Upgrade to Pro — share decks privately, control downloads, hide ads and more …

Representation Learning for Scale-free Networks: スケールフリーネットワークに対する表現学習

OpenJNY
November 09, 2018

Representation Learning for Scale-free Networks: スケールフリーネットワークに対する表現学習

OpenJNY

November 09, 2018
Tweet

More Decks by OpenJNY

Other Decks in Science

Transcript

  1. 3FQSFTFOUBUJPO-FBSOJOHGPS
    4DBMF'SFF/FUXPSLT
    εέʔϧϑϦʔωοτϫʔΫʹର͢Δදݱֶश
    ৘ใཧ޻ֶӃ৘ใ޻ֶίʔε
    Ҫ্ݚڀࣨ.ࢁޱॱ໵ʢ.ʣ

    View full-size slide

  2. 4DBMFGSFF/FUXPSL
    εέʔϧϑϦʔʢTMBDFGSFFʣωοτϫʔΫͱ͸ɺϊʔυͷ࣍਺ʢJFۙ๣ϊʔυ਺ʣͷ෼෍͕ɺ΂
    ͖৐ଇʢQPXFSMBXʣʹै͏ωοτϫʔΫͷ͜ͱ
    DGࢦ਺͕ͷ΂͖৐ଇ
    ྫʣҰ෦ͷਓ͸ඇৗʹͨ͘͞Μ༑ୡ͕͍ͯɺେଟ਺͸༑ୡ͕গͳ͍
    ස౓
    ࣍਺
    ࢦ਺ؔ਺తʹݮগ

    View full-size slide

  3. ࣍਺෼෍ͷҧ͍
    http://www.network-science.org/

    View full-size slide

  4. ैདྷͷάϥϑຒΊࠐΈख๏
    ࣗવݴޠॲཧͷք۾Ͱ༗໊ͳXPSEWFD<.JLPMPWFUBM>ΛɺϊʔυຒΊࠐΈʢ/PEF&NCFEEJOHʣ
    ʹద༻ͨ͠OPEFWFD<(SPWFSBOE-FTLPWFD>͕੒ޭΛऩΊ͍ͯΔ
    ୯ޠϊʔυ
    ηϯςϯεάϥϑ্ͷϥϯμϜ΢ΥʔΫྻ
    ·ͨɺϊʔυຒΊࠐΈ͸ɺଟ༷ମֶशͱݺ͹ΕΔ
    ඇઢܗ࣍ݩ࡟ݮख๏ͷҰ෦ͷύʔτͱͯ͠ɺੲ͔
    Β੝Μʹݚڀ͞Ε͍ͯΔλεΫͷͻͱͭ
    ϥϓϥγΞϯݻ༗Ϛοϓ๏ʢ-BQMBDJBO&JHFONBQʣ
    <#FMLJOBOE/JZPHJ>͕༗໊
    σʔλؒͷྨࣅ౓ΛΤοδͷॏΈͱͨ͠ྨࣅ౓ά
    ϥϑʢTJNJMBSJUZHSBQIʣΛߏஙͯ͠ɺ͜ͷྡ઀ߦ
    ྻͷݻ༗஋෼ղͰ௿࣍ݩຒΊࠐΈΛಘΔ
    https://www.cs.cmu.edu/~aarti/Class/10701/slides/Lecture21_2.pdf

