カーネル法：正定値カーネルの理論

- 正定値カーネルと負定値カーネル
- Bochnerの定理
- Mercerの定理

Transcript

1. Χʔωϧ๏ೖ໳ 6 ষ:ਖ਼ఆ஋Χʔωϧͷཧ࿦ Daiki Tanaka March 21, 2020
2. 6.1 ਖ਼ఆ஋Χʔωϧͱෛఆ஋Χʔωϧ 6.1.1 ෛఆ஋Χʔωϧ 6.1.2 ΧʔωϧΛੜ੒͢Δૢ࡞ 6.2 Bochner ͷఆཧ 6.3 Mercer ͷఆཧ 6.3.1 ੵ෼֩ͱੵ෼࡞༻ૉ 6.3.2 ੵ෼֩ͷ Hilbert-Schmidt ల։ 6.3.3 ਖ਼஋ੵ෼֩ͱ Mercer ͷఆཧ 8.1 ฏۉʹΑΔ֬཰෼෍ͷಛ௃͚ͮ 8.1.1 ώϧϕϧτۭؒʹ஋ΛͱΔ֬཰ม਺ 8.1.2 RKHS ʹ͓͚Δฏۉ 8.2 ֬཰෼෍Λಛ௃͚ͮΔਖ਼ఆ஋Χʔωϧ 2
3. 6.1.1 ෛఆ஋Χʔωϧ ҎԼͷΑ͏ʹఆΊΔෛఆ஋Χʔωϧ͔Βਖ਼ఆ஋ΧʔωϧΛੜ੒͢Δ͜ͱ͕Ͱ͖Δɽ ఆٛɿෛఆ஋ X:set ʹ͍ͭͯɼ : X ˆ X ! C ͕ෛఆ஋Ͱ͋Δͱ͸ɼ ͕ΤϧϛʔτతͰɼ͔ͭ X ͷ n ݸͷ೚ҙͷ఺ x1; : : : ; xn ͱ Pn i=1 ci = 0 Λຬͨ͢೚ҙͷෳૉ਺ c1; : : : ; cn 2 C ʹରͯ͠ n X i;j=1 ci — cj (xi; xj) » 0 (1) ͕੒Γཱͭ͜ͱΛ͍͏ɽ [Remark]:ෛఆ஋Χʔωϧͷఆٛ͸ਖ਼ఆ஋Χʔωϧͷఆٛ: ೚ҙͷ n 2 Nɼ೚ҙͷ x1; : : : ; xn 2 Xɼ೚ҙͷ c1; : : : ; cn 2 C ʹରͯ͠ n X i;j=1 ci — cjk (xi; xj) – 0 ͷٯූ߸Ͱ͸ͳ͍ɽ(ෛఆ஋ੑʹؔͯ͠͸੍໿:Pn i=1 ci = 0 Λຬͨ͢ ci (i = 1; : : : ; n) ʹର͚ͯͩ͠ߟ͑Δ) 3
4. 6.1.1 ෛఆ஋Χʔωϧɿෛఆ஋Χʔωϧͷྫ ໋୊ 6.1 › (1) k ͕ਖ਼ఆ஋ΧʔωϧͳΒ͹ɼ`k ͸ෛఆ஋Χʔωϧɽ › (2) ఆ਺ؔ਺͸ෛఆ஋Χʔωϧɽ › (3) ೚ҙͷؔ਺ f ʹରͯ͠ɼ (x; y) = f (x) + f (y) ͸ෛఆ஋Χʔωϧɽ [ূ໌] › (1):k ͕ਖ਼ఆ஋ΧʔωϧͰ͋Δ࣌ɼ೚ҙͷ n 2 N ɼx1; : : : ; xn 2 Xɼ c1; : : : ; cn 2 C ʹର͠ɼਖ਼஋ੑ n X i;j=1 ci — cjk (xi; xj) – 0 Λຬͨ͢ɽ͜ͷ࣌ɼk0 := `k ͸ n X i;j=1 ci — cjk0 (xi; xj) » 0 Λ೚ҙͷ c1; : : : ; cn 2 C ʹ͍ͭͯຬͨ͢͜ͱ͔Βɼ`k ͸ෛఆ஋Χʔωϧɽ 2 4
5. 6.1.1 ෛఆ஋Χʔωϧ [ূ໌] › (2):ఆ਺ؔ਺ k (´; ´) = a ͱ Pn i=1 ci = 0 Λຬͨ͢೚ҙͷෳૉ਺ c1; : : : ; cn 2 C ʹ͍ͭͯɼ n X i;j=1 ci — cjk (xi; xj) = n X i;j=1 ci — cja = n X j=1 — cj 0 @ n X i=1 cia 1 A = 0 » 0 ΑΓɼk ͸ෛఆ஋Χʔωϧɽ 2 › (3) ೚ҙͷؔ਺ f ͱ Pn i=1 ci = 0 Λຬͨ͢೚ҙͷෳૉ਺ c1; : : : ; cn 2 C ʹର͠ɼ n X i;j=1 ci — cj (xi; xj) = n X i;j=1 ci — cj (f(xi) + f(xj)) = n X j=1 — cj 0 @ n X i=1 cif(xi) 1 A + n X i=1 ci 0 @ n X j=1 — cjf(xj) 1 A = 0 » 0 ΑΓɼ ͸ෛఆ஋Χʔωϧɽ 2 5
6. 6.1.1 ෛఆ஋Χʔωϧ ໋୊ 6.2 i : X ˆ X ! C (i = 1; : : : ) Λෛఆ஋Χʔωϧͱ͢Δ࣌ɼ࣍ͷ 2 ͭͷΧʔωϧ΋ෛ ఆ஋Ͱ͋Δɽ › (1) ඇෛ݁߹:¸ 1 + ˛ 2 › (2) ۃݶ: limi!1 i (x; y) ͨͩ͠ɼ(2) Ͱ͸ۃݶ஋ͷଘࡏΛԾఆͨ͠ɽ X ্ͷෛఆ஋Χʔωϧશମ͸ดತਲ਼Ͱ͋Δɽਖ਼ఆ஋Χʔωϧͱ͸ҟͳΓɼෛఆ஋Χʔωϧͷ ੵ͸ෛఆ஋Ͱ͋Δͱ͸ݶΒͳ͍ɽ 6
7. 6.1.1 ෛఆ஋Χʔωϧ ෛఆ஋Χʔωϧͷجຊతͳྫ͸ҎԼͷ໋୊͔ΒಘΒΕΔɽ ໋୊ 6.3 ू߹ X ͔Β಺ੵۭؒ V ΁ͷࣸ૾ ffi : X ! V ʹ͍ͭͯɼ (x; y) = kffi(x) ` ffi(y)k2 ͸ X ্ͷෛఆ஋ΧʔωϧͰ͋Δɽ [ূ໌] Pn i=1 ci = 0 Λຬͨ͢೚ҙͷෳૉ਺ c1; : : : ; cn 2 C ͱ x1; : : : ; xn 2 X ʹରͯ͠ɼ n X i;j=1 ci — cj kffi(x) ` ffi(y)k2 = n X i;j=1 ci — cj hffi(x) ` ffi(y); ffi(x) ` ffi(y)i = n X i;j=1 ci — cj ˘kffi (xi)k2 + kffi (xj)k2 ` hffi (xi) ; ffi (xj)i ` hffi (xj) ; ffi (xi)i¯ = 0 + 0 ` * n X i=1 ciffi (xi) ; n X j=1 — cjffi (xj) + ` * n X j=1 cjffi (xj) ; n X j=1 ciffi (xi) + = ` ‚ ‚ ‚ ‚ ‚ ‚ n X i=1 ciffi (xi) ‚ ‚ ‚ ‚ ‚ ‚ 2 ` ‚ ‚ ‚ ‚ ‚ ‚ n X i=1 ciffi (xi) ‚ ‚ ‚ ‚ ‚ ‚ 2 » 0 □ 7
