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輪読 / topic model3.1, 3.2
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ysekky
December 22, 2015
Research
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輪読 / topic model3.1, 3.2
ysekky
December 22, 2015
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Transcript
τϐοΫϞσϧʹΑΔ౷ܭతજࡏҙຯ ղੳ 3ষ ֶशΞϧΰϦζϜ(3.1 ~ 3.2.2) Yoshifumi Seki GunosyσʔλϚΠχϯάݚڀձ #97
2015.12.21
3.1 ౷ܭతֶशΞϧΰϦζϜ ౷ܭతֶश: ؍ଌσʔλͷഎޙʹજΉϧʔϧɾنଇΛ౷ܭతʹهड़ ͠ɼσʔλ͔Βࣗಈతʹ֫ಘ͢ΔֶशͷΈ ؍ଌσʔλ ؍ଌσʔλΛੜͨ֬͠ ੜϞσϧ ɹʹͰ͖Δ͚͍ͩۙ֬ ɹΛਪఆ͢Δ
KL-divergence ౷ܭϞσϧͷۙ͞Λද͢ࢦඪ ͜ΕʹΑΓɼ ʹ͍ۙ֬Λਪఆ͢Δͱ͍͏KLΛ ࠷খԽ͢Δͱ͍͏ʹͳΔ
KL-divegenceͷ࠷খԽ ͷظΛ ͱ͢ΔͱҎԼͷΑ͏ʹల։Ͱ͖Δ ͷ߲ແࢹͰ͖ΔͷͰ࠷খԽҎԼͷΑ͏ʹͳΔ
ظܭࢉͷۙࣅ ະͳͷͰ؍ଌσʔλΛਅͷ͔ΒಘΒΕͨαϯϓϧͱͯۙ͠ࣅΛߦ͏ɽ ͜ͷํ๏࠷ਪఆ(Maximum Likelihood estimation)ͱݺΕ, ࠷ਪఆʹΑͬͯΒΕΔղΛ ͱ͢Δ
ੜϞσϧͱͯ͠ߟ͑Δ σʔλ ͷੜ֬ ࠷ਪఆ͜ͷੜ֬ͷରΛ࠷େʹ͢Δ ΛٻΊΔ͜ͱͰ ͋Δͱ͍͑Δɽ
MAPਪఆ ͱ͢Δͱੜ֬ ͱͳΓ࠷దԽҎԼ ͷΑ͏ʹ͔͚Δ աֶशΛ͙ͨΊͷਖ਼ଇԽ߲ͱͯ͠ػೳ͍ͯ͠ΔͨΊ, ࠷ਪ ఆΑΓ൚Խੑೳ͕ߴֶ͍श͕ظͰ͖Δɽ ͜ΕΛࣄޙ֬࠷େ(Maxmux a Posteriori,
MAP)ਪఆͱݺͿɽ
ࣄޙ֬ ࣄޙ֬ϕΠζͷఆཧʹΑͬͯҎԼͷΑ͏ʹٻΊΒΕΔɽ MAPਪఆ͜ͷࣄޙ͕֬࠷େͱͳΔ ΛٻΊΔͱͳΔ
ϕΠζਪఆ ࠷ਪఆMAPਪఆύϥϝʔλͷΛਪఆ͢ΔͨΊਪఆͱݺΕΔ ਪఆ͞ΕͨύϥϝʔλʹΑͬͯ৽ͨͳσʔλ ͷ༧ଌ ΛٻΊΔ͜ͱ ͕Ͱ͖Δ ͦΕʹ͍ͨͯ͠ύϥϝʔλͷࣄޙ͔֬ΒॏΈ͚͞Εͨ༧ଌΛٻΊΔํ ๏ΛϕΠζਪఆͱݺͿ ͭ·Γύϥϝʔλࣗମ֬ͱͯ͠දݱ͢Δɽ ͜ͷΑ͏ͳੵܭࢉղੳతʹٻΊΔ͜ͱ͕େͰ͖ͳ͍ͨΊɼ͜ͷۙࣅղΛͲΑ
͏ʹٻΊΔ͔ͱ͍͏ΞϧΰϦζϜ͕ଘࡏ͢Δ
