位相的データ解析の現在 - kurims.kyoto-u.ac.jp

1 * IppeiObayashiWPI AIMR,TohokuUmiversity1 , (TopologicalDataAnalysis,TDA) .TDA , , , , . , . , . *1. 1990 ,2000 TDA [2,3,4, 5,6]. 1 , 1 . [7,8] [9] , [10] , [11 , .* 980 8577 2 1 . Formne (sweepl\underline{\mathrm{i}}\mathrm{n}\ mathrm{e}) 1986 [1]. 2057 2017 34-50342 , . , 2 3 ( ) , . , [7].]

2 1 (a) (c) (MD) . , . , , 2 . ( 1(\mathrm{b})(\mathrm{d})) , 2 2 ( ) , . (b) (d) ,(d) ..\llcorner_{1}\mathrm{q}\mathrm{u}|\mat hrm{d}87654123\mathfrak{g}|\mathrm{R}\ma thrm{h} \mathrm{A}\wedge 2|5678\ovalbox{\tt\small REJECT}_{0}^{60}\infty\ovalbox{\tt\small REJECT} \mathrm{B}w320_{0}\ovalbox{\tt\small REJECT}_{0}^{60}30\not\in\S 1 ([71 )35 , . , . n ( 2(\mathrm{a})). , r ( 2(\mathrm{b})). ,2 ( ,\dim H_{1}=2 ). (a) . 2(b) .(a)\bullet\bullet\bullet\bullet\bullet\ bullet\bullet\bullet\bullet\bullet\bulle t 2 : b : d 3 , r.]

3 R . , 2 , , . , . r36 0 , . , 3 r . , b d . b (birthtime),d (deathfime), , (b,d) , (Uirth deaffipair) . \{(b_{i}, d_{\dot{ $\iota$}})\}_{i=1}^{s}\cup\{(b_{i_{\rang le}}\infty)\}_{i=s+1^{*2}}^{ $\epsilon$+t} , (x, y) (persistencediagam) . \mathbb{R}^{N} 0 N l , 0 N l .0 ,1 ,2 , . 1 1 .0 2 ,1 . , , 1 * (a) \bullet\ :\bullet0. 5\ \displaystyle \frac{B}{\circ $\omega$ \mathrm{n}} \ \ \bullet\ \bullet \underline{5}_{15} \mathrm{q}_{0} 4 (a) (b) , ( 4(\mathrm{a})) 1*20.

4 \infty *3 ..37 ( 4(\mathrm{b})) .1 1 . 4(\mathrm{a}) . (x ) 1/2 , (y ) . , , , .. , r . , ( , ) , .3 , ..\bullet , \bullet \bullet , . 5 8 . P=\{u_{i}\}_{i=1}^{N} , . 2 3 4 ( 5 ,Del(P) ). r Alp(P, r) Del(P) [3].

5 \bullet 38'\mathrm{c}\mathrm{c}\mathrm{c} 5 \bullet n , n r n (n-1) 6 3 r , , . [4] |\displaystyle \mathrm{A}\mathrm{l}\mathrm{p}(P, r)|\simeq\bigcup_{u_{i}\in P}B_{r}(u_{i}) r>0 . \simeq ,|\cdot| . r , Alp(P, r) . 6 6 r 3 / , . 39 , . r 0 , .. , $\sigma$ , r r r_{ $\sigma$} . cgal(http://cgal.\mathrm{o}\mathrm{r}\ma thrm{g}) . , . {$\sigma$_{1})..,$\sigma$_{K}} , .Condition1.\{$\sigma$_{1}, \cdots$\sigma$_{k}\} 1\leq k\leq K ,\{$\sigma$_{1},\cdots $\sigma$ X_{k} ,X_{1}\ \mapsto X_{K} , Q^{*4} H_{*}(X_{1})\rightarrow.}

6 \rightarrow H_{*}(X_{K}) .. [6] , (intervaldecomposable)\mathrm{I}[i,j)=0\ rightarrow\ldots\rightarrow 0\rightarrow Q\rightarrow 1\cdots\rightarrow 1Q\rightarrow 0\rightarrow\ldots\rightarrow 0(i j-1 ) . H_{\ell}(X_{\mathrm{i}})\rightarrow (X_{2})\rightarrow H_{\ell}(X_{3})\rightarrow 1111110 \rightarrow Q \rightarrow\sim Q\rightarrow..\oplus \oplus \oplus 0 \rightarrow Q \rightarrow 0\rightarrow..\oplus \oplus \oplus . 0\rightarrow Q ,Q\rightarrow 1Q ,Q\rightarrow 0 , . 2 . , \mathrm{I}[i,j) (i,j) .*4 , , \mathbb{Z}_{2} .40 r , . , r \{r_{ $\sigma$} | $\sigma$\in \mathrm{D}\mathrm{e}1(P)\} . r_{ $\sigma$} , ( \{$\sigma$_{1},\ldots,$\sigma$_{K}\} ), 1.]]

