persistent homology group

A brief (?) survey of "persistent homology group"

Transcript

1. ύʔγεςϯτϗϞϩδʔ܈ @ubnqf June 28, 2018
2. Outline 1. τϙϩδʔ 2. ୯ମతϗϞϩδʔ • ୯ମෳମ • ந৅୯ମෳମ • ࠯܈ • ڥք࡞༻ૉ • ϗϞϩδʔ܈ 3. ୯ମෳମͷߏங • ຺ମఆཧ • ˇ Cech ෳମ • Vietoris-Rips ෳମ • ΞϧϑΝෳମ • ํମෳମ 4. ύʔγεςϯτϗϞϩδʔ܈ 2 / 43
3. Outline 1. τϙϩδʔ 2. ୯ମతϗϞϩδʔ • ୯ମෳମ • ந৅୯ମෳମ • ࠯܈ • ڥք࡞༻ૉ • ϗϞϩδʔ܈ 3. ୯ମෳମͷߏங • ຺ମఆཧ • ˇ Cech ෳମ • Vietoris-Rips ෳମ • ΞϧϑΝෳମ • ํମෳମ 4. ύʔγεςϯτϗϞϩδʔ܈ 3 / 43
4. τϙϩδʔ τϙϩδʔʢҐ૬زԿֶʣ   ੾ΓషΓ͠ͳ͍࿈ଓมܗΛͯ͠΋ෆมͳੑ࣭ʢਤܗͷͭͳ͕ΓํʣΛ ௐ΂ΔزԿֶ   ' ' ' ' ' ' 6' 6' # connected components
   # rings
   # cavities
   1
   1
   1
   0
   1
   0
   0
   0
   1
   ▶ ࿈݁੒෼ɼϦϯάɼۭಎͷ਺͸࿈ଓมܗͯ͠΋ෆม 4 / 43
5. τϙϩδʔ • ࿈݁੒෼ɼϦϯάɼۭಎ͸ਤܗͷʮ݀ʯͱߟ͑ΒΕΔ • 0 ࣍ݩͷʮ݀ʯ ɿ࿈݁੒෼ (connected component) • 1 ࣍ݩͷʮ݀ʯ ɿϦϯά (ring) • 2 ࣍ݩͷʮ݀ʯ ɿۭಎ (cavity, void) ▶ τϙϩδʔɿ࿈ଓมܗͨ͠ͱ͖ʹ݀ͷ਺͕౳͍͠ਤܗ͸ಉ͡ͱݟͳ͢ ʮ݀ʯͬͯͲ͏΍ͬͯఆٛ͢Δͷʁ ˠ ϗϞϩδʔ܈ 5 / 43
6. τϙϩδʔ • زԿֶతର৅ X ʹରͯ͠ɼk ࣍ϗϞϩδʔ܈ Hk(X) ͸ ҎԼΛຬͨ͢ • H0 (X) ≃ Rn0 n0 : # connected components • H1 (X) ≃ Rn1 n1 : # rings • H2 (X) ≃ Rn2 n2 : # cavities . . . • Hk (X) ≃ Rnk nk : # k-dimensional holes H0(X) ' R H1(X) ' R4 H2(X) ' R X = T2 2 6 / 43
7. Outline 1. τϙϩδʔ 2. ୯ମతϗϞϩδʔ • ୯ମෳମ • ந৅୯ମෳମ • ࠯܈ • ڥք࡞༻ૉ • ϗϞϩδʔ܈ 3. ୯ମෳମͷߏங • ຺ମఆཧ • ˇ Cech ෳମ • Vietoris-Rips ෳମ • ΞϧϑΝෳମ • ํମෳମ 4. ύʔγεςϯτϗϞϩδʔ܈ 7 / 43
8. ୯ମతϗϞϩδʔ • ϗϞϩδʔ܈ͷఆٛͷྲྀΕ ୯ମෳମ
   K ࠯ෳମ
   ϗϞϩδʔ܈
   Hk(K) ୅਺Խ
   ڥքʹண໨ͯ͠ ʮ݀ʯΛநग़
   Ck(K) = ( c = nk X i=1 ✏i i ✏i 2 Z 2 ) Hk(K) = Ker@k/Im@k (C⇤(K), @⇤) @k : Ck(K) ! Ck 1(K) 8 / 43
9. ୯ମෳମ ୯ମෳମ   k ࣍ݩͷࡾ֯ܗʢ୯ମʣΛͽͬͨΓுΓ߹ΘͤͯͰ͖ΔزԿֶతର৅   0-simplex (vertex)