LDAʹ͓͚ΔఆࣜԽ • ؍ଌσʔλ: • ֤σʔλͷજࡏม: • જࡏมͷऔΓ͏Δ: • જࡏมͷऔΓ͏Δͷ֬ม: •
֤જࡏม֬ϕΫτϧ ʹجͮ͘ଟ߲ʹै͏ • \piσΟϦΫΤʹΑͬͯੜ͞ΕΔ
LDAʹ͓͚ΔఆࣜԽ • ͱ ϋΠύʔύϥϝʔλ • ͱɹ ಉ͡
LDAʹ͓͚ΔϕΠζਪఆ ҎԼͷ༧ଌΛٻΊΔͷ͕తͰ͋Δ
3.2 αϯϓϦϯάۙࣅ๏ • αϯϓϦϯάۙࣅ๏ ࣄޙ͔ΒαϯϓϦϯά͞Εͨෳͷύϥϝʔλͷฏۉʹਲ ͬͯ༧ଌΛߦ͏ • ΪϒεαϯϓϦϯά • पลԽΪϒεαϯϓϦϯά
αϯϓϦϯά͔Βͷۙࣅܭࢉ ͱͯ͠ࣄޙ͔ΒͷαϯϓϧΛSݸੜ͢Δͱɼ ͱͯۙ͠ࣅܭࢉΛߦ͏͜ͱ͕Ͱ͖Δɽ ࣄޙ͔Βͷαϯϓϧੜ͕Ͱ͖Εۙࣅܭࢉ͕ՄೳͰ͋Δ͕, αϯϓ ϧੜଟ͘ͷ߹ίετ͕ߴ͍ɽ ͜ͷΑ͏ͳ߹ʹଟ͘༻͍ΒΕΔͷ͕ΪϒεαϯϓϦϯάͰ͋Δ
ΪϒεαϯϓϦϯά ΪϒεαϯϓϦϯάͰతͷࣄޙ͔ΒͷαϯϓϧੜΛߦ͏ΘΓ ʹɼαϯϓϧͷܭࢉίετ͕͍͖݅֬Λߏ͠ɼ֬มΛ ަޓʹαϯϓϧੜ͢Δ͜ͱͰɼతͷࣄޙ͔ΒͷαϯϓϧΛੜ͢Δ LDAͰજࡏม ΛαϯϓϦϯάରͱ͢Δ͜ͱͰܭࢉίετͷ͍ ͖݅Λߏ͍ͯ͠Δɽ • ΪϒεαϯϓϦϯάͷྲྀΕ [ࣄޙ]
=> [݁߹] => [ϕΠζͷఆཧʹΑΓల։] => [ఆҼࢠΛ আ֎]
z_iͷαϯϓϦϯά ͔Β ΛऔΓআ͍ͨજࡏมू߹Λ ͱදه͢Δ Ҏ֎ͷͯ͢ͷ֬มΛطͱݻఆ͖ͯ݅֬͠ΛٻΊ Δ
• ʹؔͳ͍߲আڈͯ͠ߟ͑Δ • ݁߹֬ΛϕΠζͷఆཧͰల։͢Δ
(3.18)͔Β(3.19)ʹ͍ͭͯ • ʹ͍ͭͯల։ • ʹରͯ͠د༩͠ͳ͍ͷΛཧ -ɹ ʹ͍ͭͯల։
• ʹد༩͠ͳ͍ͷΛཧ • ʹ͍ͭͯల։
• د༩͠ͳ͍ͷΛཧ • ల։ͯ͠ ʹؔΘΔͷ, ʹؔΘΔͷ͚ͩʹ͢Δ • z_i=kʹؔΘΔͷ͚ͩʹ͢Δ
ਖ਼نԽ߲Λܭࢉ͢Δ • ࢠ͕ܭࢉͰ͖ͨͷͰɼͦΕʹ߹ΘͤͯΛઃఆ͍ͯ͠ Δɻ • z_iͷऔΓ͏ΔΛͯ͢ͱͬͯ૯Λͱ͍ͬͯΔͷͰଟཧ తʹ1ʹͳΔ
ͷ͖݅
3.2.2 पลԽΪϒεαϯϓϦϯά • Λੵফڈ͢Δ͜ͱͰ ΛαϯϓϦϯά͢Δ • ֬มͷੵআڈपลԽͱݺΕΔ͜ͱ͔ΒɼपลԽΪ ϒεαϯϓϦϯάͱݺͿ
ࣄޙͷల։
पลԽͷੵ
ੵܭࢉΛղੳతʹٻΊΔ पลԽΪϒεαϯϓϦϯάΛߦ͏ͨΊʹղੳతʹੵܭࢉ͕Ͱ͖ͳ ͚ΕͳΒͳ͍ ੵࣜࣄޙ֬ʹΑΔظܭࢉͱΈͳ͢͜ͱ͕Ͱ͖ɼ ܭࢉରͷ֬ͷڞࣄલΛ༻͍ͯࣄલΛߏ͢Δ͜ͱ ͰੵܭࢉΛղੳతʹٻΊΔ͜ͱ͕Ͱ͖Δɽ LDAͷ߹σΟϦΫϨͳͷͰ,ҎԼͷఆཧʹै͏
࠷ॳͷ͜Ζͷσʔλഁغ͢Δඞཁ͕͋Δ αϯϓϦϯάͰॳظͷࠒͷσʔλॳظʹґଘ͢ΔͷͰഁغ ͢Δඞཁ͕͋Δɽ ͜ͷظؒͷ͜ͱΛburn-in periodͱݺͿ