Support Vector Machine (SVM)

はじめての『はじめてのパターン認識』第8章
This is a general introduction of support vector machine (SVM) for machine learning, which includes hard-margin SVM, soft-margin SVM and Kernel trick method.

Transcript

1. ୈষ
   αϙʔτϕΫτϧϚγϯ αϙʔτϕΫτϧϚγϯͷಋग़ ͸͡Ίͯͷύλʔϯೝࣝྠಡձ ޻౻঑ 
2. 065-*/& ·ͱΊ ϋʔυϚʔδϯ47. ιϑτϚʔδϯ47. αϙʔτϕΫτϧϚγϯ 47. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
3. 065-*/& ·ͱΊ ϋʔυϚʔδϯ47. ιϑτϚʔδϯ47. αϙʔτϕΫτϧϚγϯ 47. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
4. 065-*/& ఆٛ ઢܗࣝผ௒ฏ໘ Ϛʔδϯ࠷େԽ৚݅ ϥάϥϯδϡͷະఆ৐਺๏ ૒ର໰୊ ྫ୊ ຕ ຕ ຕ ຕ ຕ ຕ ઢܗ෼཭Մೳͳ৔߹ͷ ϋʔυϚʔδϯ47. ·ͱΊ ຕ
5. ఆٛ ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ϋʔυϚʔδϯ47.
6. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ઢܗࣝผ௒ฏ໘ͱ
   ͦΕʹ࠷΋͍ۙσʔλͱͷڑ཭% ϋʔυϚʔδϯ47. ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ࠷େϚʔδϯߏ੒͢ΔσʔλΛ
   αϙʔτϕΫτϧͱ͍͏
7. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
   ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ϋʔυϚʔδϯ47.
8. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
   ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ σʔλΛ׬શʹ෼཭Ͱ͖Δ࣌ɼ
   ઢܗ෼཭Մೳͱ͍͏ ϋʔυϚʔδϯ47.
9. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ଟΫϥε෼ྨ͸ɼ , ݸͷରଞ47.Ͱ࣮ݱ͢Δ͜ͱ͕ଟ͍ ˔ରଞͷઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ϋʔυϚʔδϯ47.
10. 47.ͷͨΊʹೋͭͷ৚݅Λநग़͢Δ 
    ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭ͷ৚݅ 
    Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ ϋʔυϚʔδϯ47.
11. 
    ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭ͷ৚݅ 47.ͷͨΊʹೋͭͷ৚݅Λநग़͢Δ 
    Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ ϋʔυϚʔδϯ47.
12. ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ wT = (w1 w2 ) xi T = (x1i x2i ) ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ઢܗࣝผ௒ฏ໘ʹ
    ࠷΋͍ۙσʔλͱͷڑ཭% ͸ඞͣ Ҏ্ʹͳΔ d D d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ d = |wT xi + b| ∥w∥ ≥ D ϋʔυϚʔδϯ47.
13. ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ wT = (w1 w2 ) xi T = (x1i x2i ) d = |wT xi + b| ∥w∥ ≥ D ͸ඞͣ Ҏ্ʹͳΔ d D ∥w∥ = w2 1 + w2 2 ≥ 0 d∥w∥ = |wT xi + b| ≥ κ wT = (w1 w2 ) ͷ஋͸ͦΕͧΕҰఆͳͷͰ ઢܗࣝผ௒ฏ໘͸ݻఆ͞ΕΔͱ͢Δͱ ΋Ұఆͷ͸ͣͰ͋ΔɽͦͷͨΊɼڑ཭ͷࣜʹ͜ΕΛ͔͚ɼ
    ࠷ऴతͳϚʔδϯͱઢܗࣝผ௒ฏ໘͕Ұͭʹఆ·Δͱ͢Δͱ D∥w∥ = κ ͱஔ͍ͯ ڑ཭ͷ৚݅ ڑ཭ͷ৚݅ ઢܗࣝผڥքʹ࠷΋͍ۙɼ
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Ͱ͋ΔϕΫτϧ͕αϙʔτϕΫτϧ |wT xi + b| = κ ϋʔυϚʔδϯ47.
14. |wT xi + b| ≥ κ |wT xi + b| κ ≥ κ κ ⟹ |wT xi + b| ≥ 1 (wT = wT κ = ( w1 κ w2 κ ), b = b κ ) શମΛ κ ͰׂΓɼॏΈΛ࠶ఆٛ͢Δͱɽɽɽ ⟺ |wT xi + b| = { (wT xi + b) ≥ 1 (wT xi + b ≥ 0) −(wT xi + b) ≥ 1 (wT xi + b < 0) = { wT xi + b ≥ 1 (wT xi + b ≥ 0) wT xi + b ≤ − 1 (wT xi + b < 0) ڑ཭ͷ৚݅ ઢܗࣝผڥքʹ࠷΋͍ۙɼ
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Ͱ͋ΔϕΫτϧ͕αϙʔτϕΫτϧ |wT xi + b| = 1 ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ wT = (w1 w2 ) xi T = (x1i x2i ) ϋʔυϚʔδϯ47.
15. |wT xi + b| = { wT xi + b ≥ 1 (wT xi + b ≥ 0) wT xi + b ≤ − 1 (wT xi + b < 0) ࣝผਖ਼ղͷ৚݅ ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ wT = (w1 w2 ) xi T = (x1i x2i ) D D C1 C2 ti = 1 Ϋϥεͷਖ਼ղϥϕϧ Ϋϥεʹ͓͍ͯ wT xi + b ≥ 1 ti (wT xi + b) ≥ 1 ti (wT xi + b) ≥ 1 ti = − 1 Ϋϥεͷਖ਼ղϥϕϧ Ϋϥεʹ͓͍ͯ wT xi + b ≤ 1 ϋʔυϚʔδϯ47.
16. ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ wT = (w1 w2 ) xi T = (x1i x2i ) ti (wT xi + b) ≥ 1 D D C1 C2 ti = 1 Ϋϥεͷਖ਼ղϥϕϧ Ϋϥεʹ͓͍ͯ wT xi + b ≥ 1 ti (wT xi + b) ≥ 1 ti (wT xi + b) ≥ 1 ti = − 1 Ϋϥεͷਖ਼ղϥϕϧ Ϋϥεʹ͓͍ͯ wT xi + b ≤ 1 ϋʔυϚʔδϯ47. ࣝผਖ਼ղͷ৚݅
17. ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ wT = (w1 w2 ) xi T = (x1i x2i ) ti (wT xi + b) − 1 ≥ 0 D D C1 C2 ti = 1 Ϋϥεͷਖ਼ղϥϕϧ Ϋϥεʹ͓͍ͯ wT xi + b ≥ 1 ti (wT xi + b) ≥ 1 ti (wT xi + b) ≥ 1 ti = − 1 Ϋϥεͷਖ਼ղϥϕϧ Ϋϥεʹ͓͍ͯ wT xi + b ≤ 1 ϋʔυϚʔδϯ47. ࣝผਖ਼ղͷ৚݅
18. ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭͔Βಋ͍ͨࣝผΫϥε͕ਖ਼͍͠৚݅ ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ wT = (w1 w2 ) xi T = (x1i x2i ) ϋʔυϚʔδϯ47. ti (wT xi + b) − 1 ≥ 0
19. 47.ͷͨΊʹೋͭͷ৚݅Λநग़͢Δ 
    ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭ͷ৚݅ 
    Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ ϋʔυϚʔδϯ47. ti (wT xi + b) − 1 ≥ 0
20. ti (wT xi + b) − 1 ≥ 0 47.ͷͨΊʹೋͭͷ৚݅Λநग़͢Δ 
    ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭ͷ৚݅ 
    Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ ϋʔυϚʔδϯ47.
21. Ϛʔδϯɿઢܗࣝผ௒ฏ໘ͱͦΕʹ࠷΋͍ۙσʔλͱͷڑ཭% ࠷େͷϚʔδϯ Л Λୡ੒͢ΔΑ͏ͳ
    αϙʔτϕΫτϧ TW Λݟ͚ͭΔ ઢܗࣝผ௒ฏ໘ D1 D2 Ϛʔδϯ C1 C2 ρ(w, b) = D1 + D2 = min d1i + min d2i ʜϚʔδϯͷఆٛ i i ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ wT = (w1 w2 ) xi T = (x1i x2i ) d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ ϋʔυϚʔδϯ47.
22. Ϛʔδϯɿઢܗࣝผ௒ฏ໘ͱͦΕʹ࠷΋͍ۙσʔλͱͷڑ཭% ρ(w, b) = min d1i + min d2i = |wT xsv1 + b| ∥w∥ + |wT xsv2 + b| ∥w∥ Ϛʔδϯͷఆٛ = 1 ∥w∥ + 1 ∥w∥ = 2 ∥w∥ ρ(w0 , b0 ) = max ρ(w, b) = max 2 ∥w∥ ∝ max 1 ∥w∥ Ϛʔδϯ࠷େԽ৚݅ xsv1 |sv ∈ C1 xsv2 |sv ∈ C2 i i w, b w w Ϛʔδϯ࠷େ࣌ͷม਺ɹɹΛɹɹ
    ͱஔ͍ͨ w, b w0 , b0 ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ wT = (w1 w2 ) xi T = (x1i x2i ) d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ ϋʔυϚʔδϯ47.