   1-simplex (edge)
   2-simplex (triangle)
   3-simplex (tetrahedron)
    simplicial complex NOT simplicial complex 9 / 43
10. ୯ମෳମ ఆٛʢ୯ମʣ RN ಺ͷΞϑΝΠϯಠཱͳ k + 1 ݸͷ఺ p0, p1, . . . , pk ͕ͳ͢࠷খͷತू߹ { x ∈ RN x = k ∑ i=0 λipi, λi ≥ 0, k ∑ i=0 λi = 1 } (1) Λ k ୯ମ (k-simplex) ͱݺͼɼ|p0p1 · · · pk| Ͱද͢ɽ • k ୯ମ σ = |p0p1 · · · pk| ΛఆΊΔ k + 1 ݸͷ఺͔Β l + 1 (l < k) ݸͷ ఺ΛऔͬͯͰ͖Δ৽ͨͳ୯ମ τ = |p0p1 · · · pl| Λ σ ͷ໘ (face) ͱݺͼɼ τ ≺ σ Ͱද͢ p0 p1 p2 p0 p1 ⌧ 10 / 43
11. ୯ମෳମ ఆٛʢ୯ମෳମʣ RN ಺ͷ༗ݶݸͷ୯ମͷू·Γ K ͕ҎԼͷ৚݅Λຬͨ͢ͱ͖ɼK Λ ୯ମෳମ (simplicial complex) ͱݺͿɿ 1. σ ∈ K, τ ≺ σ ⇒ τ ∈ K 2. τ, σ ∈ K, (τ ∩ σ) ̸= ϕ ⇒ (τ ∩ σ) ≺ τ ∧ (τ ∩ σ) ≺ σ ·ͨɼ୯ମෳମ K ʹؚ·ΕΔ୯ମͷ࣍ݩͷ࠷େ஋Λ K ͷ࣍ݩ dimK ͱͯ͠ఆΊΔɽ p0 p1 p2 K = {|p0p1p2 |, |p0p1 |, |p0p2 |, |p1p2 |, |p0 |, |p1 |, |p2 |} 11 / 43
12. ୯ମෳମ ఆٛʢ୯ମෳମʣ RN ಺ͷ༗ݶݸͷ୯ମͷू·Γ K ͕ҎԼͷ৚݅Λຬͨ͢ͱ͖ɼK Λ ୯ମෳମ (simplicial complex) ͱݺͿɿ 1. σ ∈ K, τ ≺ σ ⇒ τ ∈ K 2. τ, σ ∈ K, (τ ∩ σ) ̸= ϕ ⇒ (τ ∩ σ) ≺ τ ∧ (τ ∩ σ) ≺ σ ·ͨɼ୯ମෳମ K ʹؚ·ΕΔ୯ମͷ࣍ݩͷ࠷େ஋Λ K ͷ࣍ݩ dimK ͱͯ͠ఆΊΔɽ • ৚݅ 2 ͸ʮ୯ମΛͽͬͨΓுΓ߹ΘͤΔํ๏ʯΛද͍ͯ͠Δ • ͭ·Γɼ֤୯ମͷுΓ߹Θͤ࡞ۀΛ୯ମͷ໘ಉ࢜ͰߦΘΕΔΑ͏ʹ੍ݶ • ୯ମෳମ K ࣗ਎͸୯ମͷ༗ݶू߹Ͱ͋Δ͕ɼK ಺ͷશ୯ମͷ࿨ू߹ |K| = ∪ σ∈K σ ΛऔΔ͜ͱͰ RN ಺ͷਤܗʢଟ໘ମʣ͕ಘΒΕΔ 12 / 43
13. ந৅୯ମෳମ • ୯ମෳମ͸ RN ಺ͷ୯ମͷू·Γͷ૊߹ͤతੑ࣭ʹண໨ͨ֓͠೦ ˠ RN ͔Β཭Εɼ૊߹ͤతੑ࣭ͷΈΛऔΓग़ͨ͠ఆٛ΋Մೳ ఆٛʢந৅୯ମෳମʣ ༗ݶू߹ V ͱ V ͷ෦෼ू߹ͷ༗ݶݸͷू·Γ Σ ͕ҎԼͷ৚݅Λຬͨ͢ͱ ͖ɼ(V, Σ) Λந৅୯ମෳମ (abstract simplicial complex) ͱݺͿɿ 1. v ∈ V, {v} ∈ Σ 2. σ ∈ Σ, τ ⊂ σ ⇒ τ ∈ Σ ͜͜ͰɼΣ ͷݩ σ = {v0, v1, . . . , vk} Λ k ୯ମͱݺͼɼdim σ = k ͱ͢Δɽ ·ͨɼσ ∈ Σ ͷ࠷େ࣍ݩΛந৅୯ମෳମͷ࣍ݩ dim(V, Σ) ͱͯ͠ఆΊΔɽ 13 / 43
14. ந৅୯ମෳମ • ୯ମෳମ K ͸ந৅୯ମෳମ (V, Σ) ΛఆΊΔɽ ٯʹɼந৅୯ମෳମ (V, Σ) Λ࣮ݱ͢Δ୯ମෳମ K Λߏ੒Ͱ͖Δɽ ▶ ୯ମෳମΛҐ૬ۭؒͱͯ͠ݟͳ͢͜ͱ͕Ͱ͖Δ ▶ ٯʹɼந৅୯ମෳମΛ RN ಺ͷ୯ମෳମͱͯ͠ѻ͑Δ K ୯ମෳମ
