1_base64="Ye/8g+BrTvK/83I4xwWRWvoI0pM=">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</latexit>

R - パーシステンス加群(表現).

<latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

ポセット P の表現の圏 repk (P ) := Fun(P, vect k)<latexit sha1_base64="eznTjOsMgIR30IHAis4udTEoyGo=">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</latexit><latexit sha1_base64="WGRly55Yl4oAhsJMVsNKVIA6UsE=">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</latexit>

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Objects: M = (M (x), M (x  y))x,y2P s.t.
• 各元 x 2 P に対し, M (x) は有限次元 k ベクトル空間.
<latexit sha1_base64="pXzWf9GCS5jmiVqpQvDaSvfatrg=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit><latexit sha1_base64="97M6diac0gYwyxZnKDKYizWF0DA=">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</latexit><latexit sha1_base64="HHZXyhatHpQ4/3po+BIavk5K1dI=">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</latexit><latexit sha1_base64="MyEfZj1WC+WmduGqtSAQcr0Y7co=">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</latexit>• 各 x  y に対し, M (x  y) : M (x) ! M (y) は k 線型写像.
• M (x  x) = idM (x) (8x 2 P ).
• M (y  z) M (x  y) = M (x  z) (8x  y  z) M (x) M (xy)/ M (y)<latexit sha1_base64="V0NMruz4TW8r9VQV2jGSl1kBEzI=">AAAD2HichVJNSxtRFD1xbE3TDxPdFAoyNFjMJryIqLgS3HQjGG1UaiTMjM845M2HMy/BMQhdFIou3XTRVQtdaJftP+imf6ALf0JxqdCNC++bTJEoHd8w8+47957z7r1zTV/YoWTsLDOgDT54OJR9lHv85Omz4XxhZDX02oHFa5YnvGDdNEIubJfXpC0FX/cDbjim4Gtma0H51zo8CG3PfSMjn286RtO1t23LkAQ18mOLE3t1wXf1qFRXaq5OQEmvS4+MqNTIF1mZxUu/a1QSo4hkLXmFTBV1bMGDhTYccLiQZAsYCOnZQAUMPmGb6BIWkGXHfo4D5IjbpihOEQahLfo26bSRoC6dlWYYsy26RdAbEFPHOPvNTtgF+8W+sT/s6r9a3VhD5RLRbva43G8MHz1f+Xsvy6FdYueGlZqzxDZm41xtyt2PEVWF1eN39j9erMwtj3dfsS/snPL/zM7YT6rA7VxaX6t8+VOK+hY6pOqTr78re319SasnpLMTo+lxqm5Vs5Ma968zkm5W94ep/9OgCJHkG6XqqvloYT81xrwnt/6O5GiqK7dn+K6xOlmuTJenq1PF+blkvrN4gZeYoBmewTxeYwk10n6PU3zHD+2t9k77oB32QgcyCWcUfUs7vga4TNhr</latexit><latexit sha1_base64="tWgVbHnnDpxQn+S0fdDt+Ner3nk=">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</latexit><latexit sha1_base64="MyEfZj1WC+WmduGqtSAQcr0Y7co=">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</latexit><latexit sha1_base64="JUdRGmVXeO/7Swwzz2GckD9R/3k=">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</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit><latexit sha1_base64="nabuYxitdXCdxQdJkJi5TdSEwf8=">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</latexit><latexit sha1_base64="TpjwO31r6XMhowCPB1Q8rOqIh+c=">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</latexit>

Morphisms: f = (fx )x2P : M ! N s.t.
• N (x  y) fx = fy M (x  y) (8x  y).
<latexit sha1_base64="p0uYbBIM0hoqpHHOJHGPINleHkA=">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</latexit>

fy

fx

✏
✏
N (xy)/ N (y)N (x)<latexit sha1_base64="PHuOgUoTg96ulV3Nlzv8Ip5BZ00=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

Example<latexit sha1_base64="0quWrkj5BffyphQD53jyjpQtK9I=">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</latexit><latexit sha1_base64="8jEBJSu9OuI5MnUpvndPaQQ3qRw=">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</latexit>

P = x<latexit sha1_base64="Jqpjm2A/TAwqK7NcLtU93oWYP4g=">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</latexit>




y
w




z



M :=

h

<latexit sha1_base64="qOtoXXAVEI73UP1qNYCgcsfz3EA=">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</latexit>

v,

1
1

i

k

2



10
00
01

k
1
0
1

k3

id

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>kPersistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>3[1 1 1]k.

<latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

ポセット P の表現の圏 repk (P ) := Fun(P, vect k)<latexit sha1_base64="eznTjOsMgIR30IHAis4udTEoyGo=">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</latexit><latexit sha1_base64="WGRly55Yl4oAhsJMVsNKVIA6UsE=">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</latexit>

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Objects: M = (M (x), M (x  y))x,y2P s.t.
• 各元 x 2 P に対し, M (x) は有限次元 k ベクトル空間.
<latexit sha1_base64="pXzWf9GCS5jmiVqpQvDaSvfatrg=">AAAD7HichVJNaxNBGH7S9aPGj6b1IngZDJUEyjIpUktBKHjxUkha0xaaEna303bIfrkzCdku+QOeBYXiQUFB/A9ePNg/4KE/QTxW8OLBdzcrEovbGXbmnXee55n3ffe1Q1cqzflpacq4dPnK1elr5es3bt6aqczObaqgHzmi7QRuEG3blhKu9EVbS+2K7TASlme7YsvuPU7vtwYiUjLwn+o4FLuedeDLfelYmlzdyuIae8Rqa7VhfYHR2nHFMxbX691kuBB3pM+aI9ZJpxZDnShTm2zUrVS5ybPBzhuN3KgiH81gttRCB3sI4KAPDwI+NNkuLCiaO2iAIyTfLhLyRWTJ7F5ghDJx+4QShLDI26P1gE47udenc6qpMrZDr7j0RcRkmOdf+Qd+xk/4R/6N//qvVpJppLHEtNtjrgi7M8/vbPy8kOXRrnH4l1UYs8Y+lrNYJcUeZp40C2fMHxy9PNtYWZ9P7vO3/DvF/4af8s+UgT/44bxrifXjAvU9DEg1pLvJqgwn6lKUj6Kzl3mLcWneac5eIe5PZTS9nL6vCv+nRQg3jzcu1E37o4ejQox9QWyTFSlTVzf+7eHzxuai2Vgyl1oPqqsreX9P4y7uoUY9/BCreIIm2qT9Cp/wBSeGb7wwjo3XY+hUKefcxsQw3v8GOa/fGA==</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit><latexit sha1_base64="97M6diac0gYwyxZnKDKYizWF0DA=">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</latexit><latexit sha1_base64="HHZXyhatHpQ4/3po+BIavk5K1dI=">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</latexit><latexit sha1_base64="MyEfZj1WC+WmduGqtSAQcr0Y7co=">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</latexit>• 各 x  y に対し, M (x  y) : M (x) ! M (y) は k 線型写像.
• M (x  x) = idM (x) (8x 2 P ).
• M (y  z) M (x  y) = M (x  z) (8x  y  z) M (x) M (xy)/ M (y)<latexit sha1_base64="V0NMruz4TW8r9VQV2jGSl1kBEzI=">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</latexit><latexit sha1_base64="tWgVbHnnDpxQn+S0fdDt+Ner3nk=">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</latexit><latexit sha1_base64="MyEfZj1WC+WmduGqtSAQcr0Y7co=">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</latexit><latexit sha1_base64="JUdRGmVXeO/7Swwzz2GckD9R/3k=">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</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit><latexit sha1_base64="nabuYxitdXCdxQdJkJi5TdSEwf8=">AAAD/XichVJLaxNRFP7S8VHjo2ndCG4uhkqyCTcibSkUCt10U2ha0xY6Jczc3rSX3Hk4MwmZBHHvH+jClUIXRXHrD3DjzpVgfoK4rODGhWcmoxKL0zvM3PP4vvOaY/tahRHno8KUceXqtevTN4o3b92+M1OandsJvW4gZFN42gv2bCuUWrmyGalIyz0/kJZja7lrd9YS/25PBqHy3CdR7MsDxzpyVVsJKyJTq7S2UYlNLZ+yQdUUKhBso9JP9bjKVv4ogyozWcVse4GlNeuzMYJlvlapzGs8PeyiUM+EMrKz6c0WGjBxCA8CXTiQcBGRrGEhpGcfdXD4ZDvAkGwBSSr1SzxDkbhdQklCWGTt0PeItP3M6pKexAxTtqAsmt6AmAzz/DM/4+f8I3/Dv/Kf/401TGMktcR022Ou9FszL+5t/7iU5dAd4fgvK7fmCG0spbUqqt1PLUkXYszvDU7Ot5e35ocP+Wv+jep/xUf8A3Xg9r6L04bcepkT/RA9iuqTb3Iq/Ym55PUTku6k1nxc0nfSs5OL+z2ZiDIn+cPc/2kRQmf1xrlxk/3oYJCLsS+pbXIiRdrq+r87fFHYeVSrL9QWGo/Lq8vZfk/jPh6gQju8iFWsYxNNin2GT/iCkfHcODXeGu/G0KlCxrmLiWO8/wWng+Xh</latexit><latexit sha1_base64="TpjwO31r6XMhowCPB1Q8rOqIh+c=">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</latexit>

