Riemann幾何学ユーザーのための情報幾何学入門

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1. Riemann زԿֶϢʔβʔͷͨΊͷ৘ใزԿֶೖ໳ தҪ ӻ࢘ʢ@enakai00ʣ 2017 ೥ 2 ݄ 4 ೔ 1 Կͷ࿩͔ͱ͍͏ͱ େֶ࣌୅ʹ෺ཧֶΛֶΜͰ͍ͯɺҰൠ૬ରੑཧ࿦ͷͨΊʹ Riemann زԿֶΛษڧͨ͠ͱ͍͏ํ͸୔ࢁ͍Δ Ͱ͠ΐ͏ɻͦͷޙɺIT ۀքʹब৬ͯ͠਺ֶͱ͸ແԑͷੜ׆Λ͍ͯͨ͠Βɺͳ͔ͥ IT ۀքͰػցֶश͕େྲྀߦ ʹͳͬͯɺվΊͯ਺ֶΛษڧ͍ͯ͠Δํ΋ଟ͍͜ͱͰ͠ΐ͏ɻͦͯ͠ɺػցֶशͷͨΊʹ౷ܭֶΛษڧ͍ͯ͠ Δͱɺͳʹ΍Βʮ৘ใزԿֶʯͱ͍͏Α͏ͳ෼໺͕͋ͬͯɺزԿֶతͳࢹ఺Ͱ౷ܭֶΛଊ͑Δ͜ͱ͕Ͱ͖Δͱ ͍͏࿩Λখࣖʹ͸͞Μͩํ΋গͳ͘ͳ͍͸ͣͰ͢ɻ ৘ใزԿֶͰ͸ɺύϥϝʔλΛ࣋ͬͨ֬཰෼෍ͷ଒ʹରͯ͠ɺͦͷύϥϝʔλΛہॴ࠲ඪͱ͢Δଟ༷ମΛߟ ͑·͢ɻͦͯ͠ɺ͜ͷଟ༷ମʹܭྔͱ઀ଓΛ༩͑ͯɺڑ཭΍ฏߦҠಈͷ֓೦Λಋೖ͠·͢ɻͭ·Γɺ ʮҟͳΔ ύϥϝʔλͷ஋Λ࣋ͭ̎ͭ֬཰෼෍Λ݁Ϳଌ஍ઢʯͳͲ͕ߟ͑ΒΕΔΑ͏ʹͳΔͷͰ͢ɻ—— ͱ͍͏Α͏ͳ ৘ใزԿֶͷ࿩Λ͸͡Ίͯখࣖʹ͸͞Μͩ࣌ɺචऀ͸ɺͳΜͱͳ͘ʮ΋΍ͬʯͱͨ͠౷ܭֶͷੈքΛزԿֶత ͳࢹ఺Ͱʮ͔ͪͬʯͱଊ͑ΒΕͦ͏ͳؾ͕ͯ͠ɺ ʮ͜Ε͸ͳΜͱ͔ͯ͠ཧղ͍ͨ͠ʂʯͱࢥͬͨͷͰ͢ɻ ʮ͢΂ ͯͷ֬཰෼෍ΛؚΉڊେͳۭؒΛߟ͑ͯɺ౷ܭֶΛେہతʹଊ͑Δ͜ͱ͕Ͱ͖ΔͷͰ͸ʁʂʯͳͲͱ͍Ζ͍Ζ ͱໝ૝΋๲ΒΈ·͢ɻ ͦͯ͠ԿΑΓɺҰൠ૬ରੑཧ࿦Ҏ֎ͷ࢖͍ಓ͕Α͘Θ͔Βͳ͔ͬͨඍ෼زԿֶ͕౷ܭֶʹԠ༻Ͱ͖Δͱ͍͏ ͷ͸*1ɺͳΜ͔ͩਓੜ͕Ұपճͬͯɺ༮͍͜ΖʹকདྷΛ੤͍͋ͬͨ༮ೃછʹ࠶ձͨ͠Α͏ͳؾ෼ʹ΋ͳΔ΋ͷ Ͱ͢ɻ͍΍ɺࢲʹ͸ͦΜͳ༮ೃછ͸͍·ͤΜ͕ɻ·͊ɺͦΜͳؾ෼Λڞ༗ͯ͘͠ΕΔ෺ཧ԰ͣ͘Εͷ IT Τϯ δχΞ͸ɺ͖ͬͱ୔ࢁ͍Δ͸ͣͰ͢ɻ —— ͱ͍͏Α͏ͳ࿩Λ౿·͑ͯɺຊߘͰ͸ɺRiemann زԿֶͷ஌ࣝΛલఏʹͯ͠ɺ৘ใزԿֶͷجૅͱͳ Δʮ૒ରฏୱͳଟ༷ମʯΛಋೖ্ͨ͠ͰɺͦͷزԿֶతͳ໘ന͕͞ײ͡ΒΕΔҰྫͱͯ͠ɺ ʮEM ΞϧΰϦζ ϜʯͷزԿֶతղऍΛઆ໌͍ͨ͠ͱࢥ͍·͢ɻ ͳ͓ɺຊߘͷٞ࿦ͷଟ͘͸ɺ[1][2] ͷ಺༰ʹج͍͍ͯ·͢ɻຊߘ͔Β͞ΒʹֶशΛਐΊ͍ͨํ͸ɺ͜ΕΒͷ ॻ੶΋ͥͻࢀߟʹ͍ͯͩ͘͠͞ɻ 2 Riemann زԿֶͷ෮श Riemann زԿֶͷ஌ࣝΛલఏʹͯ͠ʜʜͱ͸ݴͬͯ΋ɺ·ͣ͸ɺ಄Λ੔ཧ͢ΔͨΊʹ Riemann زԿֶͷओ ཁͳ֓೦Λ෮श͓͖͍ͯͨ͠ͱࢥ͍·͢ɻಛʹɺ৘ใزԿֶͰѻ͏ଟ༷ମ͸ɺ ʮܭྔͱ઀ଓΛ͍࣋ͬͯΔʯͱ *1 ΋ͪΖΜήʔδཧ࿦ͱ͔ͦͷ͋ͨΓ΋͋ΔΜͰ͕͢ɻ 1
2. ͍͏఺Ͱ͸ Riemann ଟ༷ମʹྨࣅ͍ͯ͠·͕͢ɺ࣮ࡍʹ͸ɺRiemann ଟ༷ମͱ͸ҟͳΔੑ࣭Λ͍࣋ͬͯ· ͢ɻͦͷͨΊɺ ʮRiemann ଟ༷ମ͸ɺҰൠͷଟ༷ମʹൺ΂ͯԿ͕ಛघͳͷ͔ʁʯͱ͍͏఺Λҙ͓ࣝͯ͘͜͠ͱ ͕ඞཁʹͳΓ·͢ɻ ͳ͓ɺ͜ͷઅͷ಺༰͸͋͘·Ͱ΋෮शͳͷͰɺ͢΂ͯͷࣄฑʹূ໌Λ༩͑Δ͜ͱ͸͍ͯ͠·ͤΜɻྫ͑͹ɺ ۂ཰ςϯιϧ৔ͱᎇ཰ςϯιϧ৔ʹ͍ͭͯ͸ɺΑ͋͘Δ௚ײతͳٞ࿦͸ൈ͖ʹͯ͠ɺఆٛͷΈΛఱԼΓతʹ༩ ͍͑ͯ·͢ɻ͋Β͔͡Ίྃ͝ঝ͍ͩ͘͞ɻ 2.1 ଟ༷ମ ଟ༷ମͷఆٛ͸ɺ͕͢͞ʹলུ͓͖ͯ͠·͢ɻ࠲ඪม׵͕ఆٛ͞Εͨෳ਺ͷہॴ࠲ඪܥΛ࣋ͭΞϨͰ͢ɻج ຊతʹ͸ɺC∞ ڃͷ n ࣍ݩՄඍ෼ଟ༷ମΛߟ͍͑ͯΔ΋ͷͱ͍ͯͩ͘͠͞ɻΠϯσοΫεΛ࣋ͭม਺ʹ͍ͭ ͯ͸ɺEinstein ͷ૯࿨ن໿Λద༻͠·͢ɻͨͩ͠ɺޙ΄Ͳઆ໌͢ΔΑ͏ʹɺ૒ରฏୱͳۭؒʹ͓͍ͯ͸ɺภඍ ෼ԋࢉࢠͷলུه๏ ∂i ͱ ∂i ͸ɺ ʮ૒ର࠲ඪʹΑΔภඍ෼ʯͱ͍͏ಛผͳҙຯΛ͍࣋ͬͯ·͢ɻͦͷͨΊɺҰ ൠͷہॴ࠲ඪʹΑΔภඍ෼ʹ͍ͭͯ͸ɺলུه๏͸࢖༻ͤͣʹ ∂ ∂xi ͱදه͠·͢ɻ 2.2 ઀ϕΫτϧۭؒ ہॴ࠲ඪܥΛ 1 ͭݻఆ͢Δͱɺଟ༷ମ M ্ͷ࣮਺஋ؔ਺ f(x) ʹର͢Δඍ෼ԋࢉࢠͷηοτ { ∂ ∂xi }n i=1 Λ جఈͱͯ͠ɺM ্ͷ֤఺ p ʹ͓͍ͯϕΫτϧۭؒΛߏ੒͢Δ͜ͱ͕Ͱ͖·͢ɻ͜ΕΛ఺ p ʹ͓͚Δ઀ϕΫτ ϧۭؒ Tp M ͱఆٛ͠ɺͦͷཁૉΛ઀ϕΫτϧͱݺͼ·͢ɻ͜ͷ࣌ɺ೚ҙͷ઀ϕΫτϧ v = ai ∂ ∂xi ͸ɺ࣍ͷઢ ܗੑͱϥΠϓχοπଇΛຬͨ͢͜ͱ͕෼͔Γ·͢ɻ v(af + bg) = av(f) + bv(g) (∀a, b ∈ R, ∀f, g ∈ C∞(M)) (1) v(fg) = v(f)g + fv(g) (∀f, g ∈ C∞(M)) (2) ࣮͸ɺ͜ΕΒͷੑ࣭Λຬͨ͢೚ҙͷࣸ૾͸ɺ { ∂ ∂xi }n i=1 ͷઢܗ݁߹Ͱදݱ͢Δ͜ͱ͕Ͱ͖·͢ɻैͬͯɺ্ هͷੑ࣭Λ࣋ͭࣸ૾શମͷू߹ʹઢܗԋࢉΛಋೖͨ͠΋ͷΛ઀ϕΫτϧۭؒͱఆٛ͢Δ͜ͱ΋ՄೳͰ͢ɻ ఆཧ 1 (1) ͱ (2) Λຬͨࣸ͢૾ C∞(M) −→ R ͷू߹ʹઢܗԋࢉΛಋೖͯ͠ϕΫτϧۭؒΛߏ੒ͨ͠৔߹ɺ { ∂ ∂xi }n i=1 ͸ɺ͜ͷϕΫτϧۭؒͷجఈͱͳΔɻ [ূ໌] ɹ͸͡Ίʹɺ { ∂ ∂xi }n i=1 ͸ޓ͍ʹҰ࣍ಠཱͰ͋Δ͜ͱΛࣔ͢ɻ͋Δ܎਺ {ai} Λ༻͍ͯɺ ai ∂ ∂xi = 0 Ͱ͋Δͱ͢Δ࣌ɺ͜ΕΛؔ਺ f(x) = xj ʹԋࢉ͢Δͱ͕࣍ಘΒΕΔͷͰɺ 0 = ai ∂xj ∂xi = aiδj i = aj ͢΂ͯͷ܎਺ʹ͍ͭͯ ai = 0 Ͱ͋ΓɺҰ࣍ಠཱͰ͋Δ͜ͱ͕ࣔ͞Εͨɻ ଓ͍ͯɺ(2) ʹ͓͍ͯ f = g = 1 ʢఆ਺ؔ਺ʣͱ͢Δͱɺ v(1) = v(1) + v(1) ͱͳΔͷͰɺv(1) = 0 ͕ಘΒΕΔɻैͬͯɺ(1) ͷઢܗੑΑΓɺ೚ҙͷఆ਺ؔ਺ c ʹ͍ͭͯɺv(c) = 0 ͱͳΔɻ 2
3. ࣍ʹɺ೚ҙͷؔ਺ f ∈ C∞(M) ʹ͍ͭͯɺ͋Δ఺ x0 Ͱ (1)(2) Λຬͨࣸ͢૾ v Λԋࢉͨ݁͠ՌΛݟΔɻ͸ ͡ΊʹɺTaylor ͷఆཧΑΓɺf ͸ x0 ͷۙ๣Ͱ࣍ͷΑ͏ʹల։͢Δ͜ͱ͕Ͱ͖Δɻ f(x) = f(x0 ) + ∂f ∂xi (x0 )(xi − xi 0 ) + Gij (x)(xi − xi 0 )(xj − xj 0 ) ͜Εʹ v Λԋࢉͯ͠ઢܗੑΛ༻͍Δͱɺఆ਺߲ʹର͢Δԋࢉ͸ 0 ʹͳΔ͜ͱ͔Β͕࣍ಘΒΕΔɻ v(f) = ∂f ∂xi (x0 )v(xi) + v { Gij (x)(xi − xi 0 )(xj − xj 0 ) } ࠷ޙͷ߲ΛϥΠϓχοπଇͰల։͢Δͱɺ͢΂ͯͷ߲ʹ xi − xi 0 ͱ͍͏Ҽࢠؚ͕·ΕΔ͕ɺ఺ x0 Ͱ͸ɺ͜ Ε͸ 0 ʹͳΔɻैͬͯɺ v(f) = ∂f ∂xi (x0 )v(xi) ͱͳΔ͕ɺ͜Ε͸ɺ v = v(xi) ∂ ∂xi Ͱ͋Δ͜ͱΛ͍ࣔͯ͠Δɻ೚ҙͷࣸ૾ v ͕͜ͷΑ͏ʹදݱ͞ΕΔ͜ͱ͔Βɺ { ∂ ∂xi }n i=1 ͕جఈͰ͋Δ͜ͱ͕ࣔ ͞Εͨɻ ˙ ·ͨɺҰൠʹɺϕΫτϧۭ͔ؒΒ࣮਺ R ΁ͷઢܕ൚ؔ਺ͷू߹ͱͯ͠૒ରۭ͕ؒఆٛ͞Ε·͢ɻಛʹ Tp M ͔Β࣮਺ R ΁ͷઢܕ൚ؔ਺͸ɺM ্ͷؔ਺ f ∈ C∞(M) Λ༻͍ͯɺ࣍ͷΑ͏ʹදݱ͞Ε·͢ɻ df : Tp M −→ R ɹ v −→ v(f) (3) ͜ΕΛ༨઀ϕΫτϧۭؒ (Tp M)∗ ͱఆٛ͠·͢ɻ͜ͷ࣌ɺ { dxi }n i=1 ͸ɺ { ∂ ∂xi }n i=1 ʹରͯ͠ɺ (dxi) ( ∂ ∂xj ) = δi j Λຬͨ͢૒ରجఈͱͳΔ͜ͱ͕௚઀ܭࢉͰ෼͔Γ·͢ɻ͜ͷޙ͸ɺ͍ͭ΋ͷΑ͏ʹɺr ݸͷ (Tp M)∗ ͱ s ݸͷ Tp M Λ૊Έ߹Θͤͨ௚ੵۭ͔ؒΒɺ࣮਺ R ΁ͷઢܗؔ਺ͱͯ͠ɺ(r, s) ܕςϯιϧ͕ఆٛ͞Ε·͢ɻ 2.3 ϕΫτϧ৔ લ߲Ͱ͸ɺM ্ͷ఺ p Λݻఆͯ͠ɺͦͷ఺ʹ͓͚Δ઀ϕΫτϧۭؒΛߏ੒͠·ͨ͠ɻ͞ΒʹɺM ্ͷ͢΂ ͯͷ఺ʹಉ࣌ʹ઀ϕΫτϧΛ༩͑ͨू߹ͱͯ͠ɺϕΫτϧ৔͕ఆٛ͞Ε·͢ɻ͜ͷ࣌ɺM ্ͷϕΫτϧ৔ X Λ X = vi ∂ ∂xi ͱہॴ࠲ඪදࣔͨ͠৔߹ɺvi ͸ M ্ͷ࣮਺஋ؔ਺ͱͳΓ·͢ɻ͜ΕΒ͕͢΂ͯ C∞ ڃؔ਺ ͱͳΔ৔߹ʹɺX Λ C∞ ڃϕΫτϧ৔ͱݺͼ·͢ɻM ্ͷ C∞ ڃϕΫτϧ৔શମΛ X(M) ͱද͠·͢ɻ ·ͨɺM ্ͷ࣮਺஋ؔ਺ f ∈ C∞(M) ͕༩͑ΒΕΔͱɺϕΫτϧ৔ X ͷ f ഒɺ͢ͳΘͪɺfX Λ֤఺ p ʹ͓͚Δ઀ϕΫτϧ Xp ͱؔ਺ f ͷ஋ f(p) ͷֻ͚ࢉͱͯࣗ͠વʹఆٛͰ͖·͢ɻͦͯ͠ɺ͜Εͱ͸ผʹɺϕ Ϋτϧ৔ X Λ f ʹ࡞༻ͤͯ͞ಘΒΕΔؔ਺ Xf Λ࣍ࣜͰఆٛ͢Δ͜ͱ͕Ͱ͖·͢ɻ (Xf)(p) := Xp (f) ͞Βʹɺ2 ͭͷϕΫτϧ৔ X, Y ʹରͯ͠ɺׅހੵ [X, Y ] Λ࣍ࣜͰఆٛ͢Δ͜ͱ͕Ͱ͖·͢ɻ [X, Y ]f := X(Y f) − Y (Xf) ௚઀ܭࢉ͢Δͱ෼͔ΔΑ͏ʹɺ͜Ε͸ɺ࣍ͷϥΠϓχοπଇΛຬͨ͠·͢ɻ [X, Y ](fg) = ([X, Y ]f)g + f([X, Y ]g) 3
