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Wasserstein逆FM音源

Fadis
March 18, 2021

 Wasserstein逆FM音源

逆FM音源はWasserstein距離を誤差関数に使うと良い、という話をします
これは2021年3月20日に行われた Kernel/VM探検隊online part2での発表資料です
サンプルコード: https://github.com/Fadis/wifm

Fadis

March 18, 2021
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  1. asserstein
    ٯFMԻݯ
    NAOMASA MATSUBAYASHI
    Twitter: @fadis_
    WASSERSTEIN
    INVERTED FREQUENCY MODULATION SYNTHESIZER
    https://github.com/Fadis/wifm
    αϯϓϧίʔυ
    W
    ݱࡏςετԻ੠Λ࠶ੜ͍ͯ͠·͢ɻԻ͸ฉ͍͑ͯ͜·͔͢

    View Slide

  2. asserstein
    ٯFMԻݯ
    NAOMASA MATSUBAYASHI
    Twitter: @fadis_
    WASSERSTEIN
    INVERTED FREQUENCY MODULATION SYNTHESIZER
    https://github.com/Fadis/wifm
    αϯϓϧίʔυ
    W

    View Slide

  3. FMԻݯ
    ੲͷPC΍ήʔϜػʹΑ͘ࡌ͍ͬͯͨ
    ԻָΛ૗ͰΔϋʔυ΢ΣΞ

    View Slide

  4. άϥϯυϐΞϊ
    ΞϧταοΫε
    ϏϒϥϑΥϯ
    όΠΦϦϯ
    ϚϦϯό
    ຊ෺ͷָثͷԻΛϑʔϦΤม׵

    View Slide

  5. 261.63Hz 523.25Hz 784.88Hz 1046.5Hz 1308.1Hz 1569.8Hz 1831.4Hz 2093.0Hz 2354.6Hz
    Ի֊ʹରԠ͢Δप೾਺੒෼
    άϥϯυϐΞϊ
    ΞϧταοΫε
    ϏϒϥϑΥϯ
    όΠΦϦϯ
    ϚϦϯό
    جԻ

    View Slide

  6. 261.63Hz 523.25Hz 784.88Hz 1046.5Hz 1308.1Hz 1569.8Hz 1831.4Hz 2093.0Hz 2354.6Hz
    ͦͷ੔਺ഒͷप೾਺੒෼
    άϥϯυϐΞϊ
    ΞϧταοΫε
    ϏϒϥϑΥϯ
    όΠΦϦϯ
    ϚϦϯό
    ഒԻ

    View Slide

  7. ͦͷ੔਺ഒͷप೾਺ͷ೾
    άϥϯυϐΞϊ
    ߴ͍प೾਺੒෼΄Ͳ
    ૣ͘ݮਰ͢Δ

    View Slide

  8. ͜͏ͨ͠ৼΔ෣͍Λ
    ؆୯ͳܭࢉͰਅࣅΔࣄ͕Ͱ͖Ε͹
    ؆୯ͳϋʔυ΢ΣΞͰ
    ຊ෺ͷָثͬΆ͍Ի͕ग़ͤΔ

    View Slide

  9. f (t) = Ec
    (t) sin (2πωc
    t + Em
    (t) B sin (2πωm
    t))
    ͜ͷαΠϯ೾Ͱ
    ͜ͷαΠϯ೾Λ࿪ΊΔ
    FMԻݯ

    View Slide

  10. 1ωc
    = ωm
    2ωc
    = ωm
    3ωc
    = ωm
    4ωc
    = ωm
    5ωc
    = ωm
    f (t) = Ec
    (t) sin (2πωc
    t + Em
    (t) B sin (2πωm
    t))
    6ωc
    = ωm
    ύϥϝʔλʹΑͬͯ
    ༷ʑͳप೾਺ʹ༷ʑͳେ͖͞ͷഒԻ͕ग़Δ