Morphisms: f = (fx )x2P : M ! N s.t.
• N (x  y) fx = fy M (x  y) (8x  y).
<latexit sha1_base64="p0uYbBIM0hoqpHHOJHGPINleHkA=">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</latexit><latexit sha1_base64="PHuOgUoTg96ulV3Nlzv8Ip5BZ00=">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</latexit>

fx

fy

✏
✏
N (xy)/ N (y)N (x)• repk (P ) はアーベル圏.
• repk (P ) 上で直既約分解の⼀意性が成⽴(Kull-Schmidtの定理).

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit 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sha1_base64="PbKsCenAp/UYZtbRtXcQP8j8Ceg=">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</latexit>• P が有限集合のとき, repk (P ) は隣接代数 k[P ] の加群圏と圏同値.
<latexit sha1_base64="sIjHFMt930p9fPacBAlGBE4zER4=">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</latexit>

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

ポセット P の表現の圏 repk (P ) := Fun(P, vect k)<latexit sha1_base64="eznTjOsMgIR30IHAis4udTEoyGo=">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</latexit><latexit sha1_base64="WGRly55Yl4oAhsJMVsNKVIA6UsE=">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</latexit>

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Definition（区間）Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

P の部分集合 I が区間であるとは, 次を満たすときをいう.
• (凸性) 8 x, y 2 I, 8 z 2 P, x < z < y =) z 2 I.
Ie
• (連結性) 8 x, y 2 I, 9 x = z0 , z1 , . . . , zm 1 , zm = y s.t.
zj < zj+1 or zj > zj+1 (0  j  m 1).
<latexit sha1_base64="sIjHFMt930p9fPacBAlGBE4zER4=">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</latexit><latexit sha1_base64="EvO0cSXomchaW6Bme8gSPLxYvhI=">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</latexit><latexit sha1_base64="oFQptEz8p66BxmygWScf/fLjGtU=">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</latexit><latexit sha1_base64="Wn22AwNxJqrcKjZ/GweicgHhvJ8=">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</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit><latexit sha1_base64="8Og6ZERAx0jy6udktFmoMpm2c/c=">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</latexit>

Definition（区間表現）
Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>

P の区間 I に対し, 区間表現 VI 2 repk (P ) を以下で定義する.
<latexit sha1_base64="CcZrGgh7rQ2UndpmP0gXR27GnuU=">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</latexit><latexit sha1_base64="sIjHFMt930p9fPacBAlGBE4zER4=">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</latexit><latexit sha1_base64="EvO0cSXomchaW6Bme8gSPLxYvhI=">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</latexit><latexit sha1_base64="g/u/AHRzQ23p7aRUJsLPUW3aQrI=">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</latexit>

VI (x) =(k0

if x 2 I,
and VI (x  y) =otherwise,

(

idk
0

if x, y 2 I,otherwise.

• VI は直既約表現である. 実際, End(VI ) ⇠= k が成⽴.
• VI の部分表現や商表現は区間表現の直和である.
<latexit sha1_base64="zjNKtxbNTWVD0JZOxyNJGQIP3jM=">AAAISXic1VU7aBRRFL356Mb4SaKNn2YwJEQIy1uLKIFAQAUthCRrPpANy8zsy+yw83NmknWzLKlsxFYsrBRSiKC1jY2NkNoiljYiYiERFbHwvDu72cSQNSkzy86777577j33vLf7jMCxo1iIjbb2js4jR1Ndx7qPnzh5qqe37/RM5C+Fppw2fccP5ww9ko7tyenYjh05F4RSdw1Hzhqla2p9dlmGke17d+JKIBdc3fLsRdvUY7jyvedyrh4XQ7d6wyvUhmbyty5pOdP3LK2U7+0XacGPttvI1I3+8fP3+3+s3n4z4fd1PKQcFcgnk5bIJUkexbAd0inCZ54yJCiAb4Gq8IWwbF6XVKNuYJcQJRGhw1vC28Jsvu71MFc5I0abqOLgGwKp0YB4L56LTfFOvBCfxZ89c1U5h+JSwWgkWBnkex6czf78L8rFGFOxiWrJOaZFuspcbXAP2KO6MBP88srjzezo1EB1UDwTX8D/qdgQb9GBt/zdXJuUU09a8okwd9m7t3IN3oqz2zKu0VkMRZWuUcv90BHhcGf3oGOrvGp/S7TSMmb7TjajGkgHo4E1dVoqHF2gYehXQIzJKJ1jfSgbUZqzJ/2qtQCWUt+q5/aAKjPO5Z495KnC7zDeQrTqvwSkzbOY64bwlZFBRRrIYHHOZneKUw26mMzGRVXJLJNux3ivVEZ18iOMK/APb0WG8CVMNNgK4bBCcusXNIa6g6x3DVaT0TxGpUWD9QLHJVGy3ttOjnzySHtUFtezc7nOfWkSY9VHDYm1wiFTJuGubbGPDqbRZHGfGjVVieguTrCaHQZ9mmwtYCsHU0esi0/ia/Iuf8wG+1KqwP9dxUN2jhqsVdUDaRS8rn1Q2uAuzfx7c+42Zi6nMyPpkUlcqqOUPF10gS7SEG7OKzRON2mCplFlldboJb1Krae+pX6lfieh7W11zBna8XR1/AUc9MqO</latexit><latexit sha1_base64="FpO29I3R6czqtRBv68E//fftdq8=">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</latexit><latexit sha1_base64="FpO29I3R6czqtRBv68E//fftdq8=">AAAIMnic1VVLaxNRFD59aGp8tNWNIEiwtLgo4cZFlUKhoAvdtY19QBvCzOQ2GTIvZyaJScgfEBE3UsGVgguR7ty4E9wIrrtod+pCRF1VcOPC755JmtbSmCw7IXPPPfd853znuze5umeZQSjEdl//wOCJk7GhU/HTZ86eGx4ZPb8UuCXfkIuGa7n+iq4F0jIduRiaoSVXPF9qtm7JZb14U60vl6UfmK5zN6x6MmNrecdcNw0tVK6l7J14dmRMJAU/icNGqmmMzY5c3tx6/G5zzh0deEhrlCOXDCqRTZIcCmFbpFGAzyqlSJAHX4bq8PmwTF6X1KA4sCVESURo8BbxzmO22vQ6mKucAaMNVLHw9YFM0LjYEq/ErvggXotv4s+RueqcQ3GpYtQjrPSyww8upn//F2VjDKnQRnXkHNI63WCuJrh77FFdGBG+XNvYTU8vjNcnxAvxHfyfi23xHh045V/Gy3m58KwjnwBzm71HK9firTjbHeNanYVQVOkadNwPDREWd3YfOnbKq/a3SLWOMft3sh3VQloYdayp01Ll6BxNQr8cYgxGaRzrQtmAkpw96letebCU+vlmbgeoCuNs7tlBnjr8FuPziFb9F4E0eRZyXR++CjKoSB0Z8pyz3Z3i1IAuBrOxUVUyy6jbGd4rlVGd/ABjDf7JvUgfvohJArZCWKyQ3PsFzaDuBOvdgNVmtIpRadFineG4KEo2ezvIkU8eJZ5UxK30ytpgV5qEWHVRQ2Itd8yUibgn9tgHvWk0X+hSo7YqAd3DCVaz46BPm20e2Gpv6oiP4qv4Gb0rn9NeV0rl+L+rcMzOUYu1qtqTRt7bxo7SBndp6t+b87CxdC2ZmkpOzeNSnaboGaJLdIWu4ua8TrN0m+ZoEVUK9Ig26GnsTWwn9in2JQrt72tiLtCBJ/bjL2xLwww=</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">AAADyXichVI9axtBEH3SJbGsfEiOm4AbEeGQSqxCsI0rQRqDCUh2JBlsI+5Oa+XQ3odvV0IfqHLnP+BCVYJVGP8MN/kDKfQTQkoF3LjI3OlCUEzOe9zt7Jv3ZmfmxvCEJRVj00RSe/T4yVJqOf302fMXmezKy5p0O77Jq6YrXP/A0CUXlsOrylKCH3g+121D8LrR/hD4613uS8t1Pqm+x49tveVYJ5apK4Ia2cyR4j01/Og2O4LLUa6RzbMCC1fuvlGMjDyiVXZXEhUcoQkXJjqwweFAkS2gQ9JziCIYPMKOMSTMJ8sK/RwjpEnbIRYnhk5om74tOh1GqEPnIKYM1SbdIuj1SZnDOvvOrtiMfWPX7Ae7+2+sYRgjyKVPuzHXcq+ROX+1f/ugyqZd4fNfVWzOCifYCnO1KHcvRIIqzLm+O7iY7W/vrQ/fsK/sJ+X/hU3ZDVXgdH+ZkwrfG8dEb6JLUT3yLXalt9CXuHokne0QjecFdQc127G8P51RdHNwv4z9nzoxRJRvPzZuMB9tDGI5xgO5LXYkTVNd/HeG7xu1d4XiRmGj8j5f2o7mO4U1vMZbmuFNlLCDMqrhnI9xiYm2q51qPW0wpyYTkWYVC0s7+w1mgtRX</latexit>