4. ैͬͯɺఆཧ 1 ΑΓɺ͜Ε͸઀ϕΫτϧۭؒͷཁૉʢ͢ͳΘͪɺ1 ֊ͷඍ෼ԋࢉࢠʣͰ͋Γɺ[X, Y ] ∈ X(M) ͱͳΓ·͢ɻ ಉ༷ʹɺM ্ͷ͢΂ͯͷ఺ʹಉ࣌ʹ༨઀ϕΫτϧ ω Λ༩͑ͨू߹Λ M ্ͷ 1 ࣍ඍ෼ܗࣜͱݺͼ·͢ɻ ಛʹɺω = fi dxi ͱہॴ࠲ඪදࣔͨ͠ࡍʹɺ͢΂ͯͷ fi ͕ M ্ͷ C∞ ڃؔ਺ͱͳΔ৔߹ʹɺω Λ C∞ ڃ 1 ࣍ඍ෼ܗࣜͱݺͼ·͢ɻM ্ͷ C∞ ڃ 1 ࣍ඍ෼ܗࣜશମΛ D1(M) ͱද͠·͢ɻಛʹ M ্ͷؔ਺ f ∈ C∞(M) Λ༻͍ͯɺ(3) ͷԋࢉΛ௨ͯ͠ 1 ࣍ඍ෼ܗࣜΛఆٛͨ͠৔߹ɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ df = ∂f ∂xi dxi ͜ͷؔ܎͸ɺ࣍ͷܭࢉ͔Β௚઀ʹ֬ೝ͢Δ͜ͱ͕Ͱ͖·͢ɻ ( ∂f ∂xi dxi ) ( vj ∂ ∂xj ) = ∂f ∂xi vj ∂xi ∂xj = ∂f ∂xi vjδi j = ∂f ∂xi vi = df ( vi ∂ ∂xi ) 2.4 ςϯιϧ৔ ઀ϕΫτϧɺ༨઀ϕΫτϧͱಉ༷ʹɺςϯιϧʹ͍ͭͯ΋ M ্ͷ͢΂ͯͷ఺ʹಉ࣌ʹςϯιϧΛ༩͑ͨू ߹ͱͯ͠ɺςϯιϧ৔Λఆٛ͢Δ͜ͱ͕Ͱ͖·͢ɻM ্ͷ (r, s) ܕςϯιϧ৔ F Λ F = Fi1,···,ir j1,···,js ∂ ∂xi1 ⊗ · · · ⊗ ∂ ∂xir ⊗ dxj1 ⊗ · · · ⊗ dxjs ͱہॴ࠲ඪදࣔͨ͠ࡍͷ܎਺ Fi1,···,ir j1,···,js ͕ M ্ͷ C∞ ڃؔ਺ͱͳΔ࣌ɺF Λ C∞ ڃςϯιϧ৔ͱݺͼ·͢ɻ ͜͜Ͱɺ 1 ࣍ඍ෼ܗࣜ df Λ (0, 1) ܕςϯιϧ৔ F ͱΈͳͨ͠৔߹ɺ ೚ҙͷ g, h ∈ C∞(M) ͱ v, w ∈ X(M) ʹରͯ͠ɺ࣍ͷܭࢉ͕੒Γཱͪ·͢ɻ F(gv + hw) = (df)(gv + hw) = g(vf) + h(wf) = gF(v) + hF(w) ͜Ε͸ɺ ʮҾ਺ͷؔ਺ഒʹ͍ͭͯɺؔ਺͕ͦͷ··લʹग़Δʯͱ͍͏ܗͰɺଟॏઢܗੑ͕੒Γཱͭ͜ͱΛද ͠·͢ɻಉ༷ͷܭࢉʹΑΓɺҰൠͷ (r, s) ܕςϯιϧ৔ʹ͍ͭͯ΋ಉ͡ੑ࣭͕੒Γཱͭ͜ͱ͕෼͔Γ·͢ɻٯ ʹݴ͏ͱɺM ্ͷଟॏ൚ؔ਺͕ςϯιϧ৔Ͱ͋Δ͜ͱΛ֬ೝ͢Δʹ͸ɺ ʮؔ਺ഒʹ͍ͭͯɺؔ਺͕ͦͷ··લ ʹग़Δ͜ͱʯΛνΣοΫ͢Δඞཁ͕͋Δ͜ͱΛҙຯ͍ͯ͠·͢ɻ 2.5 ܭྔςϯιϧ ଟ༷ମ M ʹܭྔςϯιϧΛಋೖ͢Δ͜ͱʹΑͬͯɺ઀ϕΫτϧۭؒʹ಺ੵΛಋೖͯ͠ɺ઀ϕΫτϧͷେ͖ ͞΍ 2 ͭͷ઀ϕΫτϧ͕੒֯͢౓Λ༩͑·͢ɻܭྔςϯιϧ g ͸ɺM ্Ͱఆٛ͞Εͨ (0, 2) ܕςϯιϧ৔ ͰɺM ্ͷ֤఺ p ʹ͓͚Δ઀ϕΫτϧۭؒʹରͯ͠ਖ਼ఆ஋Ͱରশͳ૒ઢܕܗࣜΛ༩͑·͢ɻ͜Ε͸ɺςϯι ϧ৔ g Λ࣍ͷΑ͏ʹہॴ࠲ඪදࣔͨ͠ࡍʹɺ੒෼ gij ͕ਖ਼ఆ஋ରশߦྻͱͳΔ͜ͱΛҙຯ͠·͢ɻ g = gij dxi ⊗ dxj gij = g ( ∂ ∂xi , ∂ ∂xj ) Ұൠʹɺܭྔςϯιϧͷ੒෼ΛදΘ͢ߦྻ gij ΛܭྔߦྻͱݺͼɺͦͷٯߦྻΛ gij Ͱද͠·͢ɻ͜ͷ࣌ɺ ͜ΕΒ͸͓ޓ͍ʹٯߦྻͰ͋Δ͜ͱ͔Βɺ࣍ͷؔ܎͕ࣗ໌ʹ੒Γཱͪ·͢ɻ gij gjk = δk i (4) 4
5. ܭྔςϯιϧ͸ɺM ্ͷ఺ p ͷ઀ϕΫτϧۭؒʹଐ͢Δ 2 ͭͷ઀ϕΫτϧʹ͍ͭͯ಺ੵΛ༩͑Δ΋ͷͰɺ ҟͳΔ఺্ͷ઀ϕΫτϧͷ಺ੵΛऔΔ͜ͱ͸Ͱ͖·ͤΜɻҟͳΔ఺ͷ઀ϕΫτϧΛൺֱ͢Δʹ͸ɺଟ༷ମʹΞ ϑΝΠϯ઀ଓΛಋೖͯ͠ɺ ʮ઀ϕΫτϧͷฏߦҠಈʯΛఆٛ͢Δඞཁ͕͋Γ·͢ɻ 2.6 ΞϑΝΠϯ઀ଓ ΞϑΝΠϯ઀ଓΛެཧ࿦తʹఆٛ͢Δલʹɺఆٛʹର͢Δಈػ͚ͮΛ༩͑Δ௚ײతͳٞ࿦Λࣔ͠·͢ɻہॴ ࠲ඪܥΛݻఆͯ͠ɺ఺ p ʹ͓͚ΔجఈϕΫτϧ ∂ ∂xj Λ࠲ඪ࣠ xi ͷํ޲ʹඍখྔ ∆ ͚ͩฏߦҠಈͯ͠఺ q ʹ ͖࣋ͬͯͨͱߟ͑·͢ɻϢʔΫϦουۭؒͷ௚ߦ࠲ඪܥͰ͋Ε͹ɺ͜ͷฏߦҠಈͨ͠ϕΫτϧ͸ɺ఺ q ͷجఈ ϕΫτϧ ∂ ∂xj ʹҰக͠·͕͢ɺҰൠͷଟ༷ମͰ͸ɺ͔ͦ͜ΒͷʮζϨʯ͕ଘࡏ͢ΔՄೳੑ͕͋Γ·͢ɻ͜ͷζ ϨΛߟྀͯ͠ɺ఺ q ʹฏߦҠಈͨ͠جఈϕΫτϧΛ࣍ͷΑ͏ʹදݱ͠·͢ɻ ( ∂ ∂xj )′ = ∂ ∂xj + Γ k ij ∂ ∂xk ∆ + O(∆2) ैͬͯɺ֤఺ͷجఈϕΫτϧΛཁૉͱ͢ΔʮجఈϕΫτϧ৔ʯʹ͓͍ͯɺ఺ p ͷجఈϕΫτϧ ∂ ∂xj Λ࠲ඪ ࣠ xi ํ޲ʹඍখྔ͚ͩฏߦҠಈͨ͠ࡍͷʮมԽͷׂ߹ʯ͸ɺ࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ ∇ ∂ ∂xi ∂ ∂xj := lim ∆→0 {( ∂ ∂xj )′ − ∂ ∂xj } /∆ = Γ k ij ∂ ∂xk ͜͜Ͱɺ܎਺ Γ k ij ͸ɺM ্ͷ఺ p ʹΑͬͯ஋͕มԽ͢Δؔ਺Ͱ͋Γɺہॴ࠲ඪܥ xi ʹؔ͢Δ઀ଓ܎਺ͱ ݺ͹Ε·͢ɻ·ͨɺ∇ ∂ ∂xi Ͱ༩͑ΒΕΔԋࢉΛڞมඍ෼ͱݺͼ·͢ɻ جఈϕΫτϧ͚ͩͰ͸ͳ͘ɺҰൠͷ઀ϕΫτϧʹ͍ͭͯ΋ಉ༷ͷܭࢉ͕ՄೳͰ͢ɻ఺ p ʹ͓͚Δ઀ϕΫτ ϧ vj ∂ ∂xj Λಉ͡఺ p ʹ͓͚Δ઀ϕΫτϧ wi ∂ ∂xi ͷํ޲ʹฏߦҠಈͨ͠ࡍͷมԽͷׂ߹͸ɺ࣍ͷΑ͏ʹܭࢉ͞ Ε·͢ɻ ∇wi ∂ ∂xi ( vj ∂ ∂xj ) = wi ∇ ∂ ∂xi ( vj ∂ ∂xj ) = wi ( ∂vj ∂xi ∂ ∂xj + vj∇ ∂ ∂xi ∂ ∂xj ) = wi ( ∂vj ∂xi ∂ ∂xj + vjΓ k ij ∂ ∂xk ) = wi ( ∂vk ∂xi + vjΓ k ij ) ∂ ∂xk ͜ͷܭࢉͰ͸ɺڞมඍ෼ͷઢܗੑͱϥΠϓχοπଇΛఱԼΓతʹద༻͠·͕ͨ͠ɺઌͱಉ༷ʹඍখมҐΛ· ͡Ίʹܭࢉͯ͠΋ಉ݁͡Ռ͕ಘΒΕ·͢ɻ࣮͸ɺ͜ͷઢܗੑͱϥΠϓχοπଇ͕ڞมඍ෼ͷੑ࣭Λຊ࣭తʹಛ ௃͚͓ͮͯΓɺ࣍ͷΑ͏ʹެཧ࿦తʹڞมඍ෼Λఆٛ͢Δ͜ͱ͕ՄೳʹͳΓ·͢ɻ ఆٛ 1 ଟ༷ମ M ʹ͓͍ͯɺ࣍ͷࣸ૾Λߟ͑Δɻ ∇ : X(M) × X(M) −→ X(M) (X, Y ) −→ ∇X Y ೚ҙͷ X, Y, Z ∈ X(M) ͱ೚ҙͷ f ∈ C∞(M) ʹରͯ࣍͠ͷ৚͕݅੒Γཱͭ࣌ɺࣸ૾ ∇ Λ M ͷڞมඍ෼ ͱݺͿɻ 1. ∇X (Y + Z) = ∇X Y + ∇X Z 2. ∇X (fY ) = (Xf)Y + f∇X Y 3. ∇X+Y Z = ∇X Z + ∇Y Z 5