    View full-size slide

  5. ैདྷख๏ͷ໰୊఺
    ঺հͨ͠ैདྷख๏͸ɺ఺ؒͷؔ܎ͷΑ͏ͳɺωοτϫʔ
    Ϋ͕࣋ͭہॴతͳߏ଄ͷ৘ใΛอ࣋͢ΔΑ͏ͳຒࠐΛߦ͏
    ͔͠͠εέʔϧϑϦʔੑ͸େҬతͳ৘ใͷͻͱͭ
    ݱ࣮ͷωοτϫʔΫʹසൟʹݟΒΕΔ͜ͷ৘ใΛແࢹ͢Δͷ
    ͸೗Կͳ΋ͷ͔ʁ
    ࣮ࡍɺಘΒΕͨຒΊࠐΈͰ΋ͱͷάϥϑΛ࠶ߏஙͨ͠ͱ͖ɺ
    ैདྷख๏͸IJHIEFHSFFͳϊʔυ਺͕ଟ͘ͳΔ܏޲ʹ͋Δ
    ͭ·ΓɺຒΊࠐΈۭؒͰσʔλಉ͕࢜ͻ͖ͬͭ͗ͯ͢ɺશମ
    తʹ͙ͪΌͬͱͳͬͯ͠·͍ͬͯΔ
    ʮଟ͘ͷϊʔυͷۙ͘ʹډΕΔݖརΛɺগ਺ϊʔυʹ͚ͩ༩͑Δʯ੍໿͕ඞཁ

    View full-size slide

  6. ͜ͷ࿦จͰ΍Γ͍ͨຒΊࠐΈ
    Node
    Embedding Reconstruction
    ޡϦϯΫʢΤοδʣͷ࠷খԽ
    ϢʔΫϦουڑ཭͕ᮢ஋ΑΓ
    ΋খ͚͞Ε͹ΤοδΛߏ੒

    View full-size slide

  7. Node
    Embedding Reconstruction
    ؾ࣋ͪɿ͍͍ײ͡ʹ֤ϊʔυΛݻఆ௕ϕΫτϧ
    ΁ରԠ෇͚͍ͨʢ೚ҙͷάϥϑʹରͯ͠ରԠ෇
    ͚Δؔ਺Λֶश͢ΔΘ͚Ͱ͸ͳ͍ʣ

    View full-size slide

  8. ͜ͷ࿦จͰ΍Γ͍ͨຒΊࠐΈ
    Node
    Embedding Reconstruction
    ޡϦϯΫʢΤοδʣͷ࠷খԽ
    ϢʔΫϦουڑ཭͕ᮢ஋ΑΓ
    ΋খ͚͞Ε͹ΤοδΛߏ੒

    View full-size slide

  9. ͦ΋ͦ΋εέʔϧϑϦʔੑ͸FNCFEͰ͖Δͷ͔໰୊
    ௚ײతʹ͸ɺຒΊࠐΈۭؒʹ͓͍ͯσʔλ͕ΰνϟͬͱͯ͠͠·͏͔Βɺάϥϑ࠶ߏங࣌ʹߴ࣍਺ϊʔ
    w w w
    υ͕૿͑ͯ͠·͏ͱਪଌͰ͖Δ
    ΰνϟͬͱ͍ͯ͠Δߴ࣍਺ϊʔυ͕ଟ͘ͷ௿࣍਺ϊʔυΛۙ͘ʹҾ͖دͤΔ͕ނʹɺҾ͖دͤΒΕͨ௿࣍਺ϊʔ
    υಉ࢜ʹ΋άϥϑ࠶ߏங࣌ʹΤοδ͕݁͹Εͯ͠·͏Α͏ͳঢ়ଶ
    Ͱ͸ԿΒ͔ͷ੍໿ΛՃ͑ͯɺΰνϟͬͱ͠ͳ͍Α͏ʹͰ͖Δͷ͔ʁ
    ྫ͑͹࣍ݩϢʔΫϦουۭؒʹຒΊࠐΈΛ͢ΔͳΒɺc7cͰ͢Βແཧͦ͏
    ͔ͱ͍ͬͯc7cʹରͯ͠ɺ.࣍ݩͷۭؒΛ༻ҙ͢Δͷ͸΍Γա͗ʜ
    ͦ͜Ͱɺ,࣍ݩͷϢʔΫϦουۭؒʹຒΊࠐΜͩ࣌ʹɺ໰୊ͳ͘εέʔϧϑϦʔੑΛ࠶ߏஙͰ͖ΔΑ͏
    ͳϊʔυ਺ͷ࠷େ஋ʢʹର͢ΔԼքʣʹ͍ͭͯɺཧ࿦తͳղੳΛߦͬͨ