8. 6.1.1 ෛఆ஋Χʔωϧ ਖ਼ఆ஋Χʔωϧͱෛఆ஋Χʔωϧͷؒʹ͸ີ઀ͳؔ࿈ੑ͕͋Δɽ ิ୊ 6.4 (x; y) Λू߹ X ; ্ͷ Hermite తͳΧʔωϧ ( (x; y) = (y; x)) ͱ͢Δɽ 8x0 2 X ʹରͯ͠ɼ’ ΛҎԼͷΑ͏ʹఆٛ͢Δɽ ’ (x; y) := ` (x; y) + (x; x0) + (x0; y) ` (x0; x0) ͜ͷ࣌ɼ ͕ෛఆ஋Ͱ͋Δ͜ͱͱɼ’ ͕ਖ਼ఆ஋Ͱ͋Δ͜ͱ͸ಉ஋Ͱ͋Δɽ [ূ໌] xi 2 X Λ೚ҙͷ఺ͱ͢Δɽ › (:’ Λਖ਼ఆ஋ͱ͢ΔɽPn i=1 ai = 0 Λຬͨ͢ ai 2 C ͱ͢Δͱɼਖ਼ఆ஋ੑΑΓɼ n X i;j=1 aiaj’ (xi; xj) – 0 Ͱ͋ΔɽPn i;j=1 aiaj (xi; x0) = Pn i;j=1 aiaj (x0; xj) = Pn i;j=1 aiaj (x0; x0) = 0 Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱɼ Pn i;j=1 aiaj (xi; xj) » 0 ΛಘΒΕɼ ͸ෛఆ஋Ͱ͋Δɽ 8
9. 6.1.1 ෛఆ஋Χʔωϧ [ূ໌] › ): Λෛఆ஋ͱ͢Δɽci 2 C (i = 1; : : : ; n) Λ೚ҙʹͱΓɼc0 := ` Pn i=1 ci ͱ͢Ε͹ɼ ͷෛఆ஋ੑ͔Β೚ҙͷ x0; x1; : : : ; xn 2 X ʹରͯ͠ɼ n X i=0;j=0 ci— cj (xi; xj) » 0 ͕੒Γཱͭɽ্ࣜͷࠨล͸ i = 0; j = 0 ͷ৔߹Λ֎ʹग़ͤ͹ɿ n X i=0;j=0 ci— cj (xi; xj) = n X i=0 n X j=0 ci— cj (xi; xj) = n X i;j=1 ci— cj (xi; xj) + c0 n X i=1 ci (xi; x0) + c0 n X j=1 cj (x0; xj) + jc0j2 (x0; x0) = n X i;j=1 cicj (xi; xj) ` n X i;j=1 cicj (xi; x0) ` n X i;j=1 cicj (x0; xj) + n X i;j=1 cicj (x0; x0) = ` n X i;j=1 cicj’ (xi; xj) ͱͳͬͯɼPn i;j=1 cicj’ (xi; xj) – 0 ͔Β ’ ͸ਖ਼ఆ஋Ͱ͋Δɽ 2 9
10. 6.1.1 ෛఆ஋ΧʔωϧɿΧʔωϧੜ੒ʹศརͳఆཧ ࣍ͷఆཧ͸ΧʔωϧΛܥ౷తʹੜ੒͢Δࡍʹ໾ཱͭɽ ఆཧ 6.5 Schoenberg ͷఆཧ ू߹ X ; ʹରͯ͠ : X ˆ X ! C Λ X ্ͷΧʔωϧͱ͢Δɽ͜ͷ࣌ɼ ʮ ͕ෛఆ஋ʯ () ʮexp (`t ) ͕೚ҙͷਖ਼਺ t > 0 ʹରͯ͠ਖ਼ఆ஋ʯ [ূ໌] ඍ෼ͷఆ͔ٛΒɼ8x; y 2 X ʹରͯ͠ɼ (x; y) = lim t!+0 1 ` exp (`t (x; y)) t ͕੒Γཱͭ (?)ɽ8t > 0 ʹରͯ͠ɼexp (`t (x; y)) ͕ਖ਼ఆ஋ͳΒ͹ɼ 1`exp (`t (x;y)) t ͸ෛఆ஋Ͱ͋Γɼͦͷۃݶ஋ ΋ෛఆ஋Ͱ͋Δɽ(໋୊ 6.2) ٯͷূ໌ʹ͸ ΋ෛఆ஋Ͱ͋Δ࣌ʹ t = 1 ͷ৔߹ʹ͍ͭͯ exp (`t (x; y)) ͕ਖ਼ఆ஋ Ͱ͋Δ͜ͱΛࣔͤ͹े෼ɽ೚ҙͷ x0 2 X ʹରͯ͠ ’ (x; y) := ` (x; y) + (x; x0) + (x0; y) ` (x0; x0) ͱఆٛ͢Δͱɼิ୊ 6.4 ͔Β ’ ͸ਖ਼ఆ஋Ͱ͋Γɼexp (’ (x; y)) ΋ਖ਼ఆ஋Ͱ͋Δɽ໋୊ 2.5 ͔Β exp (` (x; y)) = exp (’ (x; y)) exp (` (x; x0))exp (` (y; x0)) exp ( (x0; x0)) ͸ਖ਼ఆ஋Ͱ͋Δɽ 2 10
11. 6.1.2 ΧʔωϧΛੜ੒͢Δૢ࡞ ໋୊ 6.6 ू߹ X ্ͷෛఆ஋Χʔωϧ : X ˆ X ! C ͕ (x; x) – 0 Λຬͨ࣌͢ɼ೚ҙͷ 0 < p » 1 ʹରͯ͠ɼ (x; y)p ͸ෛఆ஋Ͱ͋Δɽ [ূ໌]8z > 0 ʹ͍ͭͯɼΨϯϚؔ਺ ` (z) Λ༻͍ͯ zp = p `(1`p) R 1 0 t`p`1 `1 ` e`tz´ dt ͱͰ͖Δ͜ͱ͔Βɼ (x; y)p = p ` (1 ` p) Z 1 0 t`p`1 “1 ` e`t (x;y)” dt Ͱ͋ΔɽSchoenberg ͷఆཧͱ໋୊ 6.1 ʹΑΓɼඃੵ෼ؔ਺͸ෆఆ஋Χʔωϧɽੵ෼͕ ϦʔϚϯ࿨ͷۃݶͰ͋Δ͜ͱʹ஫ҙ͢Ε͹ (x; y)p ΋ෆఆ஋ΧʔωϧͱͳΔ (໋୊ 6.2 Α Γෆఆ஋Χʔωϧͷඇෛ݁߹͸ෆఆ஋Χʔω)ɽ 2 11
12. 6.1.2 ΧʔωϧΛੜ੒͢Δૢ࡞ ܥ 6.7 ೚ҙͷ 0 < p » 2 ͱ ¸ > 0 ʹରͯ͠ɼ exp ``¸ kx ` ykp´ ͸ Rn ্ͷਖ਼ఆ஋ΧʔωϧͰ͋Δɽ ಛʹɼ¸ = 1; 2 ͷ࣌ɼͦΕͧΕϥϓϥεΧʔωϧٴͼΨ΢ε RBF ΧʔωϧͰ͋Δɽ 12
13. 6.1.2 ΧʔωϧΛੜ੒͢Δૢ࡞ ໋୊ 6.8 : X ˆ X ! C Λɼू߹ X ্ͷෛఆ஋Χʔωϧͱ͢Δɽ೚ҙͷ x; y 2 X ʹ͍ͭͯ (x; y) – 0 Λຬͨ࣌͢ɼ೚ҙͷ ¸ > 0 ʹ͍ͭͯ log (¸ + (x; y)) ͸ෛఆ஋ΧʔωϧͱͳΔɽ·ͨɼ (x; y) > 0 Ͱ͋Δ࣌ɼ log ( (x; y)) ͸ෛఆ஋ΧʔωϧͱͳΔɽ [ূ໌]:ੵ෼දࣔ log (1 + (x; y)) = Z 1 0 “1 ` e`t (x;y)” e`t t dt ʹΑΓɼ໋୊ 6.6 ͱಉ༷ʹඃੵ෼ؔ਺͕ෛఆ஋ΧʔωϧͰ͋Δ͜ͱ͔Β log (1 + (x; y)) ͸ෛఆ஋ΧʔωϧͰ͋Δɽ͕ͨͬͯ͠ɼ log (¸ + ) = log `1 + 1 ¸ ´ + log ¸ ΋ෛఆ஋Ͱ͋Δɽ [Remark]: ໋୊ 6.1(3): ʮ೚ҙͷؔ਺ f ʹରͯ͠ɼ (x; y) = f (x) + f (y) ͸ෛఆ஋Χʔωϧɽ ʯ ͔Βɼ (x; y) := x + y ͸ R ্ͷෛఆ஋ΧʔωϧͰ͋ΔͷͰɼ (x; y) = log (x + y) ͸ (0; 1) ্ͷෛఆ஋ΧʔωϧͰ͋Δɽ 13