6. 4. ∇fX Y = f∇X Y ͜ͷఆٛʹج͍ͮͯɺԿΒ͔ͷڞมඍ෼͕༩͑ΒΕͨͱͯ͠ɺ∇ ∂ ∂xi ∂ ∂xj Λ࣍ͷΑ͏ʹجఈϕΫτϧͰදݱ ͨ͠ࡍͷ܎਺ Γ k ij ͕઀ଓ܎਺ͱͳΓ·͢ɻ ∇ ∂ ∂xi ∂ ∂xj = Γ k ij ∂ ∂xk ͳ͓ɺఆٛ 1 ʹ͓͚Δࣸ૾ ∇ ͸ɺ(1, 2) ܕͷςϯιϧ৔Ͱ͸ͳ͍఺ʹ஫ҙ͕ඞཁͰ͢ɻ͜Ε͕ςϯιϧ৔ ͩͱͨ͠৔߹ɺ ʮ2.4 ςϯιϧ৔ʯͰઆ໌ͨ͠Α͏ʹɺ ʮҾ਺ͷؔ਺ഒʹ͓͍ͯɺؔ਺͕ͦͷ··લʹग़Δʯͱ ͍͏ੑ࣭͕੒Γཱͭඞཁ͕͋Γ·͕͢ɺ2. ͷϥΠϓχοπଇ͸͜ΕΛຬ͍ͨͯ͠·ͤΜɻ ·ͨɺ઀ଓ܎਺ͷ஋͸ɺہॴ࠲ඪܥʹґଘ͢ΔͷͰɺ࠲ඪม׵ʹ൐͏઀ଓ܎਺ͷม׵ଇΛߟ͑Δඞཁ͕͋Γ ·͢ɻ͜Ε͸ɺ{xj} ͱ {ξb} Λ 2 छྨͷہॴ࠲ඪܥͱͯ͠ɺجఈϕΫτϧͷม׵ଇ ∂ ∂xj = ∂ξb ∂xj ∂ ∂ξb ͷ྆ลʹ ∇ ∂ ∂xi Λԋࢉ͢Δ͜ͱͰಘΒΕ·͢ɻ·ͣɺࠨล͸࣍ͷΑ͏ʹͳΓ·͢ɻ ∇ ∂ ∂xi ∂ ∂xj = Γ k ij ∂ ∂xk = Γ k ij ∂ξc ∂xk ∂ ∂ξc Ұํɺӈล͸ɺఆٛ 1 ʹࣔͨ͠ੑ࣭Λ༻͍ͯܭࢉ͢Δͱ͕࣍ಘΒΕ·͢ɻ ∇ ∂ ∂xi ( ∂ξb ∂xj ∂ ∂ξb ) = ∂2ξb ∂xi∂xj ∂ ∂ξb + ∂ξb ∂xj ∇∂ξa ∂xi ∂ ∂ξa ∂ ∂ξb = ( ∂2ξc ∂xi∂xj + ∂ξb ∂xj ∂ξa ∂xi Γ c ab ) ∂ ∂ξc ͜ΕΒ྆ลͷ੒෼Λ౳ஔͯ͠ɺ྆ลʹ ∂xk ∂ξc Λԋࢉ͢Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ Γ k ij = ∂ξa ∂xi ∂ξb ∂xj ∂xk ∂ξc Γ c ab + ∂2ξc ∂xi∂xj ∂xk ∂ξc (5) ٯʹݴ͏ͱɺ(5) Λຬͨ͢ܗͰɺͦΕͧΕͷہॴ࠲ඪܥʹ͓͚Δ઀ଓ܎਺Λ༩͑Ε͹ɺͦΕʹΑͬͯɺ1 ͭ ͷڞมඍ෼͕ఆٛ͞ΕΔ͜ͱʹͳΓ·͢ɻ͜ͷΑ͏ʹɺଟ༷ମ M ʹ઀ଓ܎਺Λ༩͑ͯڞมඍ෼Λఆٛ͢Δ͜ ͱΛΞϑΝΠϯ઀ଓΛ༩͑Δͱݴ͍·͢ɻ ࠷ޙʹɺ઀ଓ܎਺ͷ࠷ޙͷ଍ΛܭྔςϯιϧͰ্͛ԼΖ͢͠Δૢ࡞Λఆ͓͖ٛͯ͠·͢ɻ·ͣɺ࣍ࣜͰ଍Λ ԼΖ͢ૢ࡞Λఆٛ͠·͢ɻ Γij,k = Γ l ij glk (6) ͜ͷ࣌ɺ(4) Λ༻͍Δͱɺ࣍ͷૢ࡞Ͱɺ࠶ͼ଍Λ্͛Δ͜ͱ͕Ͱ͖·͢ɻ Γ k ij = Γij,l glk (7) 2.7 ܭྔ઀ଓ ڞมඍ෼Λ௚ײతʹಋೖ͢Δٞ࿦ͷதͰɺϕΫτϧ৔͕༩͑ΒΕͨࡍʹɺ఺ p ͷ઀ϕΫτϧΛඍখྔ͚ͩฏ ߦҠಈͯ͠఺ q ͷ઀ϕΫτϧͱൺֱͨ͠ࡍͷࠩ෼ͱ͍͏ߟ͑ํΛ༻͍·ͨ͠ɻ͜Ε͸ݴ͍׵͑Δͱɺڞมඍ ෼͕ 0 ʹͳΔํ޲ʹ͍ͭͯ͸ɺ֤఺ͷ઀ϕΫτϧ͕͓ޓ͍ʹฏߦʹͳ͍ͬͯΔͱղऍ͢Δ͜ͱ͕Ͱ͖·͢ɻ͜ Ε͸ɺ࣍ͷΑ͏ʹఆࣜԽ͢Δ͜ͱ͕Ͱ͖·͢ɻ 6
7. ఆٛ 2 ύϥϝʔλ t Ͱද͞Εͨɺଟ༷ମ M ্ͷͳΊΒ͔ͳۂઢ C = {p(t)} ʹ͍ͭͯɺۂઢ্ͷ֤఺ʹ͓ ͚ΔҠಈํ޲ͷ઀ϕΫτϧ ˙ p(t) := ˙ xi(t) ∂ ∂xi Λߟ͑Δɻ͜͜Ͱɺxi(t) ͸ۂઢ C Λہॴ࠲ඪͰ੒෼දࣔͨ͠΋ ͷͰɺ ˙ xi(t) ͸ύϥϝʔλ t ʹΑΔඍ෼ΛදΘ͢ɻ ͜ͷۂઢ্Ͱఆٛ͞Εͨ઀ϕΫτϧ৔ Z ͕ ∇ ˙ p(t) Z = 0 (8) Λຬͨ࣌͢ɺ઀ϕΫτϧ৔ Z ͸ C ʹͦͬͯฏߦͰ͋Δͱݴ͏ɻ (8) Λہॴ࠲ඪදࣔ͢Δͱɺ௚઀ܭࢉʹΑΓɺ͕࣍ࣜಘΒΕ·͢ɻ ˙ Zk(t) + Γ k ij (p(t)) ˙ xi(t)Zj(t) = 0 (9) ͜͜ͰɺZk(t) ͸ɺۂઢ C ʹͦͬͨ઀ϕΫτϧͷ੒෼Λ t Λύϥϝʔλͱͯ͠දࣔͨ͠΋ͷͰɺ ˙ Zk(t) ͸ɺ ύϥϝʔλ t ʹΑΔඍ෼Λද͠·͢ɻ ͦͯ͠ɺ2 ͭͷ઀ϕΫτϧΛಉ͡ۂઢʹͦͬͯฏߦҠಈͤͨ࣌͞ʹɺ͜ΕΒͷ಺ੵͷ஋͕มԽ͠ͳ͍ͱ͍͏ ৚݅Λ઀ଓʹ՝͢͜ͱ͕Ͱ͖·͢ɻ͜ͷΑ͏ͳ৚݅Λຬͨ͢઀ଓ͸ɺܭྔతͰ͋Δͱݴ͍·͢ɻ͜ͷ৚݅͸ɺ ઀ଓΛఆٛ͢Δࡍʹඞਢͱ͍͏Θ͚Ͱ͸͋Γ·ͤΜ͕ɺRiemann ଟ༷ମͰ͸ɺલఏ৚݅ͷ 1 ͭͱͯ͠ɺ઀ଓ ͕ܭྔతͰ͋Δ͜ͱ͕՝ͤΒΕ·͢ɻۂઢ p(t) ʹͦͬͯฏߦͳϕΫτϧ৔ Y ͱ Z Λߟ͑Δͱɺ͜ͷ৚݅͸ ࣍ࣜͰද͞Ε·͢ɻ d dt { gij (p(t))Y i(t)Zj(t) } = 0 ࠨลͷඍ෼Λల։ͯ͠ (9) Λ༻͍Δͱɺ͜Ε͸࣍ࣜͱಉ౳ʹͳΓ·͢ɻ ( ∂gij ∂xk − Γki,j − Γkj,i ) ˙ xkY iZj = 0 ͜Ε͕೚ҙͷ xk, Y i, Zj Ͱ੒Γཱͭ͜ͱ͔Βɺ઀ଓ͕ܭྔతͰ͋Δඞཁे෼৚݅͸ɺ࣍ࣜͰ༩͑ΒΕΔ͜ ͱʹͳΓ·͢ɻ ∂gij ∂xk = Γki,j + Γkj,i (10) ͜Ε͸ɺ࣍ͷؔ܎ࣜΛہॴ࠲ඪܥͰදࣔͨ͠΋ͷʹҰக͢Δ͜ͱ͕෼͔Γ·͢ɻ ∀X, Y, Z ∈ X(M); Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇X Z) (11) 2.8 ۂ཰ςϯιϧͱᎇ཰ςϯιϧ ϢʔΫϦουۭؒͰ͸ɺดۂઢʹͦͬͯ઀ϕΫτϧΛฏߦҠಈ͢Δͱɺग़ൃ఺ʹ໭ͬͯདྷͨϕΫτϧ͸ɺ ࠷ॳͷϕΫτϧʹҰக͠·͢ɻҰํɺҰൠͷ઀ଓΛ࣋ͬͨଟ༷ମͰ͸ɺඞͣ͠΋ͦͷΑ͏ʹ͸ͳΓ·ͤΜɻ Riemann زԿֶΛ͔ͬͨ͡ࣄͷ͋Δํ͸ɺ஍ٿͷද໘্ͰϕΫτϧΛฏߦҠಈ͢Δྫ͕͙͢ʹ಄ʹࢥ͍ු͔ Ϳ͜ͱͰ͠ΐ͏ɻ ͜ΕΛඍখͳฏߦ࢛ลܗ্ͷҠಈʹ͍ͭͯදݱͨ͠΋ͷ͕ۂ཰ςϯιϧͰ͢ɻ͋Δ఺ p ʹ͓͍ͯɺ઀ϕΫτ ϧ X ͱ Y ͷํ޲ʹ޲͔ͬͨہॴ࠲ඪܥΛ༻ҙͯ͠ɺp = (x, y), (x+|X|, y), (x, y +|Y |), (x+|X|, y +|Y |) ͷ 4 ఺Λߟ͑·͢ɻ͜͜Ͱ͸ɺ࠷ॳͷ 2 ͭͷ࠲ඪ͕ X ͱ Y ͷํ޲ʹରԠ͓ͯ͠Γɺ3 ͭΊҎ߱ͷ࠲ඪ஋͸ఆ 7
8. ਤ 1 ᎇ཰ʹΑΔζϨͷൃੜ ਺ͱߟ͍͑ͯͩ͘͞ɻ఺ p ʹ͓͚Δ઀ϕΫτϧ Z Λ͜ͷ 4 ఺Λ௨ͬͯҰपͤͨ࣌͞ͷมԽͷׂ߹ʢZ ͷมԽ ʸ 4 ఺Λ௖఺ͱ͢Δ࢛֯ܗͷ໘ੵʣ͸ɺ࣍ࣜͰܭࢉ͞Ε·͢ɻ R(X, Y, Z) = ∇X (∇Y Z) − ∇Y (∇X Z) − ∇[X,Y ] Z ͜ͷࣸ૾ R ͸ɺ֤Ҿ਺ʹ͍ͭͯͷઢܗੑΛຬͨ͢ (1, 3) ܕͷςϯιϧ৔ʹͳ͓ͬͯΓɺ͜ΕΛۂ཰ςϯι ϧ৔ͱݺͼ·͢ɻ୯࿈݁ͳྖҬͰۂ཰ςϯιϧ͕͢΂ͯ 0 ͷ৔߹ɺͦͷྖҬ಺ͷดۂઢʹͦͬͯ઀ϕΫτϧ ΛฏߦҠಈ͢Δͱɺݩͷ઀ϕΫτϧʹ໭Δ͜ͱʹͳΓ·͢*2ɻ ۂ཰ςϯιϧ৔͸ɺہॴ࠲ඪܥͰ੒෼දࣔ͢Δͱ࣍ͷΑ͏ʹͳΓ·͢ɻ R l ijk = ∂Γ l jk ∂xi − ∂Γ l ik ∂xj + Γ m jk Γ l im − Γ m ik Γ l jm ࣍ʹᎇ཰ςϯιϧ৔͸ɺ࣍ͷΑ͏ʹղऍ͢Δ͜ͱ͕Ͱ͖·͢ɻ͋Δ఺ p ʹ͓͚Δ઀ϕΫτϧ X ͱ Y Λͦ ΕͧΕ૬खͷ઀ϕΫτϧʹͦͬͯฏߦҠಈͨ͠ͱ͠·͢ɻϢʔΫϦουۭؒͰ͋Ε͹ɺҠಈͨ͠઀ϕΫτϧͷ ௖఺͸ಉ͡఺Λࢦ͢͸ͣͰ͕͢ɺҰൠͷଟ༷ମͰ͸ҟͳΓ·͢ɻฏߦҠಈʹैͬͯɺ઀ϕΫτϧͷํ޲͕ճస ͯ͠ɺ௖఺ͷؒʹ։͖͕Ͱ͖ΔՄೳੑ͕͋Γ·͢ʢਤ 1ʣ ɻ͜ͷ։͖෼ʹ૬౰͢Δ઀ϕΫτϧ͸ɺ࣍ࣜͰܭࢉ͞ Ε·͢ɻ T(X, Y ) = ∇X Y − ∇Y X − [X, Y ] ͜ͷࣸ૾ T ͸ɺ(1, 2) ܕͷςϯιϧ৔ʹͳ͓ͬͯΓɺ͜ΕΛᎇ཰ςϯιϧ৔ͱݺͼ·͢ɻہॴ࠲ඪܥͰ੒෼ දࣔ͢Δͱ࣍ͷΑ͏ʹͳΓɺ઀ଓ܎਺ͷ൓ରশ੒෼ʹҰக͢Δ͜ͱ͕෼͔Γ·͢ɻ T k ij = Γ k ij − Γ k ji (12) ਤ 2 ͸ɺᎇ཰Λ࣋ͬͨฏ໘ͷྫͰɺ্Լࠨӈͷ࠲ඪ࣠ํ޲ʹ઀ϕΫτϧΛฒߦҠಈ͢Δͱɺ઀ϕΫτϧ͕ճ స͢Δ༷ࢠ͕ࣔ͞Ε͍ͯ·͢ [3]ɻͨͩ͠ɺ͜ͷฏ໘ͷۂ཰͸ 0 ʹͳ͓ͬͯΓɺ೚ҙͷดۂઢʹͦͬͯҰप͢ Δͱɺ઀ϕΫτϧͷํ޲͸΋ͱʹ໭Γ·͢ɻ *2 άϦʔϯͷੵ෼ఆཧͰূ໌Ͱ͖ΔͰ͠ΐ͏ɻ 8
9. ਤ 2 ᎇ཰ͷ͋Δฏ໘ͷྫ 2.9 ہॴฏୱੑ ઀ଓ܎਺ͷม׵๏ଇΛར༻͢Δͱɺ͋ΔҰ఺ p ʹ͓͍ͯɺΓ k ij = 0 ͱͳΔہॴ࠲ඪܥΛߏ੒͢Δ͜ͱ͕Մ ೳͰ͢ɻ͞Βʹɺۂ཰ͱᎇ཰͕ڞʹ 0 Ͱ͋Δͱ͍͏৚݅Λ՝͢ͱɺେҬతʹ Γ k ij = 0 ͱͳΔہॴ࠲ඪܥ͕ߏ ੒Ͱ͖Δ͜ͱ͕ࣔ͞Ε·͢ɻ͜ͷΑ͏ͳہॴ࠲ඪܥΛΞϑΝΠϯ࠲ඪܥͱݺͼ·͢ɻ ҰൠʹɺΞϑΝΠϯ࠲ඪܥʹ͸ɺΞϑΝΠϯม׵ͷࣗ༝౓͕͋Γ·͢ɻͭ·ΓɺΞϑΝΠϯ࠲ඪܥΛΞϑΝ Πϯม׵ͨ͠΋ͷ͸ɺ࠶ͼΞϑΝΠϯ࠲ඪܥͰ͋Γɺ͋ΔΞϑΝΠϯ࠲ඪܥ͸ɺଞͷΞϑΝΠϯ࠲ඪܥ͔Βͷ ΞϑΝΠϯม׵ͰಘΒΕ·͢ɻͳ͓ɺΞϑΝΠϯม׵ͱ͸ɺఆ਺ߦྻʹΑΔઢܗม׵ɺ͓Αͼɺఆ਺஋ʹΑΔ ฏߦҠಈΛද͠·͢ɻ 2.10 Riemann ઀ଓ ͍Α͍Α࠷ޙʹɺRiemann ଟ༷ମͷಛ௃Ͱ͋ΔɺRiemann ઀ଓΛ༩͓͖͑ͯ·͢ɻΞϑΝΠϯ઀ଓ͕ܭྔ తͰɺ͔ͭɺᎇ཰Λ࣋ͨͳ͍࣌ɺ͜ΕΛ Riemann ઀ଓͱݺͼ·͢ɻ·ͨɺRiemann ઀ଓΛ࣋ͬͨଟ༷ମΛ Riemann ଟ༷ମͱݺͼ·͢ɻͦͯ͠ɺ࣍ʹࣔ͢Α͏ʹɺRiemann ઀ଓ͸ܭྔςϯιϧ͔ΒҰҙతʹܾఆ͞Ε ·͢ɻ ·ͣɺ઀ଓ͕ܭྔతͰ͋ΔͨΊͷ৚݅ (10) ʹ͓͍ͯɺఴࣈΛ८ճͤͨ͞΋ͷΛฒ΂·͢ɻ ∂gij ∂xk = Γki,j + Γkj,i (13) ∂gki ∂xj = Γjk,i + Γji,k (14) ∂gjk ∂xi = Γij,k + Γik,j (15) ᎇ཰͕ 0 Ͱ͋Δ͜ͱ͔Βɺ(12) ΑΓ Γij,k = Γji,k Ͱ͋Δ͜ͱʹؾΛ͚ͭͯ (15) + (14) − (13) Λܭࢉ͢Δ ͱɺ͕࣍ಘΒΕ·͢ɻ Γij,k = 1 2 ( ∂gjk ∂xi + ∂gki ∂xj − ∂gij ∂xk ) (16) 9