    V = {v | v 2 K, dim v = 0} ⌃ = {{v0, . . . , vk } | |v0 · · · vk | 2 K} (V, ⌃) ந৅୯ମෳମ
    K = {|p0 · · · pk | | {v0, . . . , vk } 2 ⌃} f : V ! RN : vi 7! f(vi) = (0, . . . , 0, 1, 0, . . . , 0) := pi 14 / 43
15. ࠯܈ ఆٛʢ࠯܈ʣ n ࣍ݩ୯ମෳମ K ͷશͯͷ k ୯ମͷू·Γ Kk = {σi ∈ K | dim σi = k} ʹ͍ͭͯɼKk Ͱੜ੒͞ΕΔࣗ༝ Z/2Z Ճ܈ Ck(K) = { c = nk ∑ i=1 ϵi σi ϵi ∈ Z/2Z } (2) Λ k ࠯܈ (k-chain group) ͱݺͼɼc ∈ Ck(K) Λ k ࠯ (k-chain) ͱݺͿɽ ͳ͓ɼk < 0, n < k ⇒ Ck(K) = 0 ͱ͢Δɽ v0 v1 v2 v3 v0 v1 v2 C0(K) = {a|v0 | + b|v1 | + c|v2 |} C1(K) = {a|v0v1 | + b|v1v2 | + c|v0v2 |} C1(K) = {a|v0v1v2 |} 15 / 43
16. ࠯܈ ఆٛʢ࠯܈ʣ n ࣍ݩ୯ମෳମ K ͷશͯͷ k ୯ମͷू·Γ Kk = {σi ∈ K | dim σi = k} ʹ͍ͭͯɼKk Ͱੜ੒͞ΕΔࣗ༝ Z/2Z Ճ܈ Ck(K) = { c = nk ∑ i=1 ϵi σi ϵi ∈ Z/2Z } (3) Λ k ࠯܈ (k-chain group) ͱݺͼɼc ∈ Ck(K) Λ k ࠯ (k-chain) ͱݺͿɽ ͳ͓ɼk < 0, n < k ⇒ Ck(K) = 0 ͱ͢Δɽ • R Ճ܈ɿϕΫτϧۭؒͷҰൠԽ • ϕΫτϧͷ܎਺Λ؀ R ্ʹ੍ݶͨ͠΋ͷ • ࣗ༝Ճ܈ɿ೚ҙͷݩΛجఈͷઢܗ݁߹ͱͯ͠දͤΔՃ܈ • Z2 := Z/2Z = {[0], [1]} • ࠯܈͸ Z Ճ܈΍ R Ճ܈ͱͯ͠΋ఆٛͰ͖Δʢීว܎਺ఆཧʣ 16 / 43
17. ڥք࡞༻ૉ ఆٛʢڥք࡞༻ૉʣ ֤ 0 ≥ k ≥ n ʹରͯ͠ɼઢܗ࡞༻ૉ ∂k : Ck(K) → Ck−1(K) Λ ڥք࡞༻ૉ (boundary operator) ͱݺͼɼҎԼͰఆΊΔɿ ∂k|v0 · · · vk| = k ∑ i=0 |v0 · · · vi−1vi+1 · · · vk| (4) v0 v1 v2 v3 v0 v1 v2 v0 v1 v3 v1 v2 v3 v0 v2 v3 @ |v1v2v3 | + |v0v2v3 | + |v0v1v3 | + |v0v1v2 | |v0v1v2v3 | 17 / 43
18. ڥք࡞༻ૉ fundamental property   • ڥքͷڥք͸ͳ͍ ∂k ◦ ∂k+1 = 0 for all k (5) • ࠯ෳମɿ࠯܈ C∗(K) ͱڥք࡞༻ૉ ∂∗ ͔ΒͳΔܥྻ 0 → Cn(K) ∂n → Cn−1(K) ∂n−1 → · · · ∂2 → C1(K) ∂1 → C0(K) → 0 (6)   @2 ! @1 ! @1@2 |v0v1v2 | = @1 (|v1v2 | + |v0v2 | + |v0v1 |) = @1 |v1v2 | + @1 |v0v2 | + @1 |v0v1 | = |v1 | + |v2 | + |v0 | + |v2 | + |v0 | + |v1 | = 0 ( * 1 + 1 = 0) ' ' 18 / 43