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

<latexit sha1_base64="C3H/nc+zD7yMTNZIZrRBEYGulQI=">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</latexit>

P = (R, ) の表現の圏 repk (R)<latexit sha1_base64="NS0RpLndsu2F4fX5x4to+54xD1c=">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</latexit>

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Theorem[Crawley-Boevey ʻ15] (cf. Gabrielʼs thm for An )repk (R) の任意の直既約表現は区間表現である. したがって,
任意の表現 M 2 repk (R) は以下の形の直和に同型となる.
<latexit sha1_base64="gD4MV8KW+LdF+UwTFIjXn8O6S8w=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit><latexit sha1_base64="9KW2knOgGwhvO8anJdWx5fEQa6U=">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</latexit><latexit sha1_base64="oMfdVCFR/ZB2HeWZTizpbYDJLGE=">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</latexit>M⇠=<latexit sha1_base64="7HqN0SVdzav//cLptwr2lIdBVcc=">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</latexit>

M

VI .

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>

I2B(M )(R, ) 上の区間は以下の形のいずれか ( I = ha, biで表わす).
<latexit sha1_base64="B/Y9XHij8IqxSLDdHx9YlZGKQQM=">AAADzHichVJNa9tAEH22mtZ1ksZ2L4VeTE1KAsGsQ0hCToFeciqxEzuB2BhJXtvCq49KaxNb+NpD/0AOgUALpS39Gb30D+SQn1B6dCGXHjqSVYproqyQdvbNe7Mzo9EcYXiSsZtEUnmw8PBR6nF6cWn5yUomm6t5dt/VeVW3he2eaqrHhWHxqjSk4KeOy1VTE/xE670K/CcD7nqGbR3LocMbptqxjLahq5KgZia3VjdV2dU0vzLeqAv+Zj3fzBRYkYUrP2+UIqOAaB3a2UQZdbRgQ0cfJjgsSLIFVHj0nKEEBoewBnzCXLKM0M8xRpq0fWJxYqiE9ujbodNZhFp0DmJ6oVqnWwS9LinzWGXX7AubsO/sK/vBft8Zyw9jBLkMademWu40V949O7q9V2XSLtH9p4rNWaKN3TBXg3J3QiSoQp/qB6OLydFeZdV/yT6wn5T/e3bDvlEF1uCX/rHMK5cx0VsYUFSHfLNdOZ/pS1w9Hp3NEI3nBXUHNZuxvL+dkXRzcL8X+z9VYogo32Fs3GA+ehjFcrR7cpvtSJqmuvT/DM8btc1iabu4Xd4q7O9F853Cc7zAGs3wDvZxgENUKfY5rvAJn5XXilR8ZTylJhOR5ilmlvL2D7Kc1LI=</latexit><latexit sha1_base64="ebnzOuvLCvKR6QUjxP7Zb1ZNALc=">AAAD0HichVJNSxtRFD3JtNWmWqNCKbgZGixdSHgpRSVQENy0uyQ2KhgJbybPOOTNBzMvwSSV0q1/wEVXDXQh+i/c9A90kZ9QuozgxoV3JiMllY5vmHn3nXvOfffeuYYnrUAxNkyltUePn0xNP808m5l9PpedX9gO3LZviqrpStffNXggpOWIqrKUFLueL7htSLFjtDZD/05H+IHlOp9U1xP7Nm861oFlckVQPfvio/5er0nuNKXQ+YpR8yOzns2xPIuWft8oxEYO8Sq586kyamjAhYk2bAg4UGRLcAT07KEABo+wffQJ88myIr/AMTKkbRNLEIMT2qJvk057MerQOYwZRGqTbpH0+qTUscx+sTM2Yj/ZOfvNbv4bqx/FCHPp0m6MtcKrz5283Lp+UGXTrnD4V5WYs8IB1qNcLcrdi5CwCnOs7/ROR1vFynL/NRuwP5T/dzZkl1SB07kyf5RF5VtC9AY6FNUj32RXjib6klRPQGc7QpN5Yd1hzXYi764zim4O7w8S/ycnhozz7SbGDeejhV4ix3ggt8mOZGiqC//O8H1j+22+sJpfLb/LbRTj+Z7GEl7hDc3wGjbwASVUKfZnDHCOC62iHWlftK9jajoVaxYxsbSTWz9g1gQ=</latexit>[a, b], (a, b], [a, b), (a, b), [a, 1), (a, 1), ( 1, b], ( 1, b).
<latexit sha1_base64="U3W587Dt29Z3v3m5m+EnL13vgyI=">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</latexit>