23. Ϛʔδϯɿઢܗࣝผ௒ฏ໘ͱͦΕʹ࠷΋͍ۙσʔλͱͷڑ཭% ρ(w0 , b0 ) ⟹ ρ(w0 ) = max 1 ∥w∥ Ϛʔδϯ࠷େԽ৚݅ w ⟹ min ∥w∥ w ⟹ min 1 2 ∥w∥2 = min 1 2 wTw w min 1 2 wTw Ϛʔδϯ࠷େԽ৚݅ w ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ wT = (w1 w2 ) xi T = (x1i x2i ) d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ w ϋʔυϚʔδϯ47.
24. ఺ͱ௚ઢͷڑ཭ͷެࣜ ௚ઢ w1 x1 + w2 x2 + b = 0ͱɼ఺ (x1i , x2i ) ͷڑ཭ d ͸࣍ͷࣜͰॻ͖දͤΔɽ wT = (w1 w2 ) xi T = (x1i x2i ) d = |w1 x1i + w2 x2i + b| w2 1 + w2 2 = |wT xi + b| ∥w∥ Ϛʔδϯɿઢܗࣝผ௒ฏ໘ͱͦΕʹ࠷΋͍ۙσʔλͱͷڑ཭% Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ʢධՁؔ਺ʣ min f(w) = min 1 2 wTw w w ϋʔυϚʔδϯ47.
25. ti (wT xi + b) − 1 ≥ 0 47.ͷͨΊʹೋͭͷ৚݅Λநग़͢Δ 
    ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭ͷ৚݅ 
    Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ min f(w) = min 1 2 wTw w w ϋʔυϚʔδϯ47.
26. 47.ͷͨΊʹೋͭͷ৚݅Λநग़͢Δ 
    ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭ͷ৚݅ 
    Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ min f(w) = min 1 2 wTw w w ϋʔυϚʔδϯ47. ti (wT xi + b) − 1 ≥ 0
27. ओ໰୊ gi (w, b) = ti (wT xi + b) − 1 ≥ 0 f(w) = 1 2 wTw ධՁؔ਺ʢ࠷খԽʣ ෆ౳੍ࣜ໿৚݅ ্هͷෆ౳੍ࣜ໿৚݅ԼͰධՁؔ਺Λղ͘ʢ࠷খԽ͢Δʣ͜ͱΛओ໰୊ͱ͢Δɽ ͜͜Ͱɼෆ౳੍ࣜ໿৚݅ͱධՁؔ਺Λಉ࣌ʹຬͨ͢৽͍ؔ͠਺Λߟ͑Δͱʜ ˜ Lp (w, b, α) = 1 2 wTw − N ∑ i=1 αi (ti (wT xi + b) − 1) ϥάϥϯδϡؔ਺ ͜Ε͔ΒͷܭࢉΛ؆୯ʹ͢ΔͨΊʹ·ͣϥάϥϯδϡؔ਺Λల։͍ͯ͘͠ʜ
    ৽͍͠จࣈɹ΋ͦ͜ͰͰઆ໌͠·͢ʜ αi ϋʔυϚʔδϯ47.
28. ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) = 1 2 wTw − ∑N i=1 {αi ti (wT xi + b) − αi } = 1 2 wTw − ∑N i=1 {αi ti wT xi + αi ti b − αi } = 1 2 wTw − ∑N i=1 αi ti wT xi − ∑N i=1 αi ti b + ∑N i=1 αi = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ల։ͨ͠ϥάϥϯδϡؔ਺ ϥάϥϯδϡؔ਺ͷల։ ࣍ʹओ໰୊ͷ৚݅ͱҰக͢Δ৚݅ΛʢఱԼΓతʹʣఆ͍ٛͯ͘͠ʜ ϋʔυϚʔδϯ47.
29. ී௨ͷ࠷খԽ໰୊ͱಉ༷ʹɼม਺ʹΑΔภඍ෼
    
    Λ࢖ͬͯؔ਺ͷఈΛ୳͍ͯ͠Δɽ
    ඍ෼ͨ͠ޙʹ࠷దԽ໰୊ͷղͰ͋ΔɹɹɹΛ୅ೖͨ͠ܗɽ ϥάϥϯδϡؔ਺ ৚݅ᶃ ∂ ˜ Lp (w, b, α) ∂w w=w0 = w0 − ∑N i=1 αi ti xi = 0 w0 = ∑N i=1 αi ti xi = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) w = w0 ϋʔυϚʔδϯ47.
30. ෆ౳੍ࣜ໿৚݅Λߏ੒͢Δؔ਺ͷޯ഑ͷઢܕ݁߹ʹΑͬͯ
    ධՁؔ਺ͷޯ഑ΛදݱͰ͖Δ͜ͱΛ͍ࣔͯ͠Δɽ ϥάϥϯδϡؔ਺ ৚݅ᶃ w0 = ∑N i=1 αi ti xi ධՁؔ਺ͷඍ෼ ∂f ∂w ෆ౳੍ࣜ໿৚݅ͷඍ෼ ∂gi ∂w ∂f ∂w = ∑N i=1 αi ∂gi ∂w f(w) = 1 2 wTw gi (w, b) = ti (wT xi + b) ඍ෼ͷܗʹॻ͖׵͑Δͱʜ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ϋʔυϚʔδϯ47.
31. ෆ౳੍ࣜ໿৚݅Λߏ੒͢Δؔ਺ͷޯ഑ͷઢܕ݁߹ʹΑͬͯ
    ධՁؔ਺ͷޯ഑ΛදݱͰ͖Δ͜ͱΛ͍ࣔͯ͠Δɽ ϥάϥϯδϡؔ਺ ৚݅ᶃ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ∂f ∂w = ∑N i=1 αi ∂gi ∂w ॏΈۭؒ W f ∑N i=1 αi gi Ћͷ஋ʹΑͬͯ
    ؔ਺ͷܗ͕มΘΔ ϋʔυϚʔδϯ47.
32. ৚݅ᶄ ∂ ˜ Lp (w, b, α) ∂b = ∑N i=1 αi ti = 0 ී௨ͷ࠷খԽ໰୊ͱಉ༷ʹɼม਺ʹΑΔภඍ෼
    
    Λ࢖ͬͯؔ਺ͷఈΛ୳͍ͯ͠Δɽ
    ੍໿৚݅ʹΑͬͯੜ·Εͨม਺ʹΑΔޯ഑΋ʹ͢Δɽ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ϋʔυϚʔδϯ47.
33. ৚݅ᶅ ti (wT xi + b) − 1 ≥ 0 ੍໿৚݅ͦͷ΋ͷͰ͋Δɽ͜ͷ৚݅͸σʔλͷ਺͚ͩଘࡏ͢Δɽ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ʜ੍໿৚݅ ϋʔυϚʔδϯ47.
34. ৚݅ᶆ ϥάϥϯδϡͷະఆ৐਺ͱݺ͹ΕΔɹͷ৚݅Ͱ͋Δɽ
    ͜ͷ஋ʹΑ֤ͬͯσʔλʹ͓͚Δෆ౳੍ࣜ໿৚݅ͷॏΈ͕ܾ·Δɽ
    αi ≥ 0 ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) αi ϋʔυϚʔδϯ47.
35. ৚݅ᶆ ϥάϥϯδϡͷະఆ৐਺ͱݺ͹ΕΔɹͷ৚݅Ͱ͋Δɽ
    ͜ͷ஋ʹΑ֤ͬͯσʔλʹ͓͚Δෆ౳੍ࣜ໿৚݅ͷॏΈ͕ܾ·Δɽ
    Ճ͑ͯɼࠨਤʹ͓͚Δ੺͍ؔ਺ͷܗΛੜΈग़͢͜ͱ͕Ͱ͖Δɽ αi ≥ 0 ϥάϥϯδϡؔ਺ f ∑N i=1 αi gi = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) αi ϋʔυϚʔδϯ47.
36. ৚݅ᶇ ૬ิੑ৚݅ɽ
    ͜ΕʹΑͬͯɼϥάϥϯδϡؔ਺͸ඍ෼͢Δ·ͰݩͷධՁؔ਺ͱಉ͡ܗΛอ͓ͬͯΓɼ
    ࠷খԽͷͨΊʹඍ෼ͨ࣌͠ʹͷΈෆ౳੍ࣜ໿৚͕݅͏·͘ݱΕΔΑ͏ʹͳͬͯΔʜ αi (ti (wT xi + b) − 1) = 0 ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ϋʔυϚʔδϯ47.
37. ৚݅ᶇ αi (ti (wT xi + b) − 1) = 0 ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ૬ิੑ৚݅ɽ
    ·ͨɼɹɹͷ࣌ʹɹɹɹɹɹɹɹɹ͕੒Γཱͪɼٯ΋ಉ༷ʹ੒Γཱͭɽ
    ͢ͳΘͪɼɹɹ͸ɹ͕αϙʔτϕΫτϧͰ͋Δɹɹɹɹɹɹɹɹɹͷ࣌ʹ੒ཱ͠
    ࠷খԽʹ͓͍ͯɼαϙʔτϕΫτϧ͕ͦͷଋറʹڧ͘د༩͍ͯ͠Δ͜ͱΛ͍ࣔͯ͠Δɽ αi ≠ 0 ti (wT xi + b) − 1 = 0 αi ≠ 0 xi ti (wT xi + b) − 1 = 0 ϋʔυϚʔδϯ47.
38. Ld (w, b, α) = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi Ҏ্Λ·ͱΊͯ47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ∑N i=1 αi ti = 0 ti (wT xi + b) − 1 ≥ 0 αi ≥ 0 αi (ti (wT xi + b) − 1) = 0 w0 = ∑N i=1 αi ti xi ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ ᶃ ᶄ ᶅ ᶆ ᶇ = 0 ∑N i=1 αi ti ᶄΛద༻ͯ͠ʜ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ϋʔυϚʔδϯ47.
