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Riemann幾何学ユーザーのための情報幾何学入門

 Riemann幾何学ユーザーのための情報幾何学入門

Etsuji Nakai

April 12, 2023
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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

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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