Definition（パーシステンス図）

R := R t {±1} とおく. R 2 上の多重集合

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit><latexit sha1_base64="na1yoJbiwky/gZ2p9wzlbPTAsVY=">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</latexit><latexit sha1_base64="bUAQ17wgiYo6D2U3gd4TG44EV1E=">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</latexit><latexit sha1_base64="fv1yjLojzlxjZBBFGWRc0mqxo3c=">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</latexit>

2

D(M ) := {(a, b) 2 R | I = ha, bi 2 B(M )}を表現 M のパーシステンス図と呼ぶ.
<latexit sha1_base64="nkrrng7/fSAfzaCUtbe3kqp/CcM=">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</latexit>

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

<latexit sha1_base64="C3H/nc+zD7yMTNZIZrRBEYGulQI=">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</latexit>

P = (R, ) の表現の圏 repk (R)<latexit sha1_base64="NS0RpLndsu2F4fX5x4to+54xD1c=">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</latexit>

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Example (over k := F2 . )<latexit sha1_base64="PTwHN1m0eniUngq+2XFz61ajDSw=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit><latexit sha1_base64="7MaFQl2ehtL+1U0QKzto0jCfb2Q=">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</latexit>

2

f : K ! R, f ( t ) = t (1  t  10).
<latexit sha1_base64="rSHrjlRsGZbB+K4CFNVMwSGo3YA=">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</latexit>

7

K :=

3

<latexit sha1_base64="NpIpTakUk2NWqEwHQhPM60n2W64=">AAADvXichVLLLgRBFD2jPcdjPDYSm4kJsZrUiCASIbGR2BgMEkS6W6Ey1Y901wxj4gdsLBErEgvxGTZ+wMIniCWJjYXbPS0yRKtOd90695xb996+hiuFrxh7SjRojU3NLa1tyfaOzq5Ud0/vqu+UPJMXTEc63rqh+1wKmxeUUJKvux7XLUPyNaM4F/jXytzzhWOvqIrLtyx9zxa7wtQVQYWF9NR0ers7w7IsXOnfRi4yMojWotOTyGMTO3BgogQLHDYU2RI6fHo2kAODS9gWqoR5ZInQz3GMJGlLxOLE0Akt0nePThsRatM5iOmHapNukfR6pExjiD2yW/bKHtgde2Yff8aqhjGCXCq0GzUtd7dTJ/3L7/+qLNoV9r9VsTkr7GIyzFVQ7m6IBFWYNX356Ox1eWppqDrMrtkL5X/Fntg9VWCX38ybPF+6jIm+gzJFdclX35XDur7E1ePT2QrReF5Qd1CzFcv76oyim4P7/dj/qRNDRvlWYuMG81HEUSzH+Ce3+o4kaapzP2f4t7E6ms2NZ8fzY5nZyWi+WzGAQYzQDE9gFvNYRIFiC5ziHBfajMY1qdk1akMi0vShbmkHnyj5zrA=</latexit>

4

10

9

6

1

<latexit sha1_base64="5C+MuQ9dtCwr28Vr1czJPQR+sI8=">AAAD+3ichVJLb9NAEJ7UPEp4NIULEpcVUVG5RJsKlQoJqYgLEpemIW2lbrHWziZdZf1gdxNILf8B/gAHToAQ4nHlF3DhwhFQfwLiWCQuHBg7RihUuGvZO/vN983OjMeLlTSW0v3KjHPs+ImTs6eqp8+cPTdXmz+/YaKh9kXHj1SktzxuhJKh6FhpldiKteCBp8SmN7id+TdHQhsZhffsOBY7Ae+Hsid9bhFya7dYwO2uz1XSTu8nbBhzraOH6WLvqqvJTcISZmQ/4G5iU8JkSO4SFsgusUyJB0SztEHcWp02aL7IYaNZGHUo1lo0X2kBgy5E4MMQAhAQgkVbAQeDzzY0gUKM2A4kiGm0ZO4XkEIVtUNkCWRwRAf47eNpu0BDPGcxTa728RaFr0YlgQX6mb6mB/QjfUu/0V//jZXkMbJcxrh7E62I3bnHF9s/j1QFuFvY/asqzdlCD1byXCXmHudIVoU/0Y/2nhy0b6wvJFfoc/od839G9+kHrCAc/fBftsT605LoXRhh1Bh90115NNWXsnoMnoMcLedldWc1B6W8P52xeHN2vyn9nxwZqsh3XBo3m48B7JVyvCNym+5IFae6+e8MHzY2lhrN5cZy61p9daWY71m4BJdhEWf4OqzCHViDDsZ+BZ/gC3x1UueF88Z5N6HOVArNBZhazvvfxEHn7Q==</latexit>

8
5S " (f )r = { t 2 K | t  r}.

<latexit sha1_base64="hR0C8vXEWM6qZE+UPGy6bVpTN0g=">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</latexit>(8  r < 9)<latexit sha1_base64="vKkMR7aFlg31fk0p7oG2HEqw2Lc=">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</latexit>(r < 8)2

S " (f )r(9  r < 10)<latexit sha1_base64="3gn9vpOptFUagwlEWS3CVoybFjw=">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</latexit>(10  r)<latexit sha1_base64="Tk5HSjtiWI1BbI0zQugRT+x3usY=">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</latexit>22

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>

2<latexit sha1_base64="i41HTG22cycJzI3c9mPCr6jziC0=">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</latexit>

7

3

4

7
1

4

6

7

3
4

1
6

3
9

6

8

7
1

4

8

3

10

9

1

6

8

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>

5"<latexit sha1_base64="7dhfNV0SFMXfaw0RN+/xzSVs6dg=">AAAEm3icjVNNaxNBGH5TV1vjR9N6EUSIhkoKJUxEWump0EsRD21j2kJTw+52kiyZ3R1mN6npkrPgH/DgScGDiAcv+gO8+Ac89CeIxwpePPjMZIPE4uqEzbz7zvO887zP7DhSeFHM2Elu6px1/sL0zMX8pctXrs4W5uZ3orCnXF53QxGqPceOuPACXo+9WPA9qbjtO4LvOt11vb7b5yrywuBRPJD8wLfbgdfyXDtGqlm4tdGslhu+HXdcWyS14eOk0ZO2UuHRsNxabKrFYrNQYhVmRvFsUE2DEqVjM5zLvacGHVJILvXIJ04BxYgF2RTht09VYiSRO6AEOYXIM+uchpQHtwcUB8JGtov/Nt7202yAd10zMmwXuwg8CswiLbAv7A07ZZ/ZW/aV/fxrrcTU0FoGmJ0Rl8vm7LPrtR//ZPmYY+r8ZmVqjqlF941WD9qlyegu3BG/f/z8tLa6vZDcYa/YN+h/yU7YJ3QQ9L+7r7f49ouM6ofUR1WJtUlXnkz4ktVPhHffZLNxum/ds5+JGzsTY2e9f5R5njYQItU7yKyrv48uHU9gxjmB2UGf+jsaACuNBu2xgisRLeHpICeNO0vwjBt3lMF5iDSqkvJ0h3pNItLn1f4PhjKKhdE85rimOxeR1qXAOcp0I0EP2d5Onmget7L65x08G+zcrVSXK8tb90prq+n9nKEbdJvKuIMrtEYbtEl11H5K7+gDfbRuWuvWA+vhCDqVSznXaGJY9V9swACq</latexit>[z :=<latexit sha1_base64="jL+FKp2JsfFOq9lDXln70JvSw+0=">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</latexit>

;

の基底

3+

8+

H1 (S (f ))6+<latexit sha1_base64="tKVVIrlOhWdpy0Q9iijy+a+NmEA=">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</latexit>

[z]

7]

[z]

<latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit><latexit sha1_base64="NkhUtjniLXAHr49FH4GyTaYJljM=">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</latexit>[ 6+<latexit sha1_base64="vjHObvfKPFGytfOvmMUxLdTBgos=">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</latexit>"

5<latexit sha1_base64="tKVVIrlOhWdpy0Q9iijy+a+NmEA=">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</latexit><latexit sha1_base64="YcJ9gZ5x1YXii+YzZjaIVftqRak=">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</latexit>

H1 (S (f )r )<latexit sha1_base64="0M0p9Z2zZ4AlZ3b2AT0C/EV5kdI=">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</latexit>

5

5

··· 0
⇠
= V
<latexit sha1_base64="ivM4G8KgEZyU84K/mKjMpM4e5lE=">AAAE1HicjVO/bxMxFH4pB5TwoyksSCwn0iKQoshBqJROlVgY+4OkFbkqurs6qRXf2To7ocmRCbEwsDIwgcSAmPgPkFj4Bxj6DyAhBpCKxMLAs3MRClUPHF387vn7nr/3+RxIzpQm5KAwc8I5eer07Jni2XPnL8yV5i82lOglIa2HgotkO/AV5Symdc00p9syoX4UcLoVdO+a9a0+TRQT8X09kHQn8jsxa7PQ15hqlR54oYg7rldxG63U03Rfp15bCB0LTRUb0oXmcsVjcVsPbiyMRq7nekLynvLcY/F3KjVisNVWqUyqxA73aFDLgjJkY03MF96BB7sgIIQeREAhBo0xBx8U/ppQAwISczuQYi7BiNl1CiMoIreHKIoIH7Nd/O/gWzPLxvhuairLDnEXjk+CTBcWySfyhhySj+Qt+UJ+HVsrtTWMlgHOwZhLZWvu6eXNn/9kRThr2PvDytWsoQ3LVitD7dJmTBfhmN8fPj/cXNlYTK+RV+Qr6n9JDsgH7CDu/whfr9ONFznVd6GPVSWuTbuyP+VLXj8K3yObzceZvk3PUS5u4ozGnc3+Kvc8fUTwTO8gt675PrownMJMchznAPs039EAsdJqMB4n6IqCCj57mJPWnQp6Rq07icUxjAyqmvFMh2ZNYmTOq/MfjMQq5lbzhBPa7kKMjK4EOQ9z3Uixh3xvp0+0iLey9vcdPBo0blZrS9Wl9Vvl1ZXsfs7CFbgK1/EO3oZVuAdrUMfa7+EzfIPvTsN55Dx2noyhM4WMcwmmhvPsNzjnF24=</latexit>[8, 1)

0k
V[9, 10) .

[ 10 ]

7+

k2

9]

[1 0]

k

idk

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>···<latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

<latexit sha1_base64="C3H/nc+zD7yMTNZIZrRBEYGulQI=">AAAD5XichVLPSxtBFP7iaquxrbFeBC9isFgKYSJiS0/BXjya2KhgJMyuY7Jk9ge7k9C45B8o9FAo9dBDUfBQ+gd4VXrxH/DgnyAeLfTSQ99OVkoqXWfZnffevO+b7719pi/tUDF2mRkyhkcePBwdy44/evxkIjf5dCP02oElqpYnvWDL5KGQtiuqylZSbPmB4I4pxabZehOfb3ZEENqe+1Z1fbHj8IZr79kWVxSq517UNEdU8bpcrsi26NUcrpqBEwXC79VbC9o1zajSe17P5VmB6TV71ygmRh7JWvMmM2XUsAsPFtpwIOBCkS3BEdKzjSIYfIrtIKJYQJatzwV6yBK2TVmCMjhFW/RtkLedRF3yY85Qoy26RdIbEHIW8+yCfWM37Jx9Z1fs93+5Is0Ra+nSbvaxwq9PvJ9e/3UvyqFdofkXlapZYQ+vtFabtPs6Eldh9fGd/YOb9deV+egZO2LXpP+QXbIfVIHb+Wkdl0XlSwr7LjrE6tPZYFfeDfQlrZ6QfEdH0/PiuuOandS8284oujm+P0z9n5wyZKK3m8obz0cL+6k55j3aBjuSpaku/jvDd42NxUJxubBcXsqXSsl8j2IGc1igGX6JElaxhipxf8QJTnFmNIwPxifjcz91KJNgpjCwjK9/AH0i4Lg=</latexit>

P = (R, ) の表現の圏 repk (R)<latexit sha1_base64="NS0RpLndsu2F4fX5x4to+54xD1c=">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</latexit>

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

パーシステンス図を得るためには表現の直既約分解の計算が必要…

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>というわけではない!
実⽤上, 効率のよい計算アルゴリズムが既に実装されている.

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

e.g. 単体的複体の列から直接パーシステンス図が計算できる:
⾏列演算による標準アルゴリズムでは O(n3 ) . ( n は単体の個数)<latexit sha1_base64="/9kH/CjGtHKU60Oo7aYXQierT9E=">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</latexit><latexit sha1_base64="Tvy2DAnxVzzFxQtmnbBRTsk8teA=">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</latexit><latexit sha1_base64="bzEIIMC848SDlhRyKyNdF/MiqjQ=">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</latexit>!

cf. O(n ) for ! := matrix multiplication time < 2.371552.

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>パーシステントホモロジー解析の流れ(再訪)P ✓ RN .
X : Top/Simp.
<latexit sha1_base64="eFffrOvxDHy4XzizZFk3DyZG2Bs=">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</latexit>

⼊⼒:データ<latexit sha1_base64="+DE5GdpGFwTYoksF0/BbeclEoT4=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

fP : RN ! R.
<latexit sha1_base64="R8EfG85OlkIBsN9sI4nYsPHLtFU=">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</latexit>f : X ! R.
<latexit sha1_base64="EbF4vp3sIemb49IcP/nXr6qjCsg=">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</latexit>

S " (f ) = (S " (f )r )r2R :<latexit sha1_base64="d23q78ORUgh+5pzq/V2kDqwvCMg=">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</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>

F : X0<latexit sha1_base64="MaW40EE/LqHDhTICh6af03xZDp8=">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</latexit><latexit sha1_base64="rSySTNF50jYL6aquVPZ2aDh0uso=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">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</latexit>

X1<latexit sha1_base64="Xfqz7x57jhKAn6YUbflNt9lmENY=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">AAADwXichVJLLwNRFP5qPOvNRmIjGmLV3IogVhIbS0U9gjQz47ZuOi9zbxvV+AO2FhZYkFiIn2HjD1j4CWJJYmPhzHREqjHuZOae+53znfudM8fwLCEVY8+JFq21rb2jsyvZ3dPb1z8wOLQh3bJv8pzpWq6/ZeiSW8LhOSWUxbc8n+u2YfFNo7QU+Dcr3JfCddZV1eN7tl50REGYuiJoe1eWDckVP8wPpFiahWus2chERgrRWnEHE1nsYh8uTJRhg8OBItuCDknPDjJg8AjbQ40wnywR+jlOkCRumaI4ReiEluhbpNNOhDp0DnLKkG3SLRa9PjHHMMGe2B17Y4/snr2wzz9z1cIcgZYq7Uady718/+nI2se/LJt2hYMfVqxmhQLmQ62CtHshElRh1vmV4/O3tYXVidoku2GvpP+aPbMHqsCpvJu3Wb56EZN9HxXK6pGvsStHDX2Jq0fS2Q7R+Lig7qBmOzbuuzOKbg7ul7H/U6cIK9Jbjc0bzEcJx7Exxj/aGjuSpKnO/J7hZmNjOp2ZTc9mZ1KL89F8d2IU45iiGZ7DIpaxglzYwzNc4kpb0oTmaX49tCURcYbRsLTaFw8F0Z4=</latexit>

X2<latexit sha1_base64="2oyBNIhdNnOZoWoHDYA/Eo8q4C0=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">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</latexit>

X3<latexit sha1_base64="zbiKDVxBn2/GiAY0ijj276Pri2A=">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</latexit>✓<latexit sha1_base64="/i6FLFZNhsGugZZVOmiJZFcXFkE=">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</latexit>

Sub-level set filt.