19. ʮ݀ʯͱ͸ʁ • 1 ࣍ݩͷ৔߹ɿྠ͔͕ͬ͋Ε͹ʮ݀ʯʁ ʮ݀ʯͳ͠
    ʮ݀ʯ͋Γ
    @2 ! @1 ! ྠ͔͕ͬ͋ͬͯ ʮ݀ʯ͕͋Δ
    ྠ͔ͬ͸͋Δ͚Ͳ ʮ݀ʯ͸ͳ͍
    ▶ 1 ࣍ݩͷʮ݀ʯͱ͸ɼڥքͷͳ͍ 1 ୯ମͷू·Γʢྠ͔ͬʣͰ͋ͬͯ 2 ୯ମͷڥքʹͳ͍ͬͯͳ͍΋ͷ ˣҰൠԽ   k ࣍ݩͷʮ݀ʯͱ͸ɼڥքͷͳ͍ k ୯ମͷू·ΓͰ͋ͬͯ k + 1 ୯ମͷڥքʹͳ͍ͬͯͳ͍΋ͷ   19 / 43
20. ʮ݀ʯͱ͸ʁ   k ࣍ݩͷʮ݀ʯͱ͸ɼڥքͷͳ͍ k ୯ମͷू·ΓͰ͋ͬͯ k + 1 ୯ମͷڥքʹͳ͍ͬͯͳ͍΋ͷ   • ྠମ܈ (cyclic group) • Zk (K) = Ker ∂k (K) = {c ∈ Ck (K) | ∂k c = 0} • ڥքྠମ܈ (boundary group) • Bk (K) = Im ∂k+1 (K) = {c ∈ Ck (K) | c = ∂k+1 c′, c′ ∈ Ck+1 (K)} ɹ
    ࣍ݩͷʮ݀ʯͱ͸ɼڥքͷͳ͍ɹ
    ୯ମͷू·ΓͰ͋ͬͯ
    ɹɹɹ
    ୯ମͷڥքʹͳ͍ͬͯͳ͍΋ͷ
    ྠମ܈
    DZDMJD
    HSPVQ
    ڥքྠମ܈
    CPVOEBSZ
    HSPVQ
    Bk(K) ⇢ Zk(K) ⇢ Ck(K) * @k @k+1 = 0 Bk+1(K) Zk+1(K) Ck+1(K) Zk(K) Ck(K) Bk(K) Bk 1(K) Zk 1(K) Ck 1(K) 0 0 0 @k+1 @k Zk(K) = Ker @k = {c 2 Ck(K) | @k c = 0} Bk(K) = Im @k+1 = {c 2 Ck(K) | c = @k+1 c0, c0 2 Ck+1(K)} 20 / 43
21. ϗϞϩδʔ܈ ఆٛʢϗϞϩδʔ܈ʣ ୯ମෳମ K ͷ k ࣍ϗϞϩδʔ܈ (k-dimensional homology group) Λɼ ঎Ճ܈ Hk(K) = Zk(K)/Bk(K) = Ker ∂k/Im ∂k+1 (7) ͰఆΊΔɽ·ͨɼHk(K) ͷݩΛϗϞϩδʔྨͱݺͿɽ • k ࣍ϕον਺ βk = rank (Hk(K)) ▶ k ࣍ݩͷʮ݀ʯͷ਺Λද͢Ґ૬ෆมྔ • K ͷ֤࿈݁੒෼Λ K1, . . . , Ks ͱ͢Δͱɼશͯͷ k ʹ͍ͭͯ Hk(K) = Hk(K1) ⊕ · · · ⊕ Hk(Ks) ͕੒ཱ ▶ ୯ମෳମ K ͷϗϞϩδʔ܈͸֤࿈݁੒෼͝ͱʹٻΊΕ͹Α͍ 21 / 43
22. ϗϞϩδʔ܈ͷܭࢉ • X = {|12|, |13|, |23|, |1|, |2|, |3|} • ࠯ෳମ 0 → C2(X) ∂2 → C1(X) ∂1 → C0(X) → 0 • C2 (X) = {0} • C1 (X) = {α1 |12| + α2 |13| + α3 |23| | αi ∈ Z2 } • C0 (X) = {β1 |1| + β2 |2| + β3 |3| | βi ∈ Z2 } • ڥք࡞༻ૉ ∂2 = 0, ∂1 =   1 1 0 1 0 1 0 1 1   • Z1 (X) = Ker ∂1 =   1 1 1   ≃ Z2 • B1 (X) = Im ∂2 = 0 ∴ H1(X) = Z1(X)/B1(X) ≃ Z2 22 / 43 1 2 3 1 2 3