14. 6.1.2 ΧʔωϧΛੜ੒͢Δૢ࡞ ҎԼͷ໋୊Λ༻͍Ε͹ɼෛఆ஋Χʔωϧ͔Βਖ਼ఆ஋ΧʔωϧΛੜ੒Ͱ͖Δɽ ໋୊ 6.9 ෛఆ஋Χʔωϧ͔Βਖ਼ఆ஋ΧʔωϧΛੜ੒ ෛఆ஋Χʔωϧ ͕ Re (x; y) – 0 Λຬͨ࣌͢ɼ 1 (x; y) + a ͸ਖ਼ఆ஋ΧʔωϧͰ͋Δɽͨͩ͠ɼa ͸ਖ਼ͷఆ਺ɽ [ূ໌]:ੵ෼දࣔ 1 (x; y) + a = Z 1 0 e`t( (x;y)+a)dt ΑΓɼ໋୊ 6.6 ͱಉ༷ʹͯ͠ඃੵ෼ؔ਺ͷਖ਼ఆ஋ੑ͔Βɼਖ਼ఆ஋ੑΛ͑Δɽ 2 14
15. 6.1.2 ΧʔωϧΛੜ੒͢Δૢ࡞ ໋୊ 6.3ɼ໋୊ 6.9 ΑΓ೚ҙͷ 0 < p » 2 ʹରͯ͠ 1 1 + jx ` yjp ͸ R ্ͷਖ਼ఆ஋ΧʔωϧͰ͋Δɽ ಛʹ p = 2 ͷ৔߹͸ Cauchy ෼෍ͷີ౓ؔ਺: f (x; x0; ‚) = 1 ı‚ » 1 + “x`x0 ‚ ”2– = 1 ı ‚ (x ` x0)2 + ‚2 ͱͷྨࣅ͔Β Cauchy Χʔωϧͱݺ͹ΕΔɽ Figure: Cauchy distribution 15
16. 6.2 Bochner ͷఆཧ Rn ্ͷΧʔωϧ k ͕ฏߦҠಈෆมͰ͋Δɼͱ͸ Rn ্ͷؔ਺:ffi ͕͋ͬͯɼ k (x; y) = ffi (x ` y) ͱॻ͚Δ͜ͱͰ͋Δɽ(2 ཁૉͷࠩʹͷΈґଘ͢ΔΧʔωϧ e.g. RBF Χʔωϧ) Χʔωϧ͕ฏߦҠಈෆมͰ͋Δ͜ͱ͸ k (x; y) = k (x + z; y + z) (8z 2 Rn) ͱಉ஋Ͱ͋Δɽ ఆٛɿਖ਼஋ؔ਺ Rn ্ͷؔ਺ ffi ͕ਖ਼஋Ͱ͋Δɼͱ͸ k (x; y) := ffi (x ` y) ʹΑΓఆٛ͞ΕΔΧʔωϧ k ͕ਖ਼ఆ஋Ͱ͋Δ͜ͱΛ͍͏ɽ ఆཧ 6.10 Bochner ͷఆཧ ffi Λ Rn ্ͷෳૉ਺࿈ଓؔ਺ͱ͢Δɽ͜ͷ࣌ ffi ͕ਖ਼஋Ͱ͋Δ͜ͱͷඞཁे෼৚݅͸ɼRn ্ ͷ༗ݶͳඇෛ Borel ଌ౓ ˜ ͕͋ͬͯɼ ffi (x) = Z e p `1!Txd˜ (!) (6.1) ͱද͞ΕΔ͜ͱͰ͋Δɽ 16
17. 6.2 Bochner ͷఆཧ:ূ໌ › े෼ੑ: ffi (x) = Z e p `1!Txd˜ (!) ͱද͞ΕΔͱ͢Δɽ e p `1!T(x`y) = e p `1!Txe` p `1!Ty = e p `1!Txe p `1!Ty Ͱ͋Δ͔Β (७ڏ਺ z ʹରͯ͠ `z = z Ͱ͋Δ͜ͱͱ exp `z´ = exp (z) Λ࢖ͬͨ)ɼҎԼͷ Χʔωϧɿ K (x; y) := ffi (x ` y) = Z e p `1!Txe p `1!Tyd˜ (!) ͷඃੵ෼ؔ਺͸໋୊ 2.5(2) ͔Βਖ਼ఆ஋ΧʔωϧͰ͋ΔɽΑͬͯɼͦͷੵ෼஋ͱͯ͠ಘ ΒΕΔ K ΋ਖ਼ఆ஋ΧʔωϧͰ͋Γɼffi ͸ਖ਼஋Ͱ͋Δɽ 2 › ඞཁੑ: লུɽ Bochner ͷఆཧ͸ɼ೚ҙͷਖ਼஋࿈ଓؔ਺͕ fe p `1!Tx j ! 2 Rng ͷඇෛ݁߹ͱ͠ ͯද͞ΕΔ͜ͱΛओு͍ͯ͠Δɽ 17
18. 6.2 प೾਺ྖҬͰݟͨ RKHS, ໋୊ 2.19 ฏߦҠಈෆมͳਖ਼ఆ஋Χʔωϧ͸प೾਺ྖҬʹ͓͍ͯཅͳܗͰදݱͰ͖Δɽ(e.g. RBF ΧʔωϧɼϥϓϥεΧʔωϧ) ฏߦҠಈෆมͳΧʔωϧ K ͕ҎԼͷΑ͏ͳܗΛ΋ͭͱԾఆ͢Δɽ K (x; y) = Z e p `1!T(x`y)ȷ (!) d! ͨͩ͠ɼȷ ͸࿈ଓͰɼȷ (!) > 0; R ȷ (!) d! < 1ɽ ͜ͷ࣌ɼK Λ࠶ੜ֩ͱ͢Δ RKHSɿHK ͸ HK = ( f 2 L2 (R; dx) j Z ˛ ˛ ^ f (!)˛ ˛ 2 ȷ (!) d! < 1 ) hf; gi = Z ^ f (!) ^ g (!) ȷ (!) d! ͨͩ͠ɼ ^ f ͸ f ͷ Fourier ม׵ɿ ^ f (!) = 1 (2ı)m R f (x) e` p `1!Txd! 18
19. 6.3 Mercer ͷఆཧ ࿈ଓͳਖ਼ఆ஋Χʔωϧʹର͢Δ Mercer ͷఆཧΛಋೖ͢Δɽਖ਼ఆ஋ΧʔωϧΛੵ෼࡞༻ૉʹ ؔ͢ΔεϖΫτϧ෼ղʹΑͬͯදݱ͢ΔఆཧͰ͋Δɽ 19
20. 6.3.1 ੵ෼֩ͱੵ෼࡞༻ૉ (˙; B; —) Λଌ౓ۭؒͱ͠ɼK : ˙ ˆ ˙ ! C ΛՄଌͳΧʔωϧͱ͢Δɽ(K ͸ඞͣ͠΋ ਖ਼ఆ஋ͱ͸ݶΒͳ͍ɽ) 6.3 અͰ͸Χʔωϧ K ʹରͯ͠ৗʹҎԼͷ 2 ৐Մੵ෼ੑΛԾఆ͢Δɽ Z ˙ Z ˙ jK (x; y)j2 d— (x) d— (y) < 1 ·ͨɼL2 (˙; —)(ɿೋ৐Մੵ෼ؔ਺͔ΒͳΔۭؒ) ͸Մ෼Ͱ͋Δ (᜚ີͳ෦෼ू߹ͱͯ͠Մ ࢉू߹͕ଘࡏ͢Δ)ɿͭ·ΓՄࢉਖ਼ن௚ަجఈΛ΋ͭɼͱԾఆ͢Δɽ L2 ಺ੵ L2 (˙; —) ͷ಺ੵ (L2 ಺ੵ) ͸ hf; giL2 := Z ˙ f (x) g (x)d— (x) ·ͨɼ༠ಋ͞ΕΔϊϧϜ͸ kfkL2 := Z ˙ f (x) f (x)d— (x) !1 2 = Z ˙ jf (x)j2 d— (x) ! 1 2 Ͱ͋Δɽ 20