X4<latexit sha1_base64="dh3JEfV/QhI5AWsVbCfx23FG6Tg=">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</latexit>

Vf := Hq (S " (f )).
<latexit sha1_base64="J1INCfCjPUrqVdf8H4j0N5oYwzA=">AAAD4XichVI9bxNBEH3OAQnmI05okGhOWAGnsdYIJZElpEg0KeMYO5HicNrbrJOT9z5yt3ZwTv4BICEhGopUIFEgSlqoaPgDFPkJiDJINBTMnQ8hE3HZ093OvnlvdmZu7EA5kWbspDBlXLh4aXrmcvHK1WvXZ0tz8+3I74dCtoSv/HDL5pFUjidb2tFKbgWh5K6t5Kbde5j4NwcyjBzfe6SHgdxx+Z7ndB3BNUFW6W7b6pr1B+aadVDpuFzvC67i5uhx3OkHPAz9w1Glu7hYNa1SmVVZusyzRi0zysjWuj9XaKCDXfgQ6MOFhAdNtgJHRM82amAICNtBTFhIlpP6JUYokrZPLEkMTmiPvnt02s5Qj85JzChVC7pF0RuS0sQC+8resVP2hb1n39iv/8aK0xhJLkPa7bFWBtbss5vNn+eqXNo19v+qcnPW6GIlzdWh3IMUSaoQY/3g6NVps76xEN9hb9h3yv81O2GfqQJv8EO8bciN45zouxhQ1IB8k115MtGXvHoiOrspms9L6k5qdnN5fzqj6ebk/ij3f3JiqCzfYW7cZD56OMrl2OfkNtmRIk117d8ZPmu071VrS9Wlxv3yaj2b7xncwm1UaIaXsYo1rKNFsZ/jAz7ikyGMp8YL4+WYOlXINDcwsYzj36/F3Lg=</latexit>

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="BBIaI+9fdWyliwHK33CwaCp118U=">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</latexit>

Hq (P ) ! Hq (X1 ) ! Hq (X2 ) ! Hq (X3 ) ! Hq (X4 )<latexit sha1_base64="WH5wjWSm0xE/riZ2dAGAQnGpWjQ=">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</latexit>

Hq (F ) ⇠
=

m
M
i=1

<latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

出⼒:PD

Vf ⇠
=
<latexit sha1_base64="DXwstBcMjMvHunsiwoNdVMk8xjg=">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</latexit><latexit sha1_base64="pgfe5n1deukCq1dwv0/Gdcssusk=">AAADwHichVJNLwNRFD01Pqo+qmwkNo2GWDWvIkhXDRtLRasJ0syM15p0vsy8NtrGH7C0sWBDYiF+ho0/YNGfIJYkNhbuTEekxHiTmXffuefcd++dq9i65grGOpE+qX9gcCg6HBsZHRuPTyQmi65Vd1ReUC3dckqK7HJdM3lBaELnJdvhsqHofFeprXv+3QZ3XM0yd0TT5geGXDW1iqbKgqBSsVzZVy2zWp5IsTTzV/K3kQmMFIK1aSUieezjEBZU1GGAw4QgW4cMl549ZMBgE3aANmEOWZrv5zhFjLR1YnFiyITW6Ful016AmnT2Yrq+WqVbdHodUiYxx57YHXtlj+yePbOPP2O1/RheLk3ala6W2+X42fT2+78qg3aBo29VaM4CFaz6uWqUu+0jXhVqV99oXbxuZ7fm2vPshr1Q/teswx6oArPxpt7m+dZlSPRDNCiqTb7erpz09CWsHpfOho+G87y6vZqNUN5XZwTd7N3vhv5PmRh6kG8zNK43HzW0QjnKP7n1diRGU535OcO/jeJiOrOcXs4vpXLZYL6jmMEsFmiGV5DDBjZR8Gf3HJe4ktakI8mSjrvUvkigmULPklqfrb/Q7A==</latexit>M<latexit sha1_base64="pczTYDH7VpQtEFeTHaOfXh7dKDM=">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</latexit>

VI in repk (R).
<latexit sha1_base64="Pkhfsvg49qdA0bmIn1DLyDtejhM=">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</latexit>

I2B(Vf )VI : 区間表現.
<latexit sha1_base64="Pkhfsvg49qdA0bmIn1DLyDtejhM=">AAADu3ichVLLSsNAFD01PuujPjaCG7EorspUpIogCG50Z619gEpJ4lhjJw+SabEWf0BwqwtXCi7Ez3DjD7jwE8SlghsX3qQRqWKckMydc8+5c+/N1RxheJKxp1iH0tnV3dPbF+8fGBxKDI+MFjy75uo8r9vCdkua6nFhWDwvDSl4yXG5amqCF7Xqqu8v1rnrGba1JRsO3zXVimXsG7oqCcoVyuvl4SRLsWBN/jbSoZFEuDbskVgWO9iDDR01mOCwIMkWUOHRs400GBzCdtEkzCXLCPwcJ4iTtkYsTgyV0Cp9K3TaDlGLzn5ML1DrdIug1yXlJKbZI7tlr+yB3bFn9vFnrGYQw8+lQbvW0nKnnDgdz73/qzJplzj4VkXmLLGPxSBXg3J3AsSvQm/p68cXr7mlzenmDLtmL5T/FXti91SBVX/Tb7J88zIi+h7qFNUhX3tXjtr6ElWPR2czQKN5ft1+zWYk76szkm727/ci/6dKDBHm24iM689HFceRHO2f3No7EqepTv+c4d9GYS6VzqQy2fnkylI4372YwBRmaYYXsII1bCBPsSs4wzkulGVFVw4V0aJ2xELNGNqWUvsEzYrOmg==</latexit>

V[ai ,bi ]Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit><latexit sha1_base64="E+KGAacQUTmySREsid4aTHnxVwg=">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</latexit>

D(Vf ) :パーシステンス図.

パーシステンス図の安定性
Stability

Hmm... Are they similar to each other? PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>

Yes, they are similar by stability!

安定性(stability)

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit><latexit sha1_base64="bUAQ17wgiYo6D2U3gd4TG44EV1E=">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</latexit>

R := R t {±1} : 拡張実数

Definition（bottleneck distance）<latexit sha1_base64="sb7WE9MoaqXGzr4w5er3Uh4GdfM=">AAAD03ichVJNSxtRFD1xWpumrUnaTUs3oSGlq/AikkpWghuXJhoTMGmYGZ/JkPli5iWYDKEgrgquXXRVwYUU7I/opn+gC3+CuFRw48L7JlNKKh3fMPPuO/ec++69czXXNHzB2HliTnn0eP5J8mnq2fMXC+lM9uWW7ww8ndd1x3S8pqb63DRsXheGMHnT9bhqaSZvaP1V6W8Muecbjr0pRi5vW2rXNnYNXRUEdTJvWg65pTpoWaroaVpQm0w+LeZSnUyeFVm4cveNUmTkEa11J5uoooUdONAxgAUOG4JsEyp8erZRAoNLWBsBYR5ZRujnmCBF2gGxODFUQvv07dJpO0JtOsuYfqjW6RaTXo+UORTYb3bKrtgv9p1dsNv/xgrCGDKXEe3aVMvdTvrL642bB1UW7QK9v6rYnAV2sRzmalDubojIKvSpfjg+utqo1ArBe3bMLin/b+yc/aQK7OG1flLlta8x0XcwpKgu+Wa7sjfTl7h6fDpbIRrPk3XLmq1Y3p/OCLpZ3u/H/k+VGGaU7yg2rpyPPsaxHO2B3GY7Iqe69O8M3ze2FoulcrFcXcqvVKL5TuIt3uEDzfBHrGAN66hT7M84wRl+KHUlUPaVgyl1LhFpXmFmKYd3vqLYLw==</latexit>