23. ϗϞϩδʔ܈ͷܭࢉ • X = {|123|, |12|, |13|, |23|, |1|, |2|, |3|} • ࠯ෳମ 0 → C2(X) ∂2 → C1(X) ∂1 → C0(X) → 0 • C2 (X) = {ϵ|123| | ϵ ∈ Z2 } • C1 (X) = {α1 |12| + α2 |13| + α3 |23| | αi ∈ Z2 } • C0 (X) = {β1 |1| + β2 |2| + β3 |3| | βi ∈ Z2 } • ڥք࡞༻ૉ ∂2 =   1 1 1   , ∂1 =   1 1 0 1 0 1 0 1 1   • Z1 (X) = Ker ∂1 =   1 1 1   ≃ Z2 • B1 (X) = Im ∂2 = Z1 (X) ∴ H1(X) = Z1(X)/B1(X) ≃ 0 23 / 43 1 2 3
24. ิ଍ɿ޲͖෇͚ΒΕͨ୯ମ • ্هͰ͸ࣗ༝ Z2 Ճ܈ͱͯ͠ͷ࠯܈Λ΋ͱʹϗϞϩδʔ܈Λ ఆٛͨͨ͠Ίɼ୯ମͷ޲͖Λߟྀ͢Δඞཁ͕ͳ͔ͬͨ • ୯ମͷ޲͖Λߟྀ͢Δ৔߹͸ɼ࠯܈ʹ Z Ճ܈ͷߏ଄͕ೖΔ • ޲͖ͷҧ͍͸ϚΠφεූ߸Ͱදݱ ⟨v1 v2 v3 ⟩ = −⟨v1 v3 v2 ⟩ 2
    1
    3
    2
    1
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    hv1v2v3 i hv1v3v2 i • ޲͖෇͚ΒΕͨ୯ମ ⟨v0 · · · vk⟩ ʹର͢Δڥք࡞༻ૉ͸ҎԼͰఆٛɿ ∂k⟨v0 · · · vk⟩ = k ∑ i=0 (−1)i⟨v0 · · · vi−1vi+1 · · · vk⟩ ▶ ্ͷఆٛ͸޲͖Λߟྀͯ͠ڥքΛऔΓग़͢ૢ࡞Λදݱ 24 / 43
25. Outline 1. τϙϩδʔ 2. ୯ମతϗϞϩδʔ • ୯ମෳମ • ந৅୯ମෳମ • ࠯܈ • ڥք࡞༻ૉ • ϗϞϩδʔ܈ 3. ୯ମෳମͷߏங • ຺ମఆཧ • ˇ Cech ෳମ • Vietoris-Rips ෳମ • ΞϧϑΝෳମ • ํମෳମ 4. ύʔγεςϯτϗϞϩδʔ܈ 25 / 43
26. ୯ମෳମͷߏங • σʔλ͕༩͑ΒΕͨͱ͖ɼͲͷΑ͏ʹ୯ମෳମΛߏங͢Ε͹Α͍͔ʁ protein molecule
    point cloud
    image
    Data
    ü  Cechෳମ (Cech complex) ü  Vietoris-Ripsෳମ (Vietoris-Rips complex) ü  ΞϧϑΝෳମ (alpha complex) ü  ํମෳମ (cubical complex)
    ɾɾɾ
    26 / 43
27. ຺ମఆཧ • ͦ΋ͦ΋ɼσʔλΛ୯ମෳମͱͯ͠දݱͯ͠େৎ෉ͳͷʁ ˠ ຺ମఆཧ ఆٛʢ຺ମʣ ෦෼ू߹଒ Φ = {Bi ⊂ RN | i = 1, . . . , m} ͕෦෼ू߹ X ⊂ RN Λ ඃ෴͍ͯ͠Δঢ়گΛߟ͑ΔʀX = ∪ m i=1 Bi. ͜ͷͱ͖ɼந৅୯ମෳମ (V, Σ) Λ Φ ͷ຺ମ (nerve) ͱݺͼɼN(Φ) Ͱද͢ɽ V = {1, . . . , m} Σ =    {i0, . . . , ik} k ∩ j=0 Bij ̸= ϕ    27 / 43