    View full-size slide

  10. ݁࿦͚ͩड़΂Δͱ

    ໰୊ͳͦ͞͏
    k 10 20 50 100
    lower
    bound
    57 3325 637M 4E+17
    .@LL࣍ݩϢʔΫϦουۭؒʹ͓͍ͯɺத৺Y൒ܘЏͷ௒ٿ# Y Џ
    ͷ
    தʹɺ͓ޓ͍͕ЏҎ্཭ΕͯΔ఺Λ࠷େͰԿݸ഑ஔͰ͖Δ͔ʁ
    ʢ࣍ݩͳΒʣ࣍਺͕&ͷεʔύʔϋϒ
    ͷपΓʹɺ࣍਺͕ͷ༿ϊʔυΛ&ݸຒΊ
    ࠐΈͰ͖Δ
    େମͷݱ࣮ੈքͷωοτϫʔΫ͸ɺे෼ʹεέʔ
    ϧϑϦʔੑΛอ࣋ͨ͠··ຒΊࠐΈՄೳ
    ͪͳΈʹ
    ఆཧ͸ɺϊʔυ͕࠷େͰԿݸ഑ஔͰ͖Δ͔
    ໰୊Λɺٿॆరʢ4QIFSF1BDLJOHʣ໰୊ɺ
    ͋Δ͍͸ٿମͷ࠷ີॆర໰୊ͱͯ͠஌ΒΕΔ
    ໰୊ʹؼண͢Δ͜ͱͰূ໌͞Ε͍ͯΔɻ

    View full-size slide

  11. طଘख๏ͷ cons
    εέʔϧϑϦʔੑΛߟྀ͠ͳ͍ͱͲΜͳ໰୊͕ੜ͡Δͷ͔ʁ

    View full-size slide

  12. ϏʔόʔͱΦόϚɺ஥ྑ͠໰୊
    ैདྷख๏Ͱ͸ۙ͘ʹ
    ຒΊࠐΈ͞ΕΔ྆ऀ

    View full-size slide

  13. ϏʔόʔͱΦόϚɺ஥ྑ͠໰୊΁ͷରॲࡦ
    ैདྷख๏ͷଟ͘͸TU OEPSEFSͷQSPYJNJUZΛอͭ͜ͱʹઐ೦
    TUPSEFSQSPYJNJUZ˺ʮ༑ୡʯ౓߹͍
    OEPSEFSQSPYJNJUZ˺ʮ༑ୡͷ༑ୡʯ౓߹͍
    ͦͷ݁ՌɺεέʔϧϑϦʔωοτϫʔΫʹଘࡏ͢ΔlCJHIVCzಉ࢜͸ྨࣅ౓͕ߴ͘ͳΓ΍͍͢
    ྫʣΦόϚͷ༑ୡʹ͸ɺδϟεςΟϯɾϏʔόʔͱ༑ୡͳਓ͕ਓͰ΋ډΔ֬཰͸ߴ͍
    ྨࣅ౓͕ߴ͘ͳΓ΍͍͢ͳΒɺ࣍਺͕ߴ͍ϊʔυͷQSPYJNJUZʹϖφϧςΟΛ༩͑Ε͹ྑ͍
    جຊతํ਑ཧ࿦ղੳͷ݁Ռʹج͖ͮɺεέʔϧϑϦʔੑΛ୲อͰ͖
    ΔຒΊࠐΈΞϧΰϦζϜͷͨΊͷ࣍਺േଇʢEFHSFFQFOBMUZʣݪଇ
    ΛఏҊ͢Δɻ࣍਺േଇ͸ɺ࣍ٴͼ࣍ͷQSPYJNJUZΛ୲อ্ͨ͠Ͱɺ
    ߴ͍࣍਺Λ΋ͭ௖఺ಉ࢜ͷQSPYJNJUZʹରͯ͠േΛՊ͢ݪଇͰ͋Δɻ

