tohoku_prob_seminar_20170519

Slide for tohoku prob seminar

Transcript

1. Ґ૬తσʔλղੳͷͨΊͷΧʔωϧ๏ ૲໺ݩل (౦๺େֶେֶӃཧֶݚڀՊ਺ֶઐ߈ D2) ڞಉݚڀऀ ෱ਫ݈࣍ (౷ܭ਺ཧݚڀॴ), ฏԬ༟ষ (౦๺େֶAIMR) ౦๺େ֬཰࿦ηϛφʔ 2017/5/19

2. Motivation of this work σʔλղੳͷྲྀΕ 1.ೖྗσʔλ : γϦΧΨϥε(ɹɹ ), ͨΜͺ࣭͘, ը૾σʔλ, …
 2.਺ࣜԽ : 
 3.ಛ௃ྔநग़ :
 ɹɹɹɹɹɹମੵ, Թ౓, ѹྗ, …
 4.౷ܭղੳ : มԽ఺ݕग़, ओ੒෼෼ੳ, SVM, … SiO2 X = { xi 2 R3} f : R2!R ຊߨԋͷ಺༰
 Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Persistence weighted Gaussian kernel for topological data analysis, ICML, pp:2004-2013, 2016.
3. Motivation of this work σʔλղੳͷྲྀΕ 1.ೖྗσʔλ : γϦΧΨϥε(ɹɹ ), ͨΜͺ࣭͘, ը૾σʔλ, …
 2.਺ࣜԽ : 
 3.ಛ௃ྔநग़ :
 ɹɹɹɹɹɹମੵ, Թ౓, ѹྗ, …
 4.౷ܭղੳ : มԽ఺ݕग़, ओ੒෼෼ੳ, SVM, … SiO2 X = { xi 2 R3} f : R2!R ຊߨԋͷ಺༰
 Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Persistence weighted Gaussian kernel for topological data analysis, ICML, pp:2004-2013, 2016. ύʔγεςϯτϗϞϩδʔɺύʔγεςϯτਤ

4. −10 −5 0 5 10 15 −5 0 5 10 −15 −10 −5 0 5 10 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 30 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 30 γϦΧͷ֤ݪࢠͷ࠲ඪΛ఺ͱΈͳ͠, ͦΕΛ఺ू߹ͱͯ͠දݱ͢Δ. ͭ·Γ, σʔλͱͯ͠఺ू߹͕༩͑ΒΕ͍ͯΔͱߟ͑Δ
 (෺ཧֶత৘ใΛҰ੾ഉআ͢Δ). ͜ͷ࣌఺Ͱ͸, ӷମͱΨϥεΛݪࢠ഑ஔͷΈ͔Β۠ผ͢Δͷ͸ෆՄೳʹݟ ͑Δ(ࡐྉՊֶͷ෼໺Ͱ΋ඇৗʹࠔ೉ͳ໰୊, ΨϥεసҠ఺໰୊). Motivation of this work Ґ૬తσʔλղੳ(Topological Data Analysis, TDA) ਤܗͷ”ܗ”Λಛ௃෇͚Δσʔλղੳख๏ ӷମɹɹɹɹɹ Ψϥεঢ়ଶ ɹɹɹɹ ݻମ
5. −10 −5 0 5 10 15 −5 0 5 10 −15 −10 −5 0 5 10 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 30 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 30 γϦΧͷ֤ݪࢠͷ࠲ඪΛ఺ͱΈͳ͠, ͦΕΛ఺ू߹ͱͯ͠දݱ͢Δ. ͭ·Γ, σʔλͱͯ͠఺ू߹͕༩͑ΒΕ͍ͯΔͱߟ͑Δ
 (෺ཧֶత৘ใΛҰ੾ഉআ͢Δ). ͜ͷ࣌఺Ͱ͸, ӷମͱΨϥεΛݪࢠ഑ஔͷΈ͔Β۠ผ͢Δͷ͸ෆՄೳʹݟ ͑Δ(ࡐྉՊֶͷ෼໺Ͱ΋ඇৗʹࠔ೉ͳ໰୊, ΨϥεసҠ఺໰୊). Motivation of this work ʁ Ґ૬తσʔλղੳ(Topological Data Analysis, TDA) ਤܗͷ”ܗ”Λಛ௃෇͚Δσʔλղੳख๏ ӷମɹɹɹɹɹ Ψϥεঢ়ଶ ɹɹɹɹ ݻମ
6. Ґ૬తσʔλղੳ(Topological Data Analysis, TDA) ਤܗͷ”ܗ”Λಛ௃෇͚Δσʔλղੳख๏ −10 −5 0 5 10 15 −5 0 5 10 −15 −10 −5 0 5 10 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 30 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 30 ӷମɹɹɹɹɹ Ψϥεঢ়ଶ ɹɹɹɹ ݻମ Motivation of this work [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Y. Hiraoka et al., Hierarchical structures of amorphous solids characterized by persistent homology, PNAS, 2016. ʁ ʂ
7. Motivation of this work SiO2 X = { xi 2 R3} f : R2!R ຊߨԋͷ಺༰
 Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Persistence weighted Gaussian kernel for topological data analysis, ICML, pp:2004-2013, 2016. σʔλղੳͷྲྀΕ 1.ೖྗσʔλ : γϦΧΨϥε(ɹɹ ), ͨΜͺ࣭͘, ը૾σʔλ, …
 2.਺ࣜԽ : 
 3.ಛ௃ྔநग़ : ύʔγεςϯτϗϞϩδʔɺύʔγεςϯτਤ
 ɽ
 4.౷ܭղੳ : มԽ఺ݕग़, ओ੒෼෼ੳ, SVM, …
8. Motivation of this work SiO2 X = { xi 2 R3} f : R2!R ɾɾɾ §1 ɾɾɾ §3 ɾɾɾ §4 Χʔωϧ๏(§2)ʹΑΔϕΫτϧԽ ຊߨԋͷ಺༰
 Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Persistence weighted Gaussian kernel for topological data analysis, ICML, pp:2004-2013, 2016. σʔλղੳͷྲྀΕ 1.ೖྗσʔλ : γϦΧΨϥε(ɹɹ ), ͨΜͺ࣭͘, ը૾σʔλ, …
 2.਺ࣜԽ : 
 3.ಛ௃ྔநग़ : ύʔγεςϯτϗϞϩδʔɺύʔγεςϯτਤ
 ɽ
 4.౷ܭղੳ : มԽ఺ݕग़, ओ੒෼෼ੳ, SVM, …
9. Section 1 ύʔγεςϯτϗϞϩδʔ
10. X Xa Persistent homology ݩͷ఺σʔλͷ৘ใΛ͏·͘൓ө͢ΔΑ͏ͳ൒ܘΛબͿ͜ͱ͸೉͍͠ ύʔγεςϯτϗϞϩδʔͷಈػͷҰͭ
11. X Xa Persistent homology ݩͷ఺σʔλͷ৘ใΛ͏·͘൓ө͢ΔΑ͏ͳ൒ܘΛબͿ͜ͱ͸೉͍͠ ύʔγεςϯτϗϞϩδʔͷಈػͷҰͭ
12. X Xa Persistent homology ݩͷ఺σʔλͷ৘ใΛ͏·͘൓ө͢ΔΑ͏ͳ൒ܘΛબͿ͜ͱ͸೉͍͠ ύʔγεςϯτϗϞϩδʔͷಈػͷҰͭ
13. X Xa Persistent homology ݩͷ఺σʔλͷ৘ใΛ͏·͘൓ө͢ΔΑ͏ͳ൒ܘΛબͿ͜ͱ͸೉͍͠ ύʔγεςϯτϗϞϩδʔͷಈػͷҰͭ
14. X Xa Persistent homology ݩͷ఺σʔλͷ৘ใΛ͏·͘൓ө͢ΔΑ͏ͳ൒ܘΛબͿ͜ͱ͸೉͍͠ ύʔγεςϯτϗϞϩδʔͷಈػͷҰͭ