28. ຺ମఆཧ ຺ମఆཧ (Nerve theorem) ತด෦෼ू߹଒ Φ = {Bi ⊂ RN | i = 1, . . . , m} ͕෦෼ू߹ X ⊂ RN Λ ඃ෴͍ͯ͠Δঢ়گΛߟ͑ΔʀX = ∪ m i=1 Bi. ͜ͷͱ͖ɼX ͱ຺ମ N(Φ) ͸ϗϞτϐʔಉ஋ͱͳΔɽ B1 B2 B3 2
    1
    3
    N( ) ▶ X ͷϗϞτϐʔෆมͳྔΛௐ΂Δࡍʹ͸ɼ ʢҰൠʹ X ΑΓऔΓѻ͍͕ ༰қͳʣ຺ମʹม׵͔ͯ͠Βௐ΂Ε͹ྑ͍ 28 / 43
29. ˇ Cech ෳମ ˇ Cech ෳମ C(P, r) • RN ಺ͷ֤఺ xi Λத৺ͱͨ͠൒ܘ r ͷٿ Br(xi) ͷू·Γ Φ = {Br(xi) | xi ∈ P} ʹ͍ͭͯͷ຺ମ P = {xi ∈ RN | i = 1, . . . , m} Br(xi) = {x ∈ RN | ∥x − xi∥ ≤ r} Bi r 29 / 43
30. Vietoris-Rips ෳମ Vietoris-Rips ෳମ R(P, r) • RN ಺ͷ֤఺ xi Λத৺ͱͨ͠൒ܘ r ͷٿ Br(xi) ͷू·Γ Φ = {Br(xi) | xi ∈ P} ʹ͍ͭͯɼ௖఺ू߹Λ V = {1, . . . , m}ɼ ୯ମͷू·ΓΛ Σ = {{i0, . . . , ik} | Bis ∩ Bit ̸= ϕ, 0 ≤ s, t ≤ k} Ͱ ఆΊͨந৅୯ମෳମ (V, Σ) • ஫ҙɿR(P, r) ͸຺ମͱͯ͠ಘΒΕΔ୯ମෳମͰ͸ͳ͍ͨΊɼ ٿͷ࿨ू߹ ∪ m i=1 Br(xi) ͱͷϗϞτϐʔಉ஋ੑ͸อূ͞Εͳ͍ Bi r 30 / 43
31. ΞϧϑΝෳମ ΞϧϑΝෳମ α(P, r) • RN ಺ͷ֤఺ xi ʹର͢Δ൒ܘ r ͷٿ Br(xi) ͱϘϩϊΠྖҬ Vi = {x ∈ RN | ∥x − xi∥ ≤ ∥x − xj∥, j ̸= i} ͱͷڞ௨෦෼ Wi = Br(xi) ∩ Vi ͷू·Γ Ψ = {Wi | i = 1, . . . , m} ʹ͍ͭͯͷ຺ମ \ = Br(xi) Vi Wi ↵(P, r) 31 / 43
32. ํମෳମ ํମෳମ (cubical complex) • ୯ମʢࡾ֯ܗʣͰ͸ͳ͘ɼํମʢ࢛֯ܗʣΛ΋ͱʹͯ͠ߏ੒ͨ͠ෳମ • ը૾ͷϗϞϩδʔ܈Λܭࢉ͢Δࡍʹ࢖ΘΕΔΒ͍͠ʢϐΫηϧ͕ ࢛֯ܗͷͨΊʣ 0-cube (vertex)
    1-cube (edge)
    2-cube (square)
    3-cube (cube)
     cubical complex
    hole
    32 / 43
33. Outline 1. τϙϩδʔ 2. ୯ମతϗϞϩδʔ • ୯ମෳମ • ந৅୯ମෳମ • ࠯܈ • ڥք࡞༻ૉ • ϗϞϩδʔ܈ 3. ୯ମෳମͷߏங • ຺ମఆཧ • ˇ Cech ෳମ • Vietoris-Rips ෳମ • ΞϧϑΝෳମ • ํମෳମ 4. ύʔγεςϯτϗϞϩδʔ܈ 33 / 43
34. ύʔγεςϯτϗϞϩδʔ܈ • ෆຬɿϗϞϩδʔ܈͸σʔλͷ݀ͷ਺͸ܭࢉͰ͖Δ͕ɼ ݀ͷେ͖͞΍ܗ͸ܭࢉͰ͖ͳ͍ 0 = rank(H0(X2)) = 1 1 = rank(H1(X2)) = 2 0 = rank(H0(X1)) = 1 1 = rank(H1(X1)) = 2 34 / 43
