カーネル関数とは

Χʔωϧؔ਺ͱ͸

ࢀߟॻ੶ w ΄΅ࣸܦ͍͖ͤͯͨͩ͞·͠ ͨ w ͱͯ΋Θ͔Γқ͍಺༰Ͱײँ Ͱ͢ w IUUQTXXXJXBOBNJDPKQ CPPLCIUNM

ಛ௃நग़ʹ͍ͭͯ ↟࿥ᏣͺͼΝᤎ᷀΀ᔻṟϧόϰͽͶ͖͸ວ͚Ν ↟ͥΞ΁ᬿᔻ᧣ͼؔ๹ΤᬤͶᦉᷯͩ͝ϧόϰặ͹͟ͼ͖ ↟ΚΜີᮈ΀ၞḄᙚΤᑿͥ͘ͺΤວ͚Νͺ ↟Eᮈᡯ਺͹fᓓᙚ΁ᑤᔻṟͲ͞fϖϮϦ⑲τ⑲XͽͶ͖͸΁ᔻṟ y = wTx f(x) =
d ∑ m=1 wm xm ⋯ (1)

ಛ௃நग़ʹ͍ͭͯ ↟᤼ጽͽYͽ᷁͝ᨷΐ͵ͱᑤᔻṟมỐΤᙇͩ͸ີᮈͽࣸᓕͫΝͥͺΤ|ᴰ௃ ᱢᲇ}ͺṺΈ ↟ᴰ௃ϞδύϰΤᤎ᷀΀Κ͘ͽᔉ͡ͺͫΝ ↟ͫΝͺ ᙚ΀ᴰ௃Ϟδύϰ΁ᤎ᷀ͺͼΝ ඇઢܗม׵ ೖྗY ಛ௃ϕΫτϧП
Y ϕ(x) = (ϕ1 (x), ⋯, ϕd (x))T ϕ(x) = (x, x2, x3, ⋯, xd)T

ಛ௃நग़ʹ͍ͭͯ ↟ᴰ௃ᱢᲇͧΞͱ໖ණͽ͖͜͸fᔻṟϧόϰΤວ͚Νͺ ↟ͥ΀ວ͚ፖ΁ᔟ͝Λ΀ϖτ⑲ϸ᤹ᙣ΀ᐠᛯ͹᤹ᬰͧΞ͸͟ͱ ↟ᴰ௃ᱢᲇ΀Ϧϯψύ΁ΐͺΓΝͺᤎ᷀΀Ͷ ↟Ꮭࡶͼؔ਺ؔ๹Τ࣮ḿ͹͟Ν ↟ᡯ΀᥍ᅪY࣮͞Ϟδύϰ͹͔Νᶶ᠓͞ͼ͖ ↟ᴰ௃Ϟδύϰ࣮͞ϞδύϰͼͱΓY͞የᥙᅾΖεϮϚཋ᪦͹Δᅔ͖ fw (x) =
wTϕ(x) = d ∑ m=1 wm ϕm (x) ⋯ (2)

Χʔωϧؔ਺ͷఆٛ ↟Y Ysͺ͖͘ᤍͶ΀ม਺ͽରͩ͸ΰ⑲ϒϰؔ਺L Y Ys ΁ͯΞͰΞ ΀಺᧤΀Ẹͺͩ͸ᤎ᷀΀Κ͘ͽᨷᤅͧΞΝ ↟ ᙚ΀Ęม਺΀ၞḄᙚ͹΁fᓓᙚΤ࿥ͽᤎ᷀΀Κ͘ͽဒᘥ͹͟Ν
k(x, x′) = ϕ(x)Tϕ(x′) = d ∑ m=1 ϕm (x)ϕm (x′) ⋯ (3) k(x, x′) = d ∑ m=1 xm(x′)m ⋯ (4)

֤छͷΧʔωϧಋೖʹ͍ͭͯ ↟಺᧤ ↟ᩆᨷᲩᕥ ↟ệᖰᐠᵿď࿆ỹ΁ᕭᅊĐ

಺ੵͷΧʔωϧදݱ ↟ȃ Y ͺϖϮϦ⑲τ΀಺᧤͹͔Ν ᙚ΁fᚌᐠၞ͡΀Y Y hhhΤ᧱ ᨜ͽᕅΈͺᤎ᷀΀ṟ͹࿄ᑾͫΝͥͺ͞͹͟Ν ↟ΰ⑲ϒϰؔ਺΁ᴰ௃Ϟδύϰ΀಺᧤fଈͳᤎ᷀͹ᨷᤅͧΞΝ
↟ͥΞ΁fϖϮϦ⑲τ⑲XΤᤎ᷀΀ṟͽᷟᨷ͹͟ΝͥͺΤ᣾ዳͫΝ f(x) = ∑ i αi k(xi , x) ⋯ (5) f(x) = ∑ i αi ϕ(xi )Tϕ(x) ⋯ (6) w = ∑ i αi ϕ(xi )