    View Slide

  11. f (t) = Ec
    (t) sin (2πωc
    t + Em
    (t) B sin (2πωm
    t))
    ͜Ε ͜Ε ͜Ε ͜Ε ͜Ε΋
    ͜ΕΒͷύϥϝʔλΛௐ੔ͯ͠
    ຊ෺ͷָثͱಉ͡ഒԻΛ࣋ͭԻΛग़ͤΕ͹
    ͦͷָثͷԻʹฉ͑͜Δ

    View Slide

  12. ͜ΕΒͷύϥϝʔλΛௐ੔ͯ͠
    ຊ෺ͷָثͱಉ͡ഒԻΛ࣋ͭԻΛग़ͤΕ͹
    ͦͷָثͷԻʹฉ͑͜Δ
    f (t) = Ec
    (t) sin (2πωc
    t + Em
    (t) B sin (2πωm
    t))
    ೉ FMԻݯ͸໓Μͩ
    ͜Ε ͜Ε ͜Ε ͜Ε ͜Ε΋
    ͋·Γྑ͍Ի͕ग़ͳ͍ͱ͞Ε

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  13. αϯϓϦϯάԻΛप೾਺ղੳ
    ϥϯμϜͳҨ఻ࢠΛੜ੒
    ֤Ҩ఻ࢠΛ'.ԻݯͰԋ૗
    ֤Ҩ఻ࢠͷԻΛप೾਺ղੳ
    Ұ༷ަ伹ͱಥવมҟͰҨ఻ࢠΛੜ੒
    αϯϓϦϯάԻͱͷڑ཭Λܭࢉ
    ϧʔϨοτબ୒ͰݸମΛݫબ
    ࠷େ෼ղೳͰ
    ্ҐʹมԽ͕ݟΒΕͳ͍
    ͸͍
    ͍͍͑
    ෼ղೳΛ্͛Δ͔Ͳ͏͔Λ൑அ
    Ҩ఻తFMԻݯ
    2016೥
    https://speakerdeck.com/fadis/yi-chuan-de-fmyin-yuan
    Ҩ఻తΞϧΰϦζϜͰ
    ΑΓ༩͑ΒΕͨԻͷεϖΫτϧʹ
    ͍ۙԻʹͳΔύϥϝʔλΛ
    ੜଘͤ͞Δ
    ୈ12ճ ΧʔωϧʗVM୳ݕୂ

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  14. ຊ෺ͷϐΞϊͷԻ
    '.ԻݯͷϐΞϊͷԻ
    ͜ͷ࣮ݧͷաఔͰ
    FMԻݯͰ͸ग़ͤͳ͍ͱ͞Ε͍ͯͨ
    ϐΞϊͷԻΛͦͦ͜͜࠶ݱ͢Δύϥϝʔλ͕ݟ͔ͭͬͨ
    Ҩ఻తFMԻݯ
    2016೥
    ୈ12ճ ΧʔωϧʗVM୳ݕୂ

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  15. FMԻݯͰ͸ָثͷԻ৭Λ࠶ݱͰ͖ͳ͍
    ਓؒͷΧϯͰ͸FMԻݯͰָثͷԻ৭Λ࠶ݱͰ͖ͳ͍

    View Slide

  16. https://speakerdeck.com/fadis/niyurarufmyin-yuan
    ٯFMԻݯ
    2020೥
    FMԻݯͷࣜΛඍ෼ͯ͠
    ޯ഑๏ͰύϥϝʔλΛൃݟ͢Δ
    f (t) = A


    n=−∞
    Jn
    (B) cos (2πt (nωm
    + ωc))
    Χʔωϧ7.୳ݕୂ!ؔ੢ճ໨
    dJn
    (x)
    dx
    =
    1
    2
    (Jn−1
    (x) − Jn+1
    (x))

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  17. ࠷΋ԫ৭͘ͳΔ ͱ Λݟ͚͍ͭͨ
    ω B

