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Continuous Simplicial Neural Networks

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June 29, 2025
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Continuous Simplicial Neural Networks

第 2 回 Spatial AI 勉強会

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horiem

June 29, 2025

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Transcript

  1. $POUJOVPVT4JNQMJDJBM/FVSBM/FUXPSLT "SFG&JOJ[BEF %PSJOB5IBOPV 'SBHLJTLPT%.BMMJBSPT +IPOZ)(JSBMEP BS9JW ୈճ4QBUJBM"*ษڧձ ೥݄೔ ൃදऀງߐਖ਼৴ 3*$04

  2.  ࣗݾ঺հ • ໊લງߐਖ਼৴ • ॴଐגࣜձࣾ3*$04ɺཧݚ"*1 ٬һݚڀһ  • ઐ໳

    - ෺ཧγϛϡϨʔγϣϯ ภඍ෼ํఔࣜ ͷ ͨΊͷػցֶश - ܈ಉมχϡʔϥϧωοτϫʔΫ - άϥϑχϡʔϥϧωοτϫʔΫ (//  • ͓΋ͳ࿦จ - Horie+ ICML 2024 อଘଇΛݫີʹຬͨ͢(//  - Horie+ NeurIPS 2022 ڥք৚݅Λݫີʹຬͨ͢ (//ɺHMPCBMQPPMJOHͱγϛϡϨʔγϣϯख๏ͱͷ ؔ࿈ͷղ໌  - Horie+ ICLR 2021 ߹ಉม׵ରশੑΛݫີʹຬͨ͠ߴ ଎ͳ(//  .11%& [Brandstetter+ 2022] ఏҊख๏ [Horie & Mitsume 2024] 5FTU 5FTUΛ ճస ྖҬͷ ֎ૠ 5FTUΛ ֦େ ྲྀ଎ ࢀরղ ࢀরղ ఏҊख๏

  3.  ·ͱΊ • ୯ମෳମ ܗঢ়ͷτϙϩδʔΛ཭ࢄతʹද͢σʔλߏ଄ ʹର͢Δ৞ΈࠐΈԋࢉΛ ඍ෼ํఔࣜͷ஌ݟΛ΋ͱʹ࿈ଓԽͨ͠ $04*.0 DPOUJOVPVTTJNQMJDJBMOFVSBMOFUXPSL ΛఏҊ

    • ࣮໰୊Ͱͷ࣮ݧʹ͓͍ͯ4PU"΋͘͠͸ͦΕʹ͍ۙੑೳ • $04*.0ʹ͓͍ͯܗঢ়ͷϊΠζʹର͢Δޡࠩͷ্քΛಋग़͠ɺ࣮ݧతʹ΋֬ೝ • $04*.0͸PWFSTNPPUIJOH (//ʹΑΔܗঢ়ೝࣝͰ໰୊ͱͳΓ΍͍͢৘ใͷࢄҳ ͷ ఔ౓Λ੍ޚͰ͖Δ͜ͱΛূ໌͠ɺ࣮ݧతʹ΋֬ೝ  ୯ମෳମͷྫ[Einizade+ 20025]

  4.  ໨࣍  • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// •

    $04*.0 • ਺஋࣮ݧ • ·ͱΊ

  5.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  6.  Ϟνϕʔγϣϯ • ࣍ݩܗঢ়Λޮ཰తʹදݱ͢Δʹ͸ϝογϡ·ͨ͸ ͦͷऔΓѻ͍΍͍͢ϞσϧͰ͋Δ୯ମෳମ͕༗༻ • ࣍ݩܗঢ়ͷػցֶशʹ(//͕༻͍ΒΕ͖͕ͯͨ (//͸άϥϑͷ௖఺Ͳ͏͠ͷ૬ޓ࡞༻ͷΈΛ ߟྀ͓ͯ͠ΓΑΓߴ࣍ͷ૬ޓ࡞༻͕ݟಀ͞Ε͕ͪ -