    View full-size slide

  14. ఏҊख๏%14QFDUSBM
    2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix
    ྡ઀ߦྻ ࣍਺ߦྻ%EJBH E@ ʜ E@O

    https://www.slideshare.net/pecorarista/ss-51761860
    ॏΈ෇͖ྡ઀ߦྻʢXFJHIUFEBEKBDFODZNBUSJYʣͷྫ

    View full-size slide

  15. ఏҊख๏%14QFDUSBM
    ͩTG
    2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix
    ྡ઀ߦྻ ࣍਺ߦྻ%EJBH E@ ʜ E@O

    View full-size slide

  16. ఏҊख๏%14QFDUSBM
    ͩTG
    2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix
    ྡ઀ߦྻ ࣍਺ߦྻ%EJBH E@ ʜ E@O

    J K
    ؒͷQSPYJNJZ͕ߴ͍ʢ8@JK
    ͷ஋͕େ͖͍ʣͳΒɺຒΊࠐΈ
    ͷڑ཭Λ୹͍ͨ͘͠
    ॏཁ౓ʢ%@JʣͰεέʔϧௐઅͨ͠ޙ
    ͷۭؒͰɺ௨ৗͷݻ༗ϕΫτϧͷੑ࣭
    Λຬͨ͠
    ͯཉ͍͠ʢҰൠԽݻ༗஋໰୊ʣ
    ∀i : u⊤
    i
    Dui
    = 1
    ∀i, j(i ≠ j) : u⊤
    i
    Duj
    = 0
    Wij
    =
    1
    (di
    dj

    C′
    ij
    ࣍਺ͷߴ͞ʹରͯ͠ࢦ਺
    తͳϖφϧςΟΛ՝͢

    View full-size slide

  17. ఏҊख๏%14QFDUSBM
    ͩTG
    2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix
    ྡ઀ߦྻ ࣍਺ߦྻ%EJBH E@ ʜ E@O

    J K
    ؒͷQSPYJNJZ͕ߴ͍ʢ8@JK
    ͷ஋͕େ͖͍ʣͳΒɺຒΊࠐΈ
    ͷڑ཭Λ୹͍ͨ͘͠
    ॏཁ౓ʢ%@JʣͰεέʔϧௐઅͨ͠ޙ
    ͷۭؒͰɺ௨ৗͷݻ༗ϕΫτϧͷੑ࣭
    Λຬͨ͠
    ͯཉ͍͠ʢҰൠԽݻ༗஋໰୊ʣ
    ∀i : u⊤
    i
    Dui
    = 1
    ∀i, j(i ≠ j) : u⊤
    i
    Duj
    = 0
    LͳΒ
    Wij
    =
    1
    (di
    dj

    C′
    ij
    ࣍਺ͷߴ͞ʹରͯ͠ࢦ਺
    తͳϖφϧςΟΛ՝͢
    ৽نੑ͸8ͷ࡞Γํ

    View full-size slide

  18. ఏҊख๏%18BMLFS
    ϥϯμϜ΢ΥʔΫʹجͮ͘طଘख๏ʹ࣍਺ϖφϧςΟΛՃ͑Δ
    https://towardsdatascience.com/node2vec-embeddings-for-graph-data-32a866340fef
    ͜͜Λͪΐͬͱ͍͡Δ͚ͩ

    View full-size slide

  19. ఏҊख๏%18BMLFS
    i
    high degree
    low degree
    طଘख๏Ͱ͸ɺϥϯμϜ΢ΥʔΫͷࡍͷ
    ભҠ֬཰͸ɺۙ๣ϊʔυͰಉ֬͡཰
    ࣍਺͕ߴ͍ϊʔυ΁ͷભҠ֬཰Λখ͘͞
    ͢ΔΑ͏ͳ੍໿ΛՃ͑Δ
    i
    ߴ͍࣍਺ͷϊʔυͳΒɺଞͷϊʔυ
    ͔ΒͷϥϯμϜ΢ΥʔΫͰ͍͔ͭདྷ
    ΕΔͰ͠ΐ