39. Ld (w, α) = 1 2 wTw − ∑N i=1 αi ti wT xi + ∑N i=1 αi = 1 2 wTw − wT(∑N i=1 αi ti xi ) + ∑N i=1 αi Ҏ্Λ·ͱΊͯ47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ ɹͱؔ܎ͷͳ͍ɹΛ૯࿨ͷ֎ʹग़ͯ͠ʜ i wT ∑N i=1 αi ti = 0 ti (wT xi + b) − 1 ≥ 0 αi ≥ 0 αi (ti (wT xi + b) − 1) = 0 w0 = ∑N i=1 αi ti xi ᶃ ᶄ ᶅ ᶆ ᶇ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ϋʔυϚʔδϯ47.
40. Ld (α) = 1 2 wT 0 w0 − wT 0 (∑N i=1 αi ti xi ) + ∑N i=1 αi Ҏ্Λ·ͱΊͯ47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ ࠷దԽ৚݅ʹ͓͍ͯɹɹɹͳͷͰॻ͖׵͑ͯʜ w = w0 ∑N i=1 αi ti = 0 ti (wT xi + b) − 1 ≥ 0 αi ≥ 0 αi (ti (wT xi + b) − 1) = 0 w0 = ∑N i=1 αi ti xi ᶃ ᶄ ᶅ ᶆ ᶇ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ϋʔυϚʔδϯ47.
41. Ld (α) = 1 2 (∑N i=1 αi ti xi )T(∑N i=1 αi ti xi ) − (∑N i=1 αi ti xi )T(∑N i=1 αi ti xi ) + ∑N i=1 αi Ҏ্Λ·ͱΊͯ47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ ᶃΛద༻ͯ͠ʜ ∑N i=1 αi ti = 0 ti (wT xi + b) − 1 ≥ 0 αi ≥ 0 αi (ti (wT xi + b) − 1) = 0 w0 = ∑N i=1 αi ti xi ᶃ ᶄ ᶅ ᶆ ᶇ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) w0 = ∑N i=1 αi ti xi ϋʔυϚʔδϯ47.
42. Ld (α) = 1 2 (∑N i=1 ∑N j=1 αi αj ti tj xT i xj ) − (∑N i=1 ∑N j=1 αi αj ti tj xT i xj ) + ∑N i=1 αi Ҏ্Λ·ͱΊͯ47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ ͱॻ͚ͯɼ (∑N i=1 αi ti xi )T(∑N i=1 αi ti xi ) = (∑N i=1 ∑N j=1 αi ti xT i αj tj xj ) = (∑N i=1 ∑N j=1 αi αj ti tj xT i xj ) ∑N i=1 αi ti = 0 ti (wT xi + b) − 1 ≥ 0 αi ≥ 0 αi (ti (wT xi + b) − 1) = 0 w0 = ∑N i=1 αi ti xi ᶃ ᶄ ᶅ ᶆ ᶇ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) w0 = ∑N i=1 αi ti xi ϋʔυϚʔδϯ47.
43. Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj xT i xj + ∑N i=1 αi Ҏ্Λ·ͱΊͯ47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) ∑N i=1 αi ti = 0 ti (wT xi + b) − 1 ≥ 0 αi ≥ 0 αi (ti (wT xi + b) − 1) = 0 w0 = ∑N i=1 αi ti xi ᶃ ᶄ ᶅ ᶆ ᶇ ϋʔυϚʔδϯ47.
44. ϥάϥϯδϡؔ਺ = 1 2 wTw − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi ˜ Lp (w, b, α) = 1 2 wTw − ∑N i=1 αi (ti (wT xi + b) − 1) Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj xT i xj + ∑N i=1 αi Ҏ্Λ·ͱΊͯ47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ∑N i=1 αi ti = 0 ti (wT xi + b) − 1 ≥ 0 αi ≥ 0 αi (ti (wT xi + b) − 1) = 0 w0 = ∑N i=1 αi ti xi ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ ᶃ ᶄ ᶅ ᶆ ᶇ ਖ਼ղϥϕϧɹͱσʔλɹ͸ط஌ͳͷͰɹ͸ະఆ৐਺ɹͷΈͷؔ਺ʹͳΔɽ ti xi Ld αi ϋʔυϚʔδϯ47.
45. Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj xT i xj + ∑N i=1 αi ϥάϥϯδϡؔ਺ α = (α1 α2 . . . αN )T, t = (t1 t2 . . . tN )T, H = (Hij = ti tj xT i xj ), 1 = (1 1...1)T ͱ͢Δͱɼ Ld (α) = − 1 2 αT Hα + αT1 ͜Εʹ,,5৚݅Λߟྀ͢Δͱʜ αi (ti (wT xi + b) − 1) { ti (wT xi + b) − 1 = 0 ⟹ αi > 0 ti (wT xi + b) − 1 ≠ 0 ⟹ αi = 0 ᶅɼᶆɼᶇ ∑N i=1 αi ti = αT t = 0 ᶄ ͱͳΔͷͰɼ͜ΕΒΛ༻͍ͯ৽ͨͳ໰୊Λ૊ΈཱͯΔͱ࣍ͷ૒ର໰୊ʹͳΔ ϋʔυϚʔδϯ47.
46. α w ͜ΕʹΑΓɼओ໰୊ͷ࠷খԽ໰୊͕૒ର໰୊ͷ࠷େԽ໰୊ʹஔ͖׵Θͬͨʜ ओ໰୊Ͱ͠Α͏ͱ͍ͯͨ͜͠ͱ ૒ର໰୊Ͱ͠Α͏ͱ͍ͯ͠Δ͜ͱ ϥάϥϯδϡͷະఆ৐਺๏Ͱ͸
    ϥάϥϯδϡؔ਺ͷҌ఺ΛٻΊ͍ͯΔʜ ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0 ϋʔυϚʔδϯ47.
47. ͜ΕʹΑΓɼओ໰୊ͷ࠷খԽ໰୊͕૒ର໰୊ͷ࠷େԽ໰୊ʹஔ͖׵Θͬͨʜ α w ͜ͷ࠷దԽ໰୊Λ࣮ࡍʹղ͘ʹ͸
    4.0ͱݺ͹ΕΔ
    ߴ଎ͳΞϧΰϦζϜ͕༻͍ΒΕΔ ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0 ϋʔυϚʔδϯ47.
48. w0 = ∑N i=1 αi ti xi 4.0Λ༻͍ͯ૒ର໰୊Λղ͖ɼ࣮ࡍʹɹΛશͯٻΊऴ͑ͨΒʜ α ओ໰୊ʹ͓͚Δ,,5৚݅ͷᶃΛ༻͍ͯɹΛٻΊʜ w0 tsv (wT 0 xsv + b0 ) − 1 = 0 b0 = 1 tsv − wT 0 xsv αϙʔτϕΫτϧ TW ͢ͳΘͪɹɹɹɹɹɹɹɹͱͳ
    ɹɹɹ
    ͓ΑͼɹɹɹΛ༻͍Ε͹ɼ࠷దͳόΠΞεɹ͸ ti (wT xi + b) − 1 = 0 x = xsv t = tsv b0 ͔ΒٻΊΔɽ
    αϙʔτϕΫτϧ͸ෳ਺͋ΔͷͰͦΕΒͷฏۉͰٻΊΔ͜ͱ΋͋Δɽ ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0 ϋʔυϚʔδϯ47.
49. ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0 wT 0 w0 = ∑N i=1 ˜ αi ti xT i w0 4.0Λ༻͍ͯ૒ର໰୊Λղ͖ɼ࣮ࡍʹɹΛશͯٻΊऴ͑ͨΒʜ α ϥάϥϯδϡ৐਺ͷ࠷దղΛɹɹɹɹɹɹɹͷΑ͏ʹ͢Δͱ ˜ α = ( ˜ α1 , . . . , ˜ αN )T ,,5ᶃͷٯ ͱͯ͠࢒Δͷ͸TW༝དྷͷ΋ͷ͚ͩͳͷͰ ˜ αi ti (wT xi + b0 ) − 1 = 0 ͱͯ͠ܭࢉʜ
    = ∑N i=1 ˜ αi − b0 ∑N i=1 ti αi = ∑N i=1 ˜ αi (1 − ti b0 ) = ∑N i=1 ˜ αi ,,5ᶄ w0 = ∑N i=1 αi ti xi ∑N i=1 αi ti = 0 Dmax = 1 ∥w0 ∥ = 1 wT 0 w0 = 1 ∑N i=1 αi Αͬͯ࠷େϚʔδϯ͸ ϋʔυϚʔδϯ47.