21. 6.3.1 ੵ෼֩ͱੵ෼࡞༻ૉɿੵ෼࡞༻ૉ TK ͷఆٛͱੑ࣭ Χʔωϧ K ʹରͯ͠ɼL2 (˙; —) ্ͷઢܗ࡞༻ૉ TK : L2 (˙; —) 3 f 7! TK f 2 fg j g : ˙ ! Cg Λ (TK f) (x) := Z ˙ K (x; y) f (y) d— (y) ; `f 2 L2 (˙; —)´ (6.2) ʹΑΓఆٛ͢Δɽͨͩ͠ɼLp ͸ `R jfjp d—´ 1 p < 1 Λຬͨؔ͢਺ͷۭؒɽ ͜ͷ࣌ɼCauchy-Schwarz ͷෆ౳ࣜʹΑΓɼ Z jTK f (x)j2 d— (x) = Z (Z jK (x; y) f (y) d— (y)j )2 d— (x) = Z hK (x; ´) ; fi2 L2 d— (x) » Z kK (x; ´)k2 L2 kfk2 L2 d— (x) = Z Z jK (x; y)j2 d— (x) d— (y) kfk2 L2 < 1 Ͱ͋Δ͜ͱ͔Β TK f 2 L2 (˙; —) ͱͳΓɼ݁ہ TK : L2 (˙; —) ! L2 (˙; —) ͱͳ ΔɽTK ͸ K Λੵ෼֩ͱ͢Δੵ෼࡞༻ૉͱݺ͹ΕΔɽ 21
22. 6.3.1 ੵ෼֩ͱੵ෼࡞༻ૉɿHilbert-Schmidt ࡞༻ૉ ఆٛɿHilbert-Schmidt ࡞༻ૉ ώϧϕϧτۭؒ H1 ͔Β H2 ΁ͷ࡞༻ૉ A : H1 ! H2 ͕ Hilbert-Schmidt ࡞༻ૉͰ ͋Δͱ͸ɼ H1 ͷɼ͋Δਖ਼ن௚ަجఈ f’igI i=1 ; (I 2 N [ f1g) ʹର͠ɼҎԼͷΑ͏ʹ ఆΊΔ Hilbert-Schmidt ϊϧϜ͕ kAk2 HS := I X i=1 kA’ik2 H2 < 1 Λຬͨ͢͜ͱͰ͋Δɽ·ͨɼParseval ͷ౳ࣜΛ༻͍ΔͱɼHilbert-Schmidt ϊϧϜ͸ɼ H1; H2 ͷ೚ҙͷਖ਼ن௚ަجఈ f’igi; f jgj ʹରͯ͠ҎԼͷΑ͏ʹදͤΔɿ kAk2 HS = I X i=1 kA’ik2 H2 = I X i=1 I0 X j=1 h j; A’ii2 H2 ఆཧ 6.11ɿੵ෼࡞༻ૉ͸ Hilbert-Schmidt ࡞༻ૉ 2 ৐Մੵ෼ͳੵ෼֩ K ʹΑͬͯɼࣜ 6.2 Ͱఆ·Δੵ෼࡞༻ૉ TK ͸ Hilbert-Schmidt ࡞༻ૉͰ͋Γ kTK k2 HS = Z Z ˙ˆ˙ jK (x; y)j2 d— (x) d— (y) 22
23. ఆཧ 6.11 ূ໌ K ͷ 2 ৐Մੵ෼ੑ͔Βɼ΄ͱΜͲશͯͷ x ʹରͯ͠ɼK (x; ´) 2 L2 (˙; —) Ͱ͋Δɽ f’ig1 i=1 Λ L2 (˙; —) ͷ׬શਖ਼ن௚ަجఈͱ͢ΔͱɼParseval ͷ౳ࣜ (ؔ਺ղੳ p.111) ͔Β Z ˙ jK (x; y)j2 d— (y) = Z ˙ K (x; y) K (x; y)d— (y) `jzj2 = zz͔Β´ = kK (x; ´)k2 L2 = 1 X i=1 ˛ ˛ ˛ hK (x; ´) ; ’iiL2(˙;—) ˛ ˛ ˛ 2 (Parseval ͷ౳͔ࣜΒ) = 1 X i=1 ˛ ˛ ˛ ˛ ˛ Z ˙ K (x; y) ’i (y)d— (y) ˛ ˛ ˛ ˛ ˛ 2 `L2಺ੵͷఆ͔ٛΒ´ = 1 X i=1 ˛ ˛TK ’i (x)˛ ˛ 2 (ੵ෼࡞༻ૉͷఆ͔ٛΒ) ͕੒Γཱͭɽ˘’i ¯1 i=1 ΋ L2 (˙; —) ͷ׬શਖ਼ن௚ަجఈͰ͋Δ͜ͱ͔Β Z Z ˙ˆ˙ jK (x; y)j2 d— (x) d— (y) = 1 X i=1 ‚ ‚TK ’i ‚ ‚ 2 = kTK k2 HS Λ͑Δɽ 2 23
24. 6.3.1 ੵ෼֩ͱੵ෼࡞༻ૉ ҎԼͷఆཧ͸ఆཧ 6.11 ͷٯ͕੒Γཱͭ͜ͱΛओு͢Δɽ(Hilbert-Schmidt ࡞༻ૉʹର ͯ͠ੵ෼͕֩Ұҙʹଘࡏ͢Δ) ఆཧ 6.12 L2 (˙; —) ্ͷ೚ҙͷ Hilbert-Schmidt ࡞༻ૉ T ʹର͠ɼ2 ৐Մੵ෼ͳੵ෼֩ K (x; y) 2 L2 (˙ ˆ ˙; — ˆ —) ͕Ұҙʹଘࡏ͠ɼ T ’ = Z K (x; y) ’ (y) d— (y) (6.3) ͱͳΓɼT = TK ͕੒Γཱͭɽ ఆཧ 6.11 ٴͼఆཧ 6.12 ͔ΒɼHilbert-Schmidt ࡞༻ૉͱ 2 ৐Մੵ෼ͳੵ෼֩Λ΋ͭੵ ෼࡞༻ૉ͸ҰରҰʹରԠ͢Δɽ 24
25. ఆཧ 6.12 ূ໌ 1/3 [ূ໌]: ଘࡏੑΛࣔ͢ɽf’ig1 i=1 Λ L2 (˙; —) ͷ׬શਖ਼ن௚ަجఈͱ͠ɼ೚ҙͷ n 2 N ʹରͯ͠ Kn (x; y) := n X i=1 (T ’i) (x) ’i (y) ͱఆٛ͢Δɽ͜ͷ࣌ fKng1 n=1 ͸ L2 (˙ ˆ ˙; — ˆ —) ͷίʔγʔྻɽ࣮ࡍɼm – n ʹରͯ͠ Z Z ˙ˆ˙ jKm (x; y) ` Kn (x; y)j2 d— (x) d— (y) = Z Z ˙ˆ˙ ˛ ˛ ˛ ˛ ˛ ˛ m X i=n+1 (T ’i) (x) ’i (y) ˛ ˛ ˛ ˛ ˛ ˛ 2 d— (x) d— (y) = m X i=n+1 m X j=n+1 (T ’i; T ’j) (’j; ’i) = m X i=n+1 kT ’ik2 L2(˙;—) Ͱ͋Δ͕ɼT ͸ Hilbert-Schmidt ࡞༻ૉͳͷͰ Pm i=n+1 kT ’ik2 L2(˙;—) ! 0 (n; m ! 1) Ͱ͋ΓɼfKng1 n=1 ͸ L2 (˙ ˆ ˙; — ˆ —) ͷίʔγʔྻͱͳΔɽ 25