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  18. Ұ൪௿͍ॴΛ୳͍ͨ͠
    ࠓͷҐஔͷ܏͖Λݟͯ
    ΑΓ௿͍ํʹগ͠Ҡಈ͢Δ
    ޯ഑๏
    ΋ͬͱࠨ͔ͳ
    ͜ͷ΁ΜʹͨͲΓண͘
    ϘʔϧΛస͕ͯ͠Ұ൪௿͍ॴͰࢭ·ΔͷΛ
    ظ଴͢Δͷʹࣅ͍ͯΔ

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  19. ޯ഑๏
    ͜ͷ΁ΜʹͨͲΓண͘
    ͜͜ʹ͋Δຊ౰ͷ࠷খ஋ʹ͸
    ͨͲΓண͚ͳ͍
    ΋ͬͱࠨ͔ͳ
    ΋ͬͱӈ͔ͳ

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  20. ͜ͷঢ়ଶͰ͸ޯ഑๏ͷద༻͸ແཧ

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  21. 261.63Hz 523.25Hz 784.88Hz 1046.5Hz 1308.1Hz 1569.8Hz 1831.4Hz 2093.0Hz 2354.6Hz
    ഒԻ͸੔਺ഒͷप೾਺෇ۙʹग़Δ
    άϥϯυϐΞϊ
    ΞϧταοΫε
    ϏϒϥϑΥϯ
    όΠΦϦϯ
    ϚϦϯό
    ͸ৗʹ੔਺ʹͳΔ͜ͱʹ͢Δ
    ω =
    ωm
    ωc

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  22. ͸ৗʹ੔਺ʹͳΔ͜ͱʹ͢Δ
    ω =
    ωm
    ωc

    View Slide

  23. ͜ͷࢁΛ্ʹ௒͑ΒΕΔͱࠔΔ

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  24. ෳ਺ͷॳظ஋͔Β୳࢝͠Ίͯ࠷খ஋ʹ౸ୡͨ͠1ͭΛબͿ
    ม਺ͷ਺͕૿͑Δͱࢼߦճ਺͕ٸܹʹ૿͑Δ
    3ΦϖϨʔλҎ্ʹద༻͕ࠔ೉

    View Slide

  25. ຊ෺ͷόΠΦϦϯ
    ٯFMԻݯόΠΦϦϯ
    ࣮ࡍόΠΦϦϯͷԻʹฉ͑͜ͳ͍
    σνϡʔϯ໰୊
    ͜ͷ΁Μ͕ͬͦ͝Γফ͍͑ͯΔ 2ΦϖϨʔλ ϑΟʔυόοΫͳ͠

    View Slide

  26. ຊ෺ͷόΠΦϦϯ
    ٯFMԻݯόΠΦϦϯ
    ੔਺ഒͷप೾਺͔Βগͣ͠Εͨप೾਺ͰഒԻ͕ग़͍ͯΔ
    (ഒԻͷप೾਺ͷഒ཰)͸ৗʹ੔਺ʹͳΔ͜ͱʹ͢Δ
    ω =
    ωm
    ωc
    ͱ͍͏੍໿Λ՝ͨ͠ͷͰͣΕͨഒԻ͕શͯແࢹ͞Ε͍ͯΔ

    View Slide

  27. ޡࠩͷ෼෍͕Α͘ͳ͍
    খࡉ޻Ͱ෼෍Λྑ͘͢Δͱ
    ָثͷಛ௃ΛऔΓଛͶΔ
    ͦ΋ͦ΋ଛࣦؔ਺͕ྑ͘ͳ͍ͷͰ͸?
    ͔͜͜Β͕ࠓճͷ৽ωλ