50. ධՁؔ਺ʢ࠷খԽʣ ෆ౳੍ࣜ໿৚݅ f(x) = x2 { g1 (x) = x − 2 ≤ 0 g2 (x) = x − 3 ≤ 0 L(x, λ1 , λ2 ) = x2 − λ1 g1 (x) − λ2 g2 (x) = x2 − λ1 (x − 2) − λ2 (x − 3) ,,5৚݅͸ʜ ∂L ∂x = 2x − λ1 − λ2 = 0 ∂L ∂λ1 = − (x − 2) ≥ 0 ⟹ x − 2 ≤ 0 ∂L ∂λ2 = − (x − 3) ≥ 0 ⟹ x − 3 ≤ 0 λ1 , λ2 ≥ 0 λ1 (x − 2) = 0 λ2 (x − 3) = 0 λ1 , λ2 ≠ 0 ͷ࣌ x = 2 ⟹ g2 (2) < 0 x = 3 ⟹ g1 (3) > 0 ༗ޮ ੍໿৚͕݅ෳ਺༩͑ΒΕͨ࣌ɼ໾ʹཱ͍ͬͯΔ੍໿Λ۩ମతʹܭࢉ͢Δ໰୊Ͱ͢ɽ
    47.తʹ͸αϙʔτϕΫτϧʹؔ܎͢Δ੍໿͕໾ʹཱ͍ͬͯΔʹ౰ͨΓ·͢ɽ
    ओ໰୊ͷϥάϥϯδϡؔ਺͸ ྫ୊
    ओ໰୊ ϋʔυϚʔδϯ47.
51. ධՁؔ਺ʢ࠷খԽʣ ෆ౳੍ࣜ໿৚݅ f(x) = x2 { g1 (x) = x − 2 ≤ 0 g2 (x) = x − 3 ≤ 0 ੍໿৚͕݅ෳ਺༩͑ΒΕͨ࣌ɼ໾ʹཱ͍ͬͯΔ੍໿Λ۩ମతʹܭࢉ͢Δ໰୊Ͱ͢ɽ
    47.తʹ͸αϙʔτϕΫτϧʹؔ܎͢Δ੍໿͕໾ʹཱ͍ͬͯΔʹ౰ͨΓ·͢ɽ
    ओ໰୊ͷϥάϥϯδϡؔ਺͸ L(x, λ1 , λ2 ) = x2 − λ1 g1 (x) − λ2 g2 (x) = x2 − λ1 (x − 2) − λ2 (x − 3) ,,5৚݅͸ʜ ∂L ∂x = 2x − λ1 − λ2 = 0 ∂L ∂λ1 = − (x − 2) ≥ 0 ⟹ x − 2 ≤ 0 ∂L ∂λ2 = − (x − 3) ≥ 0 ⟹ x − 3 ≤ 0 λ1 , λ2 ≥ 0 λ1 (x − 2) = 0 λ2 (x − 3) = 0 ༗ޮ λ1 ≠ 0 λ2 = 0 ྫ୊
    ओ໰୊ ϋʔυϚʔδϯ47.
52. 
    ઢܗࣝผ௒ฏ໘ͱ֤σʔλͷڑ཭ͷ৚݅ 
    Ϛʔδϯ࠷େԽ৚݅Λຬͨؔ͢਺ ti (wT xi + b) ≥ 1 min f(w) = min 1 2 wTw w w α w ϥάϥϯδϡͷະఆ৐਺๏Λ༻͍ͯ
    ओ໰୊Λ૒ର໰୊ʹམͱ͠ࠐΜͰ
    ઢܗࣝผ௒ฏ໘ͷࣜΛٻΊΔ D D C1 C2 ·ͱΊ ϋʔυϚʔδϯࣝผث ઢܗࣝผՄೳͰ͋Δ৔߹ʹ࠷େϚʔδϯΛ
    ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥε
    ͷ෼ྨΛߦ͏ํ๏ ϋʔυϚʔδϯ47.
53. ϋʔυϚʔδϯ47. ઢܗ෼཭͕׬શʹՄೳͰ͋Δ৔߹Ͱ͔͠ద༻Ͱ͖ͳ͍ʜ Ұൠతͳσʔλ͸ઢܗ෼཭ՄೳͰͳ͍͜ͱͷํ͕ଟ͍ͷʹ
    ઢܗ෼཭ՄೳͰͳ͍৔߹͸Ͳ͏͢Δͷʜ ιϑτϚʔδϯ47. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
54. 065-*/& ·ͱΊ ϋʔυϚʔδϯ47. ιϑτϚʔδϯ47. αϙʔτϕΫτϧϚγϯ 47. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
55. 065-*/& ఆٛ ੍໿৚݅ͷ࠶ઃఆ ϥάϥϯδϡͷະఆ৐਺๏ ૒ର໰୊ ຕ ຕ ຕ ຕ ઢܗ෼཭ՄೳͰͳ͍৔߹ͷ ·ͱΊ ຕ ιϑτϚʔδϯ47.
56. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
    ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ιϑτϚʔδϯ47. ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ σʔλΛ׬શʹ෼཭Ͱ͖Δ࣌ɼ
    ઢܗ෼཭Մೳͱ͍͏ ઢܗ෼཭Ͱ͖ͳ͍σʔλͱ͸ʜ
57. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
    ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ઢܗ෼཭ՄೳͰͳ͍ྫ ࣝผ௒ฏ໘Λ௒͑ͯ
    σʔλ͕ଘࡏ͢Δ ઢܗ෼཭Ͱ͖ͳ͍σʔλͱ͸ʜ ιϑτϚʔδϯ47.
58. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
    ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ઢܗ෼཭ՄೳͰͳ͍ྫ ࣝผ௒ฏ໘Λ௒͑ͯ͸͍ͳ͍͕
    TWͷ಺ଆʹσʔλ͕ଘࡏ͢Δ ઢܗ෼཭Ͱ͖ͳ͍σʔλͱ͸ʜ ιϑτϚʔδϯ47.
59. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
    ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ઢܗ෼཭ՄೳͰͳ͍ྫ ϊΠζ࣍ୈͰΫϥε͕
    ͿΕΔ͜ͱ͕͋Δ΋ͷͨͪ ઢܗ෼཭Ͱ͖ͳ͍σʔλͱ͸ʜ ιϑτϚʔδϯ47.
60. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
    ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ઢܗ෼཭ՄೳͰͳ͍ྫ ͜ΕΒͷσʔλΛߟྀͨ͠
    ࣝผؔ਺ͷ࠶ઃఆ͕ඞཁ ઢܗ෼཭Ͱ͖ͳ͍σʔλͱ͸ʜ ιϑτϚʔδϯ47.
61. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ೋछྨͷσʔλ܈Λ
    ࠷΋ྑ͍ײ͡ʹ෼͚Δڥքઢ ઢܗࣝผ௒ฏ໘ D D Ϛʔδϯ ઢܗ෼཭ՄೳͰͳ͍ྫ ͜ΕΒͷσʔλΛߟྀͨ͠
    ੍໿৚݅ͷ࠶ઃఆ͕ඞཁ ઢܗ෼཭Ͱ͖ͳ͍σʔλͱ͸ʜ ιϑτϚʔδϯ47.
62. ੍໿৚݅ͷ࠶ઃఆ ઢܗ෼཭Մೳͳ৔߹ gi (w, b) = ti (wT xi + b) ≥ 1 ઢܗ෼཭ՄೳͰͳ͍৔߹ gi (w, b) = ti (wT xi + b) + ξi ≥ 1 D D ξi = 0 0 < ξi ≤ 1 ξi < 1 ξi = 0 +ξi σʔλʹΑͬͯҟͳΔ஋Λ࣋ͭεϥοΫม਺ɹΛಋೖͯ͠ʜ ξi ιϑτϚʔδϯ47.
63. ੍໿৚݅ͷ࠶ઃఆ ઢܗ෼཭Մೳͳ৔߹ gi (w, b) = ti (wT xi + b) ≥ 1 ઢܗ෼཭ՄೳͰͳ͍৔߹ gi (w, b) = ti (wT xi + b) + ξi ≥ 1 D D +ξi ͱͳΔΑ͏ʹ֤σʔλʹରͯ͠ɹΛܾΊΔ ξi ιϑτϚʔδϯ47. ξi = 0 0 < ξi ≤ 1 ξi < 1 ξi = 0
64. ઢܗ෼཭ՄೳͰͳ͍৔߹ gi (w, b) = ti (wT xi + b) + ξi ≥ 1 D D +ξi ੍໿৚݅ͷ࠶ઃఆ ξi = max[0,1 − ti (wT xi + b)] εϥοΫม਺ɹͱ͸ʜTWͱࣝผڥքͱͷڑ཭Λͱͨ࣌͠ʹ
    ෼ྨ͕ਖ਼͘͠TWΑΓ΋ࣝผڥք͔Βԕ͍΋ͷʹରͯ͠͸Λฦ͠
    ෼ྨ͸ਖ਼͍͕͠TWΑΓ΋ࣝผڥքʹ͍ۙ΋ͷʹରͯ͠͸ͦͷڑ཭Λฦ͠
    ෼ྨ͕ਖ਼͘͠ͳ͍΋ͷʹରͯ͠΋ಉ༷ʹͦͷڑ཭Λฦ͢Α͏ͳม਺Ͱ͋Δ ξi ιϑτϚʔδϯ47. ξi = 0 0 < ξi ≤ 1 ξi < 1 ξi = 0
65. ઢܗ෼཭ՄೳͰͳ͍৔߹ gi (w, b) = ti (wT xi + b) + ξi ≥ 1 D D ξi = 0 0 < ξi ≤ 1 ξi < 1 ξi = 0 +ξi ੍໿৚݅ͷ࠶ઃఆ ξi = max[0,1 − ti (wT xi + b)] গ͠ڧҾʹݴ͍׵͑Δͱ εϥοΫม਺ɹͱ͸ʜ47.ͷఆٛʹ্ख͘৐Βͳ͔ͬͨσʔλΛແཧ΍Γ
    TWʹԡ͠ࠐΉΑ͏ͳม਺ ξi ιϑτϚʔδϯ47.