    ࡾ֯ܗͷ໘ੵΛٻΊΔૢ࡞͸ ʮ఺ͷಛ௃ྔˠ໘ͷಛ௃ྔʯ ͷࣸ૾͕ͩ(//ͰͦͷΑ͏ͳૢ࡞͸ඇࣗ໌ • ୯ମෳମΛऔΓѻ͑Δ//΋ग़͖͍ͯͯΔ͕ (//ͷ՝୊Ͱ͋ΔPWFSTNPPUIJOH΍ ܗঢ়ͷϊΠζʹର͢Δݎ࿚ੑ͸ґવͱͯ͠՝୊ ➡ ৞ΈࠐΈԋࢉΛ࿈ଓԽ͠ਫ਼៛Խ͢Δ͜ͱͰ ͜ΕΒͷ՝୊ΛղܾͰ͖ͳ͍͔ʁ  ϝογϡͷྫ ୯ମෳମͷྫ [Einizade+ 20025]

  7.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  8.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  9.  ४උ୯ମෳମ ఆٛ ୯ମෳମTJNQMJDJBMDPNQMFY  ۭͰͳ͍༗ݶू߹ ʹର͠ɺ ͰҎԼͷ৚݅Λຬͨ͢΋ͷΛ ୯ମෳମ TJNQMJDJBMDPNQMFY

    ͱ͍͏ • ͷ֤ݩ ʹରͯ͠ɺ  • ੍ݶʹ͍ͭͯด͍ͯ͡Δɺ͢ͳΘͪ 𝒱 ( 𝒱 , 𝒳 ⊂ 2 𝒱 ∖{∅}) 𝒱 v {v} ∈ 𝒳  ఆٛ ୯ମTJNQMFY  ୯ମෳମ ʹରͯ͠ɺ Λ୯ମ TJNQMFY ͱ͍͍ɺ ͷͱ͖ɺಛʹL୯ମ LTJNQMFY ͱ͍͏ɻ ( 𝒱 , 𝒳 ) sk ∈ 𝒳 |sk | = k + 1 ୯ମ ୯ମ ୯ମ ୯ମ

  10.  ४උ઀ଓߦྻ  ఆٛ ઀ଓߦྻTJNQMFY  ୯ମෳମ ʹରͯ͠ɺL୯ମͷΈΛूΊͨ΋ͷΛ ͱॻ͘ɻߦྻ 

    ͷ ੒෼ʹ͍ͭͯɺJ൪໨ͷ L ୯ମ͕K൪໨ͷL୯ମͷ໘ʹͳ͓ͬͯΓɺ޲͖͕ἧ͍ͬͯ Δ L୯ମͷۮஔ׵Λͯ͠ઌ಄ L ྻΛൈ͖ग़ͯ͠ L ୯ମͱҰக͢Δ ͱ͖ɺٯ޲͖ͷͱ ͖ɺͦΕҎ֎ͷͱ͖ΛׂΓ౰ͯͨ΋ͷΛ઀ଓߦྻͱ͍͏ɻ ( 𝒱 , 𝒳 ) 𝒳 k Bk ∈ {−1,0,1}| 𝒳 k−1 |×| 𝒳 k | (i, j) ఆٛ ୯ମͷ޲͖  ͷݩʹॱংΛఆΊ͓ͯ͘ɻL୯ମ ΛλϓϧͱΈͳͨ͠΋ͷ  ʹରͯ͠ɺۮஔ ׵ͰঢॱʹͰ͖Δͱ͖ɺͦ͏Ͱͳ͍ͱ͖ΛׂΓ౰ͯͨ΋ͷΛ୯ମͷ޲͖ͱ͍͏ɻ 𝒱 s = (v0 , v1 , …, vk ) ఆٛ )PEHF-BQMBDJBO  Λ)PEHF-BQMBDJBOͱݺͼɺୈҰ߲ΛMPXFS-BQMBDJBO ɺୈೋ߲ΛVQQFS -BQMBDJBO ͱݺͿɻಛʹ ͸άϥϑϥϓϥγΞϯͰ͋Δɻ Lk = B⊤ k Bk + Bk+1 B⊤ k+1 Lk,d Lk,u L0 = B1 B⊤ 1

  11.  ४උ઀ଓߦྻ ۩ମྫ       

                                   

  12.  ४උ୯ମ৴߸  ఆٛ ୯ମ৴߸TJNQMJDJBMTJHOBM  Λ ࣍ݩ L୯ମ৴߸ LTJNQMJDJBM

    TJHOBM ͱݺͼɺ ͱಉҰࢹ͢Δɻ xk : 𝒳 k → ℝ xk ∈ ℝ| 𝒳 k | ஫ҙ ͸ޯ഑ɺ ͸ൃࢄɺ ͸ϥϓϥγΞϯ ʹରԠ͍ͯ͠Δɻ B⊤ 1 x0 B1 x1 L0 x0 ఆٛ %JSJDIMFUΤωϧΪʔ  Λ%JSJDIMFU ΤωϧΪʔͱݺͿɻ%JSJDIMFUΤωϧΪʔ͕খ͍͞ ΄Ͳɺ৔͕ۉҰͱͳ͍ͬͯΔɻ E(xk ) = x⊤ k Lk xk = ∥Bk xk ∥2 2 + ∥B⊤ k+1 xk ∥2 2

  13.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  14.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  15.  %JTDSFUF4JNQMJDJBM//4JNQMJDJBM'JMUFS • 4JNQMJDJBMGJMUFS ͸ҎԼͷΑ͏ʹॻ͚Δ ЋɺЌ͸ύϥϝʔλ  • )PEHF-BQMBDJBOͷ΂͖৐ͷॏͶ߹ΘͤͰؔ਺Λදݱ DG,SZMPW෦෼ۭؒ๏

     • )PEHF-BQMBDJBOΛLճ͔͚Δͱ Lϗοϓઌ·Ͱͷ৘ใΛऔΓࠐΊΔ - L ϗοϓҎ্ͷ৘ใ͸ݟ͑ͳ͍ VOEFSSFBDIJOH  - )PEHF-BQMBDJBOΛॏͶΔ΄Ͳ ৘ใ͕֦ࢄ͍͖ࣦͯ͠ΘΕͯ͠·͏ PWFSTNPPUIJOH f : ℝ| 𝒳 k | → ℝ| 𝒳 k |  l = 0 l = 1 l = 2

  16.  %JTDSFUF4JNQMJDJBM//࣍ݩͷ৔߹ • %JTDSFUF4JNQMJDJBM//ͷϨΠϠ ͔Β ΁Ϛοϐϯά ͸ɺ ߴ࣍ݩ΁ͷϓϩδΣΫγϣϯ ͱ௿࣍ݩ΁ͷϓϩδΣΫγϣϯ Λ༻͍ͯҎԼͷΑ͏ʹॻ͚Δ[Isufi+

    2022] (l − 1) l xl k,d = BT k xl k−1 xl k,u = Bk+1 xl k+1 

  17.  %JTDSFUF4JNQMJDJBM//ଟ࣍ݩͷ৔߹ • ߴ࣍ݩͷTJNQMJDJBMTJHOBMΛ ͱͯ͠ɺ%JTDSFUF4JNQMJDJBM//ͷϨΠϠ ͔Β ΁Ϛοϐϯά ͸ ҎԼͷΑ͏ʹॻ͚Δ[Isufi+ 2022]

    (l − 1) l 

  18.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  19.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  20.  $04*.0ภඍ෼ํఔࣜʹΑΔߟ࡯ • %JTDSFUF4JNQMJDJBM//Λ૚ํ޲ʹ࿈ଓԽ͠ภඍ෼ํఔࣜΛಘΔ DG/FVSBM0%&  • ఆ਺܎਺࣌ؒൃలํఔࣜͷղ͸ߦྻࢦ਺ؔ਺Ͱهड़Ͱ͖Δ

  21.  $04*.0ภඍ෼ํఔࣜʹΑΔߟ࡯  • ఆ਺܎਺࣌ؒൃలํఔࣜͷղ͸ߦྻࢦ਺ؔ਺Ͱهड़Ͱ͖Δ • ͨͩ͠ɺߦྻࢦ਺ؔ਺͸ҎԼͷΑ͏ʹఆٛ͞ΕΔ

  22.  $04*.0ϞσϧͷఆࣜԽ  • ఆ਺܎਺࣌ؒൃలํఔࣜͷղ͸ߦྻࢦ਺ؔ਺Ͱهड़Ͱ͖Δ • Ҏ্ΑΓɺ$04*.0ͷϨΠϠʔΛҎԼͷΑ͏ʹߏ੒͢Δ - ࣮ࡍʹ͸ɺߦྻࢦ਺ؔ਺Λ௿ϥϯΫۙࣅͯ͠ܭࢉ͢Δ

  23.  $04*.0ܗঢ়ͷϊΠζʹର͢Δݎ࿚ੑ • ઀ଓߦྻʹઁಈΛ༩͑ͨͱ͖ͷ༧ଌ݁ՌͷͣΕ ͷ্ք͕ಘΒΕ͍ͯΔ δXk  • ઁಈ͕े෼খ͍͞ͱ͖ɺ༧ଌ݁ՌͷͣΕ͸ઁಈʹରͯ͠ઢܗʹൺྫ

  24.  $04*.00WFS4NPPUIJOHʹର͢Δߟ࡯ ཭ࢄͷ৔߹ • %JTDSFUF4JNQMJDJBM//ͷ৔߹͸%JSJDIMFUΤωϧΪʔͷ্ք͕ )PEHF-BQMBDJBOͷݻ༗஋ͷଟ߲ࣜͱͳΔ - ೚ҙͷॏΈʹରͯ͠PWFSTNPPUIJOHΛ཈੍͢ΔͨΊʹ͸)PEHF-BQMBDJBO ͷݻ༗஋Λେ͖ͤ͘Ͷ͹ͳΒͣɺ୯ମෳମΛमਖ਼͠ͳ͚Ε͹ͳΒͳ͍ 

  25.  $04*.00WFS4NPPUIJOHʹର͢Δߟ࡯ ࿈ଓͷ৔߹ • ࿈ଓͷ৔߹͸%JSJDIMFUΤωϧΪʔͷ্քΛθϩʹऩଋ͠ͳ͍Α͏ʹ ͢Δ͜ͱ͕Մೳ 1SPQ 

  26.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  27.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  28.  ਺஋࣮ݧ࣮໰୊Ͱͷݕূ • ͞·͟·ͳ࣮໰୊Ͱ4PU"΋͘͠͸ͦΕʹ४͡ΔੑೳΛ࣮ݱ  [Einizade+ 20025]

  29.  ਺஋࣮ݧܗঢ়ͷϊΠζʹର͢Δݎ࿚ੑ • ୯ମෳମ΁ͷϊΠζ 4/3@ ͱ୯ମෳମ΁ͷϊΠζ 4/3@ Λ ͞·͟·ʹมԽͤ͞ਫ਼౓ΛධՁ -

    ͞·͟·ͳ࣍ݩͷ৘ใΛෳ߹ͤͯ͞࢖༻͍ͯ͠ΔͨΊɺ 4/3@࠷খ/43@࠷େ·ͨ͸4/3@࠷େ/43@࠷খͷͱ͖ʹੑೳͷྼԽ͕গͳ͍  [Einizade+ 20025]

  30.  ਺஋࣮ݧ0WFS4NPPUIJOHͷ੍ޚ • U ࢦ਺ؔ਺ͷݞʹͷ͍ͬͯΔύϥϝʔλ Λখ͘͢͞Δ ʙ ͜ͱʹΑͬͯ EJTDSFUFTJNQMJDJBM//ΑΓ΋PWFSTNPPUIJOHΛ཈੍͢Δ͜ͱ͕Մೳ 

    [Einizade+ 20025]

  31.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  32.  ໨࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0

    • ਺஋࣮ݧ • ·ͱΊ 

  33.  ·ͱΊ • ୯ମෳମ ܗঢ়ͷτϙϩδʔΛ཭ࢄతʹද͢σʔλߏ଄ ʹର͢Δ৞ΈࠐΈԋࢉ Λ࿈ଓԽͨ͠$04*.0 DPOUJOVPVTTJNQMJDJBMOFVSBMOFUXPSL ΛఏҊ •

    ࣮໰୊Ͱͷ࣮ݧʹ͓͍ͯ4PU"΋͘͠͸ͦΕʹ͍ۙੑೳ • $04*.0ʹ͓͍ͯܗঢ়ͷϊΠζʹର͢Δޡࠩͷ্քΛಋग़͠ɺ࣮ݧతʹ΋֬ೝ • $04*.0͸PWFSTNPPUIJOH (//ʹΑΔܗঢ়ೝࣝͰ໰୊ͱͳΓ΍͍͢৘ใͷ ࢄҳ ͷఔ౓Λ੍ޚͰ͖Δ͜ͱΛূ໌͠ɺ࣮ݧతʹ΋࣮ূ  ୯ମෳମͷྫ[Einizade+ 20025]