    View Slide

  28. ฏۉೋ৐ޡࠩ
    L =
    1
    n
    n

    i=0
    (ei
    − gi)
    2
    ຊ෺ͷप೾਺੒෼
    ੜ੒෺ͷप೾਺੒෼
    શͯͷप೾਺੒෼ʹ͍ͭͯͷޡࠩΛ଍ͯ͠ฏۉΛऔΔ
    ຊ෺ͱੜ੒෺ͷ͕ࠩ
    ਖ਼Ͱ΋ෛͰ΋ޡࠩ͸ਖ਼ʹͳΔ
    ͱ ͕ࣅ͍ͯΔఔ ͸খ͘͞ͳΔ
    e g L
    ٯFMԻݯͷଛࣦؔ਺

    View Slide

  29. ฏۉೋ৐ޡࠩ
    ຊ෺
    ੜ੒෺
    ຊ෺Ͱ͸ग़͍ͯͳ͍प೾਺ͷ੒෼͕ग़͍ͯΔ
    ຊ෺Ͱ͸ग़͍ͯΔप೾਺ͷ੒෼͕ग़͍ͯͳ͍
    ͜͜Ͱग़͍ͯΔ෼ؙ͕͝ͱޡࠩ
    ͜͜Ͱग़͍ͯͳ͍෼ؙ͕͝ͱޡࠩ
    ޡࠩ=A + B = A
    = B

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  30. ฏۉೋ৐ޡࠩ
    ຊ෺
    ੜ੒෺
    ຊ෺Ͱ͸ग़͍ͯΔप೾਺ͷ੒෼͕ग़͍ͯͳ͍
    ͜͜Ͱग़͍ͯͳ͍෼ؙ͕͝ͱޡࠩ
    ؒҧͬͨप೾਺ͰԻ͕ग़͍ͯΔΑΓ
    Ի͕ग़͍ͯͳ͍ํ͕·ͩϚγ
    ޡࠩ=B
    = B

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  31. ໼ҹͷํ޲ʹҠಈ͢Δ΄Ͳߴप೾੒෼͕ՄௌҬ֎ʹग़Δ
    = Ի͕ग़͍ͯͳ͍ͷͰϚγঢ়ଶ
    ࢁͷݪҼ

    View Slide

  32. ฏۉೋ৐ޡࠩ
    ຊ෺
    ੜ੒෺
    ޡࠩ=શ෦
    ͳ͠
    ͕มΘΔͱഒԻͷप೾਺͸Ҡಈ͢Δ͕
    ฏۉೋ৐ޡࠩͰ͸ຊ෺ͱҰக͢ΔҰॠΛআ͍ͯ
    ౳͘͠ग़͍ͯͳ͍ํ͕Ϛγ
    ω

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  33. ͜ͷ݁Ռ ํ޲͸ຊ෺ͱप೾਺͕Ұக͢Δ͔ᷮͳॠ͚ؒͩԫ৭͘ͳΔ
    ω

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  34. L =
    1
    n
    n

    i=0
    (ei
    − gi)
    2
    ฏۉೋ৐ޡ͕ࠩԻ৭ಉ࢜ͷڑ཭ΛଌΔखஈͱͯ͠ద͍ͯ͠ͳ͍ࣄ͸໌Β͔

    View Slide

  35. Wassersteinڑ཭

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  36. Wassersteinڑ཭
    W (ℙ, ℚ) = inf
    J∈𝒥(ℙ, ℚ)