7 \mathbb{I}[i,j) ,(r_{$\sigma$_{i}}, r_{$\sigma$_{j}}) , . (PI)) , , PID , *5. \searrow , . , f_{1}:V_{1}\rightarrow V_{2},f_{2}:V_{2}\rightarrow V_{3}(Vi,V_{2},V3: ,f_{1},f_{2}: ) . , .V_{1} \rightarrow f_{1} V_{2} \rightarrow f_{2} V_{3}\mathrm{k}\mathrm{e}\mathrm{r}(f_{1 })|| \rightarrow 0|| \rightarrow 0||\oplus \oplus \oplus\mathrm{k}\mathrm{e}\mathrm{r}(f_{ 2}\mathrm{o}f_{1})/\mathrm{k}\mathrm{e}\ mathrm{r}(f_{1})\rightarrow\sim\mathrm{k }\mathrm{e}\mathrm{r}(f_{2})\cap \mathrm{i}\mathrm{m}(f_{1})\rightarrow 0\oplus \oplus \oplus V_{1}/\mathrm{k}\mathrm{e}\mathrm{r}(f_{ 2}\mathrm{o}f_{1})\rightarrow\sim\mathrm {i}\mathrm{m}(f_{1})/(\mathrm{i}\mathrm{ m}(f_{1})\cap \mathrm{k}\mathrm{e}\mathrm{r}(f_{2}))\r ightarrow\sim\mathrm{i}\mathrm{m}(f_{2}\ mathrm{o}f_{1})\oplus \oplus \oplus 0 \rightarrow \mathrm{k}\mathrm{e}\mathrm{r}(f_{2})/(\ mathrm{i}\mathrm{m}(f_{1})\cap \mathrm{k}\mathrm{e}\mathrm{r}(f_{2})) \rightarrow 0\oplus\oplus\oplus 0 \rightarrow V_{2}/(\mathrm{i}\mathrm{m}(f_{1})+\math rm{k}\mathrm{e}\mathrm{r}(f_{2})) \rightarrow\sim \mathrm{i}\mathrm{m}(f_{2})/\mathrm{i}\m athrm{m}(f_{2}\mathrm{o}f_{1})]

8 \oplus \oplus \oplus 0 \rightarrow 0 \rightarrow V_{3}/\mathrm{i}\mathrm{m}(f_{2}) , . ( [12,13]) .*5 PID , \mathbb{Z} . , . , .. B \{$\sigma$_{1}, \cdots, $\sigma$_{K}\} \partial , . 1 B .. , . [14] .\displaystyle \frac{\mathrm{A}1\mathrm{g}\mathrm{o}\ma thrm{n}^{\mathrm{i}}\mathrm{t}\mathrm{h} \mathrm{m}1.\ovalbox{\tt\small REJECT}.\text{ }\mathrm{f}\mathrm{f}\mathrm{i}\text{ }\mathrm{a}\mathrm{e}(\hslash^{ $\theta$}\mathbb{Z}_{2}\emptyset\ovalbox {\tt\small REJECT}_{\mathrm{D}}^{\mathrm{a}\mathrm{ A}})}{\mathrm{f}\mathrm{o}\mathrm{r}j=1, .,n\mathrm{d}\mathrm{o}}whilethereexists i<jwithlow(i)=1\mathrm{o}\mathrm{w}(j)do addcolumnitocolumnj low(i) i 0 ( 0 low(i) ).

9 , ( ) , , . [14] .4 , 2 . [10] , [7,8] .. ,[10] .42 , ( )[4] . , r . , ..2 , .. , .\mathrm{V}\mathrm{R}(P,r)={ $\sigma$=\{v_{1_{\rangle}}\ldots,v_{k}\} \subset P|d(v_{i},v_{j})\leq rforalli,j} P=\{u_{i}\} ,d P ,r>0 .r . 2 . , .. , . (reassottment) , . ( ).

10 , . 2 , 2 .. 1 , 1 0 .. , 2 ( ) .. 1 .. 43 *6 , , . [7,8] .2 , . , ( ,LongRangeOrder ) , ( ,ShortRangeOrder ) . (\mathrm{S}\mathrm{i}\mathrm{O}_{2}) . , . , (MidiumRangeOrder) . , . , opfimalcycle[15,16] . \{C_{q}\}_{q} \partial_{q}:C_{q}\rightarrow C_{q-1} H_{q}=Z_{q}/B_{q},Z_{q}=\mathrm{k}\mathr m{e}\mathrm{r}\partial_{q},B_{q}=\mathrm {i}\mathrm{m}\partial_{q+1}.