26. ఆཧ 6.12 ূ໌ 2/3 L2 (˙ ˆ ˙; — ˆ —) ͸׬උͰ͋Γɼ2 ৐Մੵ෼ͳؔ਺ K (x; y) ͕ fKng ͷऩଋઌͱ͠ ͯ͋ΔɿKn ! K (n ! 1)ɽࣜ (6.2) ʹΑͬͯੵ෼࡞༻ૉ TK Λఆٛ͢Δ࣌ɼ TK = T Ͱ͋Δ͜ͱΛࣔͤ͹ྑ͍ɽ Kn Λੵ෼֩ͱ͢Δੵ෼࡞༻ૉΛ Tn ͱ͢Δͱɼఆཧ 6.11 ͔Β kTK ` Tnk » kTK ` TnkHS = kK ` KnkL2(˙ˆ˙;—ˆ—) Ͱ͋Δɽ͢Δͱɼ೚ҙͷ f 2 L2 (˙; —) ʹ͍ͭͯɼҰൠԽϑʔϦΤڃ਺ల։ɿf = P1 i=1 ’i hf; ’i Λ༻͍Δͱɼ kTK f ` T fk = lim n!1 kTnf ` T fk = lim n!1 ‚ ‚ ‚ ‚ ‚ ‚ n X i=1 (T ’i) hf; ’ii ` T f ‚ ‚ ‚ ‚ ‚ ‚ = lim n!1 ‚ ‚ ‚ ‚ ‚ ‚ T 0 @ n X i=1 (f; ’i) ’i 1 A ` T f ‚ ‚ ‚ ‚ ‚ ‚ ͱͳΔɽPn i=1 hf; ’ii ’i ! f (n ! 1) ͱͳΔͷͰɼ্ࣜͷ࠷ޙͷۃݶ͸ 0 ͱͳΓɼ TK = T Λ͑Δɽ□ 26
27. ఆཧ 6.12 ূ໌ 3/3 [ূ໌]: ҰҙੑΛࣔ͢ɽL2 (˙; —) ্ͷ೚ҙͷ Hilbert-Schmidt ࡞༻ૉ T ʹରͯ͠ɼҟ ͳΔ 2 ৐Մੵ෼ͳੵ෼֩ K1 (x; y) ; K2 (x; y) 2 L2 (˙ ˆ ˙; — ˆ —) ͕ଘࡏ͠ɼ T ’ = Z K1 (x; y) ’ (y) d— (y) T ’ = Z K2 (x; y) ’ (y) d— (y) ͕੒Γཱͭ͜ͱΛԾఆ͢Δɽͭ·Γɼ Z (K1 (x; y) ` K2 (x; y)) ’ (y) d— (y) = 0 Ͱ͋Δ͕ɼ͜Ε͸΄ͱΜͲશͯͷ x Ͱ K1 = K2 ͱͳΔ͜ͱΛҙຯ͠ɼԾఆʹ൓͢Δɽ □ 27
28. 6.3.2 Hermite ੑ͔Βಋ͔ΕΔ Hilbert-Schmidt ͷల։ఆཧ ੵ෼֩ K ͕ Hermite తɼ͢ͳΘͪ K (x; y) = K (y; x) Λຬͨ͢ͱ͢Δɽ͜ͷ࣌ɼK Λੵ෼֩ͱ͢Δੵ෼࡞༻ૉ TK ͸ࣗݾڞ໾࡞༻ૉͰ͋Δɽ࣮ࡍɼ hTK f; giL2(˙;—) = Z ˙ Z ˙ K (x; y) f (y) d— (y) g (x)d— (x) = Z ˙ f (x) Z ˙ K (y; x) g (x)d— (x) d— (y) = hf; TK gi Ͱ͋Δ͜ͱ͕֬ೝͰ͖Δɽࣗݾڞ໾ͳ Hilbert-Schmidt ࡞༻ૉ͸ݻ༗஋෼ղ͕ՄೳͰ͋ ΔɽTK ͷݻ༗஋ɼݻ༗ϕΫτϧ͸ TK ffi = –ffi; `– 2 C; ffi 2 L2 (˙; —)´ ʹΑΓఆٛ͞ΕΔɽࣗݾڞ໾ͳ Hilbert-Schmidt ࡞༻ૉͷݻ༗஋͸࣮਺ͱͳΓɼඇθϩݻ ༗஋ʹର͢Δݻ༗ϕΫτϧશମͷू߹ (ݻ༗ۭؒ) ͸༗ݶ࣍ݩͰ͋Δɽݻ༗ۭؒͷ࣍ݩ͚ͩ ॏෳΛڐ͠ j–1j – j–2j – ´ ´ ´ > 0 ͱ͠ɼ–i ʹରԠ͢Δݻ༗ϕΫτϧΛ ffii ͱ͢Δͱɼ fffiig ͸ L2 (˙; —) ͷਖ਼ن௚ަܥ (ۃେͰ͸ͳ͍) Ͱɼ8f 2 L2 (˙; —) ʹରͯ͠ f = 1 X i=1 hf; ffiiiL2(˙;—) ffii + ; ( 2 N (TK )) TK f = 1 X i=1 –i hf; ffiiiL2(˙;—) ffii ͱల։Ͱ͖Δ (Hilbert-Schmidt ͷల։ఆཧ)ɽ 28
29. 6.3.2 ੵ෼֩ͷ Hilbert-Schmidt ల։ ͞ΒʹɼL2 (˙ ˆ ˙; — ˆ —) ʹ͓͍ͯ K (x; y) ͸ҎԼͷΑ͏ʹల։Ͱ͖Δɽ ఆཧ 6.13 Hermite తͳੵ෼֩ K ʹର͢Δੵ෼࡞༻ૉ TK ͷඇθϩݻ༗஋ –i ͱ୯Ґݻ༗ϕΫτϧ ffii Λઌड़ͷΑ͏ʹ͢Δɽ͜ͷ࣌ɼL2 (˙ ˆ ˙; — ˆ —) ʹ͓͍ͯɼ K (x; y) = 1 X i=1 –iffii (x) ffii (y) ͷల։͕੒Γཱͭɽ Fubini ͷఆཧ ˙ Λଌ౓ۭؒͱ͠ɼf (x; y) ͕Մଌ͔ͭՄੵ෼Ͱ͋ΔͳΒ͹ɼҎԼ͕੒ཱ͢Δɽ Z ˙ Z ˙ f (x; y) dy ! dx = Z ˙ Z ˙ f (x; y) dx ! dy = Z ˙ˆ˙ f (x; y) d (x; y) 29
30. ఆཧ 6.13 ূ໌ 1/2 K ͷ 2 ৐Մੵ෼ੑ͔Βɼ΄ͱΜͲશͯͷ x ʹରͯ͠ K (x; ´) 2 L2 (˙; —) Ͱ͋Δɽ͜͜ Ͱɼੵ෼֩ͷఆٛɼٴͼݻ༗஋෼ղΛ༻͍ͯ DK (x; ´) ; ffii E L2(˙;—) = Z K (x; y) ffii (y) d— (y) = TK ffii (x) = –iffii (x) ͱͳΔ͜ͱʹ஫ҙ͢Δɽ͞Βʹɼ 2 N (TK ) ͳΒ͹ DK (x; ´) ; E L2(˙;—) = TK (x) = 0 ͱͳΓɼଌ౓ — ʹؔͯ͠ɼ΄ͱΜͲશͯͷ x ʹରͯ͠ҰൠԽϑʔϦΤڃ਺ల։Λ༻͍Δͱɿ K (x; ´) = 1 X i=1 DK (x; ´) ; ffii E L2(˙;—) ffii = 1 X i=1 –iffii (x) ffii (6.4) ͕੒ཱ͢ΔɽFubini ͷఆཧ΍ L2 ϊϧϜͷఆ͔ٛΒɼ Z Z ˛ ˛ ˛ ˛ ˛ ˛ K (x; y) ` N X i=1 –iffii (x) ffii (y) ˛ ˛ ˛ ˛ ˛ ˛ 2 d— (x) d— (y) = Z ‚ ‚ ‚ ‚ ‚ ‚ K (x; ´) ` N X i=1 –iffii (x) ffii ‚ ‚ ‚ ‚ ‚ ‚ 2 L2(˙;—) d— (x) (6.5) ͱͳΔɽ 30