35. ύʔγεςϯτϗϞϩδʔ܈ ύʔγεςϯτϗϞϩδʔ܈   • ෳମͷύϥϝʔλมԽʹ൐͏૿େྻʢϑΟϧτϨʔγϣϯʣʹ ର͢ΔϗϞϩδʔ܈ʢ࣌ؒ࣠ + ϗϞϩδʔ܈ʣ   • e.g. ˇ Cech ෳମͷϑΟϧτϨʔγϣϯ C : C(P, r(0)) ⊂ C(P, r(1)) ⊂ · · · ⊂ C(P, r(5)) ⊂ · · · t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 35 / 43
36. ύʔγεςϯτϗϞϩδʔ܈ • ୯ମෳମ Kt (t = 0, 1, . . . ) ͷϑΟϧτϨʔγϣϯ K : K0 ⊂ K1 ⊂ · · · Kt ⊂ · · · • ࣌ࠁ t Ͱͷ୯ମෳମͷ k ୯ମͷू·Γ Kt k = {σ ∈ Kt | dim σ = k} • ୯ମ σ ∈ K ͷൃੜ࣌ࠁ T(σ) = t if σ ∈ Kt \ Kt−1 ఆٛʢϑΟϧτϨʔγϣϯ K ʹର͢Δ k ࠯܈ʣ ࣗ༝ Z2 Ճ܈ Ck(Kt) = ∑ σ∈Kt k Z2σ ͷ௚࿨ Ck(K) = ⊕ t≥0 Ck(Kt) = { (c0, c1, . . . , ct, . . . ) | ct ∈ Ck(Kt) } (8) ʹ x ͷ࡞༻ x · (c0, c1, . . . ) = (0, c0, c1, . . . ) Λಋೖ͢ΔͱɼCk(K) ͸ ࣍਺෇͖ Z2[x] Ճ܈ͱͳΔɽ͜ΕΛϑΟϧτϨʔγϣϯ K ʹର͢Δ k ࠯܈ͱݺͿɽ 36 / 43
37. ύʔγεςϯτϗϞϩδʔ܈ • แؚࣸ૾ it : Ck(Kt) → Ck(K) it(σ) = (c0, c1, . . . ), ci = { σ (i = t) 0 (i ̸= t) (9) • Ck(K) ͷجఈ ξk = {eσ = iT(σ) (σ) | σ ∈ Kk} • ڥք࡞༻ૉ ∂k : Ck(K) → Ck−1(K) @k(e ) = k X i=0 ⇣ xT ( ) T ( i) ⌘ e i , 2 Kk i • ∂k−1 ◦ ∂k = 0 • Zk (K) = Ker ∂k = ⊕ t≥0 Zk (Kt) • Bk (K) = Im ∂k+1 = ⊕ t≥0 Bk (Kt) 37 / 43
38. ύʔγεςϯτϗϞϩδʔ܈ ఆٛʢύʔγεςϯτϗϞϩδʔ܈ʣ ୯ମෳମͷϑΟϧτϨʔγϣϯ K ʹର͢Δ k ࣍ύʔγεςϯτ ϗϞϩδʔ܈ (persistent homology group) Hk(K) ͸ҎԼͰ༩͑ΒΕΔɿ Hk(K) = Zk(K)/Bk(K) = ⊕ t≥0 Zk(Kt)/Bk(Kt) = ⊕ t≥0 Hk(Kt) (10) • ύʔγεςϯτϗϞϩδʔ܈ͷҰҙ෼ղఆཧ; Hk(K) ≃ ⊕ s i=1 I[bi, di] • ύʔγεςϯτ۠ؒ I[b, d]ʢb: ݀ͷൃੜ࣌ࠁɼd: ݀ͷফ໓࣌ࠁʣ ▶ શͯͷ݀ͷൃੜ࣌ࠁͱফ໓࣌ࠁΛࢦఆ͢Δ͜ͱͰʢಉܕΛআ͍ͯʣ Hk(K) ΛҰҙʹදݱՄೳ 38 / 43
39. ύʔγεςϯτϗϞϩδʔ܈ • ύʔγεςϯτϗϞϩδʔ܈ Hk(K) ͷදݱํ๏ 0 1 2 3 4
    CBSDPEF
    ֤ੜ੒ݩʢ݀ʣͷ
    CJSUI͔ΒEFBUI·Ͱ
    ઢΛҾ͘
    0 1 2 3 4
    4 3 2 1 QFSTJTUFODF
    EJBHSBN
    CJSUI
    EFBUI
    ֤ੜ੒ݩͷCJSUI