    View full-size slide

  20. σʔληοτ
    Vertex Edge |V| |E|
    Synthetic 10K 400K
    Facebook Ϣʔβ ༑ୡ 4K 88K
    Twitter Ϣʔβ ϑΥϩʔ 81K 1.76M
    Author ஶऀ ڞஶ 5K 29K
    Citation ஶऀ Ҿ༻ 48K 357K
    Mobile ొ࿥ऀ ௨࿩ 198K 1.15M

    View full-size slide

  21. ൺֱख๏
    ུশ Ҿ༻ ৄࡉ
    ϥϓϥγΞϯݻ༗
    Ϛοϓ๏
    LE
    Belkin and Niyogi
    2003
    εϖΫτϧϕʔεͷຒΊࠐΈख๏ɻ
    DP-Spectral ͱͷҧ͍͸ɺྨࣅ౓ߦྻͷߏங๏ํ๏
    Deep Walk DW
    Perozzi, Al-Rfou,
    and Skiena
    2014
    skip-gram Ϟσϧʹجͮ͘ຒΊࠐΈɻ֤௖఺͔Β10ݸͷϥϯ
    μϜ΢ΥʔΫྻʢྻ௕:40ʣΛੜ੒͢Δɻ
    DP-Walker ͱͷҧ͍͸ɺભҠ֬཰͕Ұ༷Ͱ͋Δ͜ͱ
    ఏҊख๏1 DP-Spectral - LE + ྨࣅ౓ߦྻʹ࣍਺േଇΛద༻
    ఏҊख๏2 DP-Walker - DW + ϥϯμϜ΢ΥʔΫͷભҠ֬཰ʹ࣍਺േଇΛద༻
    શख๏ڞ௨ͯ͠ɺຒΊࠐΈ࣍ݩ͸Λ࠾༻

    View full-size slide

  22. λεΫ
    ωοτϫʔΫͷ࠶ߏங
    ʲ໨తʳεέʔϧϑϦʔੑΛอ࣋ͨ͠ຒΊࠐΈ
    ʲํ๏ʳલड़௨Γʢলུʣ
    ʲධՁʳೖྗͱ࠶ߏஙωοτϫʔΫͷͦΕͧΕͷ࣍਺෼෍ΛٻΊɺϐΞιϯͷੵ཰૬ؔ܎਺
    < >Λܭࢉ͢Δʢߴ͍ͱྑ͍ʣ
    ʲඋߟʳ֤ख๏Ͱɺ͔Β·Ͱͷൣғ ࠁΈ
    ͰЏΛಈ͔͠ɺ࠷ྑ࣌ͷ૬ؔ܎਺Λใࠂ
    ϦϯΫ༧ଌ
    ʲ໨తʳϊʔυWJ WKؒʹΤοδ͕͋Δ͔ͷ༧ଌ
    ʲํ๏ʳຒΊࠐΈۭؒͰͷࠩ෼V@JV@KΛಛ௃ྔͱͯ͠ઢܗճؼϞσϧʹೖྗ͢Δ
    ʲධՁʳਫ਼౓ʢQSFDJTJPOʣɺ࠶ݱ཰ʢSFDBMMʣɺ'஋ʢ'TDPSFʣ
    ʲඋߟʳແ࡞ҝʹબ͹ΕͨϊʔυϖΞͷू߹Λ܇࿅ධՁσʔλʢͦΕͧΕશϊʔυϖΞͷ
    ʣͱ͢Δ
    ϊʔυ෼ྨ
    ʲ໨తʳϊʔυʹ෇༩͞Ε͍ͯΔϥϕϧͷ༧ଌ
    ʲํ๏ʳ༩͑ΒΕͨW@Jʹରͯ͠ɺຒΊࠐΈV@JΛಛ௃ϕΫτϧͱͯ͠ઢܗ෼ྨػʹೖྗ
    ʲධՁʳਖ਼ղ཰ʢBDDVSBDZʣ
    ʲඋߟʳσʔληοτ$JUBUJPOͰͷΈλεΫΛ࣮ߦɻϥϕϧ͸ͭͷݚڀ෼໺
    i
    P((i, j ) ∈ E ) ≜ sigmoid(w⊤(ui − ui) + b)
    j
    {Architecture
    Computer Network
    Computer Science
    Data Mining
    Theory
    Graphics
    Unknown
    i
    P(i ∈ l ) ≜ softmax(w⊤
    l
    ui
    + bl
    )