31. ఆཧ 6.13 ূ໌ 2/2 ࣜ 6.5 ͷӈลͷඃੵ෼ؔ਺͸ࣜ 6.4 ͔Β΄ͱΜͲશͯͷ x ʹରͯ͠ N ! 1 Ͱ 0 ʹऩ ଋɽ͔ͭɼ ‚ ‚ ‚ ‚ ‚ ‚ K (x; ´) ` N X i=1 –iffii (x) ffii ‚ ‚ ‚ ‚ ‚ ‚ 2 L2(˙;—) = * K (x; ´) ` N X i=1 –iffii (x) ffii; K (x; ´) ` N X i=1 –iffii (x) ffii + L2(˙;—) = Z jK (x; y)j2 d— (y) ` N X i=1 –2 i jffii (x)j2 » Z jK (x; y)j2 d— (y) ͱ N ʹΑΒͳ͍ 2 ৐Մੵ෼ؔ਺Ͱ্͔Β཈͑Δ͜ͱ͕Ͱ͖ɼ༏ऩଋఆཧʹΑͬͯࣜ (6.5) ͸ N ! 1 Ͱ 0 ʹऩଋ͢Δɽ 2 31
32. 6.3.3 ਖ਼஋ੵ෼֩ͱ Mercer ͷఆཧ ఆཧ 6.13 Ͱ͸Χʔωϧ K ͷ Hermite ੑͷΈΛԾఆͯ͠ K ͷ L2 (˙ ˆ ˙; — ˆ —) ʹ ͓͚Δల։Λࣔͨ͠ɽ͞Βʹ — ͕ίϯύΫτ Hausdorff ۭؒ ˙ ্ͷ༗ݶ Radon ଌ౓ ͰɼK ͕ ˙ ্ͷ࿈ଓͳਖ਼ఆ஋Χʔωϧͷ৔߹ʹ͸ҎԼʹࣔ͢Α͏ʹ͜ͷऩଋ͸ઈର͔ͭҰ ༷Ͱ͋Δ (Mercer ͷఆཧ)ɽ ·ͣ K ͷਖ਼ఆ஋ੑͱੵ෼࡞༻ૉ TK ͷਖ਼஋ੑ: hTK f; fiL2(˙;—) = Z ˙ Z ˙ K (x; y) f (x) f (y)d— (x) d— (y) – 0; `8f 2 L2 (˙; —)´ (6.6) ͷؔ࿈Λࣔ͢ɽ Ґ૬ۭؒ ˙ ্ͷඇෛ Borel ଌ౓ — ͷαϙʔτ:Supp (—) Λ Supp (—) := fx 2 ˙ j — (U) > 0; U ͸ x ΛؚΉ೚ҙͷ։ू߹ g ʹΑΓఆٛ͢Δɽαϙʔτ͸ดू߹Ͱ͋Δɽ·ͨɼRn ্ͷ Borel ଌ౓ — ͕࿈ଓͳ֬཰ີ ౓ p (x) Λ࣋ͭ࣌ɼ— ͷαϙʔτͱ p (x) ͷαϙʔτ͸Ұக͢Δɽ Hausdorff ্ۭؒͷ Borel ଌ౓ — ͕ Radon ଌ౓Ͱ͋Δɼͱ͸೚ҙͷίϯύΫτ༥߹ K ʹରͯ͠ — (K) < 1 Ͱɼ೚ҙͷՄଌू߹ E ʹରͯ͠ — (E) = sup f— (K) j K ͸ K ȷ E ͳΔίϯύΫτू߹ g ͕੒Γཱͭ͜ͱͰ͋Δɽ 32
33. Hausdorff ۭؒ Ґ૬ۭؒ S ͷ૬ҟͳΔ೚ҙͷ 2 ఺ɼx1; x2 ʹରͯ͠ɼU (x1) \ U (x2) = ; Λຬͨ͢։ ू߹ U (x1) ; U (x2) ͕ଘࡏ͢Δ࣌ɼS ͸ Hausdorff ۭؒɼ͋Δ͍͸ T2 ۭؒͰ͋Δͱ ͍͏ɽ 33
34. 6.3.3 ਖ਼஋ੵ෼֩ͱ Mercer ͷఆཧ ໋୊ 6.14 ˙ ΛίϯύΫτ Hausdorff ۭؒɼ— Λ ˙ ্ͷ༗ݶඇෛ Radon ଌ౓Ͱɼ Supp (—) = ˙ ͱ͢Δɽ K (´; ´) : ˙ ˆ ˙ ! C Λɼ˙ ্ͷ࿈ଓͳ Hermite తͳΧʔωϧͱ͢Δ࣌ɼK ͕ਖ਼ఆ஋ ΧʔωϧͰ͋Δඞཁे෼৚݅͸ TK ͕ਖ਼஋࡞༻ૉͰ͋Δ͜ͱͰ͋Δɽ ূ໌ɿ › ඞཁੑ:K ͕ਖ਼ఆ஋ΧʔωϧͰ͋Δͱ͢Δɽ˙ ্ͷ೚ҙͷ࿈ଓؔ਺ g : ˙ ! C ͱՄ ଌू߹ʹΑΔ ˙ ͷ෼ׂ fEign i=1 ɼ(ͨͩ͠ i j ͳΒ͹ Ei \ Ej = ;; [n i=1 Ei = ˙) ʹରͯ͠ɼK ͷਖ਼ఆ஋ੑ͔Βɼ n X i;j=1 K (xi; xj) g (xi) g (xj)— (Ei) — (Ej) – 0 ͸੒Γཱͭɽࣜ 6.6ɿhTK f; fi ͸͜ͷΑ͏ͳ࿨ͷۃݶͱͯ͠ಘΒΕΔͨΊɼඇෛͰ͋ ΔɽҰൠͷ f 2 L2 (˙; —) ʹରͯ͠͸೚ҙͷ › > 0 ʹରͯ͠ kf ` gkL2(˙;—) < › ͳΔ࿈ଓؔ਺ g ΛͱΕ͹ࣜ 6.6 ͕ࣔ͞ΕΔɽ 34
35. ໋୊ 6.14 ূ໌ 2 › े෼ੑ: TK ͕ਖ਼஋࡞༻ૉͰ͋Δͱ͢Δɽ— = 0 ͳΒ͹ࣗ໌ʹਖ਼ఆ஋ੑ͸ຬͨ͞ΕΔɽ ͦͷͨΊɼ— (˙) > 0 ͱͯ͠Α͍ɽഎཧ๏Ͱࣔͨ͢Ίʹɼ͋Δ xi 2 ˙ ͱ ci 2 C ͱ ‹ > 0 ͕͋ͬͯɼ n X i;j=1 cicjK (xi; xj) » `‹ ͕੒ΓཱͭͱԾఆ͢Δ (TK ͕ਖ਼஋࡞༻ૉͰ͋Δ࣌ʹ K ͕ਖ਼ఆ஋ΧʔωϧͰͳ͍͜ͱ ΛԾఆ͢Δ)ɽҰൠੑΛࣦΘͣɼxi xj ͱͯ͠Α͍ɽK ͷ࿈ଓੑͱ ˙ ͷ Housdorff ੑʹΑΓɼ֤ xi ͷ։ۙ๣ Ui ͕͋ͬͯɼUi \ Uj = ; (i j) ͔ͭɼ 8(x; y) 2 Ui ˆ Uj ʹରͯ͠ ˛ ˛cicjK (x; y) ` cicjK (xi; xj)˛ ˛ » ‹ 2n2 ͱͰ͖Δ (े෼খ͘͢͞Ε͹औΕΔ)ɽ͜ͷ࣌ɼSupp (—) = ˙ ͔Β — (Ui) > 0 Ͱ͋Δɽ f := n X i=1 „ ci — (Ui) « IUi 2 L2 (˙; —) ͱ͓͘ (ͨͩ͠ IUi ͸ Ui ͷఆٛؔ਺)ɽ 35
36. ໋୊ 6.14 ূ໌ 3 ͜ͷͱ͖ TK ͷਖ਼஋ੑΛνΣοΫ͢Δͱ Z Z ˙ˆ˙ K (x; y) f (x) f (y)d— (x) d— (y) » n X i;j=1 1 — (Ui) — (Uj) Z Z UiˆUj ˛ ˛cicjK (x; y) ` cicjK (xi; xj)˛ ˛ d— (x) d— (y) + n X i;j=1 Z Z UiˆUj cicj — (Ui) — (Uj) K (xi; xj) d— (x) d— (y) » ‹ 2 ` ‹ < 0 ͱͳΓɼԾఆʹໃ६͢Δɽ□ 36