2

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

R 上の多重集合 A, B の間の部分マッチングとは, 次の形の⇠
全単射 M :A
A ! B ⇢ B
<latexit sha1_base64="KwF0yRhnjvjHXtFmNwAASidr87I=">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</latexit><latexit sha1_base64="qq/2b193e6SlvzzOtrxK+ECINUQ=">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</latexit><latexit sha1_base64="UsX65lGc75/ehFUfJlWDn72uqDY=">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</latexit>

M

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit><latexit sha1_base64="UDNRVWVGHNSVPzRJQltrPQgrjYA=">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</latexit>のことである. さらに M のコスト c(M ) を以下で定める.
()|q2 q1 |c(M ) := max sup kp M (p)k1 , sup.
2
p2A
q62AtB<latexit sha1_base64="nkrrng7/fSAfzaCUtbe3kqp/CcM=">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</latexit><latexit sha1_base64="xg3HzBRCYffnPTwtNixnjzY89hk=">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</latexit>

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

ボトルネック距離とは, 次で定義される拡張擬距離:dB (A, B) := inf c(M ).
<latexit sha1_base64="qpI/oC50JMkAYKqgzlFnJcWMHM0=">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</latexit>

M“拡張” … 値に 1 を取り得る.
<latexit sha1_base64="qojQUiL7vCn9qlxmqwveaz0Ijzw=">AAADwnichVLLLgRBFD2jvd+PjcRGTIjVpEYEsZKwsMSYIUGku9VQM/3SVTMxhh+wshOsSCzEZ9j4AQufIJYkNhZu97TIEK0q3XXr1jmn7r11Dc8SUjH2lGjQGpuaW1rb2js6u7p7evv6c9It+SbPmq7l+uuGLrklHJ5VQll83fO5bhsWXzOK88H5Wpn7UrjOqqp4fMvWdx2RF6auyJXbFE5eVbZ7kyzFwjH820hHRhLRWHL7EhlsYgcuTJRgg8OBItuCDklzA2kweOTbQpV8PlkiPOc4RjtxS4TihNDJW6T/Lu02Iq9D+0BThmyTbrHo84k5jFH2yG7ZK3tgd+yZffypVQ01glgqtBo1Lve2e04GM+//smxaFfa+WbExK+QxE8YqKHYv9ARZmDV++fDsNTO7MlodY9fsheK/Yk/snjJwym/mzTJfuYxR30GZVD06q6/KQV1d4vKRtLdDbzwuyDvI2Y7FfVVG0c3B/TL2PXVCWFG8lVjdoD+KOIzFGP/EVl+RcFJnp3/28W8jN5FKT6WmlieTc7NRj7diCCMYpz6exhwWsYQs6RdwinNcaAtaQdvXZA3akIg4A6gb2tEnc9/Qag==</latexit>“擬距離” … dB (A, B) = 0 6) A = B.
<latexit sha1_base64="O+ja2A3WutC0R1VRCPUzcD4p4m8=">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</latexit>

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">AAADzXichVLLSitBED1xfEavRt0IbsINyl2FjohKVoIbdyZqVFAJM2PrHTIvpjshMepW8AdcuBAFQfEz3PgDLvwEcekFNy6snoxIlDv2MNPVp86prqopw7ctIRl7THRonV3dPb19yf6BX4NDqeGRNeFVA5OXTM/2gg1DF9y2XF6SlrT5hh9w3TFsvm5UFpR/vcYDYXnuqmz4fNvR91xr1zJ1SVA5NboleV02C4oiJHdNfpgupzIsy8KV/m7kIiODaBW84UQRW9iBBxNVOOBwIcm2oUPQs4kcGHzCttEkLCDLCv0ch0iStkosTgyd0Ap99+i0GaEunVVMEapNusWmNyBlGhPsgd2wF3bPbtkTe/tvrGYYQ+XSoN1oablfHjoZW3n9UeXQLvH3UxWbs8Qu5sJcLcrdDxFVhdnS1/ZPX1byyxPNSXbJnin/C/bI7qgCt/bPvCry5bOY6DuoUVSffO1dqbf1Ja4eQWcnRON5qm5VsxPL++iMpJvV/SL2f+rEsKN8G7Fx1XxUsB/LMX7Irb0jSZrq3NcZ/m6sTWVzM9mZ4nRmPh/Ndy/G8Rt/aIZnMY9FFFCi2A2c4xo32pJW1Q60oxa1IxFpRtG2tON3sLzWKw==</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>安定性(stability)

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit><latexit sha1_base64="bUAQ17wgiYo6D2U3gd4TG44EV1E=">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</latexit>

R := R t {±1} : 拡張実数

Definition（bottleneck distance）<latexit sha1_base64="sb7WE9MoaqXGzr4w5er3Uh4GdfM=">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</latexit>

2

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>

R 上の多重集合 A, B の間の部分マッチングとは, 次の形の⇠
全単射 M :A
A ! B ⇢ B
<latexit sha1_base64="KwF0yRhnjvjHXtFmNwAASidr87I=">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</latexit><latexit sha1_base64="qq/2b193e6SlvzzOtrxK+ECINUQ=">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</latexit><latexit sha1_base64="UsX65lGc75/ehFUfJlWDn72uqDY=">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</latexit>

M

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit><latexit sha1_base64="UDNRVWVGHNSVPzRJQltrPQgrjYA=">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</latexit>のことである. さらに M のコスト c(M ) を以下で定める.
()|q2 q1 |c(M ) := max sup kp M (p)k1 , sup.
2
p2A
q62AtB<latexit sha1_base64="nkrrng7/fSAfzaCUtbe3kqp/CcM=">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</latexit><latexit sha1_base64="xg3HzBRCYffnPTwtNixnjzY89hk=">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</latexit>

Persistence
Modules<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

ボトルネック距離とは, 次で定義される拡張擬距離:dB (A, B) := inf c(M ).
<latexit sha1_base64="qpI/oC50JMkAYKqgzlFnJcWMHM0=">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</latexit>

M“拡張” … 値に 1 を取り得る.
<latexit sha1_base64="qojQUiL7vCn9qlxmqwveaz0Ijzw=">AAADwnichVLLLgRBFD2jvd+PjcRGTIjVpEYEsZKwsMSYIUGku9VQM/3SVTMxhh+wshOsSCzEZ9j4AQufIJYkNhZu97TIEK0q3XXr1jmn7r11Dc8SUjH2lGjQGpuaW1rb2js6u7p7evv6c9It+SbPmq7l+uuGLrklHJ5VQll83fO5bhsWXzOK88H5Wpn7UrjOqqp4fMvWdx2RF6auyJXbFE5eVbZ7kyzFwjH820hHRhLRWHL7EhlsYgcuTJRgg8OBItuCDklzA2kweOTbQpV8PlkiPOc4RjtxS4TihNDJW6T/Lu02Iq9D+0BThmyTbrHo84k5jFH2yG7ZK3tgd+yZffypVQ01glgqtBo1Lve2e04GM+//smxaFfa+WbExK+QxE8YqKHYv9ARZmDV++fDsNTO7MlodY9fsheK/Yk/snjJwym/mzTJfuYxR30GZVD06q6/KQV1d4vKRtLdDbzwuyDvI2Y7FfVVG0c3B/TL2PXVCWFG8lVjdoD+KOIzFGP/EVl+RcFJnp3/28W8jN5FKT6WmlieTc7NRj7diCCMYpz6exhwWsYQs6RdwinNcaAtaQdvXZA3akIg4A6gb2tEnc9/Qag==</latexit>“擬距離” … dB (A, B) = 0 6) A = B.
<latexit sha1_base64="O+ja2A3WutC0R1VRCPUzcD4p4m8=">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</latexit>