    View full-size slide

  23. ݁ՌɿωοτϫʔΫ
    ࠶ߏஙλεΫ
    ϐΞιϯ૬ؔ܎਺ͷ%18BMLFS͸ɺ
    εϐΞϚϯ૬ؔ܎਺ͩͱ%14QFDUSBM
    ͸Ͱੑೳ޲্͕ΈΒΕͨ
    ࠷దͳЏͷ஋΋ɺ%14QFDUSBM͸
    ෇ۙͰ҆ఆ͍ͯ͠Δ
    %1WBSJBOUT͸ɺεέʔϧϑϦʔੑΛ
    ΑΓอ࣋ͨ͠··ຒΊࠐΈͰ͖Δख๏
    Ͱ͋Δͱݴ͑Δ

    View full-size slide

  24. ݁ՌɿωοτϫʔΫ࠶ߏஙλεΫ
    ຒΊࠐΈ࣍ݩ
    േଇͷڧ͞ Wij =
    1
    (didj)β
    C′
    ij
    %14QFDUSBMͷ΄͏͕ෆ҆ఆͳͷ͸ɺЌ͕௚઀໨తؔ਺ʹ૊Έࠐ·Ε͍ͯ
    Δ͔Βɻ%18BMLFS͸ϥϯμϜ΢ΥʔΫྻੜ੒ʹ͔͔͔͠ΘΒͳ͍
    4ZOUIFUJDͱ'BDFCPPLͰ࠷దͳЌ͕ҟͳΔ͕ɺ͜Ε͸േଇ͕࣍਺ͷߴ͞
    ͦͷ΋ͷʹ՝ͤΒΕ͍ͯΔ͔ΒͰɺτϙϩδʔ͕มΘΕ͹Ќͷޮ͖۩߹͍
    ͸มԽ͢Δɻ
    εέʔϧϑϦʔੑΛ୲อ͢Δͷʹे෼ͳ࣍ݩ਺͕ଘࡏ͢Δ͜ͱ͕෼͔Δɻ
    %14QFDUSBM͸ຒΊࠐΈ࣍ݩ͕গͳ͍ͱੑೳ͕ඇৗʹѱ͍͕ɺ%18BMLFS
    ͸࣍ݩ਺͕૿͑ͯ΋͋·ΓมԽͤͣɺ҆ఆͯ͠ྑ͍݁ՌɻϥϯμϜ΢Υʔ
    Ϋͷઓུ͕ޮ͍͍ͯΔͷ͕ཁҼͰ͋ΔͱਪଌͰ͖Δɻ

    View full-size slide

  25. ݁ՌϦϯΫ༧ଌ
    ͍͍ͩͨͷέʔεͰ%14QFDUSBM͕࠷
    ΋ྑ͍ੑೳ
    ࣍ݩേଇͷݪଇ͸ɺ͜ͷλεΫͷͨΊ
    ʹ௚઀ઃܭ͞Ε͍ͯΔΘ͚Ͱ͸ͳ͍͕ɺ
    ΑΓ༗ӹͳ৘ใΛଟ͘ຒΊࠐΈͰ͖ͯ
    ͍Δ݁Ռɺ޷੒੷ΛऩΊͨͱਪଌ͞Ε
    Δ