37. 6.3.3 ਖ਼஋ੵ෼֩ͱ Mercer ͷఆཧ ˙ ͱ — Λ໋୊ 6.14 ͷͱ͓Γͱ͢Δɽ࿈ଓͳਖ਼ఆ஋Χʔωϧ K ʹΑͬͯఆ·Δੵ෼֩Λ TK ͱ͢Δ࣌ɼ໋୊ 6.14 ͔Β TK ͸ਖ਼஋࡞༻ૉͰɼ೚ҙͷ f 2 L2 (˙; —) ʹରͯ͠ɼ (TK f; f) – 0 ͕੒ΓཱͭͷͰ TK ͷݻ༗஋͸ඇෛ࣮਺Ͱ͋Δɽॏෳ౓ͷͿΜ͚ͩฒ΂ͨ ਖ਼ͷݻ༗஋ͱݻ༗ϕΫτϧΛͦΕͧΕ –1 – –2 – ´ ´ ´ > 0 ٴͼ ffii Ͱද͢ɽ ఆཧ 6.15 Mercer ͷఆཧ K (x; y) = 1 X i=1 –iffii (x) ffii (y) (6.7) ͜͜Ͱɼऩଋ͸ ˙ ˆ ˙ ্ͷઈର͔ͭҰ༷ऩଋͰ͋Δɽ 37
38. Mercer ͷఆཧɿূ໌ 1 ఆཧ 6.13 ͔Βɼࣜ 6.7 ͸ L2 (˙ ˆ ˙; — ˆ —) ͷऩଋͱͯ͠੒Γཱͭɽ·ͨɼ –iffii (x) = TK ffii (x) = Z K (x; y) ffii (y) d— (y) ͔Β K ͷ࿈ଓੑ͔Β ffii ΋࿈ଓͰ͋Δɽ͜͜Ͱɼ Kn;m (x; y) := m X i=n –iffii (x) ffii (y); Rn (x; y) := K (x; y) ` K1;n`1 (x; y) ͱ͓͘ͱɼKn;m (´; ´) ͸ਖ਼ఆ஋ΧʔωϧͰ͋Δ͜ͱ͔Βɽ໋୊ 6.14 ͔Βਖ਼஋ੵ෼֩ͱͳ Δɽ·ͨɼఆཧ 6.13 ͔Β L2 (˙ ˆ ˙; — ˆ —) ʹ͓͍ͯ lim m!1 Kn;m (x; y) = 1 X i=n –iffii (x) ffii (y) = 1 X i=1 –iffii (x) ffii (y) ` n`1 X i=1 –iffii (x) ffii (y) = K (x; y) ` K1;n`1 (x; y) = Rn (x; y) Ͱ͋Γɼਖ਼஋ੵ෼֩Ͱ͋Δͱ͍͏ੑ࣭͸ L2 (˙ ˆ ˙; — ˆ —) ͷऩଋʹؔͯ͠อଘ͞ΕΔ ͜ͱ͔Βɼऩଋઌ Rn ΋ਖ਼஋ੵ෼֩Ͱ͋ΔɽRn ͕ਖ਼஋ੵ෼֩Ͱ͋Δ͜ͱ͔Βɼ໋୊ 6.14 Λ༻͍Ε͹ Rn (x; y) ͸ਖ਼ఆ஋ΧʔωϧͰ͋Δɽ 38
39. Mercer ͷఆཧɿূ໌ 2 ͜ͷ࣌ɼਖ਼ఆ஋Χʔωϧͷఆ͔ٛΒ 8x 2 ˙ ͱ 8n 2 N ʹର͠ɼRn (x; x) – 0 Ͱ͋ ΔɽK (x; x) = Rn (x; x) + K1;n`1 (x; x) ͔Βɼ K1;n`1 (x; x) = Pn`1 i=1 –i jffii (x)j2 » K (x; x) ͕੒Γཱͪɼ 1 X i=1 –i jffii (x)j2 » K (x; x) (6.8) Λ͑Δɽ·ͨɼCauchy-Schwarz ͷෆ౳͔ࣜΒɼ m X i=n ˛ ˛ ˛ –iffii (x) ffii (y) ˛ ˛ ˛ » 0 @ m X i=n –i jffii (x)j2 1 A 1 2 0 @ m X i=n –i jffii (y)j2 1 A 1 2 = Kn;m (x; x) 1 2 Kn;m (y; y) 1 2 (6.9) Ͱ͋Δɽࣜ 6.8 ͔Βɼࣜ 6.7 ͷӈลͷڃ਺͸֤ (x; y) Ͱઈରऩଋ͢Δɽ͜ͷऩଋઌΛ H (x; y) := 1 X i=1 –iffii (x) ffii (y) (6.10) ͱ͓͍ͯɼH = K ͱͳΔ͜ͱΛࣔ͢ɽ 39
40. Mercer ͷఆཧɿূ໌ 3 ·ͣɼࣜ 6.9 ͔Β m X i=n ˛ ˛ ˛ –iffii (x) ffii (y) ˛ ˛ ˛ » Kn;m (x; x) 1 2 Kn;m (y; y) 1 2 » Kn;m (x; x) 1 2 sup z2˙ K (z; z) 1 2 ͔Βɼ೚ҙʹ x 2 ˙ Λݻఆ͢Ε͹ࣜ 6.10 ͸ y 2 ˙ ʹؔͯ͠Ұ༷ʹऩଋ͢ΔɽΏ͑ʹ೚ ҙͷ x ʹର͠ɼ Z H (x; y) ffii (y) d— (y) = 1 X n=1 Z –nffin (x) ffin (y)ffii (y) d— (y) = –iffii (x) = Z K (x; y) ffii (y) d— (y) ͕੒Γཱͭ (?)ɽ͜Ε͸ L2 (˙; —) ʹ͓͍ͯ H (x; ´) = K (x; ´) Λҙຯ͢Δ͕ɼ྆ऀ͸ ࿈ଓؔ਺Ͱ͋Δ͜ͱ͔Β H (x; y) = K (x; y) ΛಘΔɽ 40
41. Mercer ͷఆཧɿূ໌ 4 ࠷ޙʹɽMercer ͷఆཧͷ͕ࣜ ˙ ˆ ˙ ্ͷҰ༷ऩଋͰ͋Δ͜ͱΛࣔ͢ɽ֤఺ऩଋͰ͋Δ͜ ͱ͸ݟͨͷͰɼಛʹ y = x ͱ͓͘ͱɼ 1 X i=1 –i jffii (x)j2 = K (x; x) ͕੒Γཱͭɽ֤ ffii (x) ͸࿈ଓؔ਺Ͱ͋Γɼࠨล͸୯ௐʹ K (x; x) ʹऩଋ͢ΔͷͰɼDini ͷఆཧ (ίϯύΫτू߹্ͷ࿈ଓؔ਺ͷ୯ௐྻ͕͋Δ࿈ଓؔ਺ʹ֤఺ऩଋ͢ΔͳΒ͹ऩଋ͸ Ұ༷) ͔Β͜ͷऩଋ͸Ұ༷Ͱ͋Δɽ͕ͨͬͯ͠ɼn; m ! 1 ͷ࣌ɼKn;m (x; x) ͸ 0 ʹ Ұ༷ऩଋ͠ɼࣜ 6.9 ͔Β Kn;m (x; y) ͸ 0 ʹҰ༷ऩଋ͢Δɽ͜Ε͸ࣜ 6.7 ͷઈର͔ͭҰ ༷ͳऩଋΛҙຯ͢Δɽ(x,y ʹΑΒͣऩଋ͢Δ͜ͱ͔Β) 2 41
42. ਖ਼ఆ஋ΧʔωϧʹରԠ͢Δ RKHS ͷཅͳදࣔ Mercer ͷఆཧΛ༻͍Δͱɼਖ਼ఆ஋ΧʔωϧʹରԠ͢Δ RKHS ͱɼͦͷ಺ੵͷཅͳදࣔΛ ༩͑Δ͜ͱ͕Ͱ͖Δɽ(2.2.2.b:༗ݶू߹্ͷ RKHS ͷදࣔͷ֦ுͱͯ͠༩͑Δ͜ͱ͕Ͱ ͖Δ) Mercer ͷఆཧͱಉ͡৚݅ͷݩɼੵ෼࡞༻ૉ TK ͷඇθϩݻ༗஋ʹରԠ͢Δ୯Ґݻ༗ϕΫτ ϧʹ N (TK ) ͷਖ਼ن௚ަجఈΛ෇͚Ճ͑ͯ L2 (˙; —) ͷ׬શਖ਼ن௚ަجఈ fffiig1 i=1 Λߏ ੒͢Δɽ͢Δͱɼfffiig1 i=1 ͸γϟ΢μʔجఈͱͳΓɼ೚ҙͷ f 2 H ͸ͦΕΒͷجఈͷઢ ܕ݁߹ͱͯ͠ɿ f = 1 X i=1 aiffii (ͨͩ͠ ai 2 R) ͱͰ͖Δɽ͜ΕΛ༻͍ͯ L2 (˙; —) ͷ෦෼ϕΫτϧۭؒ H Λ H := 8 < : f 2 L2 (˙; —) j f = 1 X i=1 aiffii; 1 X i=1 jaij2 –i < 1 9 = ; (6.11) ʹΑΓఆٛ͢Δɽ·ͨɼf = P1 i=1 aiffii 2 H ͱ g = P1 i=1 biffii 2 H ʹରͯ͠ɼ಺ ੵΛ hf; giH := 1 X i=1 aibi –i (6.12) ͱఆΊΔɽ͜ͷΑ͏ʹఆΊͨ H ͕ K Λ࠶ੜ֩ͱ͢Δ RKHS ʹͳΔ͜ͱΛࣔ͢ɽ 42