10. ٯʹ͜ͷࣜͰ઀ଓ܎਺Λ༩͑Δͱɺ࠲ඪม׵ͷެࣜΛຬ͓ͨͯ͠ΓɺRiemann ઀ଓͷ৚݅Λຬͨͨ͠઀ଓ ͕ఆٛ͞ΕΔ͜ͱ΋֬ೝͰ͖·͢ɻ ͜ͷ݁Ռ͸ɺRiemann ଟ༷ମʹ͓͍ͯ͸ɺܭྔΛ༩͑Δ͜ͱͰͦͷۭؒͷزԿֶతߏ଄͕ܾఆ͞ΕΔ͜ͱ Λද͍ͯ͠·͢ɻܭྔߦྻͷ੒෼͸جఈϕΫτϧͷ಺ੵΛදΘ͢΋ͷͰ͢ͷͰɺݴ͍׵͑Δͱɺద౰ͳہॴ࠲ ඪܥΛ͍Εͯɺ֤఺ʹ͓͚ΔجఈϕΫτϧͷ಺ੵΛௐ΂Δ͜ͱʹΑΓɺͦͷۭؒͷزԿֶతߏ଄͕׬શʹ೺Ѳ Ͱ͖Δ͜ͱΛҙຯ͠·͢ɻ 3 ૒ରฏୱͳଟ༷ମ લઅͷ஌ࣝΛ౿·͑ͯɺ૒ରฏୱͳଟ༷ମΛಋೖ͍͖ͯ͠·͢ɻ͜Ε͸ɺ ʮ૒ର઀ଓΛ࣋ͭฏୱͳଟ༷ମʯ ͱݴ͍׵͑Δ͜ͱ΋Ͱ͖·͢ɻ૒ର઀ଓʹ͓͍ͯ͸ɺܭྔతͰ͋Δ͜ͱ͕લఏͱ͞Εͣɺ(16) ͷؔ܎͕੒Γཱ ͨͳ͍͜ͱʹ஫ҙ͕ඞཁͰ͢ɻ ͳ͓ɺ͜ΕҎ߱ɺෳ਺ͷछྨͷ઀ଓ͕ొ৔͢ΔͷͰɺͦΕͧΕͷ઀ଓʹରԠͨ͠ڞมඍ෼ʢ∇ ΍ ∇∗ ͳͲʣ ͷه߸Λ༻͍ͯ઀ଓͷछྨΛදΘ͢΋ͷͱ͠·͢ɻ 3.1 ૒ର઀ଓ ઀ଓ͕ܭྔతͰ͋Δͱ͍͏৚݅Λ՝͞ͳ͍৔߹ʹɺҰൠతʹͲͷΑ͏ͳزԿֶ͕ߏ੒Ͱ͖Δ͔͸ڵຯਂ͍໰ ୊Ͱ͕͢ɺ͜ͷ৚݅Λͨͩ֎͚ͩ͢Ͱ͸͋·Γʹ΋৚͕݅ΏΔ͘ͳΔͨΊɺ͜͜Ͱ͸ɺ࣍ͷΑ͏ͳ৚݅Λߟ͑ ·͢ɻ ఆٛ 3 ܭྔ g Λ࣋ͭଟ༷ମ M ʹରͯ͠ɺ2 छྨͷ઀ଓ ∇, ∇∗ ͕ఆٛ͞Ε͓ͯΓɺ࣍ͷ৚݅Λຬͨ࣌͢ɺ઀ ଓͷ૊ (∇, ∇∗) Λ૒ର઀ଓͱݺͿɻ Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇∗ X Z) (17) (11) ͱൺֱ͢Δͱ෼͔ΔΑ͏ʹɺ͜Ε͸ɺ͋Δ఺ͷ 2 ͭͷ઀ϕΫτϧʹ͍ͭͯɺҰํΛ ∇ ͰฏߦҠಈ ͯ͠ɺ΋͏ҰํΛ ∇∗ ͰฏߦҠಈͨ͠ࡍʹɺͦͷ಺ੵ͕มԽ͠ͳ͍ͱ͍͏৚݅Λද͍ͯ͠·͢ɻ͜ͷ࣌ɺ ∇ = 1 2 (∇ + ∇∗) Ͱ৽͍͠઀ଓΛఆٛ͢Δͱɺ∇ ʹ͍ͭͯ͸ (11) ͕੒Γཱͪɺܭྔతͳ઀ଓʹͳΔ͜ͱ͕෼ ͔Γ·͢ɻ ·ͨɺ૒ର઀ଓͷͦΕͧΕͷᎇ཰ T, T∗ ʹ͍ͭͯɺ͞Βʹ࣍ͷ৚݅Λ՝͢͜ͱʹ͠·͢ɻ T + T∗ = 0 (18) ᎇ཰ͷ੒෼දࣔ (12) ΛݟΔͱɺ∇ ͷᎇ཰͸ T = 1 2 (T + T∗) ʹͳΔ͜ͱ͕෼͔Γ·͢ɻͭ·Γɺ(18) ͸ɺ ∇ ͷᎇ཰͕θϩͰ͋Δ͜ͱΛཁ੥͓ͯ͠Γɺ݁ہɺ∇ ͸ɺRiemann ઀ଓͰ͋Δ͜ͱʹͳΓ·͢ɻ 3.2 ૒ରฏୱͳۭؒ (17) Λຬͨ͢૒ର઀ଓͰ͸ɺͦΕͧΕͷ઀ଓʹର͢Δۂ཰ͱᎇ཰ʹ͍ͭͯɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ ఆཧ 2 ૒ର઀ଓ (∇, ∇∗) ʹ͓͍ͯɺ∇ ʹؔ͢Δۂ཰ R ͕ 0 Ͱ͋Δ͜ͱͱɺ∇∗ ʹؔ͢Δۂ཰ R∗ ͕ 0 Ͱ͋ Δ͜ͱ͸ಉ஋Ͱ͋Δɻ 10
11. [ূ໌] ɹ఺ p Ͱ઀ϕΫτϧ A ͱ B Λऔͬͯɺ͜ΕΒΛʢ୯࿈݁ྖҬ্ͷʣดۂઢʹͦͬͯɺA ͸ ∇ ʹؔ͢ ΔฏߦҠಈɺB ͸ ∇∗ ʹؔ͢ΔฏߦҠಈΛߦ͏ɻ఺ p ʹ໭͖ͬͯͨࡍͷ઀ϕΫτϧΛ A′ ͓Αͼ B′ ͱ͢Δ ͱɺ(17) ΑΓ͕࣍੒Γཱͭɻ g(A, B) = g(A′, B′) ैͬͯɺ R = 0 ͱ͢Δͱɺ A = A′ ΑΓɺ ೚ҙͷ A ʹ͍ͭͯɺ g(A, B) = g(A, B′)ɺ ͢ͳΘͪɺ g(A, B−B′) = 0 ͕੒ཱ͢Δɻܭྔͷਖ਼ఆ஋ੑ͔Βɺ͜Ε͸ɺB = B′ɺ͢ͳΘͪɺR′ = 0 Λҙຯ͢Δɻ ˙ ͜Εͱಉ༷ʹɺ(18) ͷ৚݅ͷԼͰ͸ɺ∇ ʹؔ͢Δᎇ཰ T ͕ 0 Ͱ͋Δ͜ͱͱɺ∇∗ ʹؔ͢Δᎇ཰ T∗ ͕ 0 Ͱ ͋Δ͜ͱ͕ಉ஋ʹͳΔ͜ͱ΋෼͔Γ·͢ɻͭ·Γɺ∇ ͕ʢR = 0, T = 0 ͱ͍͏ҙຯͰʣฏୱͳΒ͹ɺ∇∗ ΋ ʢR∗ = 0, T∗ = 0 ͱ͍͏ҙຯͰʣฏୱʹͳΔͷͰ͢ɻ͜ͷΑ͏ʹɺ૒ର઀ଓΛ͓࣋ͬͯΓɺ͞Βʹɺ྆ํͷ઀ ଓʹ͍ͭͯฏୱͳۭؒΛʮ૒ରฏୱͳۭؒʯͱݺͿ͜ͱʹ͠·͢ɻ ૒ରฏୱͳۭؒͰ͸ɺͦΕͧΕͷ઀ଓʹ͍ͭͯΞϑΝΠϯ࠲ඪܥɺ͢ͳΘͪɺେҬతʹ Γ k ij = 0 ͱͳΔہ ॴ࠲ඪܥ͕ଘࡏ͠·͢ɻͨͩ͠ɺҰํͷ઀ଓʹର͢ΔΞϑΝΠϯ࠲ඪܥ͸ɺଞํͷ઀ଓʹର͢ΔΞϑΝΠϯ࠲ ඪܥʹ͸ͳΓ·ͤΜɻ߲࣍Ͱ͸ɺ૒ରฏୱͳۭؒͷߏ੒ํ๏Λઆ໌͠ͳ͕Βɺ2 ͭͷΞϑΝΠϯ࠲ඪܥͷؒʹ ͋Δؔ܎Λͻ΋ͱ͍͍͖ͯ·͢ɻ 3.3 ૒ରฏୱͳۭؒͷߏ੒ Ұൠͷ૒ରฏୱͳۭؒΛௐ΂Δલʹɺ૒ରฏୱͳۭؒΛ۩ମతʹߏ੒͢Δख๏Λ঺հ͠·͢ɻ ͸͡Ίʹɺ಺ੵ΋઀ଓ΋࣋ͨͳ͍ଟ༷ମ M Λ༻ҙͯ͠ɺ͜ͷ্ʹద౰ͳ࠲ඪܥ θ = {θi}n i=1 ͱɺ͜ͷ࠲ඪ ܥʹ͓͚Δඍ෼Մೳͳತؔ਺ ψ(θ) Λ༻ҙ͠·͢ɻψ(θ) ͕ತؔ਺Ͱ͋Δ͜ͱ͔Βɺϔοηߦྻ͸ਖ਼ఆ஋ʹͳ ΔͷͰɺ͜ΕΛ͜ͷۭؒͷܭྔͱͯ͠ಋೖ͠·͢ɻ gij (θ) := ∂2 ∂θi ∂θj ψ(θ) (19) ͞Βʹɺ͜ͷ࠲ඪܥ͕ΞϑΝΠϯ࠲ඪܥʹͳΔΑ͏ʹɺ઀ଓ ∇ Λಋೖ͠·͢ɻͭ·Γɺ͜ͷ࠲ඪܥͰ͸ɺ઀ ଓ܎਺͸͢΂ͯ 0 ʹͳΔ΋ͷͱఆٛ͠·͢ɻ ଓ͍ͯɺ૒ର࠲ඪͷํΛ༻ҙ͠·͢ɻψ(θ) ͕ತؔ਺Ͱ͋Δ͜ͱ͔Βɺ࣍ͷϧδϟϯυϧม׵ʹΑͬͯɺ૒ ର࠲ඪ η ͱ૒ରತؔ਺ φ(η) Λఆٛ͢Δ͜ͱ͕Ͱ͖·͢ɻ ηi := ∂i ψ(θ) (20) φ(η) := max θ′ { θ′iηi − ψ(θ′) } (21) ҎԼͷٞ࿦Ͱ͸ɺθ ͱ η ͸ (20) ͷؔ܎Ͱޓ͍ͷؔ਺ʹͳ͍ͬͯΔͱཧղ͠·͢ɻϧδϟϯυϧม׵ͷੑ࣭ ͱͯ͠ɺ࣍ͷ૒ରؔ܎͕੒Γཱͭ͜ͱ΋ূ໌͞Ε·͢ɻ θi = ∂iφ(η) ψ(θ) = max η′ { θiη′ i − φ(η′) } ψ(θ) + φ(η) = θiηi ͜͜Ͱɺ∂i ͱ ∂i ͸ɺͦΕͧΕ θi ͓Αͼ ηi ʹؔ͢Δภඍ෼Λද͠·͢ɻ͜ΕΒͷূ໌͸ [4] Λࢀরͯ͘͠ ͍ͩ͞ɻ 11
12. ͦͯ͠ɺ૒ର࠲ඪ η = {ηi }n i=1 ͕ΞϑΝΠϯ࠲ඪܥʹͳΔΑ͏ʹɺ઀ଓ ∇∗ Λಋೖ͠·͢ɻ͜ͷ࣌ɺ઀ଓͷ ૊ (∇, ∇∗) ͸ɺ(17) Λຬͨ͢૒ର઀ଓʹͳΔ͜ͱ͕ূ໌͞Ε·͢ɻ [ূ໌] ɹ (17) ͸ɺہॴ࠲ඪܥͰදࣔ͢Δͱ࣍ࣜʹͳΔɻ ∂k gij = Γkj,i + Γ∗ kj,i θ ࠲ඪܥͰ͸ɺ઀ଓ܎਺ Γ ͸͢΂ͯ 0 ʹͳΔͷͰɺ͜Ε͸࣍ࣜͱಉ஋Ͱ͋Γɺ͜ΕΛূ໌͢Ε͹Α͍ɻ ∂k gij = Γ∗ kj,i (22) Ұํɺη ࠲ඪܥͰ͸ Γ∗ ͷ੒෼͸ 0 ʹͳΔ͜ͱ͔Βɺ (5) Λ༻͍ͯɺΓ∗ ͷ੒෼Λ η ࠲ඪܥ͔Β θ ࠲ඪܥʹ ม׵͢ΔެࣜΛॻ͖Լ͢ͱ࣍ʹͳΔɻ Γ∗ k ij = ∂2ηl ∂θi∂θj ∂θk ∂ηl ͜͜Ͱ gkm = ∂k ∂m ψ(θ) = ∂ηm ∂θk Λ྆ลʹ͔͚Δͱ ɹʢࠨลʣ= Γ∗ k ij gkm = Γ∗ ij,m ɹʢӈลʣ= ∂2ηl ∂θi∂θj ∂θk ∂ηl ∂ηm ∂θk = ∂2ηl ∂θi∂θj δl m = ∂2ηm ∂θi∂θj = ∂i ∂j ∂m ψ(θ) = ∂i gmj ͜ΕͰ (22) ͕ࣔ͞Εͨɻ ˙ ͜ͷূ໌ͷதͰ࢖ͬͨؔ܎ࣜ gij = ∂ηi ∂θj ͸ɺgij ͕࠲ඪม׵ η → θ ͷϠίϏߦྻʹͳ͍ͬͯΔ͜ͱΛࣔͯ͠ ͓Γɺͦͷٯߦྻ͸ٯม׵ͷϠίϏߦྻ gij = ∂θi ∂ηj Ͱ༩͑ΒΕΔ͜ͱʹͳΓ·͢ɻ͜Ε͸ͪΐ͏Ͳɺ࠲ඪม׵ ͷެࣜͱ gij , gij ʹΑΔ଍ͷ্͛Լ͕͛Ұக͢Δ͜ͱΛ͍ࣔͯ͠·͢ɻ ηi = gij θj, θi = gijηj (23) ·ͨɺ∂i ͸ θi ʹΑΔภඍ෼ɺ∂i ͸ ηi ʹΑΔภඍ෼ͱఆٛ͠·͕ͨ͠ɺ͜ΕΒͷภඍ෼ԋࢉࢠʹ͍ͭͯ΋ ଍ͷ্͛Լ͕͛ՄೳͱͳΓ·͢ɻ ∂i = gij∂j , ∂i = gij ∂j (24) ͢Δͱ࣍ͷܭࢉͰࣔ͞ΕΔΑ͏ʹɺͦΕͧΕͷ࠲ඪܥʹ͓͚ΔجఈϕΫτϧ {∂i }n i=1 ͱ {∂i}n i=1 ͸ޓ͍ʹ ௚ߦ͢Δ࠲ඪܥʹͳ͍ͬͯΔ͜ͱ΋෼͔Γ·͢ɻ g(∂i , ∂j) = g(∂i , gjk∂k ) = gjkgik = δj i (25) ͔͜͜Βɺ∂i ͱ ∂j ͷ಺ੵ͸࠲ඪʹґଘ͠ͳ͍ఆ਺ͱ͍͏͜ͱʹͳΓ·͕͢ɺ͜Ε͸ɺ૒ର઀ଓͷҙຯΛߟ ͑Δͱ͔֬ʹ੒Γཱͭ΂͖ࣄ࣮Ͱ͢ɻ࠲ඪܥ θ ͸઀ଓ ∇ ʹର͢ΔΞϑΝΠϯ࠲ඪܥͰ͢ͷͰɺఆ਺੒෼ͷ઀ ϕΫτϧ৔ ai∂i ͸઀ଓ ∇ ʹؔͯ͠ฒߦͳϕΫτϧͷू·ΓʹͳΓ·͢ɻಉ༷ʹɺఆ਺੒෼ͷ઀ϕΫτϧ৔ bj ∂j ͸઀ଓ ∇∗ ʹؔͯ͠ฒߦͰ͢ɻैͬͯɺ૒ର઀ଓͷ৚݅ΑΓɺ͜ΕΒͷ֤఺Ͱͷ಺ੵ͸͢΂ͯಉ͡஋ʹ ͳΔ͸ͣͰɺͦΕ͕ g(ai∂i , bj ∂j) = aibi Ͱ༩͑ΒΕΔ͜ͱʹͳΓ·͢ɻ ݟํΛม͑Δͱɺ ͜ͷߏ੒ํ๏ͷϙΠϯτ͸ɺ (19) ͱ (20) ʹू໿͞ΕΔͱݴ͑·͢ɻ͜ΕΒ͔Βɺ gij = ∂ηi ∂θj ɺ ͢ͳΘͪɺ ʮܭྔ = ϠίϏߦྻʯͱ͍͏ؔ܎͕ੜ·Εͯɺͦͷ݁Ռɺ(25) ͷΑ͏ʹͦΕͧΕͷฏߦҠಈͰਖ਼ن ௚ߦੑʢ͢ͳΘͪܭྔʣ͕อଘ͞ΕΔΑ͏ͳجఈϕΫτϧ͕࡞ΒΕͨͱ͍͏Θ͚Ͱ͢ɻ 12
13. Ҏ্͔Βɺ೚ҙͷ࠲ඪܥ θ ͱ೚ҙͷತؔ਺ ψ(θ) ͕͋Ε͹ɺ͔ͦ͜Βϧδϟϯυϧม׵Λ௨ͯ͠ɺࣗવͳܗ Ͱɺ૒ରฏୱͳۭؒͱͦΕͧΕͷ઀ଓʹର͢ΔΞϑΝΠϯ࠲ඪܥ͕ߏ੒Ͱ͖Δ͜ͱ͕෼͔Γ·ͨ͠ɻߥͬΆ͘ ݴ͑͹ɺತؔ਺ͷ਺͚ͩ૒ରฏୱͳۭ͕ؒ͋Δͱ͍͏Θ͚Ͱ͢ɻ ͦΕͰ͸ٯʹɺ೚ҙͷ૒ରฏୱͳۭ͕ؒ͋ͬͨ࣌ʹɺͦΕΛಋ͘Α͏ͳತؔ਺Λݟ͚ͭΔ͜ͱ͸Ͱ͖ΔͷͰ ͠ΐ͏͔ʁɹ͜Ε͕ՄೳͰ͋Δ͜ͱΛ߲࣍Ͱ͍͖ࣔͯ͠·͢ɻ 3.4 ૒ରฏୱͳۭؒʹରԠ͢Δತؔ਺͕ଘࡏ͢Δ͜ͱͷূ໌ (19)(20) ͔Β (25) ʹࢸΔಓےΛٯʹͨͲΔ͜ͱͰ͜ΕΛূ໌͠·͢ɻ͜͜Ͱ͸ɺ೚ҙͷ૒ରฏୱͳଟ༷ମ M ͕͋Δͱͯ͠ɺͦͷܭྔͱ૒ର઀ଓΛ·ͱΊͯ (g, ∇, ∇∗) ͱදه͠·͢ɻ ิ୊ 1 ૒ର઀ଓ (∇, ∇∗) Λ࣋ͭ૒ରฏୱͳଟ༷ମ M ʹରͯ͠ɺ࣍Λຬͨ͢Α͏ͳɺ઀ଓ ∇ ʹؔ͢ΔΞ ϑΝΠϯ࠲ඪܥ θ ͱ઀ଓ ∇∗ ʹؔ͢ΔΞϑΝΠϯ࠲ඪܥ η ΛऔΔ͜ͱ͕Ͱ͖Δɻ g(∂i , ∂j) = δj i (26) ͜͜Ͱɺ ∂i := ∂ ∂θi , ∂i := ∂ ∂ηi ͱ͢Δɻ [ূ໌] ɹ M ͕૒ରฏୱͰ͋Δ͜ͱ͔Βɺ∇ ʹؔ͢ΔΞϑΝΠϯ࠲ඪܥ x ͱ ∇∗ ʹؔ͢ΔΞϑΝΠϯ࠲ඪܥ y ͕ଘࡏ͢ΔɻM ্ͷ఺ p Λݻఆͯ͠ɺ఺ p ʹ͓͚Δ಺ੵ஋Λ༻͍ͯɺఆ਺ߦྻ G Λ࣍Ͱఆٛ͢Δɻ Gij = g (( ∂ ∂xi ) p , ( ∂ ∂yj ) p ) ͜͜Ͱ৽͍͠࠲ඪܥ θ = {θi}n i=1 ͱ η = {ηi }n i=1 Λ࣍Ͱఆٛ͢Δɻ θi = xi ηi = Gij yj ΞϑΝΠϯ࠲ඪܥͷఆ਺ߦྻ G ʹΑΔҰ࣍ม׵͸࠶ͼΞϑΝΠϯ࠲ඪܥʹͳΔͷͰɺη ͸ ∇∗ ʹؔ͢ΔΞ ϑΝΠϯ࠲ඪܥͰ͋Δɻ͜ΕΒ͕ (26) ͷؔ܎Λຬͨ͢͜ͱΛࣔ͢ɻ ·ͣɺ఺ p ʹ͓͍ͯߟ͑Δͱɺ͕࣍ࣗ໌ʹ੒Γཱͭɻ g (( ∂ ∂θi ) p , ( ∂ ∂ηj ) p ) = g (( ∂ ∂xi ) p , ( G−1 jk ∂ ∂yk ) p ) = G−1 jk Gik = δj i Ұํɺ ∂ ∂θi ͱ ∂ ∂ηj Λఆ਺܎਺ 1 Λ࣋ͬͨఆ਺੒෼ͷ઀ϕΫτϧ৔ͱߟ͑Δͱɺθ ͱ η ͕ΞϑΝΠϯ࠲ඪܥ Ͱ͋Δ͜ͱ͔Βɺ͜ΕΒ͸ɺͦΕͧΕɺ∇ ͱ ∇∗ ʹؔͯ͠ฏߦͳ઀ϕΫτϧ৔ͱݴ͑Δɻैͬͯɺ૒ର઀ଓͷ ఆٛΑΓ͢΂ͯͷ఺Ͱ಺ੵ͕ಉ͡ʹͳΔ͜ͱ͔Βɺ͢΂ͯͷ఺Ͱ (26) ͕੒Γཱͭ͜ͱ͕ݴ͑Δɻ ˙ ิ୊ 2 ิ୊ (1) Λຬͨ͢࠲ඪܥ θ, η ʹ͍ͭͯɺ͕࣍੒Γཱͭɻ gij = g(∂i , ∂j ) = ∂ηi ∂θj (27) gij = g(∂i, ∂j) = ∂θi ∂ηj (28) 13
14. [ূ໌] ɹϠίϏߦྻΛ༻͍ͨม਺ม׵ͷެࣜΑΓɺ ∂i = ∂ηk ∂θi ∂k ͱͳΔ͜ͱʹ஫ҙ͢Δͱɺ(26) Λ༻͍͕ͯ࣍੒Γཱͭɻ g(∂i , ∂j ) = g ( ∂ηk ∂θi ∂k, ∂j ) = ∂ηk ∂θi δk j = ∂ηj ∂θi g ͷରশੑΑΓɺ͜Ε͸ (27) ʹ౳͍͠ɻ(28) ΋ಉ༷ͷܭࢉʹͳΔɻ ˙ ͜ΕͰɺܭྔߦྻ gij ͕ม਺ม׵ θ → η ͷϠίϏߦྻʹͳΔ͜ͱ͕ূ໌͞ΕͨͷͰɺܭྔߦྻʹΑΔఴࣈ ͷ্͛Լ͛ (23)(24) ͕ՄೳʹͳΓ·͢ɻ ิ୊ 3 ิ୊ 1 Λຬͨ͢࠲ඪܥ θ, η ʹ͍ͭͯɺ2 ͭͷತؔ਺ ψ(θ), φ(η) ͕ଘࡏͯ͠ɺ͕࣍੒ཱ͢Δɻ ηi = ∂i ψ(θ) (29) θi = ∂iφ(η) (30) ψ(θ) + φ(η) − θiηi = 0 (31) [ূ໌] ɹ (27) ʹ͓͍ͯ g ͷରশੑΑΓɺ∂j ηi = ∂i ηj ͱͳΔɻ͕ͨͬͯ͠ɺϙςϯγϟϧؔ਺ͷଘࡏఆཧΑ Γɺηi = ∂i ψ(θ) ͱͳΔؔ਺ ψ(θ) ͕ଘࡏ͢Δɻθi = ∂iφ(η) ʹ͍ͭͯ΋ಉ༷ɻͨͩ͠ɺψ(θ) ͱ φ(η) ʹ͸ɺ ೚ҙͷఆ਺ΛՃ͑Δࣗ༝౓͕ଘࡏ͢Δɻ ͜ͷ࣌ɺ(27)(28) ΑΓɺψ(θ) ͱ φ(η) ͷϔοηߦྻ͸ɺܭྔͷߦྻʢ͓ΑͼͦͷٯߦྻʣʹҰக͢Δɻ ∂i ∂j ψ(θ) = ∂i ηj = gij (32) ∂i∂jφ(θ) = ∂iθj = gij (33) ैͬͯɺϔοηߦྻ͸ਖ਼ఆ஋ߦྻͰ͋Γɺψ(θ) ͱ φ(η) ͸ತؔ਺ʹͳ͍ͬͯΔɻ ͞Βʹɺ࣍ͷܭࢉ͔Βɺؔ਺ ψ(θ) + φ(η) − θiηi ͷશඍ෼͸ 0 ʹͳΔ͜ͱ͕Θ͔Δɻ d ( ψ(θ) + φ(η) − θiηi ) = (∂i ψ)dθi + (∂iφ)dηi − ηi dθi − θidηi = 0 ैͬͯɺ͜ͷؔ਺͸ఆ਺ؔ਺Ͱ͋Γɺψ(θ) ʹ೚ҙͷఆ਺ΛՃ͑Δࣗ༝౓Λ༻͍ͯɺψ(θ) + φ(η) − θiηi = 0 ʹͰ͖Δɻ ˙ ͜ΕΒͷิ୊ʹΑΓɺ೚ҙͷ૒ରฏୱͳۭؒ M ʹରͯ͠ɺತؔ਺ ψ(θ) ʹΑΔϧδϟϯυϧม׵Ͱ݁ͼͭ ͍ͨɺ૒ରΞϑΝΠϯ࠲ඪܥ͕ߏ੒Ͱ͖Δ͜ͱ͕෼͔Γ·ͨ͠ɻ Ҏ্ͷٞ࿦ΛৼΓฦΔͱɺ ʮ૒ରతͳҙຯͰܭྔ͕อଘ͞ΕΔʯ͜ͱͱɺ ʮ૒ରతͳҙຯͰͷΞϑΝΠϯ࠲ඪ ܥʢ૒ରΞϑΝΠϯ࠲ඪܥʣ͕ଘࡏ͢Δʯ͜ͱ͔Βɺ ʮܭྔʹϠίϏߦྻʯͱ͍͏ܭྔʹର͢Δڧ͍റΓ͕ಘΒ Εͨ͜ͱ͕෼͔Γ·͢ɻ͜Ε͸ɺRiemann ଟ༷ମʹ͓͍ͯʮܭྔͷอଘʯͱʮΞϑΝΠϯ࠲ඪܥͷଘࡏʯͱ͍ ͏৚͔݅Βʮܭྔ ∼ ୯Ґߦྻʯ ʢͭ·ΓϢʔΫϦουۭؒʣͱ͍͏റΓ͕ಘΒΕͨ͜ͱʹରԠ͢Δͱߟ͑ΒΕ ·͢ɻ Riemann ଟ༷ମͷ৔߹ɺϢʔΫϦουۭؒʹͳͬͯ͠·͑͹ͦΕҎ্ͷٞ࿦ͷ޿͕Γ͸͋Γ·ͤΜ͕ɺ૒ ରฏୱͳۭؒͷ৔߹͸ɺ ʮܭྔʹϠίϏߦྻʯʹ͓͍ͯɺܭྔͷରশੑ͔ΒϠίϏߦྻͷରশੑͱ͍͏Մੵ෼ ৚͕݅ੜ·Εͯɺ͔ͦ͜Βϙςϯγϟϧؔ਺ ψ(θ) ͷଘࡏͱϧδϟϯυϧม׵ʹΑΔ࠲ඪܥͷܨ͕Γ͕ੜΈग़ ͞Εͨ͜ͱʹͳΓ·͢ɻ 14
15. ਤ 3 Bregman μΠόʔδΣϯε 3.5 Bregman μΠόʔδΣϯε ຊઅͷ࠷ޙʹɺ૒ରฏୱͳଟ༷ମͷܭྔతͳੑ࣭Λಛ௃͚ͮΔɺBregman μΠόʔδΣϯεΛಋೖ͓ͯ͠ ͖·͢ɻ͜Ε͸ɺϢʔΫϦουۭؒʹ͓͚Δʮڑ཭ͷೋ৐ʯΛ֦ுͨ֓͠೦Ͱ͋Γɺ૒ରฏୱͳۭؒʹ͓͚Δ ʮ֦ுϐλΰϥεͷఆཧʯ͕ಋ͔Ε·͢ɻ લ߲ͷٞ࿦ʹΑΓɺ૒ରฏୱͳଟ༷ମ M ʹ͸ɺରԠ͢Δತؔ਺ ψ(θ) ͕ଘࡏ͢Δ͜ͱ͕෼͔Γ·ͨ͠ɻM ্ͷ 2 ఺ p ͱ q ʹରͯ͠ɺBregman μΠόʔδΣϯεΛ࣍ࣜͰఆٛ͠·͢ɻ D(p || q) := {ψ(θ) − ψ(θ′)} − ∂i ψ(θ′)(θ − θ′)i (34) ͜͜Ͱɺθ ͱ θ′ ͸ɺͦΕͧΕɺp ͱ q ʹରԠ͢Δ࠲ඪ θ ͷ஋ͱ͠·͢ɻ͜Ε͸ɺਤ 3 ͷΑ͏ʹɺ఺ q ͔Β ఺ p ʹҠಈͨ͠ࡍͷ ψ(θ) ͷ૿Ճ෼ͱɺ఺ q ʹ͓͚Δ઀ઢͷ૿Ճ෼ͷࠩΛද͍ͯ͠·͢ɻψ(θ) ͕ತؔ਺Ͱ͋ Δ͜ͱ͔Βɺ͜Ε͸ɺ͔ͳΒͣਖ਼ͷ஋ʹͳΓɺp = q ͷ͚࣌ͩ 0 ʹͳΓ·͢ɻ ͞Βʹɺ(29) ͱ (31) Λ༻͍Δͱ࣍ͷΑ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ D(p || q) = ψ(θ) + φ(η′) − θiη′ i (35) ͜͜Ͱɺη′ ͸ɺ఺ q ʹରԠ͢Δ η ࠲ඪͷ஋Ͱ͢ɻ(35) ͷදࣜ͸ɺ૒ର࠲ඪܥΛΞϑΝΠϯม׵ͯ͠΋มԽ ͠ͳ͍͜ͱ͕ܭࢉͰ֬ೝͰ͖·͢ɻ θ ͱ ηɺ͓Αͼɺψ(θ) ͱ φ(η) ͷ໾ׂΛೖΕସ͑ͯɺ(34) Ҏ߱ͷٞ࿦Λ܁Γฦ͢͜ͱʹΑΓɺ࣍ͷ૒ରμΠ όʔδΣϯεΛఆٛ͢Δ͜ͱ΋ՄೳͰ͢ɻ D∗(p || q) = φ(η) + ψ(θ′) − θ′iηi (36) ͜͜Ͱɺη ͸఺ p ʹରԠ͢Δ η ࠲ඪͷ஋Ͱɺθ′ ͸఺ q ʹରԠ͢Δ θ ࠲ඪͷ஋ʹͳΓ·͢ɻ͜ͷ࣌ɺ࣍ͷؔ ܎͕੒Γཱͭ͜ͱ͕͙͢ʹ෼͔Γ·͢ɻ D(p || q) = D∗(q || p) (37) 15