    View full-size slide

  26. ݁Ռϊʔυ෼ྨ
    %14QFDUSBM͸ϥϓϥγΞϯݻ༗Ϛοϓ๏Λɺ%1
    8BMLFS͸%FFQ8BMLΛ౗ͨ͠ɻ
    %14QFDUSBM͸҆ఆͯ͠޷੒੷Λ࢒͍ͯ͠Δɻ
    ඪ४ภࠩ΋%14QFDUSBMͷ΄͏͕গͳ͔ͬͨ
    %14QFDUSBM
    -&
    %FFQ8BML
    ϊʔυ෼ྨλεΫͷਖ਼ղ཰ʢBDDVSBDZʣΛใࠂͨ͠ද
    ֤Ϛε͸σʔληοτͰͷBDDVSBDZͷฏۉ஋Λ͍ࣔͯ͠Δ

    View full-size slide

  27. ײ૝
    ʮ୯ʹطଘख๏ʹߴ࣍਺ͷϊʔυʹϖφϧςΟΛ՝͚ͩ͢ʯͱ͍͏ࢸͬͯγϯϓϧͳख๏Ͱɺগ͠ݞ͔͢͠Λ৯
    Βͬͨɻ
    ͱ͸͍͑)JHIPSEFSQSPYJNJUZΛ֫ಘ͢ΔຒΊࠐΈख๏͸ɺաڈʹ͋·Γͳ͍ͷͰ༗ӹ
    ʢϊʔυຒΊࠐΈʹର͢ΔʣεϖΫτϧ෼ղख๏͸ɺύϥϝʔλʢFHຒΊࠐΈ࣍ݩ਺ʣʹରͯ͠ඇৗʹ҆ఆ͠
    ͳ͍͜ͱ͕Θ͔ͬͨͷ͸େ͖ͳऩ֭ɻ
    ϥϯμϜ΢ΥʔΫख๏͕҆ఆ͍ͯ͠Δͷ͸ɺඇઢܗͳQSPYJNJUZΛ௿࣍ݩͳۭؒͰ্खʹଊ͑ΒΔ͔ΒʁʢXPSEWFDͷڧΈ
    Λ׆༻Ͱ͖͍ͯΔͷ͸ؒҧ͍ͳͦ͞͏ʣ
    εέʔϧϑϦʔੑͷอ࣋ͷͨΊʹ͸ɺ࣍਺ϖφϧςΟҎ֎ʹ΋खஈ͸͋Γͦ͏ɻ
    ҰൠͷϊʔυຒΊࠐΈख๏ʹద༻Ͱ͖ΔɺεέʔϧϑϦʔੑΛ֫ಘ͢Δҝͷϝλઓུख๏͕"""*Ͱൃද͞Ε͍ͯΔ
    ͪ͜Β͸άϥϑΛ֊૚ߏ଄ʹ෼ׂ͠େہతߏ଄ͷຒΊࠐΈΛ֫ಘ͍ͯ͠Δ
    LDPSFΛ࢖ͬͯάϥϑΧʔωϧΛվળͨ͠ϝλઓུͱࣅ͍ͯͯɺLDPSFΛ࢖ͬͨϊʔυຒΊࠐΈͰ΋εέʔϧϑϦʔੑ͕֫
    ಘͰ͖ͦ͏ʢ͜ͷख๏ͷ࿦จ͕ൃද͞ΕΔͷ͸࣌ؒͷ໰୊ͳؾ΋͢Δ͕ʜʣ
    /JLPMFOU[PT (JBOOJT FUBM"%FHFOFSBDZ'SBNFXPSLGPS(SBQI4JNJMBSJUZ*+$"*
    $IFO )BPDIFO FUBM)"31IJFSBSDIJDBMSFQSFTFOUBUJPOMFBSOJOHGPSOFUXPSLT"""*

    View full-size slide