66. ઢܗ෼཭ՄೳͰͳ͍৔߹ gi (w, b) = ti (wT xi + b) + ξi ≥ 1 D D ξi = 0 0 < ξi ≤ 1 ξi < 1 ξi = 0 +ξi ੍໿৚݅ͷ࠶ઃఆ ξi = max[0,1 − ti (wT xi + b)] ٙ໰ɿTWʹԡ͠ࠐΜͰେৎ෉ͳͷ͔ʜ ͋͘·Ͱ΋ઢܗࣝผ௒ฏ໘Λಋग़͢ΔͨΊͷ֦ுͳͷͰ໰୊ͳ͍
    ࣝผ͢Δࡍ͸ٻΊͨɹͱɹΛ࢖ͬͯΫϥεΛܾΊΔɽ w0 b0 ιϑτϚʔδϯ47.
67. ઢܗ෼཭ՄೳͰͳ͍৔߹ gi (w, b) = ti (wT xi + b) + ξi ≥ 1 D D ξi = 0 0 < ξi ≤ 1 ξi < 1 ξi = 0 +ξi ੍໿৚݅ͷ࠶ઃఆ ξi = max[0,1 − ti (wT xi + b)] ͱͳΔͷͰɼ͜ͷ࿨͕,ҎԼͰ͋ΔͳΒޡࣝผ͸,ݸະຬʹͳΔɽ ξi > 1 ޡࣝผͨ͠΋ͷͷεϥοΫม਺ Λ,ूΊΔͱ∑ i∈{1,2,...n} ξi > K ͜͜·Ͱͷ৘ใͰϥάϥϯδϡະఆ৐਺๏ʹ͓͚Δओ໰୊Λߏஙͯ͠ΈΔʜ ιϑτϚʔδϯ47.
68. ઢܗ෼཭ՄೳͰͳ͍৔߹ gi (w, b) = ti (wT xi + b) + ξi ≥ 1 D D ξi = 0 0 < ξi ≤ 1 ξi < 1 ξi = 0 +ξi ੍໿৚݅ͷ࠶ઃఆ ξi = max[0,1 − ti (wT xi + b)] ξi = f+ (1 − ti (wT xi + b)) ͱॻ͚Δɹɹ͸Ұൠతʹ f+ (x) = { x (x > 0) 0 (x ≤ 0) f+ ͱॻ͘͜ͱ͕Ͱ͖ɼͦͷάϥϑͷܗ͔ΒIJOHFؔ਺ͱݺ͹ΕΔ ιϑτϚʔδϯ47.
69. f(w, ξ) = 1 2 wTw + C∑N i=1 ξi ओ໰୊ ti (wT xi + b) + ξi ≥ 1, ξi ≥ 0 ධՁؔ਺ʢ࠷খԽʣ ෆ౳੍ࣜ໿৚݅ ෆ౳੍ࣜ໿৚݅ʹθϩҎ্ͷɹ͕ଘࡏ͢ΔͷͰ͜ͷ஋࣍ୈͰͲΜͳɹͰ΋
    ۃ୺ʹݴ͑͹ɹɹɹʹͯ͠΋৚͕݅੒Γཱͬͯ͠·͏͜ͱ͕ى͜Γ͏Δʜ
    ͦ͜ͰɼධՁؔ਺ɹɹɹʹɹʹ༝དྷ͢ΔϖφϧςΟɹɹɹɹΛ՝͢͜ͱͰ
    ɹ͕খ͘͞ͳΓ͗͢Δ͜ͱΛ๷͍Ͱ͍Δʜ f(w, ξ) ξi C∑N i=1 ξi w w = 0 w ξi ͜ͷ࣌ϖφϧςΟ߲ʹ͋Δɹ͸՝͢ΔϖφϧςΟͷڧ͞ΛܾΊΔ஋Ͱ͋Γ
    ਓ͕ؒ೚ҙʹܾΊΔύϥϝʔλ ϋΠύʔύϥϝʔλ Ͱ͋Δɽ C C∑N i=1 ξi ιϑτϚʔδϯ47.
70. f(w, ξ) = 1 2 wTw + C∑N i=1 ξi ओ໰୊ ti (wT xi + b) − 1 + ξi ≥ 0, ξi ≥ 0 ධՁؔ਺ʢ࠷খԽʣ ෆ౳੍ࣜ໿৚݅ ্هͷෆ౳੍ࣜ໿৚݅ԼͰධՁؔ਺Λղ͘ʢ࠷খԽ͢Δʣ͜ͱΛओ໰୊ͱ͢Δɽ ઢܗ෼཭Մೳͳ৔߹ͱಉ༷ʹϥάϥϯδϡؔ਺Λߟ͑Δͱʜ ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ϥάϥϯδϡؔ਺ ͜Ε΋ઢܗ෼཭Մೳͳ৔߹ͱಉ༷ʹ·ͣϥάϥϯδϡؔ਺Λల։͍ͯ͘͠ʜ ιϑτϚʔδϯ47.
71. ల։ͨ͠ϥάϥϯδϡؔ਺ ϥάϥϯδϡؔ਺ͷల։ ࣍ʹओ໰୊ͷ৚݅ͱҰக͢Δ৚݅ΛʢఱԼΓతʹʣఆ͍ٛͯ͘͠ʜ ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 {αi ti (wT xi + b) − αi + αi ξi } − ∑N i=1 μi ξi = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 {αi ti wT xi + αi ti b − αi + αi ξi } − ∑N i=1 μi ξi = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi ti wT xi − ∑N i=1 αi ti b + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ιϑτϚʔδϯ47.
72. ৚݅ᶃɼᶄ͸ઢܗࣝผՄೳͳ৔߹ͱಉ͡ܗʹͳΔɽ ϥάϥϯδϡؔ਺ ৚݅ᶃ ∂ ˜ Lp (w, b, α, ξ, μ) ∂w w=w0 = w0 − ∑N i=1 αi ti xi = 0 w0 = ∑N i=1 αi ti xi = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ৚݅ᶄ ∂ ˜ Lp (w, b, α, ξ, μ) ∂b = ∑N i=1 αi ti = 0 1 2 wTw −∑N i=1 αi ti wT xi −b∑N i=1 αi ti ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ιϑτϚʔδϯ47.
73. ϥάϥϯδϡະఆ৐਺ͷɹͱɹͷ৚݅ ৚݅ᶇ ͔ΒϖφϧςΟͷڧ͞ΛܾΊΔɹ͕
    ɹͷൣғΛԡ͚͍͑ͭͯ͞Δ৚݅Λಋ͖ग़ͤΔɽ ϥάϥϯδϡؔ਺ ৚݅ᶅ ∂ ˜ Lp (w, b, α, ξ, μ) ∂ξi = C − αi − μi = 0 C − αi = μi ≥ 0 ⟹ C − αi ≥ 0 C ≥ αi ≥ 0 αi μi C αi = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi C∑N i=1 ξi ∑N i=1 αi ξi − ∑N i=1 μi ξi ιϑτϚʔδϯ47.
74. ϥάϥϯδϡؔ਺ ৚݅ᶆ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ti (wT xi + b) − 1 + ξi ʜ੍໿৚݅ ti (wT xi + b) − 1 + ξi ≥ 0 ੍໿৚݅ͦͷ΋ͷͰ͋Δɽ͜ͷ৚݅͸σʔλͷ਺͚ͩଘࡏ͢Δɽ ιϑτϚʔδϯ47.
75. ϥάϥϯδϡؔ਺ ৚݅ᶇ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ξi αi μi ξi ξi ξi ξi ξi αi μi ϥάϥϯδϡͷະఆ৐਺Ͱ͋Δɹɼ͓ΑͼεϥοΫม਺ɹͷ৚݅Ͱ͋Δɽ
    ͜ͷ஋ʹΑ֤ͬͯσʔλʹ͓͚Δෆ౳੍ࣜ໿৚݅ͷॏΈ͕ܾ·Δɽ
    ξi ≥ 0, αi ≥ 0, μi ≥ 0 αi μi ξi αi αi αi ιϑτϚʔδϯ47.
76. ϥάϥϯδϡؔ਺ ৚݅ᶈ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi αi (ti (wT xi + b) − 1 + ξi ) = 0 αi (ti (wT xi + b) − 1 + ξi ) ૬ิੑ৚݅ɽ
    ·ͨɼɹɹͷ࣌ʹɹɹɹɹɹɹɹɹɹɹ͕੒Γཱͪɼٯ΋ಉ༷ʹ੒Γཱͭɽ
    ͢ͳΘͪɼɹɹ͸ɹ͕αϙʔτϕΫτϧͰ͋Δɹɹɹɹɹɹɹɹɹɹͷ࣌ʹ੒ཱ͠
    ࠷খԽʹ͓͍ͯɼαϙʔτϕΫτϧ͕ͦͷଋറʹڧ͘د༩͍ͯ͠Δ͜ͱΛ͍ࣔͯ͠Δɽ αi ≠ 0 ti (wT xi + b) − 1 + ξi = 0 αi ≠ 0 xi ti (wT xi + b) − 1 + ξi = 0 ti (wT xi + b) − 1 + ξi = 0 ιϑτϚʔδϯ47.
77. ϥάϥϯδϡؔ਺ ৚݅ᶉ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi μi ξi = 0 αi (ti (wT xi + b) − 1 + ξi ) ૬ิੑ৚݅ɽ
    ·ͨɼɹɹͷ࣌ʹɹɹɹ͕੒Γཱͪɼٯ΋ಉ༷ʹ੒Γཱͭɽ
    μi ≠ 0 ξi = 0 ιϑτϚʔδϯ47.
78. ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi αi (ti (wT xi + b) − 1 + ξi ) ৚݅ᶅ C − αi = μi ≥ 0 ⟹ C − αi ≥ 0 ⟺ C ≥ αi ≥ 0 Λ࢖͏ͱ৚݅ᶉ͸ʜ μi ξi = 0 { μi = 0 (ξi > 0) ⟹ C − αi = μi = 0 ⟹ C − αi = 0 ⟺ C = αi ≥ 0 ξi = 0 (μi > 0) ⟹ C − αi = μi > 0 ⟹ C − αi > 0 ⟺ C > αi ≥ 0 C = αi C > αi μi = 0 μi > 0 ιϑτϚʔδϯ47.
79. ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi αi (ti (wT xi + b) − 1 + ξi ) ৚݅ᶉ D D μi ξi = 0 { μi = 0 (ξi > 0) ⟹ C − αi = μi = 0 ⟹ C − αi = 0 ⟺ C = αi ≥ 0 ξi = 0 (μi > 0) ⟹ C − αi = μi > 0 ⟹ C − αi > 0 ⟺ C > αi ≥ 0 μi = 0 (ξi > 0) ⟹ C − αi = μi = 0 ⟹ C − αi = 0 ⟺ C = αi ≥ 0 ιϑτϚʔδϯ47.
80. ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi αi (ti (wT xi + b) − 1 + ξi ) ৚݅ᶉ D D +ξi D D μi ξi = 0 { μi = 0 (ξi > 0) ⟹ C − αi = μi = 0 ⟹ C − αi = 0 ⟺ C = αi ≥ 0 ξi = 0 (μi > 0) ⟹ C − αi = μi > 0 ⟹ C − αi > 0 ⟺ C > αi ≥ 0 μi = 0 (ξi > 0) ⟹ C − αi = μi = 0 ⟹ C − αi = 0 ⟺ C = αi ≥ 0
    
    ͕ ্ݶαϙʔτϕΫτϧͨͪ αi ιϑτϚʔδϯ47.
81. ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi αi (ti (wT xi + b) − 1 + ξi ) ৚݅ᶉ D D μi ξi = 0 { μi = 0 (ξi > 0) ⟹ C − αi = μi = 0 ⟹ C − αi = 0 ⟺ C = αi ≥ 0 ξi = 0 (μi > 0) ⟹ C − αi = μi > 0 ⟹ C − αi > 0 ⟺ C > αi ≥ 0 ξi = 0 (μi > 0) ⟹ C − αi = μi > 0 ⟹ C − αi > 0 ⟺ C > αi ≥ 0 ιϑτϚʔδϯ47.
82. ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi αi (ti (wT xi + b) − 1 + ξi ) ৚݅ᶉ D D μi ξi = 0 { μi = 0 (ξi > 0) ⟹ C − αi = μi = 0 ⟹ C − αi = 0 ⟺ C = αi ≥ 0 ξi = 0 (μi > 0) ⟹ C − αi = μi > 0 ⟹ C − αi > 0 ⟺ C > αi ≥ 0 ξi = 0 (μi > 0) ⟹ C − αi = μi > 0 ⟹ C − αi > 0 ⟺ C > αi ≥ 0 D D αi > 0 +ξi
    
    ͕ ࣗ༝αϙʔτϕΫτϧͨͪ αi ιϑτϚʔδϯ47.
83. ઢܗࣝผՄೳͰͳ͍47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ w0 = ∑N i=1 αi ti xi ᶃ ∑N i=1 αi ti = 0 ᶄ ᶄΛద༻ͯ͠ʜ ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ᶆ ti (wT xi + b) − 1 + ξi ≥ 0 C − αi = μi ᶅ ᶇ ξi ≥ 0, αi ≥ 0, μi ≥ 0 ᶇ αi (ti (wT xi + b) − 1 + ξi ) = 0 ᶈ μi ξi = 0 ᶉ Ld (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ∑N i=1 αi ti = 0 ιϑτϚʔδϯ47.
84. ઢܗࣝผՄೳͰͳ͍47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ w0 = ∑N i=1 αi ti xi ᶃ ∑N i=1 αi ti = 0 ᶄ ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ᶆ ti (wT xi + b) − 1 + ξi ≥ 0 C − αi = μi ᶅ ᶇ ξi ≥ 0, αi ≥ 0, μi ≥ 0 ᶇ αi (ti (wT xi + b) − 1 + ξi ) = 0 ᶈ μi ξi = 0 ᶉ Ld (w, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ∑N i=1 αi ξi + ∑N i=1 μi ξi = ∑N i=1 (αi + μi )ξi = ∑N i=1 Cξi = C∑N i=1 ξi ᶅ͔Β −∑N i=1 αi ξi − ∑N i=1 μi ξi ͳͷͰ ιϑτϚʔδϯ47.
85. ઢܗࣝผՄೳͰͳ͍47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ w0 = ∑N i=1 αi ti xi ᶃ ∑N i=1 αi ti = 0 ᶄ ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ᶆ ti (wT xi + b) − 1 + ξi ≥ 0 C − αi = μi ᶅ ᶇ ξi ≥ 0, αi ≥ 0, μi ≥ 0 ᶇ αi (ti (wT xi + b) − 1 + ξi ) = 0 ᶈ μi ξi = 0 ᶉ −∑N i=1 αi ti wT xi + ∑N i=1 αi − C∑N i=1 ξi ∑N i=1 αi ξi + ∑N i=1 μi ξi = ∑N i=1 (αi + μi )ξi = ∑N i=1 Cξi = C∑N i=1 ξi ᶅ͔Β ͳͷͰ −C∑N i=1 ξi Ld (w, α, ξ) = 1 2 wTw + C∑N i=1 ξi C∑N i=1 ξi ιϑτϚʔδϯ47.
86. ઢܗࣝผՄೳͰͳ͍47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ w0 = ∑N i=1 αi ti xi ᶃ ∑N i=1 αi ti = 0 ᶄ ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ᶆ ti (wT xi + b) − 1 + ξi ≥ 0 C − αi = μi ᶅ ᶇ ξi ≥ 0, αi ≥ 0, μi ≥ 0 ᶇ αi (ti (wT xi + b) − 1 + ξi ) = 0 ᶈ μi ξi = 0 ᶉ Ld (w, b, α) = 1 2 wTw − ∑N i=1 αi ti wT xi + ∑N i=1 αi ∑N i=1 αi ξi + ∑N i=1 μi ξi = ∑N i=1 (αi − μi )ξi = ∑N i=1 Cξi = C∑N i=1 ξi ᶅ͔Β ͳͷͰ ιϑτϚʔδϯ47.
87. ઢܗࣝผՄೳͰͳ͍47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ w0 = ∑N i=1 αi ti xi ᶃ ∑N i=1 αi ti = 0 ᶄ ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ᶆ ti (wT xi + b) − 1 + ξi ≥ 0 C − αi = μi ᶅ ᶇ ξi ≥ 0, αi ≥ 0, μi ≥ 0 ᶇ αi (ti (wT xi + b) − 1 + ξi ) = 0 ᶈ μi ξi = 0 ᶉ Ld (w, b, α) = 1 2 wTw − ∑N i=1 αi ti wT xi + ∑N i=1 αi ઢܗࣝผՄೳͳ৔߹ͷϥάϥϯδϡؔ਺ͱಉ͡ܗ͕ग़ͯ͘ΔͷͰʜ ιϑτϚʔδϯ47.
88. ઢܗࣝผՄೳͰͳ͍47.࠷దԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ͕ಘΒΕͨͷͰɼݩͷϥάϥϯδϡؔ਺Λ͜ΕΒͷ৚݅Ͱஔ͖׵͑ͯΈΔͱʜ w0 = ∑N i=1 αi ti xi ᶃ ∑N i=1 αi ti = 0 ᶄ ϥάϥϯδϡؔ਺ = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi ti wT xi − b∑N i=1 αi ti + ∑N i=1 αi − ∑N i=1 αi ξi − ∑N i=1 μi ξi ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi − ∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ᶆ ti (wT xi + b) − 1 + ξi ≥ 0 C − αi = μi ᶅ ᶇ ξi ≥ 0, αi ≥ 0, μi ≥ 0 ᶇ αi (ti (wT xi + b) − 1 + ξi ) = 0 ᶈ μi ξi = 0 ᶉ ઢܗࣝผՄೳͳ৔߹ͱશ͘ಉ͡มܗͰҎԼͷ͕ࣜಘΒΕΔ Ld (α) = − 1 2 αT Hα + αT1 ιϑτϚʔδϯ47.
89. α w ͋ͱ͸ɹͷɹʹΑΔൣғࢦఆͷ৚݅Λ੝ΓࠐΜͩ૒ର໰୊ʹམͱ͠ࠐΜͰ
    4.0ͱ͍͏ϥΠϒϥϦΛ༻͍ͯ࠷దԽ໰୊Λղ͚͹ྑ͍ ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0, C ≥ αi ≥ 0 αi C ιϑτϚʔδϯ47.
90. ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0, C ≥ αi ≥ 0 4.0Λ༻͍ͯ૒ର໰୊Λղ͖ɼ࣮ࡍʹɹΛશͯٻΊऴ͑ͨΒʜ α w0 = ∑N i=1 αi ti xi ઢܗࣝผՄೳͳ΋ͷͱಉ༷ʹओ໰୊ʹ͓͚Δ,,5৚݅ͷᶃΛ༻͍ͯɹΛٻΊʜ w0 ιϑτϚʔδϯ47.
91. ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0, C ≥ αi ≥ 0 ࣗ༝αϙʔτϕΫτϧ TW ͢ͳΘͪɹɹɹɹɹɹɹɹͱͳΔ
    ɹɹɹ
    ͓ΑͼɹɹɹΛ༻͍Ε͹ɼ࠷దͳόΠΞεɹ͸ ti (wT xi + b) − 1 = 0 xi = xsv ti = tsv b0 ti (wT xi + b) − 1 = 0 ti (wT xi + b) − 1 = 0 ti (wT xi + b) − 1 = 0 ti (wT xi + b) − 1 = 0 ti (wT 0 xi + b) − 1 = 0 tsv (wT 0 xsv + b0 ) − 1 = 0 b0 = 1 tsv − wT 0 xsv ˠྫ୊ ͔ΒٻΊΔɽ ࣗ༝αϙʔτϕΫτϧΛ༻͍ͨཧ༝͸σʔλຖʹҟͳΔɹ͕ͳ͘࢖͍΍͍ͨ͢Ί
    ɹɹɹɹɹɹɹɹɹɹͱॻ͚Δ্ݶαϙʔτϕΫτϧ͸࢖͍ʹ͍͘ ξi ti (wT xi + b) − 1 + ξi = 0 4.0Λ༻͍ͯ૒ର໰୊Λղ͖ɼ࣮ࡍʹɹΛશͯٻΊऴ͑ͨΒʜ ιϑτϚʔδϯ47.
92. ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ Ld (α) = − 1 2 αT Hα + αT1 αT t = 0, C ≥ αi ≥ 0 wT 0 w0 = ∑N i=1 ˜ αi ti xT i w0 4.0Λ༻͍ͯ૒ର໰୊Λղ͖ɼ࣮ࡍʹɹΛશͯٻΊऴ͑ͨΒʜ α ϥάϥϯδϡ৐਺ͷ࠷దղΛɹɹɹɹɹɹɹͷΑ͏ʹ͢Δͱ ˜ α = ( ˜ α1 , . . . , ˜ αN )T ,,5ᶃͷٯ ͱͯ͠࢒Δͷ͸TW༝དྷͷ΋ͷ͚ͩͳͷͰ ˜ αi ti (wT 0 xi + b0 ) − 1 = 0 ͱͯ͠ܭࢉʜ
    = ∑N i=1 ˜ αi − b0 ∑N i=1 ti αi = ∑N i=1 ˜ αi (1 − ti b0 ) = ∑N i=1 ˜ αi ,,5ᶄ w0 = ∑N i=1 αi ti xi ∑N i=1 αi ti = 0 Dmax = 1 ∥w0 ∥ = 1 wT 0 w0 = 1 ∑N i=1 αi Αͬͯ࠷େϚʔδϯ͸ ιϑτϚʔδϯ47.
93. D D ιϑτϚʔδϯ47. ઢܗࣝผՄೳͰͳ͍৔߹ʹ࠷େϚʔδϯΛ
    ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥε
    ͷ෼ྨΛߦ͏ํ๏ ϥάϥϯδϡͷະఆ৐਺๏Λ༻͍ͯ
    ओ໰୊Λ૒ର໰୊ʹམͱ͠ࠐΜͰ
    ઢܗࣝผ௒ฏ໘ͷࣜΛٻΊΔ ιϑτϚʔδϯࣝผث D D +ξi εϥοΫม਺Λಋೖͯؒ͠ҧͬͨ΋ͷ΍
    ؒҧ͍ͦ͏ͳ΋ͷΛTWʹԡ͠ࠐΉ ϖφϧςΟΛઃ͚ͨϥάϥϯδϡؔ਺Λ࡞Δ ˜ Lp (w, b, α, ξ, μ) = 1 2 wTw + C∑N i=1 ξi −∑N i=1 αi (ti (wT xi + b) − 1 + ξi ) − ∑N i=1 μi ξi ·ͱΊ
94. ϋʔυϚʔδϯ47. ઢܗ෼཭͕׬શʹՄೳͰ͋Δ৔߹Ͱ͔͠ద༻Ͱ͖ͳ͍ʜ Ұൠతͳσʔλ͸ઢܗ෼཭ՄೳͰͳ͍͜ͱͷํ͕ଟ͍ͷʹ
    ઢܗ෼཭ՄೳͰͳ͍৔߹͸Ͳ͏͢Δͷʜ ιϑτϚʔδϯ47. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
95. 065-*/& ·ͱΊ ϋʔυϚʔδϯ47. ιϑτϚʔδϯ47. αϙʔτϕΫτϧϚγϯ 47. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
96. 065-*/& ఆٛ ૒ର໰୊ ಈܘجఈؔ਺Χʔωϧ ຕ ຕ ຕ ·ͱΊ ຕ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
97. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ઢܗ෼཭͸ແཧͦ͏͚ͩͲɼͳΜ͔ͩڥքઢ͸Ҿ͚ͦ͏ʜ x2 x1
98. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ઢܗ෼཭͸ແཧͦ͏͚ͩͲɼͳΜ͔ͩڥքઢ͸Ҿ͚ͦ͏ʜ ପԁܗͷڥքઢͰ෼ྨͰ͖Δ x2 x1
99. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ઢܗ෼཭͸ແཧͦ͏͚ͩͲɼͳΜ͔ͩڥքઢ͸Ҿ͚ͦ͏ʜ ପԁܗͷڥքઢͰ෼ྨͰ͖Δ ࠨͷΑ͏ͳ෼෍ਤΛ࣮ࡍʹ࡞ͬͯΈͯ
    σʔλͷ෼෍ͷ࢓ํ͕Θ͔ͬͯͳ͍ͱ
    ڥքઢͷܗΛܾΊΒΕͳ͍ʜ σʔλͷ࣍ݩ͕େ͖͘ͳͬͯ͘Δͱ
    ͦ΋ͦ΋άϥϑͷՄࢹԽ͕ແཧͰ͸ʜ x2 x1
100. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ઢܗ෼཭͸ແཧͦ͏͚ͩͲɼͳΜ͔ͩڥքઢ͸Ҿ͚ͦ͏ʜ ೋ࣍ݩ͔Βࡾ࣍ݩʹಛ௃ྔΛ૿΍͢ͱ
     ઢܗࣝผ௒ฏ໘Λॻ͚ͯͦ͏ʜʁ x1 x2 x2 1 + x2 2 x2 x1 ࠨਤͷத৺Λʹͨ͠
     σʔλ·Ͱͷڑ཭ͷ࣠࡞Δ
101. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ઢܗ෼཭͸ແཧͦ͏͚ͩͲɼͳΜ͔ͩڥքઢ͸Ҿ͚ͦ͏ʜ ઢܗ෼཭ՄೳͰͳ͍σʔλͰ΋ಛ௃ྔΛ૊Έ߹Θͤͨߴ͍࣍ݩ΁ͷࣸ૾Ͱ
     ઢܗ෼཭ՄೳʹͳΔ͜ͱ͕͋Δ x1 x2 x2 x2 1 + x2 2 x1
102. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ Ͱ΋͜ͷߴ࣍ݩ΁ͷࣸ૾΋ʜ x1 x2 x2 1 + x2 2 ݩͷ෼෍ਤΛ࣮ࡍʹ࡞ͬͯΈͯ
     σʔλͷ෼෍ͷ࢓ํ͕Θ͔ͬͯͳ͍ͱ
     ͲΜͳ৽͍͠ಛ௃Λ࡞Ε͹͍͍͔Θ͔Βͳ͍ʜ Ͳ͏͠Α͏΋ͳ͍͠ɼ࢓ํͳ͍ͷͰ
     Ұ୴ͲΜͳಛ௃ྔΛ࡞Δ͔Ͳ͏͔͸ஔ͍ͱ͍ͯ
     ಛ௃ྔͷඇઢܗ΁ͷม׵ͷҰൠతͳܗΛߟ͑Δʜ
103. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ಛ௃ྔͷඇઢܗม׵ͷҰൠతͳܗ x1 x2 x2 1 + x2 2 x = (x1 x2 ) ඇઢܗม׵ φ φ(x) = (φ0 (x) = 1, φ1 (x), . . . , φM (x))T = (1, x1 , x2 , x2 1 , x2 2 , x1 x2 , . . . )T ྫ
     
104. ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ ఆٛ ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ಛ௃ྔͷඇઢܗม׵ͷҰൠతͳܗ φ x = (x1 x2 ) ⟹ φ(x) = (φ0 (x) = 1, φ1 (x), . . . , φM (x))T ͸ಛ௃ྔΛूΊͨϕΫτϧͳͷͰ͜Ε·Ͱಉ༷ʹઢܗࣝผ௒ฏ໘ͷࣜ͸ φ(x) w0 φ0 + w1 φ1 + w2 φ2 + . . . + wM φM = wTφ ͜͜ͰɼιϑτϚʔδϯ47.ʹ͓͚Δ૒ର໰୊Λࢥ͍ग़͢ͱʜ max Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj xT i xj + ∑N i=1 αi s . t . αT t = 0, C ≥ αi ≥ 0 α
105. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ αT t = 0, C ≥ αi ≥ 0 Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj xT i xj + ∑N i=1 αi σʔλɹɹ͕ιϑτϚʔδϯ47.Ͱઢܗ෼཭ՄೳͰ͋ΔͱԾఆ͢Ε͹
     ্هͷධՁؔ਺ʹɹɹɹɹΛ୅ೖ͢Δ͜ͱ͕Ͱ͖ͯ xi = φ(xi ) Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj φ(xi )Tφ(xj ) + ∑N i=1 αi ͔͠͠ɼಛ௃ྔͷม׵ͷ࢓ํɹ͕Θ͔Βͳ͚Ε͹͜ͷؔ਺͸࢖͑ͳ͍͜ͱʹͳΔʜ ͦ͜Ͱɼɹɹɹɹɹͷܗʹ஫໨͠ɼ͜ͷ಺ੵΛදݱͰ͖ΔΧʔωϧؔ਺
     ͱݺ͹ΕΔؔ਺ɹɹɹɹΛɹɹɹɹɹͱஔ׵͢Δ͜ͱͰಉ͡ܭࢉΛදݱ͢Δͱʜ
     φ(xi )Tφ(xj ) φ(xi )Tφ(xj ) K(xi , xj ) φ(x) φ Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj K(xi , xj ) + ∑N i=1 αi
106. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ αT t = 0, C ≥ αi ≥ 0 Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj K(xi , xj ) + ∑N i=1 αi ্هͷΑ͏ͳϥάϥϯδϡؔ਺Λ࡞Δ͜ͱ͕Ͱ͖Δ
     ۩ମతͳΧʔωϧؔ਺ʹ͸ҎԼͷΑ͏ͳ΋ͷ͕ڍ͛ΒΕΔ ଟ߲ࣜΧʔωϧ ಈܘجఈؔ਺ Ψ΢ε Χʔωϧ Kp (xi , xj ) = (α + xT i xj )p Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2)
107. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ૒ର໰୊ αT t = 0, C ≥ αi ≥ 0 Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj K(xi , xj ) + ∑N i=1 αi ্هͷΑ͏ͳϥάϥϯδϡؔ਺Λ࡞Δ͜ͱ͕Ͱ͖Δ
     ۩ମతͳΧʔωϧؔ਺ʹ͸ҎԼͷΑ͏ͳ΋ͷ͕ڍ͛ΒΕΔ ଟ߲ࣜΧʔωϧ ಈܘجఈؔ਺ Ψ΢ε Χʔωϧ Kp (xi , xj ) = (α + xT i xj ) Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(−γ∥xi − xj ∥2)
108. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ಈܘجఈؔ਺ Ψ΢ε Χʔωϧ ݩ͸಺ੵͳͷͰσʔλಉ࢜ͷྨࣅ౓ ࣅ͍ͯΕ͹ ࣅ͍ͯͳ͚Ε͹ Λ
     දݱ͍ͯ͠ΔΘ͚͕ͩɼͦΕΛσʔλಉ࢜ͷڑ཭Ͱ࠶ݱͯ͠Δؔ਺ɽ
     ɹ͸ਓؒͷखͰઃఆ͢ΔϋΠύʔύϥϝʔλͰ͋Δɽ Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(−γ∥xi − xj ∥2) γ
109. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ಈܘجఈؔ਺ Ψ΢ε Χʔωϧ Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(−γ∥xi − xj ∥2) Ψ΢εΧʔωϧΛల։͢Δͱʜ Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(− 1 2σ2 (xi − xj )T(xi − xj )) = exp(− 1 2σ2 xT i xi − 1 2σ2 xT j xj + 1 σ2 xi xj )) = exp(− 1 2σ2 xT i xi )exp(− 1 2σ2 xT j xj )exp( 1 σ2 xT i xj ) = g(xi )g(xj )exp( K1 (xi , xj ) σ2 ) g(xi ) = exp(− 1 2σ2 xT i xi ) g(xj ) = exp(− 1 2σ2 xT j xj ) K1 (xi , xj ) = xT i xj
110. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ಈܘجఈؔ਺ Ψ΢ε Χʔωϧ Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(−γ∥xi − xj ∥2) Ψ΢εΧʔωϧΛల։͢Δͱʜ Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(− 1 2σ2 (xi − xj )T(xi − xj )) = exp(− 1 2σ2 xT i xi − 1 2σ2 xT j xj + 1 σ2 xi xj )) = exp(− 1 2σ2 xT i xi )exp(− 1 2σ2 xT j xj )exp( 1 σ2 xT i xj ) = g(xi )g(xj )exp( K1 (xi , xj ) σ2 ) exp( K1 (xi , xj ) σ2 ) ଟ߲ࣜΧʔωϧ෦෼ʹ͍ͭͯ
     ΋͏গ͠۷ΓԼ͛ͯΈΔ
111. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ಈܘجఈؔ਺ Ψ΢ε Χʔωϧ Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(−γ∥xi − xj ∥2) Ψ΢εΧʔωϧ಺ͷଟ߲ࣜΧʔωϧ෦෼Λల։͢Δͱʜ exp( K1 (xi , xj ) σ2 ) = exp( xT i xj σ2 ) = exp( ∑d k=1 xik xjk σ2 ) = exp(∑d k=1 xik σ xjk σ ) = ∏d k=1 exp( xik σ xjk σ ) exp( xik σ xjk σ ) f(x) = 1 0! f(0) + 1 1! f′(0)x + 1 2! f′′(0)x2 + 1 3! f′′′(0)x3 + . . . = ∑∞ n=0 1 n! f(n)(0)xn Λ΋͏͢͜͠มܗ͢ΔͨΊʹϚΫϩʔϦϯల։Λಋೖ͢Δ ∑∞ n=0 1 n! f(n)(0)xn f(x) = exp(x) ͱ͢Δͱʜ exp(x) = 1 0! exp(0) + 1 1! exp(0)x + 1 2! exp(0)x2 + 1 3! exp(0)x3 + . . . = ∑∞ n=0 1 n! xn
112. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ಈܘجఈؔ਺ Ψ΢ε Χʔωϧ Kσ (xi , xj ) = exp(− 1 2σ2 ∥xi − xj ∥2) = exp(−γ∥xi − xj ∥2) Ψ΢εΧʔωϧ಺ͷଟ߲ࣜΧʔωϧ෦෼Λల։͢Δͱʜ exp( K1 (xi , xj ) σ2 ) = exp( xT i xj σ2 ) = exp( ∑d k=1 xik xjk σ2 ) = exp(∑d k=1 xik σ xjk σ ) = ∏d k=1 exp( xik σ xjk σ ) exp(x) = 1 0! exp(0) + 1 1! exp(0)x + 1 2! exp(0)x2 + 1 3! exp(0)x3 + . . . = ∑∞ n=0 1 n! xn exp( xik σ xjk σ ) = ∑∞ n=0 1 n! ( xik σ xjk σ )n = 1 + xik σ xjk σ + 1 2 ( xik σ xjk σ )2 + 1 6 ( xik σ xjk σ )3 + 1 24 ( xik σ xjk σ )4 + . . . = (1 xik σ 1 2 ( xik σ )2 1 6 ( xik σ )3 1 2 6 ( xik σ )4 . . . )(1 xjk σ 1 2 ( xjk σ )2 1 6 ( xjk σ )3 1 2 6 ( xjk σ )4 . . . )T = xT ik xjk ಛ௃ྔ͸ɼɹͱɹʹ༝དྷ͢Δ্ʹ࣍ݩ͕ɹ͔Βɹ·Ͱͷແݶେͷ಺ੵʹͳΔɽ xi xj 0 ∞ (1 xik σ 1 2 ( xik σ )2 1 6 ( xik σ )3 1 2 6 ( xik σ )4 . . . )(1 xjk σ 1 2 ( xjk σ )2 1 6 ( xjk σ )3 1 2 6 ( xjk σ )4 . . . )T ແݶݸͷಛ௃
113. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾ ઢܗࣝผՄೳͰͳ͍৔߹ʹ࠷େϚʔδϯΛ
     ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥε
     ͷ෼ྨΛߦ͏ํ๏ ಈܘجఈؔ਺ΧʔωϧΛ༻͍Δͱ
     ແݶ࣍ݩͷಛ௃ϕΫτϧʹͳΔͨΊ
     ඞͣઢܗ෼཭ՄೳͳσʔλʹͳΔʁ Χʔωϧ๏ ߴ࣍ݩʹσʔλΛ֦ு͢Δ͜ͱͰ
     ઢܗ෼཭Մೳʹ͢Δ Χʔωϧؔ਺Ͱ಺ੵΛදݱͨ͠
     ϥάϥϯδϡؔ਺Λ࡞Δ ·ͱΊ x2 x1 x1 x2 x2 1 + x2 2 φ Ld (α) = − 1 2 ∑N i=1 ∑N j=1 αi αj ti tj K(xi xi ) + ∑N i=1 αi
114. 065-*/& ·ͱΊ ϋʔυϚʔδϯ47. ιϑτϚʔδϯ47. αϙʔτϕΫτϧϚγϯ 47. ߴ࣍ݩຒΊࠐΈ ඇઢܗಛ௃ࣸ૾
115. ·ͱΊ Χʔωϧ๏ ιϑτϚʔδϯ ϋʔυϚʔδϯ ࠷େϚʔδϯΛ࣮ݱ͢Δઢܗࣝผ௒ฏ໘Λج४ʹ̎Ϋϥεͷ෼ྨΛߦ͏ํ๏ αϙʔτϕΫτϧϚγϯ 47. Ұൠઢܗ෼཭ՄೳͰͳ͍৔߹ Ұൠઢܗ෼཭ՄೳͰ͋Δ৔߹