಺ੵͷΧʔωϧදݱ ↟ ᙚ͹΁|ᚌᐠၞ͡΀}ᨨY@J͹΀ΰ⑲ϒϰؔ਺΀ᔻṟẸ͹ͩ͝ͼ͖ ↟κϸϜϰᨨ͹΀ᔻṟẸ΀ṟ͹ᔉ͡ͱΓͽ΁Ħ ↟͔ΝᴰᎭͼᩆ᳀ặΤḝͥ͘ͺ͹fᤎ᷀΀Κ͘ͼဒᘥ͞൱ၙͺͼΝ ↟ϯϜϱρϸτ⑲ᨷሑďᕭᅊĐf-ϓϰϥΤᑿ͘ ↟OE΀ᦉᷯfX΀ႚΡΜͽEΤᠭ͖Νͥͺ͹ϖϮϦ⑲τ਺ΤවΛͫͥͺ͞͹͟Ν ↟FY ᮈ΀༜ṹᥭᠭḄࠐΑ΀ၞḄᙚ΁Θ͘ͽᮈᡯ਺͞਺ᯭͺͼΝͱΓfޮໝΤൃἿͫΝ
f(x) = n ∑ i=1 αi k(x(i), x) = n ∑ i=1 αi d ∑ m=1 (x(i))mxm ⋯ (7)

ਖ਼ఆ஋ੑ͔ΒͷΧʔωϧؔ਺ ಋೖ ↟಺᧤΀ᕥ᭳͝Λfΰ⑲ϒϰؔ਺΁ᴰ௃ᅗ͹฼ͱͺ͟΀YͺYs΀ᇮᑾ ႷΤᶡͩ͸͖Νͺວ͚ΛΞΝ ↟ჯͪፖḟΤḟ͖͸͖Νͺ͟ͽႝ͟͡ͼΜfᬿ༜ͫΝͺͽͼΝ ↟ᝂͽYͺYs΀ᇮᑾႷΤᶡͫΚ͘ͼؔ਺L Y Ys ͔͞Νͺ͟ͽfͯΞΤ ΰ⑲ϒϰؔ਺ͺͩ͸ᑿ͚ͼ͖͝Ħ
↟|ᩆᨷᲩᕥ}Τຬͱͭ΂ZFT

ਖ਼ఆ஋ੑ ↟͔Νؔ਺L Y Ys ͞ᩆᨷᲩ͹͔Νͺ΁f᥅᣾΀O෪΀ᨨY@ hhh Y@O ͝ΛຌᒿͧΞΝᤎ᷀΀ḝᅾďεϮϥḝᅾĐ͔͞Ν ↟ͥΞͽͶ͖͸ęᮈṟᙚ͞ᓝͽᑤᐔ͹͔Νͺ͟fᤎ᷀΀ᙚ͞᥅᣾΀O ᮈϞδύϰǯ
ǯ@ hhh ǯ@O ?5ͽͶ͖͸ᕧΜርͶͥͺ͹͔Νďᩆͩ ͡΁ጪᩆᨷᲩᕥĐ K = k(x1 , x1 ) k(x2 , x1 ) ⋯ k(xn , x1 ) k(x1 , x2 ) k(x2 , x2 ) ⋯ k(xn , x2 ) ⋮ ⋮ ⋱ ⋮ k(x1 , xn ) k(x2 , xn ) ⋯ k(xn , xn ) n ∑ i=1 n ∑ j=1 αi αj Kij ≥ 0 ⋯ (8)

