X ্ͷ࠶ੜ֩ώϧϕϧτۭؒ (Reproducing Kernrl Hilbert Space, RKHS) ͱ ɺX ্ͷ͔ؔΒͳΔώϧϕϧτۭؒ H Ͱɺҙͷ x 2 X ʹରͯ͋͠Δؔ kx 2 H : X ! R ͕ଘࡏͯ͠ɺҎԼͷ࠶ੜੑΛຬͨ͢Λ͍͏ɻ hf; kxiH = f (x) (8f 2 H : X ! R) (3) ·ͣ x ͕༩͑ΒΕͯɺH ͷதͷҙͷؔ f ʹରͯ͠ɺͦΕͧΕͷ x ʹ্͍ͭͯهΛຬͨ ؔ͢ kx ͕ଘࡏ͢Δɺͱ͍͏Πϝʔδ ࠶ੜ֩ ্هͷఆٛͷ kx ʹରͯ͠ɺk (y; x) = kx (y) ʹΑͬͯఆ·ΔΧʔωϧ k : X ˆ X ! R Λ࠶ੜ֩ͱݺͿɻ 5
Λ Hk ͱ͢ΔɻX ͔Β Hk ͷࣸ૾ ˘ : X ! Hk Λ x 7! k (´; x) (4) ʹΑͬͯఆٛ͢Δɻ͢Δͱɺ࠶ੜੑͷఆ͔ٛΒɺ h˘ (x) ; ˘ (y)iHk = (˘ (x)) (y) = k (x; y) (5) ͜Ε͕·͞ʹΧʔωϧ๏ͷཁͱͳΔΈʹͳ͍ͬͯͯɺΧʔωϧτϦοΫͱݺΕΔɻ Χʔωϧ k Ͱܭࢉͨ͠ (k (x; y)) k ʹରԠ͢Δ RKHS(ߴ࣍ݩ) Ͱɺؔؒͷੵ Λͱ͍ͬͯΔ͜ͱʹ૬͢Δɻؔؒͷੵͱྫ͑ɿ Z 1 `1 f (x) g (x) dx (6) 7