15. ͜ͷߨԋΛ௨ͯ͠, ɹɹɹɹΛڑ཭ۭؒͱ͢Δ. Persistent homology ఆٛ Ґ૬ۭؒͷྻ(΋͘͠͸, ୯ମෳମͷྻ)ɹɹɹɹɹɹɹɹɹ ɹ ͕ ೚ҙͷɹɹɹʹର͠ɹɹɹɹͱͳ͍ͬͯΔͱ͖, ΛϑΟϧτϨʔγϣϯ ͱ͍͏. F (M, dM ) a  b Fa ⇢ Fb ྫ1 ༗ݶݸͷ఺ू߹ɹɹɹɹɹɹɹɹɹɹɹ ͔Β࡞ΒΕΔٿϞσϧ ɹɹɹɹɹɹ ͸ filtration ɹɹɹɹɹɹ Λ࡞Δ. (લͷεϥΠυͷྫ) ͨͩ͠, ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ. ͜͜Ͱ͸, ٿϞσϧϑΟϧτϨʔγϣϯͱ͍͏. ※ Cech complex ΍ alpha complex ͱhomotopy ಉ஋. B ( x ; a ) := { y 2 M | dM ( x, y )  a } B(X) = {Xa } Xa := S i B ( xi; a ) X = { x1, . . . , xn } ⇢ M F = {Fa | a = 1, . . . , N}
16. Persistent homology ྫ2 ؔ਺ɹɹɹɹ ͔Βఆ·Δ αϒϨϕϧू߹ϑΟϧτϨʔγϣϯΛ ɹɹɹɹɹɹɹɹɹɹɹ ͱͯ͠ఆΊΔ. ※ ɹɹɹɹɹɹɹɹ ͱ͢Δͱɹɹɹɹɹɹɹɹ Ͱ͋Δ. f : M!R Sub(f) = {f 1(( 1, a])} f ( z ) = inf dM ( z, xi) f 1(( 1, a]) = Xa
17. Persistent homology ྫ2 ؔ਺ɹɹɹɹ ͔Βఆ·Δ αϒϨϕϧू߹ϑΟϧτϨʔγϣϯΛ ɹɹɹɹɹɹɹɹɹɹɹ ͱͯ͠ఆΊΔ. ※ ɹɹɹɹɹɹɹɹ ͱ͢Δͱɹɹɹɹɹɹɹɹ Ͱ͋Δ. f : M!R Sub(f) = {f 1(( 1, a])} f ( z ) = inf dM ( z, xi) f 1(( 1, a]) = Xa f 0 255 άϨʔεέʔϧը૾ͷαϒϨϕϧू߹
18. Persistent homology ఆٛ ϑΟϧτϨʔγϣϯɹɹɹɹɹɹɹɹɹɹɹ͔Βఆ·Δܥྻ ΛύʔγεςϯτϗϞϩδʔ(persistent homology)ͱ͍͏. ͨͩ͠, ɹɹɹ͸ɹ࣍ݩͷମɹ ܎਺ϗϞϩδʔ܈ɹɹɹɹɹ , ࣹ͸แؚؔ ܎͔Β༠ಋ͞ΕΔ΋ͷͱ͢Δ.ɹ H(Fi) Hq(Fi; K) K q H(F) : H(F1) ! H(F2) ! · · · ! H(FN ) F = {Fa | a = 1, . . . , N}
19. Persistent homology ఆٛ ϑΟϧτϨʔγϣϯɹɹɹɹɹɹɹɹɹɹɹ͔Βఆ·Δܥྻ ΛύʔγεςϯτϗϞϩδʔ(persistent homology)ͱ͍͏. ͨͩ͠, ɹɹɹ͸ɹ࣍ݩͷମɹ ܎਺ϗϞϩδʔ܈ɹɹɹɹɹ , ࣹ͸แؚؔ ܎͔Β༠ಋ͞ΕΔ΋ͷͱ͢Δ.ɹ H(Fi) Hq(Fi; K) K q H(F) : H(F1) ! H(F2) ! · · · ! H(FN ) ※1 : ఺ू߹ ɹ͕༗ݶू߹ͷ৔߹΍ؔ਺ɹ ͕Morse ؔ਺ͷ৔߹͸ରԠ͢Δ ϑΟϧτϨʔγϣϯͷ homotopy ͷมԽ͸༗ݶճʹͳΔ. (tameness) ※2 : ௚࿨ʹͨ͠ɹ -ϕΫτϧۭؒɹɹɹɹɹɹɹɹɹ ͸࡞༻Λద੾ʹ ఆΊΔͱଟ߲ࣜ؀ɹɹ ্ͷՃ܈ͱͳΔ. ࣍Ͱ঺հ͢Δ෼ղఆཧͷ͜ͱ Λߟ͑Δͱ, PID্ͷ༗ݶੜ੒Ճ܈ͷ෼ղఆཧͷͨΊʹ, ܎਺ɹ ͸ମ Ͱ͋Δ͜ͱ͕๬·͍͠[ฏԬ, 16]. X f K K[t] PH(F) = L i H(Fi) K F = {Fa | a = 1, . . . , N}
20. Persistent homology ύʔγεςϯτϗϞϩδʔ͸ɹ - quiverͷදݱͱݟͳͤΔ. ͜ͷͱ͖, Krull-Remak-SchimidtͱGabrielͷఆཧʹΑΓ࣍ͷΑ͏ʹҰҙʹ෼ղͰ͖Δ: AN H(F) ⇠ = L i2I I[bi , di] I[bi, di] : 0 ! 0 ! · · · ! 0 ! K 1 ! K 1 ! · · · 1 ! K ! 0 ! · · · ! 0 Xa ྫ : ٿϞσϧϑΟϧτϨʔγϣϯͷύʔγεςϯτϗϞϩδʔ ൒ܘύϥϝʔλ 0 ! K ! K ! K ! 0 0 ! 0 ! K ! 0 ! 0 bi di H(B(X)) ⇠ = I[2, 5] I[3, 4]
21. Persistent homology ύʔγεςϯτϗϞϩδʔ͸ɹ - quiverͷදݱͱݟͳͤΔ. ͜ͷͱ͖, Krull-Remak-SchimidtͱGabrielͷఆཧʹΑΓ࣍ͷΑ͏ʹҰҙʹ෼ղͰ͖Δ: AN H(F) ⇠ = L i2I I[bi , di] I[bi, di] : 0 ! 0 ! · · · ! 0 ! K 1 ! K 1 ! · · · 1 ! K ! 0 ! · · · ! 0 Xa ྫ : ٿϞσϧϑΟϧτϨʔγϣϯͷύʔγεςϯτϗϞϩδʔ ൒ܘύϥϝʔλ 0 ! K ! K ! K ! 0 0 ! 0 ! K ! 0 ! 0 bi di H(B(X)) ⇠ = I[2, 5] I[3, 4]
22. Persistent homology ύʔγεςϯτϗϞϩδʔ͸ɹ - quiverͷදݱͱݟͳͤΔ. ͜ͷͱ͖, Krull-Remak-SchimidtͱGabrielͷఆཧʹΑΓ࣍ͷΑ͏ʹҰҙʹ෼ղͰ͖Δ: AN H(F) ⇠ = L i2I I[bi , di] I[bi, di] : 0 ! 0 ! · · · ! 0 ! K 1 ! K 1 ! · · · 1 ! K ! 0 ! · · · ! 0 Xa ྫ : ٿϞσϧϑΟϧτϨʔγϣϯͷύʔγεςϯτϗϞϩδʔ ൒ܘύϥϝʔλ 0 ! K ! K ! K ! 0 0 ! 0 ! K ! 0 ! 0 bi di H(B(X)) ⇠ = I[2, 5] I[3, 4]
23. Persistent homology ύʔγεςϯτϗϞϩδʔ͸ɹ - quiverͷදݱͱݟͳͤΔ. ͜ͷͱ͖, Krull-Remak-SchimidtͱGabrielͷఆཧʹΑΓ࣍ͷΑ͏ʹҰҙʹ෼ղͰ͖Δ: AN H(F) ⇠ = L i2I I[bi , di] I[bi, di] : 0 ! 0 ! · · · ! 0 ! K 1 ! K 1 ! · · · 1 ! K ! 0 ! · · · ! 0 Xa ྫ : ٿϞσϧϑΟϧτϨʔγϣϯͷύʔγεςϯτϗϞϩδʔ ൒ܘύϥϝʔλ 0 ! K ! K ! K ! 0 0 ! 0 ! K ! 0 ! 0 bi di H(B(X)) ⇠ = I[2, 5] I[3, 4]
24. Persistent homology ύʔγεςϯτϗϞϩδʔ͸ɹ - quiverͷදݱͱݟͳͤΔ. ͜ͷͱ͖, Krull-Remak-SchimidtͱGabrielͷఆཧʹΑΓ࣍ͷΑ͏ʹҰҙʹ෼ղͰ͖Δ: AN H(F) ⇠ = L i2I I[bi , di] I[bi, di] : 0 ! 0 ! · · · ! 0 ! K 1 ! K 1 ! · · · 1 ! K ! 0 ! · · · ! 0 Xa ྫ : ٿϞσϧϑΟϧτϨʔγϣϯͷύʔγεςϯτϗϞϩδʔ ൒ܘύϥϝʔλ 0 ! K ! K ! K ! 0 0 ! 0 ! K ! 0 ! 0 bi di H(B(X)) ⇠ = I[2, 5] I[3, 4]