ΧʔωϧτϦοΫ ↟طͽᩆᨷᬰᕥΤຬͱͫͥͺ͞Ρ͝͵͸͖Νؔ਺Τᑿ͚΂f಺᧤΀ຌᒿ͞Ᏽ᠓ͽͼΝ ΀͹΁Ħďᤎ᷀΀ἦᥥ΀ζ⑲ξͽͶ͖͸Đ ↟ᴰ௃Ϟδύϰ΀಺᧤ͺͩ͸ᨷᤅͩͱΰ⑲ϒϰ΁|ᩆᨷᲩ}͹͔Ν ↟ᝂͽf᥅᣾΀ᩧᨷᲩؔ਺L Y Ys ͞ରশ L Y
Ys L Ys Y ͹͔Ξ΂ͯΞ΁|ᴰ௃Ϟδύϰ΀಺ ᧤}͹͔Ν ↟ͥΞ΁ᦉᷯͽΚ͵͸΁዗ᷟᮈᡯ΀ᴰ௃Ϟδύϰ΀಺᧤͞ඦ୯ͼؔ਺ͽͼΝᦉᷯ͞ ͔Ν ↟ͶΐΜfᩆᨷᲩᕥΤΰ⑲ϒϰ΀ᨷᤅͺͫΞ΂ຌᒿᅗΤවΛͭΝͥͺ͞͹͟Ν ↟ͥΞΤ|ΰ⑲ϒϰύϯψδ}ͺ͖͘

Ψ΢εΧʔωϧ ↟༺ମᆚ ↟ᩆྥᐠᵿ΀ጓႷؔ਺ͺჯͪṟ ↟αΫξΰ⑲ϒϰͺṺΈͥͺΔ͔Ν ↟ͥΞ΁ᴰ௃Ϟδύϰණ΀ฌሜ΀಺᧤ͺͩ͸ᶡḿ͹͟Ν ↟ᩆᨷᲩᕥΔຬͱͫďᕭᅊĐ k(x, x′) = exp(−β∥x
− x′∥2) ⋯ (9)

ଟ߲ࣜΧʔωϧ ↟αΫξΰ⑲ϒϰͺჯ༷ͽfΚ͡ᑿΡΞΝ ↟Qᥦ᝔਺ D∕ ↟᤼ፖfͥ΀ᙚΤᤎ᷀΀ṟͽ᧾෺ͫΝͺE?Qί⑲υ⑲΀ಠርͼḄ΀Ẹ ͺͩ͸ᔉͣΝ k(x, x′) = (xT
x′+ c)P ⋯ (10) k(x, x′) = p ∑ m=0 ( p m) { d ∑ l=1 (xl x′ l ) } m cp−m ⋯ (11)

֤Χʔωϧͷಛ௃ w ϊΠζΛ৐ͤͨਖ਼ݭ೾ʹର͢ΔϑΟο ςΟϯάΛ֤ΧʔωϧͰ࣮ࢪ w ্ஈͰ͸Y< >ͷൣғͷϑΟοςΟ ϯάΛ֬ೝ͍ͯ͠Δ͕େ͖ͳࠩҟ͸ ݟΒΕͳ͍ w
͔͠͠ɺԼஈͰ͸ֶशσʔλͷൣ ғ֎Λ༧ଌͤ͞Δͱɺଟ߲ࣜ͸ൃ ࢄํ޲ɺ3#'͸΁ऩଋ͢Δํ޲΁ ͱ޲͔͏ w IUUQTHJTUHJUIVCDPN LJDIJOPTVLFZ

カーネル関数とは

カーネル関数とは

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Χʔωϧؔ਺ͱ͸

ࢀߟॻ੶ w ΄΅ࣸܦ͍͖ͤͯͨͩ͞·͠ ͨ w ͱͯ΋Θ͔Γқ͍಺༰Ͱײँ Ͱ͢ w IUUQTXXXJXBOBNJDPKQ CPPLCIUNM

ಛ௃நग़ʹ͍ͭͯ ↟࿥ᏣͺͼΝᤎ᷀΀ᔻṟϧόϰͽͶ͖͸ວ͚Ν ↟ͥΞ΁ᬿᔻ᧣ͼؔ๹ΤᬤͶᦉᷯͩ͝ϧόϰặ͹͟ͼ͖ ↟ΚΜີᮈ΀ၞḄᙚΤᑿͥ͘ͺΤວ͚Νͺ ↟Eᮈᡯ਺͹fᓓᙚ΁ᑤᔻṟͲ͞fϖϮϦ⑲τ⑲XͽͶ͖͸΁ᔻṟ y = wTx f(x) =