16. ਤ 4 ֦ுϐλΰϥεͷఆཧ Bregman μΠόʔδΣϯε͸ɺp ͱ q ʹ͍ͭͯର৅Ͱ͸ͳ͍఺ʹ஫ҙ͍ͯͩ͘͠͞ɻ ࣍ʹɺη′ = η + dη ͷ৔߹Λߟ͑ͯ (35) Λ dη ͷ 2 ࣍ͷ߲·Ͱల։ͯ͠Έ·͢ɻ(30)(31) ΑΓ 1 ࣍ͷ߲͕ 0 ʹͳΓɺ2 ࣍ͷ߲ʹ͍ͭͯ͸ɺ(33) ͔Β࣍ͷؔ܎͕ࣜಘΒΕ·͢ɻ D(p || q) = ∂i∂jφ(η)dηi dηj = gijdηi dηj ͭ·Γɺඍখมҟʹର͢Δ Bregman μΠόʔδΣϯε͸ɺܭྔ g ʹΑΔඍখڑ཭ʹҰக͍ͯ͠·͢ɻ ࠷ޙʹɺBregman μΠόʔδΣϯεʹؔ͢Δɺ֦ுϐλΰϥεͷఆཧΛࣔ͠·͢ɻ·ͣɺਤ 4 ʹ͓͍ͯɺ ఺ p ͱ఺ q ͸ɺύϥϝʔλ t Λ࣋ͭ࣍ͷۂઢͰ݁͹Ε͍ͯΔͱ͠·͢ɻ θi = θi(p) + t∆i (0 ≤ t ≤ 1) ͜͜ʹɺ∆i ͸ɺ࠲ඪʹґଘ͠ͳ͍ఆ਺ͱ͠·͢ɻ͜Ε͸ɺ͜ͷۂઢͷ઀ϕΫτϧ͸ɺθ ࠲ඪܥʹ͓͍ͯ܎਺ ͕ҰఆͷϕΫτϧ ∆i∂i Ͱ͋Γɺθ ࠲ඪܥʹ͓͚Δଌ஍ઢʢ∇ ଌ஍ઢʣͰ͋Δ͜ͱΛҙຯ͠·͢ɻಛʹ t = 1 ͷ৔߹Λߟ͑Δͱɺ͕࣍੒Γཱͪ·͢ɻ θi(q) = θi(p) + ∆i ಉ༷ʹɺ఺ q ͱ఺ r ͸ɺύϥϝʔλ t Λ࣋ͭ࣍ͷۂઢͰ݁͹Ε͍ͯΔͱ͠·͢ɻ ηi = ηi (q) + t∆∗ i (0 ≤ t ≤ 1) ͜Ε͸ɺ͜ͷۂઢ͕ η ࠲ඪܥʹ͓͚Δଌ஍ઢʢ∇∗ ଌ஍ઢʣͰ͋Δ͜ͱΛҙຯ͓ͯ͠Γɺt = 1 ͷ৔߹Λߟ ͑Δͱɺ͕࣍੒Γཱͪ·͢ɻ ηi (r) = ηi (q) + ∆∗ i ͜͜Ͱɺ(35) Λ༻͍ͯܭࢉ͢Δͱɺ࣍ͷؔ܎͕੒Γཱͭ͜ͱ͕෼͔Γ·͢ɻ D(p || q) + D(q || r) − D(p || r) = − { θi(p) − θi(q) } {ηi (q) − ηi (r)} = −∆i∆∗ i ैͬͯɺ2 ͭͷଌ஍ઢ͕௚ߦ͓ͯ͠Γɺ∆i∆∗ i = 0 ͱͳΔ৔߹ɺ࣍ͷؔ܎͕੒Γཱͭ͜ͱ͕෼͔Γ·͢ɻ D(p || q) + D(q || r) = D(p || r) 16
17. ͜Ε͕ɺ֦ுϐλΰϥεͷఆཧʹଞͳΓ·ͤΜɻ఺ r ͕ ∇∗ ଌ஍ઢ্Λಈ͘ࡍʹɺD(p || r) ͕࠷খʹͳΔ ͷ͸ɺr = q ͷ৔߹Ͱ͋Γɺ͜Ε͸ɺ఺ p ͔Β ∇∗ ଌ஍ઢʹରͯ͠ԼΖͨ͠ਨઢʢ∇∗ ଌ஍ઢʹ௚ߦ͢Δ ∇ ଌ஍ઢʣͱͷަ఺Ͱ͋Δ͜ͱΛҙຯ͠·͢ɻ ૒ରμΠόʔδΣϯεʹ͍ͭͯ΋ಉٞ͡࿦Λల։͢Δ͜ͱ͕ՄೳͰɺ࣍ͷ֦ுϐλΰϥεͷఆཧ͕ಘΒΕ ·͢ɻ D∗(p || q) + D∗(q || r) = D∗(p || r) ͨͩ͠ɺμΠόʔδΣϯεͱ૒ରμΠόʔδΣϯεͰ͸ɺθ ͱ ηɺ͢ͳΘͪɺ∇ ͱ ∇∗ ͷ໾ׂ͕ೖΕସΘͬ ͍ͯ·͢ɻͪ͜Βͷ৔߹ɺp ͔Β q ͸ ∇∗ ଌ஍ઢͰ݁͹Ε͓ͯΓɺq ͔Β r ͸ɺ∇ ଌ஍ઢͰ݁͹Ε͍ͯΔඞ ཁ͕͋Γ·͢ɻ Ҏ্ͷؔ܎ΛվΊͯɺఆཧͱͯ͠·ͱΊ͓͖ͯ·͢ɻ ఆཧ 3 ૒ରฏୱͳଟ༷ମ M ্ͷ 3 ఺ p, q, r ʹ͍ͭͯɺp ͱ q Λ݁Ϳ ∇ ଌ஍ઢɺ͓Αͼɺq ͱ r Λ݁Ϳ ∇∗ ଌ஍ઢ͕ޓ͍ʹ௚ߦ͍ͯ͠Δ৔߹ɺ࣍ͷؔ܎͕੒Γཱͭɻ D(p || q) + D(q || r) = D(p || r) ఆཧ 4 ૒ରฏୱͳଟ༷ମ M ্ͷ 3 ఺ p, q, r ʹ͍ͭͯɺp ͱ q Λ݁Ϳ ∇∗ ଌ஍ઢɺ͓Αͼɺq ͱ r Λ݁Ϳ ∇ ଌ஍ઢ͕ޓ͍ʹ௚ߦ͍ͯ͠Δ৔߹ɺ࣍ͷؔ܎͕੒Γཱͭɻ D∗(p || q) + D∗(q || r) = D∗(p || r) 4 ࢦ਺ܕ෼෍଒ͷ૒ରฏୱߏ଄ ͜͜·Ͱɺ७ਮʹزԿֶతͳࢹ఺Ͱɺ૒ରฏୱͳଟ༷ମΛߏ੒͖ͯ͠·ͨ͠ɻ͜͜ͰɺύϥϝʔλΛ࣋ͬͨ ֬཰෼෍ͷ଒Λଟ༷ମͱΈͳ͢͜ͱͰɺزԿֶͱ౷ܭֶͷܨ͕ΓΛੜΈग़͠·͢ɻಛʹɺࢦ਺ܕ෼෍଒ͱݺ͹ ΕΔయܕతͳ֬཰෼෍଒Ͱ͸ɺ֬཰ີ౓ͷن֨Խఆ਺͔ΒಘΒΕΔತؔ਺ ψ(θ) Λ༻͍ͯ૒ରฏୱͳߏ଄Λಋ ೖ͢Δ͜ͱ͕ՄೳͰɺBregman μΠόʔδΣϯε͕ KLʢΧϧόοΫɾϥΠϒϥʔʣμΠόʔδΣϯεʹҰக ͢Δ͜ͱ͕ࣔ͞Ε·͢ɻ 4.1 ࢦ਺ܕ෼෍଒ X Λ֬཰ม਺ͱͯ͠ɺ ͦͷ֬཰ີ౓͕ɺ ύϥϝʔλ θ = {θi }n i=1 Λ༻͍ͯɺ ࣍ࣜͰ༩͑ΒΕΔ΋ͷͱ͠·͢ɻ p(X | θ) = 1 Z(θ) exp { θiki (X) + r(X) } (38) ͜ͷΑ͏ͳ֬཰෼෍ͷ଒Λࢦ਺ܕ෼෍଒ͱݺͼ·͢ɻਖ਼ن෼෍΍཭ࢄ෼෍ͳͲɺଟ͘ͷ֬཰෼෍଒͕ࢦ਺ܕ ෼෍଒Ͱ͋Δ͜ͱ͕஌ΒΕ͍ͯ·͢ɻ ͜͜Ͱɺxi = ki (X) Ͱ৽ͨͳ֬཰ม਺Λఆٛͯ͠ɺdµ(x) = exp {r(X)} dX Ͱɺx = {xi }n i=1 ͷଌ౓Λఆ ٛ͢Δͱɺ(38) ͸࣍ͷΑ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ p(X | θ)d(X) = 1 Z(θ) exp(θixi )dµ(x) 17
18. ͜ͷ࣌ɺZ(θ) ͸ɺp(x | θ) ͷશ֬཰͕ 1 ͱ͍͏৚͔݅Β࣍ͷΑ͏ʹॻ͖ද͞Ε·͢ɻ Z(θ) = ∫ exp(θixi )dµ(x) (39) ͜ͷޙ͸ɺଌ౓ dµ(x) Λ༻͍Δલఏͷ΋ͱʹɺ࣍Ͱఆٛͨ͠ඪ४ܗ p(x | θ) Λ༻͍ͯٞ࿦ΛਐΊ·͢ɻ p(x | θ) := 1 Z(θ) exp(θixi ) (40) ͸͡Ίʹɺؔ਺ ψ(θ) Λ࣍ࣜͰఆٛ͠·͢ɻ ψ(θ) := log Z(θ) = log ∫ exp(θixi )dµ(x) (41) ͜ͷޙͰࣔ͢Α͏ʹɺψ(θ) ͸ತؔ਺ʹͳ͓ͬͯΓɺ͜ΕΛ༻͍ͯɺࢦ਺ܕ෼෍଒͕ߏ੒͢Δଟ༷ମʹ૒ର ฏୱͷߏ଄͕ಋೖ͞Ε·͢ɻ ·ͣɺψ(θ) Λඍ෼͢Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ ∂i ψ(θ) = 1 Z(θ) ∫ xi exp(θkxk )dµ(x) = ∫ xi p(x | θ)dµ(x) = E[x] (29) Λࢥ͍ग़͢ͱɺ∂i ψ(θ) ͸૒ର࠲ඪ ηi ʹରԠ͢Δ΋ͷͰͨ͠ɻ͜Ε͸ɺθ Λݻఆͨ࣌͠ɺରԠ͢Δ૒ର ࠲ඪ η ͷ஋͸ɺx ͷظ଴஋ E[x] ʹҰக͢Δ͜ͱΛҙຯ͠·͢ɻ ηi = ∂i ψ(θ) = E[xi ] (42) ∂i ψ(θ) Λ͞Βʹඍ෼͢Δͱɺ͕࣍ಘΒΕ·͢ɻ ∂i ∂j ψ(θ) = −1 {Z(θ)}2 ∫ xj exp(θkxk )dµ(x) ∫ xi exp(θkxk )dµ(x) + 1 Z(θ) ∫ xj xi exp(θkxk )dµ(x) = − ∫ xj p(x | θ)dµ(x) ∫ xi p(x | θ)dµ(x) + ∫ xj xi p(x | θ)dµ(x) = −E[xj ]E[xi ] + E[xj xi ] = Cov[xi , xj ] ैͬͯɺؔ਺ ψ(θ) ͷϔοηߦྻ͸ x ͷڞ෼ࢄߦྻ Cov[xi , xj ] ʹҰக͓ͯ͠Γɺਖ਼ఆ஋ߦྻͰ͋Δ͜ͱ͕ ෼͔Γ·͢ɻ͜ΕͰɺؔ਺ ψ(θ) ͸ತؔ਺͕͋Δ͜ͱ͕ࣔ͞Ε·ͨ͠ɻ·ͨɺ(32) Λࢥ͍ग़͢ͱɺ͜ͷ݁Ռ ͸ɺ͜ͷۭؒͷܭྔߦྻ gij ͕ڞ෼ࢄߦྻʹҰக͢ΔΛҙຯ͍ͯ͠·͢ɻ ͞Βʹɺ(39) ͱ (40) Λ༻͍Δͱɺ௚઀ܭࢉͰ͕࣍ࣔ͞Ε·͢ɻ ∂i log p(x | θ) = xi − E[xi ] ैͬͯɺ࣍ͷؔ܎͕੒Γཱͭ͜ͱʹͳΓ·͢ɻ gij = Cov[xi , xj ] = E [∂i log p(x | θ)∂j log p(x | θ)] ͜ͷ࠷ޙͷදࣜ͸ɺ౷ܭֶʹ͓͍ͯɺFisher ৘ใߦྻͱͯ͠஌ΒΕ͍ͯΔ΋ͷʹͳΓ·͢ɻ ࠷ޙʹɺࢦ਺ܕ෼෍଒Ͱ͸ɺBregman μΠόʔδΣϯεʢਖ਼֬ʹ͸૒ରμΠόʔδΣϯεʣ͕ KL μΠόʔ δΣϯεʹҰக͢Δ͜ͱΛࣔ͠·͢ɻ·ͣɺҰൠʹɺ2 ఺ p = p(x | θ) ͱ q = p(x | θ′) ʹؔ͢Δ KL μΠόʔ δΣϯε͸࣍ࣜͰ༩͑ΒΕ·͢ɻ KL(p || q) = ∫ p(x | θ) log p(x | θ) p(x | θ′) dµ(x) 18
