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カーネル関数とは
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kichinosukey
January 14, 2019
Science
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カーネル関数とは
kichinosukey
January 14, 2019
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Transcript
Χʔωϧؔͱ
ࢀߟॻ੶ w ΄΅ࣸܦ͍͖ͤͯͨͩ͞·͠ ͨ w ͱͯΘ͔Γқ͍༰Ͱײँ Ͱ͢ w IUUQTXXXJXBOBNJDPKQ CPPLCIUNM
ಛநग़ʹ͍ͭͯ ↟ᏣͺͼΝᤎ᷀ᔻṟϧόϰͽͶ͖ວ͚Ν ↟ͥΞᬿᔻ᧣ͼؔΤᬤͶᦉᷯͩ͝ϧόϰặ͟ͼ͖ ↟ΚΜີᮈၞḄᙚΤᑿͥ͘ͺΤວ͚Νͺ ↟EᮈᡯfᓓᙚᑤᔻṟͲ͞fϖϮϦ⑲τ⑲XͽͶ͖ᔻṟ y = wTx f(x) =
d ∑ m=1 wm xm ⋯ (1)
ಛநग़ʹ͍ͭͯ ↟ጽͽYͽ᷁͝ᨷΐ͵ͱᑤᔻṟมỐΤᙇͩີᮈͽࣸᓕͫΝͥͺΤ|ᴰ ᱢᲇ}ͺṺΈ ↟ᴰϞδύϰΤᤎ᷀Κ͘ͽᔉ͡ͺͫΝ ↟ͫΝͺ ᙚᴰϞδύϰᤎ᷀ͺͼΝ ඇઢܗม ೖྗY ಛϕΫτϧП
Y ϕ(x) = (ϕ1 (x), ⋯, ϕd (x))T ϕ(x) = (x, x2, x3, ⋯, xd)T
ಛநग़ʹ͍ͭͯ ↟ᴰᱢᲇͧΞͱ໖ණͽ͖͜fᔻṟϧόϰΤວ͚Νͺ ↟ͥວ͚ፖᔟ͝Λϖτ⑲ϸ᤹ᙣᐠᛯ᤹ᬰͧΞ͟ͱ ↟ᴰᱢᲇϦϯψύΐͺΓΝͺᤎ᷀Ͷ ↟ᏝࡶͼؔؔΤ࣮ḿ͟Ν ↟ᡯ᥍ᅪY࣮͞Ϟδύϰ͔Νᶶ᠓͞ͼ͖ ↟ᴰϞδύϰ࣮͞ϞδύϰͼͱΓY͞የᥙᅾΖεϮϚཋ᪦Δᅔ͖ fw (x) =
wTϕ(x) = d ∑ m=1 wm ϕm (x) ⋯ (2)
Χʔωϧؔͷఆٛ ↟Y Ysͺ͖͘ᤍͶมͽରͩΰ⑲ϒϰؔL Y Ys ͯΞͰΞ ᧤Ẹͺͩᤎ᷀Κ͘ͽᨷᤅͧΞΝ ↟ ᙚĘมၞḄᙚfᓓᙚΤͽᤎ᷀Κ͘ͽဒᘥ͟Ν
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 ↟ệᖰᐠᵿď࿆ỹᕭᅊĐ
ੵͷΧʔωϧදݱ ↟ȃ Y ͺϖϮϦ⑲τ᧤͔Ν ᙚfᚌᐠၞ͡Y Y hhhΤ᧱ ͽᕅΈͺᤎ᷀ṟ࿄ᑾͫΝͥͺ͟͞Ν ↟ΰ⑲ϒϰؔᴰϞδύϰ᧤fଈͳᤎ᷀ᨷᤅͧΞΝ