ಛ௃நग़ʹ͍ͭͯ ↟᤼ጽͽYͽ᷁͝ᨷΐ͵ͱᑤᔻṟมỐΤᙇͩ͸ີᮈͽࣸᓕͫΝͥͺΤ|ᴰ௃ ᱢᲇ}ͺṺΈ ↟ᴰ௃ϞδύϰΤᤎ᷀΀Κ͘ͽᔉ͡ͺͫΝ ↟ͫΝͺ ᙚ΀ᴰ௃Ϟδύϰ΁ᤎ᷀ͺͼΝ ඇઢܗม׵ ೖྗY ಛ௃ϕΫτϧП

Χʔωϧؔ਺ͷఆٛ ↟Y Ysͺ͖͘ᤍͶ΀ม਺ͽରͩ͸ΰ⑲ϒϰؔ਺L Y Ys ΁ͯΞͰΞ ΀಺᧤΀Ẹͺͩ͸ᤎ᷀΀Κ͘ͽᨷᤅͧΞΝ ↟ ᙚ΀Ęม਺΀ၞḄᙚ͹΁fᓓᙚΤ࿥ͽᤎ᷀΀Κ͘ͽဒᘥ͹͟Ν

֤छͷΧʔωϧಋೖʹ͍ͭͯ ↟಺᧤ ↟ᩆᨷᲩᕥ ↟ệᖰᐠᵿď࿆ỹ΁ᕭᅊĐ

಺ੵͷΧʔωϧදݱ ↟ȃ Y ͺϖϮϦ⑲τ΀಺᧤͹͔Ν ᙚ΁fᚌᐠၞ͡΀Y Y hhhΤ᧱ ᨜ͽᕅΈͺᤎ᷀΀ṟ͹࿄ᑾͫΝͥͺ͞͹͟Ν ↟ΰ⑲ϒϰؔ਺΁ᴰ௃Ϟδύϰ΀಺᧤fଈͳᤎ᷀͹ᨷᤅͧΞΝ

ਖ਼ఆ஋ੑ ↟͔Νؔ਺L Y Ys ͞ᩆᨷᲩ͹͔Νͺ΁f᥅᣾΀O෪΀ᨨY@ hhh Y@O ͝ΛຌᒿͧΞΝᤎ᷀΀ḝᅾďεϮϥḝᅾĐ͔͞Ν ↟ͥΞͽͶ͖͸ęᮈṟᙚ͞ᓝͽᑤᐔ͹͔Νͺ͟fᤎ᷀΀ᙚ͞᥅᣾΀O ᮈϞδύϰǯ

ΧʔωϧτϦοΫ ↟طͽᩆᨷᬰᕥΤຬͱͫͥͺ͞Ρ͝͵͸͖Νؔ਺Τᑿ͚΂f಺᧤΀ຌᒿ͞Ᏽ᠓ͽͼΝ ΀͹΁Ħďᤎ᷀΀ἦᥥ΀ζ⑲ξͽͶ͖͸Đ ↟ᴰ௃Ϟδύϰ΀಺᧤ͺͩ͸ᨷᤅͩͱΰ⑲ϒϰ΁|ᩆᨷᲩ}͹͔Ν ↟ᝂͽf᥅᣾΀ᩧᨷᲩؔ਺L Y Ys ͞ରশ L Y

Ψ΢εΧʔωϧ ↟༺ମᆚ ↟ᩆྥᐠᵿ΀ጓႷؔ਺ͺჯͪṟ ↟αΫξΰ⑲ϒϰͺṺΈͥͺΔ͔Ν ↟ͥΞ΁ᴰ௃Ϟδύϰණ΀ฌሜ΀಺᧤ͺͩ͸ᶡḿ͹͟Ν ↟ᩆᨷᲩᕥΔຬͱͫďᕭᅊĐ k(x, x′) = exp(−β∥x

ଟ߲ࣜΧʔωϧ ↟αΫξΰ⑲ϒϰͺჯ༷ͽfΚ͡ᑿΡΞΝ ↟Qᥦ᝔਺ D∕ ↟᤼ፖfͥ΀ᙚΤᤎ᷀΀ṟͽ᧾෺ͫΝͺE?Qί⑲υ⑲΀ಠርͼḄ΀Ẹ ͺͩ͸ᔉͣΝ k(x, x′) = (xT

֤Χʔωϧͷಛ௃ w ϊΠζΛ৐ͤͨਖ਼ݭ೾ʹର͢ΔϑΟο ςΟϯάΛ֤ΧʔωϧͰ࣮ࢪ w ্ஈͰ͸Y< >ͷൣғͷϑΟοςΟ ϯάΛ֬ೝ͍ͯ͠Δ͕େ͖ͳࠩҟ͸ ݟΒΕͳ͍ w