25. ఆٛ ύʔγεςϯτϗϞϩδʔͷ෼ղɹɹɹɹɹɹɹɹɹɹ ͔Βఆ·Δଟॏ ू߹ Λύʔγεςϯτਤ(persistence diagram)ͱ͍͏. Persistent homology H(F) ⇠ = L i2I I[bi , di] Xa ྫ : ٿϞσϧϑΟϧτϨʔγϣϯͷύʔγεςϯτਤ H(B(X)) ⇠ = I[2, 5] I[3, 4] D(B(X)) = {(2, 5), (3, 4)} D(F) = {(bi, di) | i 2 I} 
    
    
    
    
    
    
    
    
    
    
     
    
    
    
     
26. ύʔγεςϯτਤͷಛ௃ Persistent homology 1 : ֤఺ɹɹɹɹɹɹɹɹ Λ(ൃੜ࣌ؒ, ফ໓࣌ؒ)ͷ૊ͱݟͳ͢ͱ, ͦͷ ੜଘ࣌ؒ ɹɹɹɹɹɹɹ͸ϗϞϩδʔͷੜ੒ݩͷϩόετੑΛද͢. 2 : ଟ͘ͷ৔߹, ର֯ઢʹ͍ۙ఺͸ϊΠζͱݟͳ͢(ੜଘ͕࣌ؒ୹͍). 3 : ൃੜͯ͠ফ໓͢Δੜ੒ݩ͸Ұͭͱ͸ݶΒͳ͍ͨΊ, ύʔγεςϯτਤ͸ ͷଟॏू߹Ͱ͋Δ. pers( x ) = d b x = ( b, d ) 2 D (F) R2
27. [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ӷମɹɹɹɹɹ Ψϥεঢ়ଶ ɹɹɹɹ ݻମ ύʔγεςϯτਤͷಛ௃ Persistent homology 1 : ֤఺ɹɹɹɹɹɹɹɹ Λ(ൃੜ࣌ؒ, ফ໓࣌ؒ)ͷ૊ͱݟͳ͢ͱ, ͦͷ ੜଘ࣌ؒ ɹɹɹɹɹɹɹ͸ϗϞϩδʔͷੜ੒ݩͷϩόετੑΛද͢. 2 : ଟ͘ͷ৔߹, ର֯ઢʹ͍ۙ఺͸ϊΠζͱݟͳ͢(ੜଘ͕࣌ؒ୹͍). 3 : ൃੜͯ͠ফ໓͢Δੜ੒ݩ͸Ұͭͱ͸ݶΒͳ͍ͨΊ, ύʔγεςϯτਤ͸ ͷଟॏू߹Ͱ͋Δ. pers( x ) = d b x = ( b, d ) 2 D (F) R2
28. [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ӷମɹɹɹɹɹ Ψϥεঢ়ଶ ɹɹɹɹ ݻମ ύʔγεςϯτਤͷಛ௃ Persistent homology 1 : ֤఺ɹɹɹɹɹɹɹɹ Λ(ൃੜ࣌ؒ, ফ໓࣌ؒ)ͷ૊ͱݟͳ͢ͱ, ͦͷ ੜଘ࣌ؒ ɹɹɹɹɹɹɹ͸ϗϞϩδʔͷੜ੒ݩͷϩόετੑΛද͢. 2 : ଟ͘ͷ৔߹, ର֯ઢʹ͍ۙ఺͸ϊΠζͱݟͳ͢(ੜଘ͕࣌ؒ୹͍). 3 : ൃੜͯ͠ফ໓͢Δੜ੒ݩ͸Ұͭͱ͸ݶΒͳ͍ͨΊ, ύʔγεςϯτਤ͸ ͷଟॏू߹Ͱ͋Δ. pers( x ) = d b x = ( b, d ) 2 D (F) R2 ϊΠζ
29. [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ӷମɹɹɹɹɹ Ψϥεঢ়ଶ ɹɹɹɹ ݻମ ύʔγεςϯτਤͷಛ௃ Persistent homology 1 : ֤఺ɹɹɹɹɹɹɹɹ Λ(ൃੜ࣌ؒ, ফ໓࣌ؒ)ͷ૊ͱݟͳ͢ͱ, ͦͷ ੜଘ࣌ؒ ɹɹɹɹɹɹɹ͸ϗϞϩδʔͷੜ੒ݩͷϩόετੑΛද͢. 2 : ଟ͘ͷ৔߹, ର֯ઢʹ͍ۙ఺͸ϊΠζͱݟͳ͢(ੜଘ͕࣌ؒ୹͍). 3 : ൃੜͯ͠ফ໓͢Δੜ੒ݩ͸Ұͭͱ͸ݶΒͳ͍ͨΊ, ύʔγεςϯτਤ͸ ͷଟॏू߹Ͱ͋Δ. pers( x ) = d b x = ( b, d ) 2 D (F) R2 ϊΠζ ಛ௃త
30. [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ӷମɹɹɹɹɹ Ψϥεঢ়ଶ ɹɹɹɹ ݻମ ύʔγεςϯτਤͷಛ௃ Persistent homology 1 : ֤఺ɹɹɹɹɹɹɹɹ Λ(ൃੜ࣌ؒ, ফ໓࣌ؒ)ͷ૊ͱݟͳ͢ͱ, ͦͷ ੜଘ࣌ؒ ɹɹɹɹɹɹɹ͸ϗϞϩδʔͷੜ੒ݩͷϩόετੑΛද͢. 2 : ଟ͘ͷ৔߹, ର֯ઢʹ͍ۙ఺͸ϊΠζͱݟͳ͢(ੜଘ͕࣌ؒ୹͍). 3 : ൃੜͯ͠ফ໓͢Δੜ੒ݩ͸Ұͭͱ͸ݶΒͳ͍ͨΊ, ύʔγεςϯτਤ͸ ͷଟॏू߹Ͱ͋Δ. pers( x ) = d b x = ( b, d ) 2 D (F) R2 ϊΠζ ಛ௃త 500ݸ 15000ݸ
31. 1 : σʔλ 2 : ϑΟϧτϨʔγϣϯ 3 : ύʔγεςϯτϗϞϩδʔ 4 : ύʔγεςϯτਤ ͜͜·Ͱͷ·ͱΊ Persistent homology ύʔγεςϯτਤΛܭࢉ͢Διϑτ΢ΣΞ͸։ൃ͞Ε͍ͯΔ: HomCloud : http://www.wpi-aimr.tohoku.ac.jp/hiraoka_labo/homcloud.html B(X), Sub(f), . . . X = { x1, . . . , xn } , f : M !R , . . . H(B(X)), H(Sub(f)), . . . D(B(X)), D(Sub(f)), . . .
32. 1 : σʔλ 2 : ϑΟϧτϨʔγϣϯ 3 : ύʔγεςϯτϗϞϩδʔ 4 : ύʔγεςϯτਤ ͜͜·Ͱͷ·ͱΊ Persistent homology ύʔγεςϯτਤΛܭࢉ͢Διϑτ΢ΣΞ͸։ൃ͞Ε͍ͯΔ: HomCloud : http://www.wpi-aimr.tohoku.ac.jp/hiraoka_labo/homcloud.html B(X), Sub(f), . . . X = { x1, . . . , xn } , f : M !R , . . . H(B(X)), H(Sub(f)), . . . D(B(X)), D(Sub(f)), . . . dI dB, dWp ͦΕͧΕͷڑ཭ͷԼͰ࿈ଓੑΛٞ࿦Ͱ͖Δ (زԿηϛφʔͰߨԋ༧ఆ, 10/10) dH, k·k 1
33. ౷ܭతҐ૬తσʔλղੳΛ໨ࢦͯ͠ Statistical TDA ύʔγεςϯτਤͷมԽΛݟΔ͜ͱ Ͱ, Կ͔ݴ͑Δ͔΋͠Εͳ͍. ΋͠, ӷମ-ΨϥεసҠ఺ʹ͍ͭͯؔ ࿈Λݟग़ͤΔͱͱͯ΋خ͍͠. ӷମ͔ΒΨϥε΁ͷมԽ
34. ౷ܭతҐ૬తσʔλղੳΛ໨ࢦͯ͠ Statistical TDA ύʔγεςϯτਤͷมԽΛݟΔ͜ͱ Ͱ, Կ͔ݴ͑Δ͔΋͠Εͳ͍. ΋͠, ӷମ-ΨϥεసҠ఺ʹ͍ͭͯؔ ࿈Λݟग़ͤΔͱͱͯ΋خ͍͠. ӷମ͔ΒΨϥε΁ͷมԽ
35. ౷ܭతҐ૬తσʔλղੳΛ໨ࢦͯ͠ Statistical TDA ύʔγεςϯτਤͷมԽΛݟΔ͜ͱ Ͱ, Կ͔ݴ͑Δ͔΋͠Εͳ͍. ΋͠, ӷମ-ΨϥεసҠ఺ʹ͍ͭͯؔ ࿈Λݟग़ͤΔͱͱͯ΋خ͍͠. ਺ཧతʹ৘ใΛऔΓग़ͨ͢Ίʹ, ౷ܭతͳ࿮૊ΈΛ࡞Γ͍ͨ. ӷମ͔ΒΨϥε΁ͷมԽ
36. ౷ܭతҐ૬తσʔλղੳΛ໨ࢦͯ͠ Statistical TDA ύʔγεςϯτਤͷมԽΛݟΔ͜ͱ Ͱ, Կ͔ݴ͑Δ͔΋͠Εͳ͍. ΋͠, ӷମ-ΨϥεసҠ఺ʹ͍ͭͯؔ ࿈Λݟग़ͤΔͱͱͯ΋خ͍͠. ਺ཧతʹ৘ใΛऔΓग़ͨ͢Ίʹ, ౷ܭతͳ࿮૊ΈΛ࡞Γ͍ͨ. Χʔωϧ๏ʹΑΔύʔγεςϯτਤͷ౷ܭཧ࿦Λߏ੒͢Δ. ӷମ͔ΒΨϥε΁ͷมԽ
37. Section 2 Χʔωϧ๏
38. Kernel method σʔλͷू߹Λɹ ɹɹɹɹɹɹɹ ͱ͢Δ(ޙͰ, ύʔγεςϯτਤͷ ू·ΓΛࢥ͏). ͜͜Ͱ͸, ू߹Ҏ֎ͷߏ଄͸Կ΋૝ఆ͠ͳ͍. ͨͩ, ౷ܭత ͳ͜ͱΛߟ͑Δࡍʹ͸͋Δఔ౓ͷߏ଄͕ཉ͍͠. D = {D1, . . . , Dn } ฏۉ? 1 n P Di