    ∥x − y∥dJ (x, y)
    ຊ෺ͷ෼෍
    ੜ੒෺ͷ෼෍
    ͷ͋Δαϯϓϧ͔Β
    ͷ͋Δαϯϓϧ΁ͷڑ཭


    ࠷΋ޮ཰ͷྑ͍༌ૹํ๏Λ༻͍ͨ৔߹ͷ
    2ͭͷ෼෍ͷྨࣅ౓ΛଌΔڑ཭
    ͷ͋Δαϯϓϧ͔Β
    ͷ͋Δαϯϓϧ΁ͷ༌ૹྔ


    ༌ૹྔ ڑ཭ͷ૯࿨
    ×

    View Slide

  37. Wassersteinڑ཭
    0
    0.25
    0.5
    0.75
    1
    0 1 2 3 4 5
    0
    0.25
    0.5
    0.75
    1
    0 1 2 3 4 5
    ℙ ℚ
    W (ℙ, ℚ) = × 1+ × 1+ × 2
    ͷঢ়ଶ͔Β ͷঢ়ଶʹ͢Δҝʹ
    ӡ͹ͳ͚Ε͹ͳΒͳ͍෺ ӡͿڑ཭ ͷ૯࿨
    ℙ ℚ
    ×
    ͜ͷ஋͕খ͍͞ఔ2ͭͷ෼෍͸ࣅ͍ͯΔͱݴ͑Δ

    View Slide

  38. ਺ֶతʹͪΌΜͱͨ͠આ໌΍ޮ཰ͷྑ͍ٻΊํ
    http://www.stat.cmu.edu/~larry/=sml/
    36-708 Statistical Methods for Machine Learning (CMUͷߨٛ)ͷࢿྉ
    ͱ͔
    https://arxiv.org/abs/1701.07875
    Arjovsky, Martin, Soumith Chintala, and Léon Bottou. "Wasserstein generative adversarial
    networks." International conference on machine learning. PMLR, 2017.
    Wasserstein GAN
    Wassersteinڑ཭Λ࢖ͬͯϞʔυ่յΛ๷͙GANʹ͍ͭͯͷ࿦จ
    Optimal Transport and Wasserstein Distance
    ͋ͨΓݟͯ

    View Slide

  39. Wassersteinڑ཭ͷੌ͍ॴ
    ຊ෺
    ੜ੒෺
    ฏۉೋ৐ޡࠩϚϯ
    ੜ੒෺͸Կ΋͔΋ؒҧ͍ͬͯΔ
    ੜ੒෺ͷ ͸ཁΒͳ͍ॴʹੜ͍͑ͯΔͷͰແ͍ํ͕ྑ͍
    WassersteinϚϯ
    ੯͍͠
    ੜ੒෺ͷ ͸͋ͱ΋͏গ͠ࠨʹ͋ͬͨΒ׬ᘳͩͬͨͷʹ
    ΄Μͷগ͠ͷࠩ

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  40. ਓ͕ؒݟͯ΋࠷దͳ ͱ ͷҐஔ͕ݟ͑ΔϨϕϧ
    B ω

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  41. ϑʔϦΤม׵
    Wassersteinڑ཭ؔ਺ Τϯϕϩʔϓਪఆ
    ຊ෺ͷָثͷԻ

    ਖ਼نԽ
    ຊ෺ͷԻͱੜ੒ͨ͠Իͷڑ཭L ԻྔA
    प೾਺ྖҬFMԻݯ
    มௐͷڧ͞B 2ͭͷαΠϯ೾ͷप೾਺ൺω
    ·ͣద౰ͳ ͱ Ͱ ΛٻΊͯ
    B ω L

    View Slide

  42. ϑʔϦΤม׵
    प೾਺ྖҬFMԻݯͷඍ෼
    Wassersteinڑ཭ؔ਺ͷඍ෼ Τϯϕϩʔϓਪఆ
    ຊ෺ͷָثͷԻ

    ਖ਼نԽ
    มௐͷڧ͞B 2ͭͷαΠϯ೾ͷप೾਺ൺω
    ຊ෺ͷԻͱੜ੒ͨ͠Իͷڑ཭L ԻྔA
    Adam
    ޡࠩٯ఻೻Ͱ ͔Β ͱ ΛͲ͏मਖ਼͢΂͖͔ΛٻΊΔ
    L B ω