19. ·ͨɺ(40) ͱ (41) Λ༻͍Δͱɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ log p(x | θ) = θixi − log Z(θ) = θixi − ψ(θ) log p(x | θ′) = θ′ixi − log Z(θ′) = θ′ixi − ψ(θ′) ͜ΕΒͷؔ܎ࣜͱ (42) Λ༻͍Δͱɺ࣍ͷΑ͏ʹܭࢉ͢Δ͜ͱ͕Ͱ͖·͢ɻ KL(p || q) = ∫ p(x | θ) { θixi − ψ(θ) } dµ(x) − ∫ p(x | θ) { θ′ixi − ψ(θ′) } dµ(x) = (θi − θ′i)E[xi ] − ψ(θ) + ψ(θ′) = ψ(θ′) − ψ(θ) − ∂i ψ(θ)(θ′i − θi) (43) ࠷ޙͷදࣜΛ Bregman μΠόʔδΣϯεͷఆٛ (34) ͱൺֱ͢Δͱɺ࣍ͷؔ܎͕੒Γཱͭ͜ͱ͕෼͔Γ ·͢ɻ KL(p || q) = D(q || p) (44) ͭ·ΓɺKL μΠόʔδΣϯε͸ɺp ͱ q ͷॱংΛೖΕସ͑ͨ Bregman μΠόʔδΣϯεʹҰக͍ͯ͠· ͢ɻ΋͘͠͸ɺ(37) ΑΓɺ૒ରμΠόʔδΣϯεʹҰக͍ͯ͠Δͱݴͬͯ΋Α͍Ͱ͠ΐ͏ɻ KL(p || q) = D∗(p || q) (45) ͳ͓ɺࢦ਺ܕ෼෍଒ʹ͓͍ͯ͸ɺ∇ ଌ஍ઢΛ e-ଌ஍ઢɺ∇∗ ଌ஍ઢΛ m-ଌ஍ઢͱݺͿ͜ͱ͕͋Γ·͢ɻ 5 EM ΞϧΰϦζϜͷزԿֶతղऍ ຊઅͰ͸ɺ૒ରฏୱͳଟ༷ମͷԠ༻ͱͯ͠ɺજࡏม਺Λ࣋ͭϞσϧͷ࠷໬ਪఆʹ༻͍ΒΕΔ EM ΞϧΰϦ ζϜΛزԿֶతʹղऍ͢Δͱ͍͏ྫΛ঺հ͠·͢ɻ 5.1 EM ΞϧΰϦζϜ ͜͜Ͱ͸ɺҰൠతͳ EM ΞϧΰϦζϜͷ෮शΛߦ͍·͢ɻજࡏม਺ Z Λ࣋ͭ֬཰Ϟσϧ p(X, Z | θ) ʹ͓ ͍ͯɺજࡏม਺Λফڈͨ͠ɺ؍ଌՄೳͳม਺ͷपล෼෍ʹର͢Δର਺໬౓Λߟ͑·͢ɻ log p(X | θ) = log ∑ Z p(X, Z | θ) (46) ͜ΕΛۃେʹ͢Δύϥϝʔλ θ Λܾఆ͢Δ͜ͱ͕ɺEM ΞϧΰϦζϜͷ໨తͰ͢ɻ ͪͳΈʹɺ͜ͷ໰୊ʹಛผͳΞϧΰϦζϜ͕ඞཁͱͳΔཧ༝͸ɺp(X, Z | θ) ͕ i.i.d. ͳ؍ଌσʔλͷ֬཰ͷ ੵͱͯ͠දݱ͞ΕΔ৔߹Λߟ͑Δͱ෼͔Γ·͢ɻજࡏม਺͕ଘࡏ͠ͳ͍৔߹ɺର਺໬౓͸ɺݸʑͷ֬཰ͷର਺ ͷ࿨ʹ෼ղ͞ΕΔͷͰɺղੳతͳܭࢉΛൺֱత؆୯ʹਐΊΔ͜ͱ͕Ͱ͖·͢ɻ͔͠͠ͳ͕Βɺ(46) ͷ৔߹ɺର ਺ͷதʹજࡏม਺ʹ͍ͭͯͷ࿨ؚ͕·Ε͍ͯΔͨΊɺͦͷΑ͏ͳ෼ղ͕Ͱ͖·ͤΜɻͦͷͨΊɺର਺ͷதʹ͋ Δ࿨ΛͳΜΒ͔ͷํ๏Ͱର਺ͷ֎ʹग़͢Α͏ͳܭࢉख๏͕ඞཁͰ͋Γɺ͜Ε͕ EM ΞϧΰϦζϜʹଞͳΓ· ͤΜɻ ઌʹ݁࿦Λड़΂ΔͱɺEM ΞϧΰϦζϜ͸࣍ͷΑ͏ͳखଓ͖ʹͳΓ·͢ɻ͸͡Ίʹɺθold Λద౰ͳॳظ஋ͱ ͯ͠ɺ q0 (Z) := p(Z | X, θold ) (47) 19
20. ͷԼʹɺ Q(θ) = ∑ Z q0 (Z) log p(X, Z | θ) (48) Λ࠷େԽ͢Δ θ = θnew ΛٻΊ·͢ɻ͜ͷࡍɺ(48) ͸ର਺ͷ֎ʹ࿨͕ग़͍ͯΔͷͰɺݩͷ໰୊ʹൺ΂ͯɺղੳ తͳܭࢉ͕༰қʹͳ͍ͬͯΔ͜ͱ͕෼͔Γ·͢ɻ ͜ͷ࣌ɺඞͣɺ࣍ͷෆ౳͕ࣜ੒ཱ͢Δ͜ͱ͕ূ໌͞Ε·͢ɻ log p(X | θnew ) ≥ log p(X | θold ) (49) ͦ͜Ͱɺθnew Λ৽ͨʹ θold ͱ্ͯ͠هͷܭࢉΛ܁Γฦ͢͜ͱͰɺlog p(X | θ) Λۃେʹ͢Δ θ ͕ಘΒΕ· ͢ɻͳ͓ɺq0 (Z) ͸؍ଌσʔλ X Λಘͨޙͷજࡏม਺ Z ͷࣄޙ෼෍ͰɺQ(θ) ͸ࣄޙ෼෍ͷԼʹ͓͚Δର਺ ໬౓ͷظ଴஋ͱղऍ͢Δ͜ͱ͕ՄೳͰ͢ɻ ͦΕͰ͸ɺ(49) ͷূ໌Λ༩͑·͢ɻ·ͣɺ৚݅෇͖֬཰ͷఆٛΑΓ͕࣍੒Γཱͪɺ p(X | θ) = p(X, Z | θ) p(Z | X, θ) ྆ลͷର਺ΛऔΔͱɺ͕࣍ಘΒΕ·͢ɻ log p(X | θ) = log p(X, Z | θ) − log p(Z | X, θ) ͜͜Ͱɺ೚ҙͷ֬཰෼෍ q(Z) Λ྆ลʹֻ͚ͯ Z ͷ࿨ΛͱΔͱɺ ∑ Z q(Z) = 1 ʹ஫ҙͯ͠ɺԼه͕ಘΒΕ ·͢ɻ log p(X | θ) = L(q, θ) + KL(q || p) (50) ͜͜ʹɺ L(q, θ) = ∑ Z q(Z) log p(X, Z | θ) q(Z) KL(q || p) = − ∑ Z q(Z) log p(Z | X, θ) q(Z) ͜͜ͰɺKL(q || p) ͸ɺKL μΠόʔδΣϯεͳͷͰɺKL(q || p) ≥ 0ɺ͔ͭɺq(Z) = p(Z | X, θ)ʢ͢ͳΘ ͪɺq(Z) ͕ Z ͷࣄޙ෼෍ʹҰக͢Δʣͷ࣌ͷΈʹ KL(q || p) = 0 ͱͳΓ·͢ɻ ͦ͜Ͱɺಛʹ q(Z) = p(Z | X, θ) ͷ৔߹Λߟ͑ΔͱɺKL(q || p) = 0 ΑΓɺ࣍ͷؔ܎͕ࣜಘΒΕ·͢ɻ log p(X | θ) = L(q, θ) = ∑ Z q(Z) log p(X, Z | θ) − ∑ Z q(Z) log q(Z) ͜Ε͸೚ҙͷ θ ʹ͍ͭͯ੒Γཱͪ·͕͢ɺ͜͜Ͱɺಛఆͷ θ = θold ʹݻఆ͢Δͱɺ(47) ͱ (48) Ͱఆٛͨ͠ ه߸ q0 (Z) ͱ Q(θ) Λ༻͍ͯɺ࣍ͷΑ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ log p(X | θold ) = ∑ Z q0 (Z) log p(X, Z | θold ) − ∑ Z q0 (Z) log q0 (Z) = Q(θold ) − ∑ Z q0 (Z) log q0 (Z) (51) 20
21. Ұํɺಉ͡ q0 (Z) ͷԼʹɺҰൠͷ θ ʹରͯ͠ (50) ΛվΊͯܭࢉ͠·͢ɻ(50) ͸ɺ೚ҙͷ q(Z) ʹ͍ͭͯ੒ ΓཱͭͷͰɺಛʹ q(Z) = q0 (Z) ͷ৔߹ΛվΊͯߟ͑Δͱ͍͏Θ͚Ͱ͢ɻ log p(X | θ) = L(q0 , θ) + KL(q0 || p) = ∑ Z q0 (Z) log p(X, Z | θ) − ∑ Z q0 (Z) log q0 (Z) + KL(q0 || p) = Q(θ) − ∑ Z q0 (Z) log q0 (Z) + KL(q0 || p) (52) KL(q0 || p) ≥ 0 ʹ஫ҙͯ͠ (51) ͱ (52) Λൺֱ͢Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ Q(θ) ≥ Q(θold ) ⇒ log p(X | θ) ≥ log p(X | θold ) ͜ΕͰɺ(49) ͕ূ໌͞Ε·ͨ͠ɻ 5.2 ࠷୹ڑ཭Λ༻͍ͨਪఆํ๏ ࢦ਺ܕ෼෍଒ͷྫͰݟͨΑ͏ʹɺ৘ใزԿֶͰ͸ɺ֬཰෼෍଒͕ߏ੒͢Δଟ༷ମΛݚڀର৅ͱ͠·͢ɻ͜ͷ ࣌ɺ͋ΒΏΔ֬཰෼෍ΛؚΉۭؒ S Λߟ͑ΔͱɺಛఆͷܗࣜͰද͞ΕΔ֬཰෼෍଒ M ͸ɺͦͷதͷ෦෼ۭ ؒΛߏ੒͢Δͱߟ͑ΒΕ·͢ɻ͜ͷ࣌ɺ؍ଌσʔλΛݩʹ֬཰෼෍଒ M ͷύϥϝʔλΛਪఆ͢Δͱ͍͏ߦҝ ͸ɺ؍ଌσʔλʹ࠷΋͍ۙ఺Λ෦෼ۭؒͷத͔Β୳͢΋ͷͱߟ͑Δ͜ͱ͕Ͱ͖·͢ɻ͜͜Ͱ͸ɺલ߲Ͱѻͬͨ જࡏม਺ΛؚΉ֬཰෼෍Λྫʹͯ͠ɺ͜ͷߟ͑ํΛਖ਼֬ʹड़΂͍͖ͯ·͢ɻ ͸͡Ίʹɺ ؍ଌՄೳͳม਺ X ͱજࡏม਺ Z ʹର͢Δಉ࣌֬཰෼෍ʹ͍ͭͯɺ ߟ͑͏Δ͢΂ͯͷ෼෍ p(X, Z) ΛूΊۭͨؒ S Λ༻ҙ͠·͢ɻ͜ͷதͰಛʹɺύϥϝʔλ θ Ͱಛ௃͚ͮΒΕͨϞσϧͷ෼෍ p(X, Z | θ) Λ ूΊΔͱɺ͜Ε͸ɺۭؒ S ͷ෦෼ۭؒ M Λߏ੒͠·͢ɻ͜ΕΛʮϞσϧۭؒʯͱݺͼ·͢ɻ Ұํɺ؍ଌσʔλ {Xn }N n=1 ͕༩͑ΒΕͨ৔߹ɺ͜ͷ؍ଌσʔλ͕ಘΒΕΔ֬཰͕ 1 ʹͳΔʢ͜ͷ؍ଌσʔ λʹΦʔόʔϑΟοςΟϯάͨ͠ʣ֬཰෼෍͕ߏ੒Ͱ͖·͢ɻ ˆ q(X, Z) = N ∏ n=1 δ(X − Xn )q(Z) ͜͜ʹɺq(Z) ͸ ∑ Z q(Z) = 1 Λຬͨ͢೚ҙͷؔ਺Ͱ͢ɻ͜ͷΑ͏ͳ ˆ q(X, Z) Λ͢΂ͯूΊͨ΋ͷ͸ɺ΍ ͸Γɺۭؒ S ͷ෦෼ۭؒ D ͱͳΓ·͢ɻ͜ΕΛʮσʔλۭؒʯͱݺͼ·͢ɻ જࡏม਺Λ࣋ͨͳ͍Ϟσϧͷ৔߹ɺ؍ଌσʔλͷ֬཰Λ 1 ʹ͢Δ෼෍͸ 1 ͭʹܾ·ΔͷͰɺ͜Ε͸ɺS ্ ͷ 1 ఺ q Λද͠·͢ɻͦ͜Ͱɺq ͔ΒϞσϧۭؒ M ʹԿΒ͔ͷҙຯͰʮਖ਼ࣹӨʯͯ͠ಘΒΕΔ M ্ͷ఺Λ ਪఆ෼෍ͱͯ͠࠾༻͢Δ͜ͱ͕ՄೳʹͳΓ·͢ɻ ʮਖ਼ࣹӨʯͷҙຯʢͭ·Γɺۭؒ S ʹಋೖ͢ΔزԿֶߏ଄ʣ ͷͱΓํʹΑͬͯɺਪఆํ๏͕มΘΔ͜ͱʹͳΓ·͢ɻ Ұํɺࠓͷ৔߹͸ɺજࡏม਺͕͋ΔͨΊʹɺ؍ଌσʔλʹରԠ͢Δ෼෍͸ 1 ͭʹܾ·Βͣɺ෦෼ۭؒ D Λ ߏ੒͠·ͨ͠ɻͦ͜Ͱɺσʔλۭؒ D ͱϞσϧۭؒ M ͷʮ࠷୹ڑ཭ʯΛ࣮ݱ͢Δ 2 ఺Λݟ͚ͭग़ͯ͠ɺۭ ؒ M ଆͷ఺Λਪఆ෼෍ͱͯ͠࠾༻͢Δͱ͍͏ํ๏͕ߟ͑ΒΕ·͢ɻ ͜͜Ͱɺಛʹʮڑ཭ʯͱͯ͠ɺσʔλ্ۭؒͷ఺͔ΒϞσϧ্ۭؒͷ఺ʹର͢Δ KL μΠόʔδΣϯεΛ࠾ ༻͠·͢ɻ KL (ˆ q(X, Z) || p(X, Z | θ)) = KL ( N ∏ n=1 δ(X − Xn )q(Z) || p(X, Z | θ) ) 21
22. ্هΛ θ ͓Αͼ q(Z) ͷؔ਺ʢ൚ؔ਺ʣͱݟͳͯ͠ɺ͜ΕΛ࠷খʹ͢Δ θ Λਪఆύϥϝʔλͱͯ͠࠾༻͠· ͢ɻ͜ͷޙͰࣔ͢Α͏ʹɺ࣮͸ɺ͜ΕʹΑͬͯಘΒΕΔ θ ͸Լهͷ֬཰Λۃେʹ͠·͢ɻ͢ͳΘͪɺEM Ξϧ ΰϦζϜͱಉ͡࠷໬ਪఆʹͳ͍ͬͯΔ͜ͱ͕ূ໌͞ΕΔ͜ͱʹͳΓ·͢ɻ log p(X | θ) = log ∑ Z p(X, Z | θ) 5.3 em ΞϧΰϦζϜ ͦΕͰ͸ɺલ߲Ͱड़΂ͨɺσʔλۭ͔ؒΒϞσϧۭؒ΁ͷʮ࠷୹ڑ཭ʯΛ࣮ݱ͢Δ 2 ఺ q ∈ D ͱ p ∈ M Λ ൃݟ͢Δखଓ͖ʢem ΞϧΰϦζϜʣΛઆ໌͠·͢ɻ͜Ε͸ɺEM ΞϧΰϦζϜͷزԿֶόʔδϣϯͰ࣍ͷΑ ͏ͳ܁Γฦ͠ૢ࡞ʹͳΓ·͢ɻ ͸͡ΊʹɺϞσϧۭؒ M ͷҰ఺ pold Λ೚ҙʹબ୒ͯ͠ɺσʔλۭؒ D ্Ͱɺ͜ͷ఺΁ͷ KL μΠόʔ δΣϯε͕࠷খʹͳΔ఺Λ୳͠·͢ɻ q = arg min q∈D KL(q || pold ) = arg min q∈D D(pold || q) (53) 2 ͭ໨ͷ౳ࣜͰ͸ɺKL μΠόʔδΣϯεͱ Bregman μΠόʔδΣϯεͷؔ܎ (44) Λ༻͍͍ͯ·͢ɻఆཧ 3ʢ֦ுϐλΰϥεͷఆཧʣΛࢥ͍ग़͢ͱɺ఺ q ͸ɺ఺ pold ͔Βσʔλۭؒ D ʹ߱Ζͨ͠ ∇ ଌ஍ઢʹΑΔਨ ઢʢ఺ q ʹ͓͚Δ ∇∗ ଌ஍ઢํ޲ͷ઀ฏ໘ʹ௚ߦ͢Δ ∇ ଌ஍ઢʣͷ଍ʹͳΔ͜ͱ͕෼͔Γ·͢ɻ͜ͷૢ࡞Λ ʮe-ࣹӨʯͱݺͼ·͢ɻ ଓ͍ͯɺϞσϧۭؒ M ্Ͱɺ఺ q ͔Βͷ KL μΠόʔδΣϯε͕࠷খʹͳΔ఺Λ୳͠·͢ɻ pnew = arg min p∈M KL(q || p) = arg min p∈M D∗(q || p) (54) 2 ͭ໨ͷ౳ࣜͰ͸ɺؔ܎ (45) Λ༻͍͍ͯ·͢ɻఆཧ 4 Λࢥ͍ग़͢ͱɺ఺ pnew ͸ɺ఺ q ͔ΒϞσϧۭؒ M ΁ͷ ∇∗ ଌ஍ઢʹΑΔਨઢʢ఺ pnew ʹ͓͚Δ ∇ ଌ஍ઢํ޲ͷ઀ฏ໘ʹ௚ߦ͢Δ ∇∗ ଌ஍ઢʣͷ଍ʹͳΔ͜ ͱ͕෼͔Γ·͢ɻ͜ͷૢ࡞Λʮm-ࣹӨʯͱݺͼ·͢ɻ ͜ͷޙɺpnew Λ৽ͨʹ pold ͱͯ͠ɺ্هͷʮe-ࣹӨʯͱʮm-ࣹӨʯͷૢ࡞Λ܁Γฦ͠·͢ɻͦΕͧΕͷࣹ ӨͰ KL μΠόʔδΣϯε͸୯ௐʹݮগ͍͖ͯ͠·͢ͷͰɺ࠷ऴతʹɺσʔλۭؒ D ͔ΒϞσϧۭؒ M ΁ ͷ KL μΠόʔδΣϯεΛۃখʹ͢Δ఺ͷ૊ʹऩଋ͢Δ͜ͱʹͳΓ·͢ʢਤ 5ʣ ɻͳ͓ɺҎ্ͷٞ࿦Ͱ͸ɺࢦ ਺ܕ෼෍଒Λલఏͱͯ͠ɺKL μΠόʔδΣϯεͱ Bregman μΠόʔδΣϯεͷؔ܎Λར༻͠·ͨ͠ɻԾʹ ͜ͷؔ܎͕੒Γཱͨͳ͍৔߹Ͱ΋ɺଌ஍ઢͱͯ͠ͷղऍ͕Ͱ͖ͳ͍఺Λͷ͚ͧ͹ɺ্هͷܭࢉखଓ͖ͦͷ΋ͷ ͸ਖ਼͘͠੒Γཱͪ·͢ɻ 5.4 EM ΞϧΰϦζϜͱͷରԠ (53) ͱ (54) ͷܭࢉΛ࣮ࡍʹߦͬͯݟΔͱɺͦΕͧΕɺEM ΞϧΰϦζϜͷ 2 ͭͷૢ࡞ɺ͢ͳΘͪɺࣄޙ෼෍ q0 (Z) = p(Z | {Xn }, θold ) 22
23. ਤ 5 em ΞϧΰϦζϜ ͷܭࢉʢE εςοϓʣͱɺࣄޙ෼෍ͷԼͰͷର਺໬౓ͷظ଴஋ Q(θ) = ∑ Z q0 (Z) log p({Xn }, Z | θ) ͷ࠷େԽʢM εςοϓʣʹҰக͢Δ͜ͱ͕෼͔Γ·͢ɻैͬͯɺem ΞϧΰϦζϜ͸ɺEM ΞϧΰϦζϜͱಉ ݁͡Ռʢ࠷໬ਪఆʣΛಋ͘͜ͱ͕ূ໌͞Ε·͢ɻ ͦΕͰ͸ɺ(53) ͱ (54) Λ࣮ࡍʹܭࢉͯ͠Έ·͢ɻ·ͣ͸ɺ(53) ʹؚ·ΕΔ KL μΠόʔδΣϯεΛ۩ମత ʹܭࢉ͠·͢ɻ KL(q || pold ) = ∫ ∑ Z { N ∏ n=1 δ(X − Xn )q(Z) log ∏ N n=1 δ(X − Xn )q(Z) p({Xn }, Z | θold ) } dX = ∑ Z q(Z) {log q(Z) − log p({Xn }, Z | θold )} ଋറ৚݅ ∑ Z q(Z) = 1 Λߟྀͯ͠ɺϥάϥϯδϡͷະఆ৐਺߲ΛՃ͑ͯ q(Z) ͷม෼ΛऔΓ·͢ɻ δ { KL(q || pold ) + λ( ∑ Z q(Z) − 1) } = ∑ Z [{log q(Z) − log p({Xn }, Z | θold ) + 1 + λ} δq(Z)] ্ه͕೚ҙͷ δq(Z) ʹରͯ͠ 0 ʹͳΔ͜ͱ͔Βɺ q(Z) = e−(λ+1)p({Xn }, Z | θold ) = e−(λ+1)p(Z | {Xn }, θold )p({Xn } | θold ) (55) ͞Βʹଋറ৚݅ΑΓɺ 1 = ∑ Z q(Z) = e−(λ+1) ∑ Z p(Z | {Xn }, θold )p({Xn } | θold ) = e−(λ+1)p({Xn } | θold ) ͜Ε͔Βܾ·Δ e−(λ+1) Λ (55) ʹ୅ೖͯ͠ɺ q(Z) = p({Xn }, Z | θold ) p({Xn } | θold ) = p(Z | {Xn }, θold ) 23
24. ͜Ε͸ɺEM ΞϧΰϦζϜʹ͓͚Δ E εςοϓʢࣄޙ෼෍ q0 (Z) ͷܭࢉʣͱಉ͡ܭࢉʹͳΓ·͢ɻ ଓ͍ͯɺ্هͰܾ·Δ q(Z) = q0 (Z) Λ༻͍Δ΋ͷͱͯ͠ɺ(54) ʹؚ·ΕΔ KL μΠόʔδΣϯεΛܭࢉ͠ ·͢ɻ KL(q0 || p) = ∫ ∑ Z { N ∏ n=1 δ(X − Xn )q0 (Z) log ∏ N n=1 δ(X − Xn )q0 (Z) p(X, Z | θ) } dX = ∑ Z q0 (Z) log q0 (Z) − ∑ Z q0 (Z) log p({Xn }, Z | θ) (56) ্هͷୈ 1 ߲͸ θ ʹґଘ͠ͳ͍͜ͱʹ஫ҙ͢Δͱɺ(56) Λ࠷খʹ͢Δ θ ΛٻΊΔ͜ͱ͸ɺ Q(θ) = ∑ Z q0 (Z) log p({Xn }, Z | θ) Λ࠷େʹ͢Δ θ ΛٻΊΔ͜ͱʹଞͳΒͣɺEM ΞϧΰϦζϜʹ͓͚Δ M εςοϓʢࣄޙ෼෍ͷԼͰͷର਺໬ ౓ͷظ଴஋ͷ࠷େԽʣͱಉ͜͡ͱʹͳΓ·͢ɻ ͜ΕͰɺem ΞϧΰϦζϜ͕ EM ΞϧΰϦζϜͱಉ౳Ͱ͋Δ͜ͱ͕ূ໌͞Ε·ͨ͠ɻ ࢀߟจݙ [1]ʮ৘ใزԿֶͷجૅ (਺ཧ৘ใՊֶγϦʔζ)ʯ౻ݪ জ෉ʢஶʣ຀໺ॻళ [2]ʮ৘ใزԿֶͷ৽ల։ʯ؁ར ढ़ҰʢஶʣαΠΤϯεࣾ [3] ᎇ཰͕ 0 Ͱͳ͍ʮฏ໘ʯͷྫ http://enakai00.hatenablog.com/entry/2016/12/18/215906 [4] Legendre ม׵ͷ·ͱΊ http://enakai00.hatenablog.com/entry/2015/05/07/153209 24