39. Kernel method σʔλͷू߹Λɹ ɹɹɹɹɹɹɹ ͱ͢Δ(ޙͰ, ύʔγεςϯτਤͷ ू·ΓΛࢥ͏). ͜͜Ͱ͸, ू߹Ҏ֎ͷߏ଄͸Կ΋૝ఆ͠ͳ͍. ͨͩ, ౷ܭత ͳ͜ͱΛߟ͑Δࡍʹ͸͋Δఔ౓ͷߏ଄͕ཉ͍͠. D = {D1, . . . , Dn } ϕΫτϧۭؒɹ ͱϕΫτϧԽͷࣸ૾ɹɹɹɹɹΛ༻ҙ͢Δ. ͦ͏͢Δͱ, ฏۉͳͲΛͦͷϕΫτϧۭؒͰߟ͑Δ͜ͱ͕Ͱ͖Δ: H : D!H 1 n P (Di) 2 H ฏۉ? 1 n P Di
40. Kernel method σʔλͷू߹Λɹ ɹɹɹɹɹɹɹ ͱ͢Δ(ޙͰ, ύʔγεςϯτਤͷ ू·ΓΛࢥ͏). ͜͜Ͱ͸, ू߹Ҏ֎ͷߏ଄͸Կ΋૝ఆ͠ͳ͍. ͨͩ, ౷ܭత ͳ͜ͱΛߟ͑Δࡍʹ͸͋Δఔ౓ͷߏ଄͕ཉ͍͠. D = {D1, . . . , Dn } ϕΫτϧۭؒɹ ͱϕΫτϧԽͷࣸ૾ɹɹɹɹɹΛ༻ҙ͢Δ. ͦ͏͢Δͱ, ฏۉͳͲΛͦͷϕΫτϧۭؒͰߟ͑Δ͜ͱ͕Ͱ͖Δ: H : D!H 1 n P (Di) 2 H ฏۉ? 1 n P Di ߟ͑Δ΂͖͜ͱ 1 : ͦͷϕΫτϧԽͷۭؒͱࣸ૾͸਺ֶతʹѻ͍΍͍͢΋ͷ 2 : σʔλղੳͷࡍʹྑ͍ੑ࣭Λࣔ͢΋ͷ 3 : ܭࢉίετ͕ΑΓগͳ͍΋ͷ
41. Kernel method σʔλͷू߹Λɹ ɹɹɹɹɹɹɹ ͱ͢Δ(ޙͰ, ύʔγεςϯτਤͷ ू·ΓΛࢥ͏). ͜͜Ͱ͸, ू߹Ҏ֎ͷߏ଄͸Կ΋૝ఆ͠ͳ͍. ͨͩ, ౷ܭత ͳ͜ͱΛߟ͑Δࡍʹ͸͋Δఔ౓ͷߏ଄͕ཉ͍͠. D = {D1, . . . , Dn } ϕΫτϧۭؒɹ ͱϕΫτϧԽͷࣸ૾ɹɹɹɹɹΛ༻ҙ͢Δ. ͦ͏͢Δͱ, ฏۉͳͲΛͦͷϕΫτϧۭؒͰߟ͑Δ͜ͱ͕Ͱ͖Δ: H : D!H 1 n P (Di) 2 H ฏۉ? 1 n P Di ߟ͑Δ΂͖͜ͱ 1 : ͦͷϕΫτϧԽͷۭؒͱࣸ૾͸਺ֶతʹѻ͍΍͍͢΋ͷ 2 : σʔλղੳͷࡍʹྑ͍ੑ࣭Λࣔ͢΋ͷ 3 : ܭࢉίετ͕ΑΓগͳ͍΋ͷ ͍͍ؔ਺(ਖ਼ఆ஋Χʔωϧ) ɹɹɹɹ Λ༻ҙ͢Δͱ͜Ͱղܾ͢Δ. K : D ⇥ D!R
42. Kernel method ਖ਼ఆ஋ΧʔωϧΛ ɹɹɹɹ ͱ͢Δ. 1 : ͦͷϕΫτϧԽͷۭؒͱࣸ૾͸਺ֶతʹѻ͍΍͍͢΋ͷ 2 : σʔλղੳͷࡍʹྑ͍ੑ࣭Λࣔ͢΋ͷ 3 : ܭࢉίετ͕ΑΓগͳ͍΋ͷ K : D ⇥ D!R
43. Kernel method ਖ਼ఆ஋ΧʔωϧΛ ɹɹɹɹ ͱ͢Δ. 1 : ͦͷϕΫτϧԽͷۭؒͱࣸ૾͸਺ֶతʹѻ͍΍͍͢΋ͷ 2 : σʔλղੳͷࡍʹྑ͍ੑ࣭Λࣔ͢΋ͷ 3 : ܭࢉίετ͕ΑΓগͳ͍΋ͷ K : D ⇥ D!R ਖ਼ఆ஋Χʔωϧɹ ͸ࣗવʹώϧϕϧτۭؒɹ Λؔ਺ۭؒɹɹ ͷ ෦෼ۭؒͱͯ͠ఆΊΔ. ϕΫτϧԽ͸࣍ͷΑ͏ʹఆ·Δ. K HK C(D) (Di) = K(·, Di) 2 HK ⇢ C(D)
44. Kernel method ਖ਼ఆ஋ΧʔωϧΛ ɹɹɹɹ ͱ͢Δ. 1 : ͦͷϕΫτϧԽͷۭؒͱࣸ૾͸਺ֶతʹѻ͍΍͍͢΋ͷ 2 : σʔλղੳͷࡍʹྑ͍ੑ࣭Λࣔ͢΋ͷ 3 : ܭࢉίετ͕ΑΓগͳ͍΋ͷ K : D ⇥ D!R ଟ͘ͷ৔߹, ౷ܭղੳͰඞཁͱͳΔͷ͸ߦྻ ɹɹɹɹ Ͱ͋Δ. (K(Di, Dj))i,j=1,...,n ౷ܭղੳख๏ 0 B @ K(D1, D1) · · · K(D1, Dn) . . . ... . . . K(Dn, D1) · · · K(Dn, Dn) 1 C A D Di K ਖ਼ఆ஋Χʔωϧɹ ͸ࣗવʹώϧϕϧτۭؒɹ Λؔ਺ۭؒɹɹ ͷ ෦෼ۭؒͱͯ͠ఆΊΔ. ϕΫτϧԽ͸࣍ͷΑ͏ʹఆ·Δ. K HK C(D) (Di) = K(·, Di) 2 HK ⇢ C(D)
45. Kernel method ਖ਼ఆ஋ΧʔωϧΛ ɹɹɹɹ ͱ͢Δ. 1 : ͦͷϕΫτϧԽͷۭؒͱࣸ૾͸਺ֶతʹѻ͍΍͍͢΋ͷ 2 : σʔλղੳͷࡍʹྑ͍ੑ࣭Λࣔ͢΋ͷ 3 : ܭࢉίετ͕ΑΓগͳ͍΋ͷ K : D ⇥ D!R ଟ͘ͷ৔߹, ౷ܭղੳͰඞཁͱͳΔͷ͸ߦྻ ɹɹɹɹ Ͱ͋Δ. (K(Di, Dj))i,j=1,...,n ౷ܭղੳख๏ 0 B @ K(D1, D1) · · · K(D1, Dn) . . . ... . . . K(Dn, D1) · · · K(Dn, Dn) 1 C A D Di K (ແݶ࣍ݩͷՄೳੑ΋) (1࣍ݩ) ϕΫτϧͦͷ΋ͷ͸ෆඞཁͰ, ͦͷΧʔωϧͷ஋ͷΈॏཁ. ਖ਼ఆ஋Χʔωϧɹ ͸ࣗવʹώϧϕϧτۭؒɹ Λؔ਺ۭؒɹɹ ͷ ෦෼ۭؒͱͯ͠ఆΊΔ. ϕΫτϧԽ͸࣍ͷΑ͏ʹఆ·Δ. K HK C(D) (Di) = K(·, Di) 2 HK ⇢ C(D)
46. ɹɹɹɹͷ৔߹ͷਖ਼ఆ஋Χʔωϧͷྫ • ઢܗΧʔωϧ • ଟ߲ࣜΧʔωϧ • Ψ΢εRBFΧʔωϧ ⌦ = Rd kL( x, y ) = h x, y i kP ( x, y ) = (h x, y i + c )d kG( x, y ) = e k x y k2 2 2 ఆٛ ू߹ɹ ্ͷਖ਼ఆ஋Χʔωϧ ɹ ɹɹɹɹɹ ͱ͸࣍Λຬͨؔ͢਺Ͱ͋Δ. • ରশͰ͋Δ: • ೚ҙͷɹɹɹɹɹɹ ʹରͯ͠, ߦྻɹɹɹɹɹ
 ͕൒ਖ਼ఆ஋Ͱ͋Δ. (͜ͷߦྻΛάϥϜߦྻͱ͍͏) k : ⌦ ⇥ ⌦!R ⌦ k ( x, y ) = k ( y, x ) x1, . . . , xn 2 ⌦ ( k ( xi, xj))i,j=1,··· ,n Kernel method Χʔωϧ๏ʹ͍ͭͯ͸[෱ਫ, 14]Λࢀর
47. Moore-Aronszajn ͷఆཧ ਖ਼ఆ஋Χʔωϧɹ͸࣍Λຬͨ͢࠶ੜ֩ώϧϕϧτۭؒ (reproducing kernel Hilbert space, RKHS) ΛҰҙʹఆΊΔ. • ೚ҙͷ ɹʹରͯ͠, ɹɹɹɹ ɹ͸ ɹ ͷݩͰ͋Δ. • ϕΫτϧͷுΔۭؒ ͸ɹɹͰ᜚ີͰ͋Δ. • ೚ҙͷɹ ɹ ͱɹɹɹ ʹରͯ͠, ɹɹɹɹɹ ͕੒ཱ͢Δ(࠶ੜੑ). k Hk x 2 ⌦ k (· , x ) : ⌦!R Hk Hk x 2 ⌦ f 2 Hk h f, k (· , x )iHk = f ( x ) Span{ k (· , x ) | x 2 ⌦} Kernel method
48. Moore-Aronszajn ͷఆཧ ਖ਼ఆ஋Χʔωϧɹ͸࣍Λຬͨ͢࠶ੜ֩ώϧϕϧτۭؒ (reproducing kernel Hilbert space, RKHS) ΛҰҙʹఆΊΔ. • ೚ҙͷ ɹʹରͯ͠, ɹɹɹɹ ɹ͸ ɹ ͷݩͰ͋Δ. • ϕΫτϧͷுΔۭؒ ͸ɹɹͰ᜚ີͰ͋Δ. • ೚ҙͷɹ ɹ ͱɹɹɹ ʹରͯ͠, ɹɹɹɹɹ ͕੒ཱ͢Δ(࠶ੜੑ). k Hk x 2 ⌦ k (· , x ) : ⌦!R Hk Hk x 2 ⌦ f 2 Hk h f, k (· , x )iHk = f ( x ) Span{ k (· , x ) | x 2 ⌦} Kernel method ࠶ੜੑ͔Β಺ੵ͸࣍ͷΑ͏ʹܭࢉ͞ΕΔ: ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ h k (· , y ) , k (· , x )iHk = k ( x, y ) σʔλɹɹɹͷϕΫτϧදݱ͸ɹɹɹɹɹɹͰ͋Δ(RKHS ϕΫτϧ). x 2 ⌦ k (· , x ) 2 Hk