    View Slide

  43. ϑʔϦΤม׵
    प೾਺ྖҬFMԻݯͷඍ෼
    Wassersteinڑ཭ؔ਺ͷඍ෼ Τϯϕϩʔϓਪఆ
    ຊ෺ͷָثͷԻ

    ਖ਼نԽ
    มௐͷڧ͞B 2ͭͷαΠϯ೾ͷप೾਺ൺω
    ຊ෺ͷԻͱੜ੒ͨ͠Իͷڑ཭L ԻྔA
    Adam
    ޡࠩٯ఻೻Ͱ ͔Β ͱ ΛͲ͏मਖ਼͢΂͖͔ΛٻΊΔ
    L B ω
    Wassersteinڑ཭ؔ਺͸
    ղੳతʹඍ෼Ͱ͖ͳ͍

    View Slide

  44. std::tuple< std::vector< T >, std::vector< T > > backward(
    const std::vector &A,
    const std::vector &AWeights,
    const std::vector &B,
    const std::vector &BWeights,
    T dist, T delta, T wdelta
    ) {
    std::vector dAWeights( AWeights.size() );
    std::vector dA( A.size() );
    #pragma omp parallel for
    for( size_t i = 0; i < A.size(); ++i ) {
    auto modif_ = A;
    modif_[ i ] += delta;
    auto modified_dist = forward_x( modif_, AWeights, B, BWeights );
    dA[ i ] = ( modified_dist - dist ) / delta;
    }
    #pragma omp parallel for
    for( size_t i = 0; i < AWeights.size(); ++i ) {
    auto modif_ = AWeights;
    modif_[ i ] += wdelta;
    auto modified_dist = forward_x( A, modif_, B, BWeights );
    dAWeights[ i ] = ( modified_dist - dist ) / wdelta;
    }
    return std::make_tuple( dA, dAWeights );
    }
    ղੳతʹඍ෼Ͱ͖ͳ͍ͳΒ
    ਺஋తʹඍ෼͢Ε͹ྑ͍

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  45. ɺ Ͱ࢝ΊΔ ෳ਺ͷॳظ஋͔Βࢼ͢ඞཁ͸ͳ͍
    B = 5 ω = 5

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  46. ݁Ռ

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  47. ຊ෺ͷόΠΦϦϯ
    ٯFMԻݯόΠΦϦϯ
    ͜ͷ΁Μ͕ͬͦ͝Γফ͍͑ͯΔ
    ࣮ࡍόΠΦϦϯͷԻʹฉ͑͜ͳ͍
    2ΦϖϨʔλ ϑΟʔυόοΫͳ͠

    View Slide

  48. ຊ෺ͷόΠΦϦϯ
    Wasserstein
    ٯFMԻݯόΠΦϦϯ
    όΠΦϦϯͷԻʹฉ͑͜Δ
    2ΦϖϨʔλ ϑΟʔυόοΫͳ͠

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  49. ຊ෺ͷϐΞϊ
    ਓ͕ؒ࡞ͬͨFMԻݯϐΞϊ
    80೥୅ʙ90೥୅ͷPCήʔϜͱ͔ͰΑ͘໐ͬͯͨ΍ͭ
    2ΦϖϨʔλ ϑΟʔυόοΫͳ͠

    View Slide

  50. ຊ෺ͷϐΞϊ
    ٯFMԻݯͰ࡞ͬͨFMԻݯϐΞϊ
    2ΦϖϨʔλ ϑΟʔυόοΫͳ͠

    View Slide

  51. ຊ෺ͷϐΞϊ
    WassersteinٯFMԻݯͰ
    ࡞ͬͨFMԻݯϐΞϊ
    2ΦϖϨʔλ ϑΟʔυόοΫͳ͠

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  52. ՝୊

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  53. $ time wav2fm2 -i Piano.ff.C4.aiff -d 1 -n 60 -t 40
    ...
    real 4m43.748s
    user 17m5.382s
    sys 0m3.239s
    $ time wav2fm -i Piano.ff.C4.aiff -d 1 -n 60
    ...
    real 3m27.189s
    user 3m29.071s
    sys 0m0.036s
    ٯFMԻݯͰϐΞϊͷύϥϝʔλΛ୳͢
    WassersteinٯFMԻݯͰϐΞϊͷύϥϝʔλΛ୳͢
    ΍͸Γ਺஋ඍ෼͸஗͍
    Intel Core i5-6500 (Gentoo Linux)Λ࢖༻