    C
    ͱ EFBUI
    E
    Λ࣍ݩฏ໘্ͷ ఺
    C
    E
    ͱͯ͠දݱ
    ü  ୹͍ઢ
    ˠ
    OPJTZ
    ü  ௕͍ઢ
    ˠ
    SPCVTU
    ü  ର֯ઢʹ͍ۙ఺
    ˠ
    OPJTZ
    ü  ର֯ઢ͔Βԕ͍఺
    ˠ
    SPCVTU
    t = 0 t = 1 t = 2 t = 3 t = 4 39 / 43
40. Ԡ༻ྫɿλϯύΫ࣭ͷߏ଄ղੳ • λϯύΫ࣭ͷѹॖ཰ʹ૬ؔͷ͋ΔྔΛύʔγεςϯτਤ͔Βநग़ assumption 1
    ѹॖ཰͸಺෦ͷۭಎʹؔ܎
    assumption 2
    assumption 3
    assumption 4
    ण໋ͷ୹͍ੜ੒ݩ͸τϙϩδΧϧͳ ϊΠζͰ͋Γɼѹॖ཰ʹد༩͠ͳ͍
    ີͳݪࢠ഑ஔ͸ܗঢ়͕มԽ͠ʹ͍͘
 ʢѹॖ཰΁ͷد༩͕খ͍͞ʣ
    ԁ౵ྖҬͷ࣠ํ޲ͷ௕͕͞௕͍৔߹ ΑΓ޿͍಺෦ྖҬΛ࣋ͭͨΊѹॖ཰ ΁ͷد༩͸େ͖͍
    ୹໋ͷੜ੒ݩ
    ௕໋ͷੜ੒ݩ
    ૄͳݪࢠ഑ஔ
    ີͳݪࢠ഑ஔ
    40 / 43
41. Ԡ༻ྫɿλϯύΫ࣭ͷߏ଄ղੳ • લทͷԾఆΛ౿·͑ɼҎԼͷྔ Cp Λಋೖ Cp(l1, u1, l2, u2, δ) = |PD2(l2, u2, δ)| |PD1(l1, u1, δ)| with PDk(lk, uk, δ) = {(pb, pd) ∈ PDk | pb ∈ [lk, uk]} with Nk(δ) = {(pb, pd) ∈ PDk | pd − pb ≤ δ} • ݁ՌɿCp ͱ࣮ݧ஋ͷ૬ؔ • http://www2.math.kyushu-u.ac.jp/ ∼nishii/FujitsuWS/Hiraoka.pdf ΑΓҾ༻ 41 / 43 ѹॖ཰ C p ͷؒ
42. Ԡ༻ྫɿλϯύΫ࣭ͷߏ଄ղੳ • λϯύΫ࣭ͷߏ଄ʹج͍ͮͯਐԽܥ౷थΛߏங • 3 छྨͷௗྨٴͼ 4 छྨͷᄡೕྨͷϔϞάϩϏϯσʔλʹ͍ͭͯ ύʔγεςϯτਤΛܭࢉ͠ɼͦͷؒͷڑ཭Λ΋ͱʹܥ౷थΛߏங • 2 ͭͷϑΟϧτϨʔγϣϯ K, L ʹର͢Δύʔγεςϯτਤͷ ڑ཭ʢϘτϧωοΫڑ཭ʣ͸ҎԼͰܭࢉ d(K, L) = inf γ sup p∈P Dk(K) ∥p − γ(p)∥∞ where PDk (K) := PDk (K) ∪ {(a, a) | a ∈ R} • ݁Ռ 42 / 43 Πϯί
    ΠϯυΨϯ
    Ξώϧ
    ϊ΢αΪ
    ϒλ
    ϠΪ
    ωζϛ
    
43. ࢀߟจݙ • ฏԬ༟ষʮλϯύΫ࣭ߏ଄ͱτϙϩδʔɿύʔγεςϯτϗϞϩδʔ ܈ೖ໳ʯڞཱग़൛, 2013. • G. Carlsson, “Topology and data,” Bulletin of the American Mathematical Society 46.2, pp. 255–308, 2009. • F. Chazal and B. Michel, “An introduction to topological data analysis: fundamental and practical aspects for data scientists,” arXiv preprint, arXiv:1710.04019, 2017. 43 / 43