49. ͪΐͬͱ༨ஊ… ͳͥΧʔωϧ๏͸ศརͳͷ͔. Kernel method Representer ఆཧ σʔλͷ૊ɹɹɹɹɹɹɹɹ ɹɹɹ, ɹ ্ͷΧʔωϧ , جఈؔ਺ ɹɹɹɹɹɹɹɹ , ୯ௐ૿Ճؔ਺ɹɹɹɹɹɹʹର͠, ਖ਼ଇԽ߲෇͖ଛࣦ ؔ਺࠷খԽ໰୊ ͷ࠷దղ͸ɹɹɹɹɹ ɹɹͷܗͰද͞ΕΔ. ଟ͘ͷ໰୊͸͜ͷλΠϓͷ࠷దԽ໰୊Λղ͘͜ͱʹؼண͞ΕΔ. ͦͷ࣌, Represneter ఆཧ͸࠷దղ(͔͠΋ແݶ࣍ݩύϥϝʔλ)ΛσʔλͷுΔۭؒ ͷத͔Βݟ͚ͭΒΕΔ͜ͱΛอূ͍ͯ͠Δ!! ࠓྲྀߦΓͷdeep learning(ଟ૚ χϡʔϥϧωοτϫʔΫ)Ͱ͸࠷దղʹؔ͢Δ໰୊͕͋ͬͨ. X k {hi : X!R}i=1,...,m {(Xi, Yi) 2 X ⇥ Y}i=1,...,N : [0, 1)!R min f2Hk,c2Rm L 0 @{Xi }, {Yi }, {f(Xi) + m X j=1 cjhj(Xj)} 1 A + (kfkHk ) f = PN i=1 ↵ik(·, Xi)
50. ہॴίϯύΫτϋ΢ευϧϑۭؒɹ ্ͷ༗ݶූ߸෇͖ϥυϯଌ౓શମͷ ۭؒΛ ɹɹͱ͢Δ. ͜ͷ࣌, ଌ౓͸RKHSͷݩͱͯ͠දࣔͰ͖Δ: ͜ͷରԠͷ͜ͱΛΧʔωϧຒΊࠐΈ(kernel embedding)ͱ͍͏. µ 7! Ek( µ ) := Z k (· , x ) dµ ( x ) 2 Hk Mb(⌦) 3 Mb(⌦) ⌦ σʔλۭؒɹ ্ͷଌ౓(෼෍)΋ϕΫτϧԽ͢Δ͜ͱ͕Ͱ͖Δ. ⌦ Kernel method ※1 : ੵ෼͸Bochnerੵ෼(Banachۭؒʹ஋Λ࣋ͭؔ਺ͷੵ෼) ※2 : ܦݧ෼෍ʹΑΔຒΊࠐΈ͸Χʔωϧฏۉͱ΋͍͏: ※3 : ैདྷ͸, ೋͭͷ֬཰෼෍͕ಉ͔͡Ͳ͏͔ΛRKHSͰߟ͑ΔͨΊʹ ߟҊ͞Εͨ: µX = 1 n P xi ) Ek ( µX ) = 1 n P k (· , xi ) P = Q , Ek(P) = Ek(Q)
51. Ұൠʹ, ͸ɹ ্ͷٖڑ཭ΛఆΊΔ. Mb(⌦) Kernel method dk(µ, ⌫) = kEk(µ) Ek(⌫)kHk (µ, ⌫ 2 Mb(⌦))
52. Ұൠʹ, ͸ɹ ্ͷٖڑ཭ΛఆΊΔ. Mb(⌦) Kernel method dk(µ, ⌫) = kEk(µ) Ek(⌫)kHk (µ, ⌫ 2 Mb(⌦)) ໋୊ [Sriperumbudur et al., 11] ਖ਼ఆ஋Χʔωϧɹɹɹɹɹɹɹ ͕ C0-universalΛຬͨ͢ͱ͢Δ. ͜ͷͱ͖, ΧʔωϧຒΊࠐΈɹ ͸୯ࣹʹͳΓ, ͕ͨͬͯ͠ɹ ͸ڑ཭ΛఆΊΔ. k : ⌦ ⇥ ⌦!R Ek dk ※1 : C0-universalͱ͸ରԠ͢ΔRKHS͕ແݶԕ఺Ͱফ͑Δ࿈ଓؔ਺ͷͳ͢ ۭؒʹ᜚ີʹೖ͍ͬͯΔ͜ͱ: ※2 : ઢܗΧʔωϧɹɹɹɹɹɹɹ ͸C0-universalͰͳ͘, ɹɹΨ΢εΧʔωϧɹɹɹɹɹɹɹɹ͸C0-universalͰ͋Δ. Hk ⇢ C0(⌦) kL( x, y ) = h x, y i kG( x, y ) = e k x y k2 2 2 HkL ⇠ = Rd, dim HkG = 1 (⌦ = Rd)
53. ໨త : ύʔγεςϯτਤͷΧʔωϧΛ࡞Δ. ͜Ε͔ΒͷྲྀΕ 1 : ύʔγεςϯτਤΛଌ౓ͱݟͳ͢. 2 : ΧʔωϧຒΊࠐΈʹΑΓύʔγεςϯτਤͷRKHSϕΫτϧΛಘΔ. 3 : ͦͷRKHSϕΫτϧΛσʔλͱݟͳ͠, ΧʔωϧΛߏ੒͢Δ. ὎ ௨ৗͷΧʔωϧ๏(SVMʹΑΔ෼ྨ, ओ੒෼෼ੳ, มԽ఺ݕग़, …) ͜ͷͱ͖ʹ, ྑ͍ੑ࣭(ύʔγεςϯτਤʹಛԽ, ୯ࣹੑ, ࿈ଓੑ, …)Λ ࣋ͭΑ͏ʹߏ੒͢Δ. Kernel method ※ : 3ͰRKHSͷ಺ੵΛ࢖͏͜ͱΛઢܗΧʔωϧΛߏ੒͢Δ͜ͱͱಉ͡.
54. Section 3 ύʔγεςϯτਤͷਖ਼ఆ஋Χʔωϧ
55. Birth Death Birth Death ύʔγεςϯτਤ ͸ɹ ͷଟॏू߹Ͱ͋Δ͕, ֤఺ɹɹ Λ Diracͷdelta ଌ౓ɹ ʹஔ͖׵͑Δ͜ͱͰ, ଌ౓දࣔ ΛಘΔ. x Kernel on persistence diagram
56. ※ : ର֯ઢʹ͍ۙ఺͸ϊΠζͰ͋Δͱ ɹ ݟͳ͍ͨ͠. Birth Death Birth Death ύʔγεςϯτਤ ͸ɹ ͷଟॏू߹Ͱ͋Δ͕, ֤఺ɹɹ Λ Diracͷdelta ଌ౓ɹ ʹஔ͖׵͑Δ͜ͱͰ, ଌ౓දࣔ ΛಘΔ. x Kernel on persistence diagram
57. ద੾ͳॏΈؔ਺ Λಋೖ͢Δ͜ͱͰ, ύʔγεςϯτਤͷ ଌ౓දࣔΛॏΈ෇͖ͷ΋ͷͱͯ͠ಘΔ: Birth Death Birth Death w : R2!R ύʔγεςϯτਤ ͸ɹ ͷଟॏू߹Ͱ͋Δ͕, ֤఺ɹɹ Λ Diracͷdelta ଌ౓ɹ ʹஔ͖׵͑Δ͜ͱͰ, ଌ౓දࣔ ΛಘΔ. x Kernel on persistence diagram
58. Birth Death ੑ࣭1 Ψ΢εΧʔωϧΛબ୒͢Δͱ, C0-universalੑ͔ΒΧʔωϧຒΊࠐΈ͸ ୯ࣹʹͳΔ. (ࠞ߹Ψ΢ε෼෍ͷΑ͏ͳܗ) Birth Death ύʔγεςϯτਤͷଌ౓දࣔ Ͱdelta ଌ౓ΛΧʔωϧͰஔ͖׵͑Δ͜ ͱ͸, ΧʔωϧຒΊࠐΈΛͱΔ͜ͱʹର Ԡ͢Δ: Kernel on persistence diagram
59. ※ : ύʔγεςϯτਤɹɹ ͷ఺ͷݸ਺͕͓Αͦɹͱ͢Δͱ, ಺ੵΛٻΊΔ
 ͨΊʹ͸ɹɹ ճɹɹɹΛܭࢉ͢Δඞཁ͕͋Δ.
 ὎ Ψ΢εΧʔωϧʹରͯ͠͸, Random Fourier features 
 [Rahimi and Recht, 07]ʹΑΔߴ଎ۙࣅΞϧΰϦζϜʹΑΓ
 ܭࢉ࣌ؒ͸ɹ ʹؔ͠ઢܗʹམͪΔ. k Ek ( µw D ) Ek ( µw E )k2 Hk = X x 2 D X x 02 D w ( x ) w ( x 0) k ( x, x 0) + X y 2 E X y 02 E w ( y ) w ( y 0) k ( y, y 0) 2 X x 2 D X y 2 E w ( x ) w ( y ) k ( x, y ) RKHSϕΫτϧɹɹɹ ΛܭࢉػͰ௚઀දࣔ͢Δ͜ͱ͸ࠔ೉͕ͩ, ಺ੵ͸ܭ ࢉͰ͖Δ(ΧʔωϧτϦοΫ): ڑ཭΋಺ੵ͔Βܭࢉ͞ΕΔ: Ek(µw D ) Kernel on persistence diagram
60. Ψ΢εΧʔωϧɹɹɹ ɹɹ ʹؔ͢Δߟ࡯ RKHSϕΫτϧɹɹ ͸ɹɹɹɹ ͷؔ܎͔Βؔ਺ ͱ΋ݟΕΔ. ͦͯ͠, ॏΈؔ਺ ɹͷੑ࣭͕ྑ͚Ε͹ɹɹ ͷ಺ੵ͕ೖΔ: Ek(µw D ) Kernel on persistence diagram
61. Ψ΢εΧʔωϧɹɹɹ ɹɹ ʹؔ͢Δߟ࡯ RKHSϕΫτϧɹɹ ͸ɹɹɹɹ ͷؔ܎͔Βؔ਺ ͱ΋ݟΕΔ. ͦͯ͠, ॏΈؔ਺ ɹͷੑ࣭͕ྑ͚Ε͹ɹɹ ͷ಺ੵ͕ೖΔ: Ek(µw D ) Ψ΢εؔ਺ͷੑ࣭ͱͯ͠ൺྫͷؔ܎ࣜ ͕͋ΔͨΊ, RKHSͷ಺ੵͱ(ఆ਺ഒΛআ͍ͯ)ಉ͡ʹͳΔ. ܭࢉޮ཰͸Χʔωϧ๏ͷํ͕ѹ౗తʹྑ͍. Kernel on persistence diagram
62. ύʔγεςϯτਤͷਖ਼نԽʹؔ͠ ύʔγεςϯτਤͷॏΈ෇͚͢Δલͷଌ౓දࣔͱͯ͠ɹɹɹɹɹɹɹɹΛ ಘͨ. ͜Ε͸ύʔγεςϯτਤͷ఺ͷݸ਺ʹґଘ͢ΔͷͰ, ύʔγεςϯτ ਤ͕ೋͭҎ্͋ͬͨΒͦΕΒͷશ࣭ྔ(total mass)ͷ஋͸ҟͳΔ. ਖ਼نԽ͢Δͱtotal mass͸ಉ͡ʹͳΔ, ಛʹ֬཰ଌ౓ͱݟͳͤΔͨΊ֬཰࿦ ΍࠷ద༌ૹͳͲͷཧ࿦͕࢖͑Δ. ͔͠͠, ͦΕ͸ϊΠζͷӨڹʹҾͬுΒΕͯ͠·͏ͨΊಘࡦͰ͸ͳ͍. Kernel on persistence diagram ˜ µ D = 1 | D | P x 2 D x Birth Death Birth Death ࣅͯΔ?