↟ͥΞfϖϮϦ⑲τ⑲XΤᤎ᷀ṟͽᷟᨷ͟ΝͥͺΤዳͫΝ 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@Jΰ⑲ϒϰؔᔻṟẸͩ͝ͼ͖ ↟κϸϜϰᨨᔻṟẸṟᔉ͡ͱΓͽĦ ↟͔ΝᴰᎭͼᩆ᳀ặΤḝͥ͘ͺfᤎ᷀Κ͘ͼဒᘥ͞൱ၙͺͼΝ ↟ϯϜϱρϸτ⑲ᨷሑďᕭᅊĐf-ϓϰϥΤᑿ͘ ↟OEᦉᷯfXႚΡΜͽEΤᠭ͖ΝͥͺϖϮϦ⑲τΤවΛͫͥͺ͟͞Ν ↟FY ᮈ༜ṹᥭᠭḄࠐΑၞḄᙚΘ͘ͽᮈᡯ͞ᯭͺͼΝͱΓfޮໝΤൃἿͫΝ
f(x) = n ∑ i=1 αi k(x(i), x) = n ∑ i=1 αi d ∑ m=1 (x(i))mxm ⋯ (7)
ਖ਼ఆੑ͔ΒͷΧʔωϧؔ ಋೖ ↟᧤ᕥ᭳͝Λfΰ⑲ϒϰؔᴰᅗͱͺ͟YͺYsᇮᑾ ႷΤᶡ͖ͩΝͺວ͚ΛΞΝ ↟ჯͪፖḟΤḟ͖͖Νͺ͟ͽႝ͟͡ͼΜfᬿ༜ͫΝͺͽͼΝ ↟ᝂͽYͺYsᇮᑾႷΤᶡͫΚ͘ͼؔL Y Ys ͔͞Νͺ͟ͽfͯΞΤ ΰ⑲ϒϰؔͺͩᑿ͚ͼ͖͝Ħ
↟|ᩆᨷᲩᕥ}ΤຬͱͭZFT
ਖ਼ఆੑ ↟͔ΝؔL Y Ys ͞ᩆᨷᲩ͔Νͺf᥅O෪ᨨY@ hhh Y@O ͝ΛຌᒿͧΞΝᤎ᷀ḝᅾďεϮϥḝᅾĐ͔͞Ν ↟ͥΞͽͶ͖ęᮈṟᙚ͞ᓝͽᑤᐔ͔Νͺ͟fᤎ᷀ᙚ͞᥅O ᮈϞδύϰǯ
ǯ@ hhh ǯ@O ?5ͽͶ͖ᕧΜርͶͥͺ͔Νďᩆͩ ͡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͵͖ΝؔΤᑿ͚f᧤ຌᒿ͞Ᏽ᠓ͽͼΝ Ħďᤎ᷀ἦᥥζ⑲ξͽͶ͖Đ ↟ᴰϞδύϰ᧤ͺͩᨷᤅͩͱΰ⑲ϒϰ|ᩆᨷᲩ}͔Ν ↟ᝂͽf᥅ᩧᨷᲩؔL Y Ys ͞ରশ L Y
Ys L Ys Y ͔ΞͯΞ|ᴰϞδύϰ ᧤}͔Ν ↟ͥΞᦉᷯͽΚ͵ᷟᮈᡯᴰϞδύϰ᧤͞ඦ୯ͼؔͽͼΝᦉᷯ͞ ͔Ν ↟ͶΐΜfᩆᨷᲩᕥΤΰ⑲ϒϰᨷᤅͺͫΞຌᒿᅗΤවΛͭΝͥͺ͟͞Ν ↟ͥΞΤ|ΰ⑲ϒϰύϯψδ}ͺ͖͘
ΨεΧʔωϧ ↟༺ମᆚ ↟ᩆྥᐠᵿጓႷؔͺჯͪṟ ↟αΫξΰ⑲ϒϰͺṺΈͥͺΔ͔Ν ↟ͥΞᴰϞδύϰණฌሜ᧤ͺͩᶡḿ͟Ν ↟ᩆᨷᲩᕥΔຬͱͫďᕭᅊĐ k(x, x′) = exp(−β∥x
− x′∥2) ⋯ (9)
ଟ߲ࣜΧʔωϧ ↟αΫξΰ⑲ϒϰͺჯ༷ͽfΚ͡ᑿΡΞΝ ↟Qᥦ D∕ ↟ፖfͥᙚΤᤎ᷀ṟͽ᧾ͫΝͺE?Qί⑲υ⑲ಠርͼḄẸ ͺͩᔉͣΝ 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 ϊΠζΛͤͨਖ਼ݭʹର͢ΔϑΟο ςΟϯάΛ֤ΧʔωϧͰ࣮ࢪ w ্ஈͰY< >ͷൣғͷϑΟοςΟ ϯάΛ֬ೝ͍ͯ͠Δ͕େ͖ͳࠩҟ ݟΒΕͳ͍ w
͔͠͠ɺԼஈͰֶशσʔλͷൣ ғ֎Λ༧ଌͤ͞Δͱɺଟ߲ࣜൃ ࢄํɺ3#'ऩଋ͢Δํ ͱ͔͏ w IUUQTHJTUHJUIVCDPN LJDIJOPTVLFZ