A, B
Persistence
Diagrams
A, B<latexit sha1_base64="KwF0yRhnjvjHXtFmNwAASidr87I=">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</latexit><latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit><latexit sha1_base64="KwF0yRhnjvjHXtFmNwAASidr87I=">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</latexit>安定性(stability)

PH Analysis<latexit sha1_base64="DrO/mJ3acVG3NJ029YWppBxvOHU=">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</latexit>

X : 位相空間. f, g : X ! R : 関数.
<latexit sha1_base64="fdpJMJs63I0P5HsiVZ88g1YVT1Y=">AAAD1XichVJNSxtRFD1x2hpjW6NuBDdDQ0oXJbyUouIq0I1Lk5gPSCTMTF7ikPli5iUYQxZCceGmyy5cWSiluNHf0I1/wIU/oXRpwU0XvW8ypaTSyRtm3n3n3nPevXeu7llmIBi7Tcwpjx4/mU8upBafPnu+lF5eqQZu3zd4xXAt16/rWsAt0+EVYQqL1z2fa7Zu8Zreeyf9tQH3A9N19sTQ4/u21nXMjmlogqBWer3zuqs2pY6j1tWmcNWmrYkDXR+Vxq10huVYuNSHRj4yMojWrrucKKKJNlwY6MMGhwNBtgUNAT0N5MHgEbaPEWE+WWbo5xgjRdw+RXGK0Ajt0bdLp0aEOnSWmkHINugWi16fmCqy7IZ9ZXfsml2w7+zXf7VGoYbMZUi7PuFyr7V0ula+n8myaRc4+MuKzVmgg60wV5Ny90JEVmFM+IOjj3fl7VJ29JJ9Yj8o/3N2y75RBc7gp/G5yEtnMeptDEjVI990Vw6n+hJXT0BnO0Tj42TdsmY7Nu5PZwTdLO8PYv+nRhFWlO8wVlfORw9HsTH6jNymO5Kiqc7/O8MPjeqbXH4jt1F8mylsR/OdxDpe4BXN8CYK2MEuKqR9jC+4xJVSU8bKe+VkEjqXiDirmFrKh99Mo9hY</latexit><latexit sha1_base64="Jf9VRQTRT/R1H8X4ScJ2Y4jz/Uw=">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</latexit><latexit sha1_base64="41ObN+C0qH8K4S5+5dfR5eNelro=">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</latexit>"

Data<latexit sha1_base64="SkTkL10QvgotkFClpWeVaIiLTX0=">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</latexit>"

Vf := Hq (S (f )), Vg := Hq (S (g)) 2 repk (R).

D(Vf ), D(Vg ) ✓ R 2 : パーシステンス図（多重集合）.
<latexit sha1_base64="e+UrpGbF6+ioUdC2u/oA1XEA9OY=">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</latexit>

Theorem（安定性定理）

Filtration<latexit sha1_base64="2oxs3oDbiOiASMRcB5zhFiYkOlw=">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</latexit>dB (D(Vf ), D(Vg ))  kf
<latexit sha1_base64="lapHm3zB4yJZOVSGuBxsdt2/IDs=">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</latexit>

gk1 .

g(x)|.

Persistence
Modules

に対し, 以下が成⽴.

Persistence
Diagrams<latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit>

kf<latexit sha1_base64="daruxsFLb476mJTGvBs1KgptvK0=">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</latexit>

gk1 := sup |f (x)<latexit sha1_base64="6zbsJboUpUXtVtiPadH0V8UsklU=">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</latexit>

x2X

Corollary0

有限部分集合 P, P ✓ R<latexit sha1_base64="VKzyRZYjL9XU2DdHmmB/z7AbkXo=">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</latexit>NdB (D(VfP ), D(VfP 0 ))  kfP<latexit sha1_base64="3Q3MRPvMVsFhssBMtvAHqPO8edI=">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</latexit><latexit sha1_base64="liu8sXtNFum6EQ7hoXc1AZRoc/E=">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</latexit><latexit sha1_base64="JPKR3CR/8yJkLbRe+gtE8bJ1rh4=">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</latexit>f P 0 k1 .

まとめパーシステントホモロジー解析はデータの「かたち」を捉える解析⼿法.
情報をパーシステンス図として出⼒.
「かたち」＝ ”⽳”の持続性(パーシステンス)
逆解析により”⽳”の特定・解釈が可能.
表現論が⾃然に現れる.
摂動に対する安定性定理が成り⽴つ.
ただし, 外れ値に対しては頑強ではない.
特に材料科学への応⽤.
最近では機械学習との組合せ.
〜 Special Thanks 〜

資料提供 多⽥駿介さん(神⼾⼤D3)
with HomCloud and python.

パーシステントホモロジー解析のマルチパラメータ化(⾼次元化)に向けて
半径パラメータ r
時間パラメータ t<latexit sha1_base64="rw02LOQTE5wGwGCvpONSwwhOBm8=">AAADuXichVJNLwNRFD01Pqq+2UhsRFOxal5FkNpIbCzbUppUIzPjYXS+Mu+1UY0/wJZYWJFYiJ9h4w9Y+AliSWJj4c50REqMN5l59517zn333rmaaxpCMvYU61A6u7p74r2Jvv6BwaHhkdEN4dQ8nRd1x3S8kqYKbho2L0pDmrzkely1NJNvatUV379Z554wHHtdNlxesdQ929g1dFUSlPe2h5MszYI1+dvIhEYS4co5I7E8trADBzpqsMBhQ5JtQoWgp4wMGFzCKmgS5pFlBH6OYyRIWyMWJ4ZKaJW+e3Qqh6hNZz+mCNQ63WLS65FyEin2yG7ZK3tgd+yZffwZqxnE8HNp0K61tNzdHjoZX3v/V2XRLrH/rYrMWWIXi0GuBuXuBohfhd7S148uXteyhVRzml2zF8r/ij2xe6rArr/pN3leuIyIvoM6RXXJ196Vw7a+RNUj6GwFaDTPr9uv2YrkfXVG0s3+/SLyf6rEMMN8G5Fx/fmo4iiSo/2TW3tHEjTVmZ8z/NvYmE1n5tPz+bnkcjac7zgmMIUZmuEFLGMVORQpNscpznCuLCmqsq8ctKgdsVAzhraliE+sec36</latexit>

e.g. 時系列PH解析<latexit sha1_base64="3WQZ3OTlbnRSDHnUsbdDqlQu3VQ=">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</latexit>

⼀般に区間では表わせない.

“パーシステンス図”が定義できない！

表現の分類すら困難(wild表現型).

“⽳”の解釈が困難！

P = RN 上の表現は<latexit sha1_base64="W2WvO+8fO6Ju7Af/mGbi5ptIT+k=">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</latexit>

現在は表現論的な観点から性質や不変量を研究.
<latexit sha1_base64="hvsDZTwJE23a95sT/1kkOzjnrBI=">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</latexit>

repk (P ) 上の表現論へパーシステントホモロジー解析のマルチパラメータ化(⾼次元化)に向けて安定性定理の代数的理解
インターリービング距離

区間表現の理論
区間近似・区間分解次元

Answer is…

Exercise.

Answer is…

逆解析

Exercise.