63. ྲྀΕͷ෮श 0 : σʔλͷऔಘͱύʔγεςϯτਤͷܭࢉ 1 : ύʔγεςϯτਤΛଌ౓ͱݟͳ͢.
 2 : ΧʔωϧຒΊࠐΈʹΑΓύʔγεςϯτਤͷRKHSϕΫτϧΛಘΔ.
 3 : ͦͷRKHSϕΫτϧΛσʔλͱݟͳ͠, ΧʔωϧΛߏ੒͢Δ.
 ࣮ݧͰ͸RKHS্ͷΨ΢εΧʔωϧΛ࢖༻͢Δ. Kernel on persistence diagram
64. ྲྀΕͷ෮श 0 : σʔλͷऔಘͱύʔγεςϯτਤͷܭࢉ 1 : ύʔγεςϯτਤΛଌ౓ͱݟͳ͢.
 2 : ΧʔωϧຒΊࠐΈʹΑΓύʔγεςϯτਤͷRKHSϕΫτϧΛಘΔ.
 3 : ͦͷRKHSϕΫτϧΛσʔλͱݟͳ͠, ΧʔωϧΛߏ੒͢Δ.
 ࣮ݧͰ͸RKHS্ͷΨ΢εΧʔωϧΛ࢖༻͢Δ. ॏΈؔ਺ͱΧʔωϧ͸ ࣗ༝ʹઃܭͰ͖Δ Kernel on persistence diagram
65. ੜଘ࣌ؒɹɹɹɹɹɹɹɹɹɹɹ ͸ͦͷେখ͕ݩͷزԿߏ଄ͷେ͖͞Λ ද͍ͯ͠Δͱࢥ͑Δ. ͍͔ͭ͘ॏΈؔ਺ΛఏҊ͢Δ. ͨͩ͠, ɹɹɹɹ ͸ύϥϝʔλͰ͋Γঢ়گ΍σʔλʹԠͯ͡ద੾ ʹఆΊΔ(Cross-Validation, heuristic, ..). ͜ͷ࿮૊ΈͰ͸ (ཧ࿦্, ࣮༻্)”ྑ͍” ॏΈؔ਺ΛఆΊΔ͜ͱ͕େࣄ Kernel on persistence diagram
66. ఆཧ [Kusano, Fukumizu, Hiraoka, 16] ίϯύΫτ෦෼ۭؒɹɹɹɹ ͷ༗ݶ෦෼ू߹ɹɹ ʹରͯ͠, ΋͠
 ɹɹɹɹɹ ͳΒ͹, ͕࣍੒ཱ͢Δ: ͨͩ͠,ɹɹ ͸ɹ ʹґଘ͢Δఆ਺Ͱ͋Γ, M ⇢ Rd p > d + 1 X, Y ॏΈؔ਺ͱΧʔωϧΛద੾ʹબ୒͢Δͱ, σʔλͷઁಈʹ҆ఆͳRKHS ϕΫτϧΛಘΔ(㱻 ౷ܭղੳ݁Ռ͸ϊΠζʹେ͖͘Өڹ͞Εͳ͍). EkG (µwarc Dq (X) ) EkG (µwarc Dq (Y ) ) HkG  L(M, d; C, p, )dH(X, Y ) L ( M, d ; C, p, ) = p 2 p p ddiam( M ) + 4 p ( p 1) p 1 d ! CM diam( M ) p 1 dC dH ( X, Y ) = max ⇢ max x 2X min y 2Y dM (x , y) , max y 2Y min x 2X dM (x , y) CM M Kernel on persistence diagram
67. ূ໌ͷུ֓ ͍͔ͭ͘ͷ҆ఆੑఆཧΛ༻͍Δ: 1 : Hausdorff distance stability [Cohen-Stenier et al., 07, Chazal et al., 14] 2 : Interleaving distance stability [Bauer and Lesnick., 15, Chazal et al., 09] 3 : Bottleneck distance stability [Kusano, Fukumizu, Hiraoka, 16] dI(H(B(X)), H(B(Y )))  dH(X, Y ) dB(D(F), D(F0))  dI(H(F), H(F0)) EkG (µwarc Dq(X) ) EkG (µwarc Dq(Y ) ) HkG  L(M, d; C, p, )dB(D(X), D(Y )) Kernel on persistence diagram
68. ଟ͘ͷ৔߹, ౷ܭղੳͰඞཁͱͳΔͷ͸ߦྻ ɹɹɹɹ Ͱ͋Δ. (K(Di, Dj))i,j=1,...,n ౷ܭղੳख๏ 0 B @ K(D1, D1) · · · K(D1, Dn) . . . ... . . . K(Dn, D1) · · · K(Dn, Dn) 1 C A D Di K ΧʔωϧͱॏΈؔ਺ͱͯ͠ɹɹɹɹͷ૊ΈΛར༻ͨ͠RKHS্ͷΨ΢ε ΧʔωϧΛ persistence weighted Gaussian kernel (PWGK) ͱݺͿ: ͨͩ͠, ͸෼ࢄύϥϝʔλ. ύʔγεςϯτਤ্ʹΧʔωϧ͕ఆ·ͬͨͷͰ, ͲΜͳΧʔωϧ๏Ͱ΋ ద༻Մೳ (ػցֶशͷ࿮૊Έͷߏ੒ऴྃ). KPWG( D, E ) = exp ✓ 1 2 ⌧2 kEkG ( µwarc D ) EkG ( µwarc E ) k2 HkG ◆ (kG, warc) ⌧ Kernel on persistence diagram
69. Section 4 ػցֶश΁ͷԠ༻
70. ύʔγεςϯτਤͷϕΫτϧԽʹؔ͢ΔઌߦݚڀͱͷൺֱΛߦ͏. Persistence landscape [Bubenik, 15]
 ※1 : Persistence landscapeɹ ͸ɹɹɹɹɹ ؔ਺Ͱ͋Δ. ※2 : ܭࢉྔ͸ੵ෼ܭࢉ͕͋Δ෼PWGKΑΓଟ͍. Persistence scale-space kernel (PSSK) [Reininghaus et al., 15]
 ※1 : Persistence scale-space kernel͸PWGKͷҰྫͱ΋ݟͳͤΔ: ※2 : ॏΈؔ਺͸ݻఆ͞Ε͍ͯΔ. ※3 : Bottleneck distance stability Λຬͨ͞ͳ͍. Comparison with other vectorizations wpss( x ) = ( 1 x 2 D 1 x 2 D ⇤ D⇤ = {(d, b) | (b, d) 2 D} KPSS(· , D ) := X x 2 D [ D ⇤ wpss( x ) kG (· , x ) = D D 7! D( k, t ) = k -th largest value of max { min {t bi, di t}, 0 } D 7! D( y ) = P x 2 D e k x y k2 4 t e k x ¯ y k2 4 t (¯ y = ( y2, y1 ) for y = ( y1, y2 )) L2(N ⇥ R) D
71. ύʔγεςϯτਤͷϕΫτϧԽʹؔ͢ΔઌߦݚڀͱͷൺֱΛߦ͏. Persistence image [Adams et al., 17] ͨͩ͠,ɹɹɹɹɹɹɹ ɹɹɹͱ͠, ɹ ͸ɹΛฏۉʹ΋ͭ֬཰෼෍. ※1 : Persistence image͸ຊݚڀख๏ͷ཭ࢄԽͱࢥ͑Δ: ɹɹɹɹɹ ͱ͢Δͱ, ɹɹɹ ͸ɹɹɹ ΛάϦου ɹɹɹɹɹɹɹɹ ্Ͱੵ෼ͨ͠΋ͷ ※2 : ߦྻͳͷͰ, ༗ݶ࣍ݩ
 ϕΫτϧͱݟͳ͢ͱ ܭࢉ͕ૣ͍. ※3 : ϝογϡαΠζ͸ద੾ʹ બͿඞཁ͕͋Δ. Comparison with other vectorizations z( x ) = k ( x, z ) (ai 1, ai] ⇥ (aj 1, aj] Ek(µw D ) PI(D)i,j D 7! PI( D ) i,j = P x 2 D w ( x ) R Pi,j x ( z ) dz Pi,j := (ai 1, ai] ⇥ (aj 1, aj] z z M = 8 M = 20 M = 100
72. Synthesized data σʔλͷ࡞੒ ɹ : த৺ɹ ɹɹ൒ܘɹ ͷԁ͔Β ɹ ݸ఺Λऔͬͨ΋ͷʹ
 “randomness”ΛೖΕͨ΋ͷ ɹ : த৺ɹɹ ൒ܘɹ ͷԁ͔Β ɹ ݸ఺Λऔͬͨ΋ͷ. ఺σʔλɹ ͸ɹ ʹ֬཰0.5Ͱɹ Λ෇͚Ճ͑ͨ΋ͷͱ͠, 200σʔλɹɹɹɹɹ ੜ੒͢Δ. S2 S1 S1 S1 S2 r1 = 1 + 8W 2 (W ⇠ N(0, 1)), x1 = y1 = 1.5r1, N1 : a random integer with equal probability in ( d⇡r 2 e , 4⇡r), ( x1, y1) r1 N1 (0, 0) 0.2 10 X S1 S2 X1, . . . , X200
73. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α β γ Synthesized data S1 S2 Z0 Z1 ೋ஋෼ྨͷͨΊͷϥϕϧΛ ɹɹɹɹɹɿൃੜ1ҎԼফ໓4Ҏ্ͷੜ੒ݩ͕͋Δ(੨ͷྖҬʹ఺͕͋Δ) ɹɹɹɹɹɿখ͍͞ԁɹ ͕ଘࡏ͢Δ. ͔Βఆ·Δɹɹɹɹɹɹɹɹɹ ɹɹ ʹΑͬͯఆΊΔ. ਖ਼͘͠෼ྨ͢ΔͨΊʹ͸, େ͖͍݀΋খ͍݀͞΋ݟΔඞཁ͕͋Δ. Z0 = 1 Z1 = 1 S2 Y = 2XOR(Z0, Z1 ) 1
74. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ S1 S2 ༩͑ΒΕͨσʔλ ରԠ͢Δύʔγεςϯτਤ Synthesized data 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (0, 0) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (0, 1) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (1, 1) α 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 βγ (Z0, Z1) = (1, 0) Y = 1 Y = 1 Y = 1 Y = 1
75. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ S1 S2 ༩͑ΒΕͨσʔλ ରԠ͢Δύʔγεςϯτਤ Synthesized data 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (0, 0) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (0, 1) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (1, 1) α 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 βγ (Z0, Z1) = (1, 0) Y = 1 Y = 1 Y = 1 Y = 1
76. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ S1 S2 ༩͑ΒΕͨσʔλ ରԠ͢Δύʔγεςϯτਤ Synthesized data 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (0, 0) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (0, 1) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 α βγ (Z0, Z1) = (1, 1) α 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V1 V2 βγ (Z0, Z1) = (1, 0) Y = 1 Y = 1 Y = 1 Y = 1
77. Support vector machine αϙʔτϕΫλʔϚγϯ(support vector machine, SVM)ͷ෮श ϥϕϧ෇͖σʔλɹɹɹ ͕༩͑ΒΕͨ࣌ʹ, ൑ผؔ਺ ͷਖ਼ෛʹΑΓϥϕϧͷΘ͔Βͳ͍σʔλɹ ͷϥϕϧɹ Λਪఆ͢Δ. ܎਺ɹɹɹ͸ϥάϥϯδϡະఆ৐਺๏ͳͲͰٻΊΔ. ΧʔωϧͷੑೳࠩΛ෼ྨͷਖ਼౴཰ͰଌΔ 1 : ڭࢣσʔλͱͯ͠100ݸɹɹɹ ɹɹɹɹΛऔΓग़͢. 2 : ൑ผؔ਺ɹɹ Λڭࢣσʔλͱ༻ҙͨ͠Χʔωϧɹ ͔Β࡞੒͢Δ. 3 : ςετσʔλɹɹɹɹɹɹ ͔Βਪఆͨ͠ϥϕϧɹɹɹɹɹɹͱ, 
 ਅͷϥϕϧɹɹɹɹɹɹͱͷࠩͷݸ਺Λܭࢉ͠ਖ਼౴཰Λܭࢉ͢Δ. {(Xi, Yi)} y(X) = P i aiYiK(D(Xi), D(X)) + b ˆ Y X {ai }, b {(Xi, Yi)}i=1,...,100 y(X) {Yi }i=101,...,200 {ˆ Yi }i=101,...,200 {Xi }i=101,...,200 K
78. SVM results SVMʹΑΔ൑ผͷਖ਼౴཰ͱ༻͍ͨΧʔωϧͷද Linear Gaussian kernel weight w arc (p = 1) 67.6 ± 5.45 88.6 ± 2.19 w arc (p = 5) 68.4 ± 6.30 87.2 ± 4.71 k G w arc (p = 10) 68.8 ± 5.58 86.6 ± 5.02 w pers 55.4 ± 9.34 67.0 ± 11.7 w one 59.8 ± 5.54 64.0 ± 5.00 w arc (p = 5) 50.8 ± 8.16 51.4 ± 3.28 k L w pers 50.8 ± 8.16 46.4 ± 4.15 w one 50.8 ± 8.16 50.4 ± 5.17 PSSK 67.6 ± 9.71 79.2 ± 8.72 Persistence landscape 51.0 ± 7.51 47.2 ±2.04 Persistence image M = 100 w pers 54.0 ± 13.7 58.8 ± 10.3 M = 20 w arc (p = 5) 48.4 ± 4.15 52.0 ± 8.45 M = 50 w arc (p = 5) 48.0 ± 8.03 92.0 ± 2.00 M = 100 w arc (p = 5) 61.0 ± 14.1 91.6 ± 5.02
79. SVM results SVMʹΑΔ൑ผͷਖ਼౴཰ͱ༻͍ͨΧʔωϧͷද Linear Gaussian kernel weight w arc (p = 1) 67.6 ± 5.45 88.6 ± 2.19 w arc (p = 5) 68.4 ± 6.30 87.2 ± 4.71 k G w arc (p = 10) 68.8 ± 5.58 86.6 ± 5.02 w pers 55.4 ± 9.34 67.0 ± 11.7 w one 59.8 ± 5.54 64.0 ± 5.00 w arc (p = 5) 50.8 ± 8.16 51.4 ± 3.28 k L w pers 50.8 ± 8.16 46.4 ± 4.15 w one 50.8 ± 8.16 50.4 ± 5.17 PSSK 67.6 ± 9.71 79.2 ± 8.72 Persistence landscape 51.0 ± 7.51 47.2 ±2.04 Persistence image M = 100 w pers 54.0 ± 13.7 58.8 ± 10.3 M = 20 w arc (p = 5) 48.4 ± 4.15 52.0 ± 8.45 M = 50 w arc (p = 5) 48.0 ± 8.03 92.0 ± 2.00 M = 100 w arc (p = 5) 61.0 ± 14.1 91.6 ± 5.02 PWGK͕Ͳͷख๏ΑΓ΋ྑ͍෼ྨਫ਼౓Λࣔ͢. ॏΈؔ਺͕ॏཁͳ໾ׂΛ Ռͨ͢. (ύϥϝʔλ͸10-fold cross-validationͰٻΊͨ.)
80. ӷମ Ψϥε Q1 : ͲͷΠϯσοΫε(Թ౓)ɹͰύʔγεςϯτਤ͸େ͖͘ มԽ͍ͯ͠Δ͔ Q2 : ͦͷԹ౓͸ӷମ-ΨϥεసҠͱؔ࿈͕͋Δ͔ Liquid-glass transition problem D1 D80 D` D`+1 · · · · · · ` γϦΧͷԹ౓Λม͑Δຖʹݪࢠ഑ஔΛऔΓग़͠, ͦΕͧΕͷ
 ύʔγεςϯτਤɹɹɹɹɹɹɹɹ Λܭࢉ͢Δ. {D` | ` = 1, . . . , n}
81. ༩͑ΒΕͨܥɹɹɹɹɹɹɹɹɹͷมԽ఺Λਪఆ͢ΔͨΊͷख๏ͱͯ͠ kernel discriminant Fisher ratio (KFDR) [Harchaoui et al., 09] Λ ༻͍Δ. ͜Ε͸࣍Ͱ༩͑ΒΕΔؔ਺Ͱ͋Δ. ͜͜Ͱ, {D` | ` = 1, . . . , n} ※1 : ͸ਖ਼ଇԽύϥϝʔλͰٯߦྻ͕औΕΔΑ͏ʹখ͘͞બͿ. ※2 : KFDR͸άϥϜߦྻͷΈ͔ΒܭࢉՄೳͰ͋Δ. KFDR(`) = `(n `) n ⌘T ⇢ 1 n G(In Q) + In 1 G⌘ Liquid-glass transition problem 1 l1 ` = (1, . . . , 1 | {z } ` , 0, . . . , 0)T , 1 l2 n ` = (0, · · · , 0, 1, . . . , 1 | {z } n ` )T Q = 1 ` 1 l1 ` (1 l1 ` )T + 1 n ` 1 l2 n ` (1 l2 n ` )T , ⌘ = 1 ` 1 l1 ` + 1 n ` 1 l2 n ` G = (K(Di, Dj))i,j=1,...,n
82. ਺ֶతʹ͸, ࣍ͷྔͰ༩͑ΒΕΔ΋ͷͰ͋Δ: ͨͩ͠, ඪຊฏۉ΍ඪຊ૬ޓڞ෼ࢄ࡞༻ૉ͸࣍ͷΑ͏ʹ༩͑ΒΕΔ. Liquid-glass transition problem ˆ µi:j = 1 j i + 1 j X `=i K(·, D`), ˆ ⌃i:j = 1 j i + 1 j X `=i (K(·, D`) ˆ µi:j) ⌦ (K(·, D`) ˆ µi:j) ෼ࢄͷิਖ਼ ฏۉͷͣΕ ΠϯσοΫεɹͷલޙͰฏۉΛܭࢉ͠, ͦͷͣΕ͕Ұ൪େ͖͍ͱ͜Ζ ɹɹɹɹɹɹɹɹɹ ΛมԽ఺ͱͯ͠ਪఆ͢Δख๏Ͱ͋Δ. ` `⇤ = argmax`KFDR( ` ) KFDR(`) = `(n `) n ✓ ` n ˆ ⌃1:` + n ` n ˆ ⌃`+1:n + I ◆ 1 2 (ˆ µ`+1:n ˆ µ1:`) HK KFDR(`) = `(n `) n ✓ ` n ˆ ⌃1:` + n ` n ˆ ⌃`+1:n + I ◆ 1 2 (ˆ µ`+1:n ˆ µ1:`) HK
83. Liquid-glass transition problem 1FSTJTUFODF
    JNBHF 144, 18(, ύʔγεςϯτਤʹΑΔมԽ఺ݕग़ͷ݁Ռ͸ ΛมԽ఺ͱͯ͠ਪఆͨ͠. Ұํ, ෺ཧֶతʹ͸ɹɹɹɹ ͷൣғͰӷମ-Ψϥ εసҠ͕ى͍ͬͯ͜Δͱݚڀ͞Ε͍ͯΔɽ ɹ὎ ӷମ-ΨϥεసҠͱύʔγεςϯτਤͷมԽΛ౷ܭతʹಛ௃෇͚ͨ.ɹɹ 39 (PWGK), 33 (PSSK), 64 (Persistence image) ` 2 [35, 50]
84. Χʔωϧओ੒෼෼ੳ(PCA)ͷ2࣍ݩͱ3࣍ݩࣹӨਤ Ϋϥελʔߏ଄΋PWGKͷํ͕ΑΓ͸͖ͬΓͱݟ͑Δ 㱻 ύʔγεςϯτਤ͕ೋͭͷߏ଄Λ͸͖ͬΓ෼཭͍ͤͯ͞Δ ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ(ͱࢥ͑Δ. ٞ࿦ͷ༨஍͋Γ…) Liquid-glass transition problem 1FSTJTUFODF
    JNBHF 144, 18(,
85. Other applications • คମͷॆర཰ — มԽ఺ݕग़
 TDAͰਪఆ͞ΕͨมԽ఺͸ crystal packing ͱ random packingΛ ෼ׂ͢Δॆర཰ʹରԠ͍ͯ͠Δ. σʔλ͸[Saadatfar et al., 17]ΑΓ. • λϯύΫ࣭ͷߏ଄ (from PDB) — SVMʹΑΔ෼ྨ
 Binding and non-binging drug 100.0%
 The taut and relaxed forms of hemoglobin 88.9%
 ([Xia et al., 14] ΑΓߴ͍෼ྨੑೳΛ͍ࣔͯ͠Δ)
86. Relations to probability theory Persistence landscape [Bubenik, 15] ͸
 Ͱఆٛ͞ΕΔ ɹɹɹؔ਺Ͱ͋Δ. INRIAͷFrédéric ChazalΒͷάϧʔϓ͔Βpersistence landscapeͷத৺ۃ ݶఆཧ, Bootstrap๏, ৴པ۠ؒ, subsampling๏, ऩଋϨʔτʹؔ͢Δݚڀ͕ ఏҊ͞Ε͍ͯΔ(Banachۭؒʹ஋Λ࣋ͭ֬཰ม਺ͷཧ࿦ͷԠ༻). PWGK΋Banachۭؒʹ஋Λ࣋ͭ֬཰ม਺ͱݟͳͤΔ + Χʔωϧ๏Ͱ஌Β Ε͍ͯΔཧ࿦Λద༻Ͱ͖Δ. D 7! D( k, t ) = k -th largest value of max { min {t bi, di t}, 0 } L2(N ⇥ R)