43. H ͕ਖ਼ఆ஋ΧʔωϧʹରԠ͢Δ RKHS Ͱ͋Δ͜ͱɿূ໌ 1 ·ͣ H ͕ώϧϕϧτۭؒͰ͋Δ (׬උͰ͋Δ) ͜ͱΛࣔ͢ɽ ffng1 n=1 Λ H ͷ Cauchy ྻͱ͢Δɽࣜ 6.11 ͔Β fn = P1 i=1 ¸n;iffii ͱ͓͘͜ͱ͕ Ͱ͖ͯɼP1 i=1 j¸n;ij2 –i < 1 Ͱ͋ΔͷͰɼ tn := ( ¸n;i p–i )1 i=1 ͸਺ྻۭؒ l2 ͷ Cauchy ྻͰ͋Δɽl2 ͸׬උͳͷͰɼ͋Δ t = f˛ig1 i=1 2 l2 ͕ଘࡏ ͠ɼtn ! t (n ! 1) Ͱ͋Δɽ͜ͷ࣌ɼ¸˜ i := p–i˛i ͱ͢Δͱɼ 1 X i=1 ˛ ˛¸˜ i ˛ ˛ 2 –i = 1 X i=1 j˛ij2 < 1 `f˛ig1 i=1 2 l2͔Β´ 1 X i=1 ˛ ˛¸n;i ` ¸˜ i ˛ ˛ 2 –i ! 0 ͕੒Γཱͭɽ͢Δͱɼf = P1 i=1 ¸˜ i ffii 2 H ʹରͯ͠ɼkfn ` fkH ! 0 ΛಘΔɽ 43
44. H ͕ਖ਼ఆ஋ΧʔωϧʹରԠ͢Δ RKHS Ͱ͋Δ͜ͱɿূ໌ 2 ࣍ʹɼH ͕ K Λ࠶ੜ֩ʹ࣋ͭ͜ͱΛࣔ͢ɽ K (´; x) = P1 i=1 –iffii (´) ffii (x) ʹ͓͍ͯɼMercer ͷఆཧ͔Β P1 i=1 –i jffii (x)j2 < 1 Ͱ͋ΔͷͰ K (´; x) 2 H Ͱ͋Δɽ8f = P1 i=1 aiffii 2 H ʹର͠ɼ hf; K (´; x)iH = 1 X i=1 ¸i–iffii (x) –i = 1 X i=1 ¸iffii (x) = f (x) ͱͳΓɼ࠶ੜੑ͕֬ೝͰ͖ͨɽ 2 Ҏ্͔Βɼਖ਼ఆ஋ΧʔωϧʹରԠ͢Δ RKHS ͸ࣜ 6.11 Ͱ༩͑ΒΕΔ H ʹҰக͠ɼͦͷ ಺ੵ͸ࣜ 6.12 ͷΑ͏ʹɼڃ਺ͱͯ͠༩͑ΒΕΔɽ 44
45. ώϧϕϧτ্ۭؒͷ֬཰ม਺ͷฏۉ Hɿώϧϕϧτۭؒ (ؔ਺ۭؒ)ɼF ɿH ্ͷ֬཰ม਺ɽͨͩ͠ɼE[kF k] < 1ɽ ͜ͷ࣌ɼf 2 H ʹରͯ͠ H ্ͷઢܗ൚ؔ਺ ffiF : H ! R ΛҎԼͷΑ͏ʹఆΊΔɿ ffiF (f) := E[hf; F i] Ϧʔεͷදݱఆཧ͔Βɼ೚ҙͷ f 2 H ʹରͯ͠ɼ͋Δ mF 2 H ͕ଘࡏ͠ɼ hf; mF i = ffiF (f) ͕੒ΓཱͭɽΑͬͯ ffiF (f) = E[hf; F i] = hf; mF i (8.1) Λ͑Δɽ͜ͷ mF Λ֬཰ม਺ F ͷฏۉͱݺͼɼE[F ] Ͱද͢ɽ͜ͷ࣌ɼ E[hf; F i] = hf; mF i = hf; E[F ]i ͱͳΓɼฏۉͱ಺ੵͷૢ࡞͸ަ׵ՄೳͰ͋Δɽ 45
46. RKHS ʹ͓͚Δฏۉ (X; B)ɿՄଌۭؒɼXɿX ʹ஋ΛͱΔ֬཰ม਺ɼRKHSɿ(Hk; k) Λߟ͑Δɽͨͩ͠ɼ E[ pk (X; X)] < 1 ΛԾఆ͢Δɽ ಛ௃ࣸ૾ ˘ (x) = k (´; x) ʹରͯ͠ɼ࠶ੜੑ͔Β k˘ (X)k2 = hk (´; X) ; k (´; X)i = hkX ; kX i = kX (X) = k (X; X) ͕੒Γཱͭ͜ͱʹ஫ҙ͢Ε͹ɼE k˘ (X)k < 1 ͱͳΓɼલ߲ͷԾఆΛຬͨ͠ɼ֬཰ม਺ ˘ (X) ͷฏۉ mk X ͕ଘࡏ͢Δɽ͜ͷ࣌ɼmk X ΛɼX ͷ Hk ʹ͓͚ΔฏۉɼͱݺͿɽࣜ (8.1) ͓Αͼ࠶ੜੑ͔Βɼ೚ҙͷf 2 H ʹରͯ͠ ˙f; mk X ¸ = E[hf; ˘ (X)i] = E[hf; K (X; ´)i] = E[f (X)] (8.2) ͱͳΓɼ೚ҙͷ f ʹରͯ͠ظ଴஋ E[f (X)] ͕ f ͱ mk X ͷ಺ੵͰද͞ΕΔɽ ฏۉ mk X ͷཅͳදࣔΛٻΊΔɽmk X 2 H ͔Βɼ೚ҙͷ y 2 X ʹ͍ͭͯɼ࠶ੜੑΛ༻͍ Δͱ mk X (y) = ˙mk X ; k (´; y)¸ = hE (˘ (X)) ; k (´; y)i = E hk (´; X) ; k (´; y)i = E[k (X; y)] (8.8) ͱͳͬͯɼฏۉ mk X ͸Χʔωϧؔ਺ͷظ଴஋ͱͯ͠༩͑ΒΕΔɽ 46
47. RKHS ʹ͓͚Δฏۉͷਪఆྔ X1; : : : ; Xn Λ P ʹै͏ i.i.d. αϯϓϧͱ͠ɼk Λ X ্ͷՄଌͳΧʔωϧɽmk X ͷਪఆ ྔ ^ mk (n) ΛҎԼͷΑ͏ʹఆٛ͢Δɿ ^ mk (n) := 1 n n X i=1 k (´; Xi) ͜Ε͕ mk X ͷෆภਪఆྔͰ͋Δ͜ͱ͸໌Β͔ɽ 47
48. ಛੑతͳਖ਼ఆ஋Χʔωϧ ಛੑతͳΧʔωϧ (X; BX ):ՄଌۭؒɼP:֬཰ଌ౓શମͱ͢ΔɽX ্ͷ༗ք͔ͭՄଌͳਖ਼ఆ஋Χʔωϧ k ͕ ಛੑతͰ͋Δɼͱ͸ɼࣸ૾ɿ P ! Hk; P 7! mk P ͕୯ࣹͰ͋Δ͜ͱΛ͍͏ɽ͜͜Ͱɼmk P ͸෼෍ P Λ΋ͭ X ্ͷ֬཰ม਺ͷ Hk ʹ͓͚Δ ฏۉΛද͢ɽ ͜ͷఆٛ͸ɼ೚ҙͷ f 2 Hk ʹ͍ͭͯ EX‰P [f (X)] = EX‰Q[f (X)] ) P = Q ͱಉ஋Ͱ͋Δɽ 48