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  54. Sinkhorn Distances: Lightspeed Computation of Optimal Transport
    Cuturi, Marco. "Sinkhorn distances: lightspeed computation of optimal transport." NIPS.
    Vol. 2. No. 3. 2013.
    https://papers.nips.cc/paper/2013/hash/af21d0c97db2e27e13572cbf59eb343d-
    Abstract.html
    Wassersteinڑ཭ͷۙࣅ
    Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance
    Luise, Giulia, et al. "Differential properties of sinkhorn approximation for learning with
    wasserstein distance." arXiv preprint arXiv:1805.11897 (2018).
    Wassersteinڑ཭ͷۙࣅͷඍ෼
    https://arxiv.org/abs/1805.11897
    ্ख͘΍Ε͹͜ͷ΁Μ͕࢖͑Δ͔΋

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  55. Λ੔਺ʹἧ͑Δ΂ָ͖ثͱἧ͑ͯ͸͍͚ͳָ͍ث͕͋Δ
    ω
    WassersteinٯFMԻݯͰ͸ Λ੔਺ʹറΖ͏ͱ͢Δޯ഑͸ੜ͡ͳ͍ͷͰ
    ͸ඇ੔਺ͷ஋͕ग़΍͍͢
    ω
    ω

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  56. Λ੔਺ʹἧ͑Δ΂ָ͖ثͱἧ͑ͯ͸͍͚ͳָ͍ث͕͋Δ
    ω
    WassersteinٯFMԻݯͰ͸ Λ੔਺ʹറΖ͏ͱ͢Δޯ഑͸ੜ͡ͳ͍ͷͰ
    ͸ඇ੔਺ͷ஋͕ग़΍͍͢
    ω
    ω
    όΠΦϦϯͷΑ͏ʹഒԻ͕੔਺ഒʹͳ͍ͬͯͳ͍ࣄ͕
    ຊ෺Β͠͞Λग़ָ͢ث΋͋Ε͹
    τϥϯϖοτͷΑ͏ʹഒԻ͕੔਺ഒʹͳ͍ͬͯͳ͍ͱ
    ຊ෺Β͕͠͞ग़ͳָ͍ث΋͋Δ

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  57. ຊ෺ͷτϥϯϖοτ
    WassersteinٯFMԻݯͰ
    ࡞ͬͨFMԻݯτϥϯϖοτ
    2ΦϖϨʔλ ϑΟʔυόοΫͳ͠
    Λ੔਺஋ʹݻఆ͠ͳ͍
    ω

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  58. ຊ෺ͷτϥϯϖοτ
    WassersteinٯFMԻݯͰ
    ࡞ͬͨFMԻݯτϥϯϖοτ
    2ΦϖϨʔλ ϑΟʔυόοΫͳ͠
    Λ1ʹݻఆ͢Δ
    ω

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  59. Λಛఆͷ஋ʹݻఆͨ͠ํ͕ྑ͍݁Ռ͕ಘΒΕΔ͔
    ݻఆ͠ͳ͍ํ͕ྑ͍݁Ռ͕ಘΒΕΔ͔Λ
    ࣗಈͰ൑அ͢Δखஈ͸ࠓͷͱ͜Ζແ͍
    ω

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  60. ·ͱΊ
    FMԻݯͰຊ෺ͷָثΒ͍͠ԻΛग़͢ύϥϝʔλΛ
    ޯ഑๏ Ͱ୳͢ͱ͖͸
    Wassersteinڑ཭
    Λ࢖͏ͱྑ͍

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