87. Future work ॏΈؔ਺͸Ͳ͏ఆΊΔ͔. ྫ͑͹Ψϥεͷύʔγεςϯτਤ΋ॏෳ౓Λlogεέʔϧ͍ͯ͠ΔͨΊ, ዞ ҙతͳૢ࡞͕ೖ͍ͬͯΔ. logεέʔϧΛ͠ͳ͍ͱେྔͷର֯ઢ෇ۙͷ఺(ϊ Πζ)ʹҾͬுΒΕͯಛ௃͕ݟ͑ͯ͜ͳ͍. ҆ఆੑఆཧ͕੒ཱ͢ΔΑ͏ͳॏΈؔ਺Ͱ͋ͬͯཉ͍͠. ὎ ཧ࿦తʹྑ͍ੑ࣭Λ࣋ͭؔ਺, arctanҎ֎ͷܗ? σʔλʹґଘͯ͠ॏΈ͸มԽ͢Δͱ΋ࢥ͑Δ. ὎ persistence ʹґଘͤ͞Δͷ͸ݹ͍? [˚ A2] [˚ A2] −0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Multiplicity 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
88. ྲྀΕͷ෮श 0 : σʔλͷऔಘͱύʔγεςϯτਤͷܭࢉ 1 : ύʔγεςϯτਤΛଌ౓ͱݟͳ͢.
 2 : ΧʔωϧຒΊࠐΈʹΑΓύʔγεςϯτਤͷRKHSϕΫτϧΛಘΔ.
 3 : ͦͷRKHSϕΫτϧΛσʔλͱݟͳ͠, ΧʔωϧΛߏ੒͢Δ.
 ࣮ݧͰ͸RKHS্ͷΨ΢εΧʔωϧΛ࢖༻͢Δ. ྑ͍ॏΈؔ਺ͱΧʔωϧΛ ఆΊΔͱ࿈ଓ୯ࣹʹͳΔ. (§3ͷεϥΠυΑΓ) Conclusion
89. • ຊߨԋͷ಺༰
 [Kusano et al., 16] Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Persistence weighted Gaussian kernel for topological data analysis, ICML, pp:2004-2013, 2016.
 [Kusano et al., ] Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Kernel method for persistence diagrams via kernel embedding and weight factor. in preparation.
 • ύʔγεςϯτਤͷܭࢉιϑτ΢ΣΞ
 HomCloud : http://www.wpi-aimr.tohoku.ac.jp/hiraoka_labo/homcloud.html • ύʔγεςϯτϗϞϩδʔʹ͍ͭͯ
 [ฏԬ, 13] ฏԬ༟ষ. λϯύΫ࣭ߏ଄ͱτϙϩδʔɿύʔγεςϯτϗϞϩδʔ܈ೖ໳. ڞཱग़൛. 2013. • Χʔωϧ๏ʹ͍ͭͯ
 [෱ਫ, 10] ෱ਫ݈࣍. Χʔωϧ๏ೖ໳--ਖ਼ఆ஋ΧʔωϧʹΑΔσʔλղੳ. ே૔ग़൛. 2010. References
90. References • Ψϥεͷύʔγεςϯτਤ
 [Hiraoka et al., 16] Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G Escolar, Kaname Matsue, and Yasumasa Nishiura. Hierarchical structures of amorphous solids characterized by persistent homology. Proceedings of the National Academy of Sciences, 113(26):7035– 7040, 2016. • คମͷύʔγεςϯτਤ
 [Saadatfar et al., 17] Mohammad Saadatfar, Hiroshi Takeuchi, Vanessa Robins, Nicolas Francois, and Yasuaki Hiraoka. Pore configuration landscape of granular crystallisation. In press, 2017. • [Sriperumbudur et al., 11] Bharath K. Sriperumbudur, Kenji Fukumizu, and Gert R. G. Lanckriet. Universality, characteristic kernels and RKHS embedding of measures. The Journal of Machine Learning Research, 12:2389–2410, 2011. • [Rahimi and Recht, 07] Ali Rahimi and Benjamin Recht. Random features for large- scale kernel machines. In Advances in neural information processing systems, pages 1177–1184, 2007.
91. • [Cohen-Stenier et al., 07] David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1): 103–120, 2007. • [Chazal et al., 14] Frédéric Chazal, Vin de Silva, and Steve Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193–214, 2014. • [Bauer and Lesnick., 15] Ulrich Bauer and Michael Lesnick. Induced matchings and the algebraic stability of persistence barcodes. Journal of Computational Geometry, 6(2):162–191, 2015. • [Chazal et al., 09] Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas Guibas, and Steve Oudot. Proximity of Persistence Modules and their Diagrams. Proceedings of the twenty-fifth annual symposium on Computational geometry. 237— 246. 2009. • [Bubenik, 15] Peter Bubenik. Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(1):77–102, 2015. • [Reininghaus et al., 15] Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning. In In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(CVPR), pages 4741–4748, 2015. References
92. • [Adams et al., 17] Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Ship- man, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8):1–35, 2017. • [Harchaoui et al., 09] Zaid Harchaoui, Eric Moulines, and Francis R. Bach. Kernel change-point analysis. In Advances in Neural Information Processing Systems, pages 609–616, 2009. • [Xia et al., 14] Kelin Xia and Guo-Wei Wei. Persistent homology analysis of protein structure, flexibility, and folding. International journal for numerical methods in biomedical engineering, 30 (8):814–844, 2014. References
93. ༻͍ͨΧʔωϧͱॏΈؔ਺ʹؚ·ΕΔύϥϝʔλͷܾΊํ͸ [Gretton et al., 07]ͷܾΊํΛࢀߟʹͨ͠: C` = median{pers( x ) | x 2 D` } ( D`) = median{k xi xj k | xi, xj 2 D`, i < j } C = median{C` | ` = 1, · · · , n} = median{ (D`) | ` = 1, · · · , n} kG( x, y ) = e k x y k2 2 2 , warc( x ) = arctan( C pers( x )p) ॏΈؔ਺ͷϕΩɹʹؔͯ͠͸҆ఆੑఆཧͷԾఆɹɹɹɹ ͱ༻͍Δσʔλ ͸ɹ ͔Βऔ͍ͬͯΔͨΊ, ɹɹͱݻఆ͢Δ. ɹΛม͑ͯ΋PWGKͷ෼ྨ ਫ਼౓͕େ͖͘มԽ͠ͳ͍͜ͱ͸SVMͷ࣮ݧͰ֬ೝࡁΈ. p > d + 1 R3 p = 5 p p Parameters
94. Random Fourier features Random Fourier features [Rahimi and Recht. 07]ͷ࢓૊Έ͸ٯFourier ม׵Ͱ͋Δ. e k x y k2 2 2 = Z R2 e p 1 z ( x y ) 2 2⇡ e 2 2 k z k2 dz ⇡ 1 M M X a =1 e p 1 za ( x y ) = 1 M M X a =1 e p 1 zaxe p 1 zay z1, . . . , zM ⇠ N(0, 2I2) X x 2 D X y 2 E w ( x ) w ( y ) e k x y k2 2 2 ⇡ 1 M M X a =1 X x 2 D w ( x ) e p 1 zax ! 0 @ X y 2 E w ( y ) e p 1 zay 1 A ⇤ ※ : ෳૉڞ໾ RKHSϕΫτϧͷ಺ੵ͸࣍ͷΑ͏ʹۙࣅ͞ΕΔ: