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Continuous Simplicial Neural Networks
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horiem
June 29, 2025
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Continuous Simplicial Neural Networks
第 2 回 Spatial AI 勉強会
horiem
June 29, 2025
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Transcript
$POUJOVPVT4JNQMJDJBM/FVSBM/FUXPSLT "SFG&JOJ[BEF %PSJOB5IBOPV 'SBHLJTLPT%.BMMJBSPT +IPOZ)(JSBMEP BS9JW ୈճ4QBUJBM"*ษڧձ ݄ ൃදऀງߐਖ਼৴ 3*$04
ࣗݾհ • ໊લງߐਖ਼৴ • ॴଐגࣜձࣾ3*$04ɺཧݚ"*1 ٬һݚڀһ • ઐ
- ཧγϛϡϨʔγϣϯ ภඍํఔࣜ ͷ ͨΊͷػցֶश - ܈ಉมχϡʔϥϧωοτϫʔΫ - άϥϑχϡʔϥϧωοτϫʔΫ (// • ͓ͳจ - Horie+ ICML 2024 อଘଇΛݫີʹຬͨ͢(// - Horie+ NeurIPS 2022 ڥք݅Λݫີʹຬͨ͢ (//ɺHMPCBMQPPMJOHͱγϛϡϨʔγϣϯख๏ͱͷ ؔ࿈ͷղ໌ - Horie+ ICLR 2021 ߹ಉมରশੑΛݫີʹຬͨ͠ߴ ͳ(// .11%& [Brandstetter+ 2022] ఏҊख๏ [Horie & Mitsume 2024] 5FTU 5FTUΛ ճస ྖҬͷ ֎ૠ 5FTUΛ ֦େ ྲྀ ࢀরղ ࢀরղ ఏҊख๏
·ͱΊ • ୯ମෳମ ܗঢ়ͷτϙϩδʔΛࢄతʹද͢σʔλߏ ʹର͢ΔΈࠐΈԋࢉΛ ඍํఔࣜͷݟΛͱʹ࿈ଓԽͨ͠ $04*.0 DPOUJOVPVTTJNQMJDJBMOFVSBMOFUXPSL ΛఏҊ
• ࣮Ͱͷ࣮ݧʹ͓͍ͯ4PU"ͦ͘͠Εʹ͍ۙੑೳ • $04*.0ʹ͓͍ͯܗঢ়ͷϊΠζʹର͢Δޡࠩͷ্քΛಋग़͠ɺ࣮ݧతʹ֬ೝ • $04*.0PWFSTNPPUIJOH (//ʹΑΔܗঢ়ೝࣝͰͱͳΓ͍͢ใͷࢄҳ ͷ ఔΛ੍ޚͰ͖Δ͜ͱΛূ໌͠ɺ࣮ݧతʹ֬ೝ ୯ମෳମͷྫ[Einizade+ 20025]
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// •
$04*.0 • ࣮ݧ • ·ͱΊ
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
Ϟνϕʔγϣϯ • ࣍ݩܗঢ়Λޮతʹදݱ͢Δʹϝογϡ·ͨ ͦͷऔΓѻ͍͍͢ϞσϧͰ͋Δ୯ମෳମ͕༗༻ • ࣍ݩܗঢ়ͷػցֶशʹ(//͕༻͍ΒΕ͖͕ͯͨ (//άϥϑͷͲ͏͠ͷ૬ޓ࡞༻ͷΈΛ ߟྀ͓ͯ͠ΓΑΓߴ࣍ͷ૬ޓ࡞༻͕ݟಀ͞Ε͕ͪ -
ࡾ֯ܗͷ໘ੵΛٻΊΔૢ࡞ ʮͷಛྔˠ໘ͷಛྔʯ ͷࣸ૾͕ͩ(//ͰͦͷΑ͏ͳૢ࡞ඇࣗ໌ • ୯ମෳମΛऔΓѻ͑Δ//ग़͖͍ͯͯΔ͕ (//ͷ՝Ͱ͋ΔPWFSTNPPUIJOH ܗঢ়ͷϊΠζʹର͢Δݎ࿚ੑґવͱͯ͠՝ ➡ ΈࠐΈԋࢉΛ࿈ଓԽ͠ਫ਼៛Խ͢Δ͜ͱͰ ͜ΕΒͷ՝ΛղܾͰ͖ͳ͍͔ʁ ϝογϡͷྫ ୯ମෳମͷྫ [Einizade+ 20025]
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
४උ୯ମෳମ ఆٛ ୯ମෳମTJNQMJDJBMDPNQMFY ۭͰͳ͍༗ݶू߹ ʹର͠ɺ ͰҎԼͷ݅Λຬͨ͢ͷΛ 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
४උଓߦྻ ఆٛ ଓߦྻTJNQMFY ୯ମෳମ ʹରͯ͠ɺL୯ମͷΈΛूΊͨͷΛ ͱॻ͘ɻߦྻ
ͷ ʹ͍ͭͯɺJ൪ͷ L ୯ମ͕K൪ͷL୯ମͷ໘ʹͳ͓ͬͯΓɺ͖͕ἧ͍ͬͯ Δ L୯ମͷۮஔΛͯ͠ઌ಄ L ྻΛൈ͖ग़ͯ͠ L ୯ମͱҰக͢Δ ͱ͖ɺٯ͖ͷͱ ͖ɺͦΕҎ֎ͷͱ͖ΛׂΓͯͨͷΛଓߦྻͱ͍͏ɻ ( 𝒱 , 𝒳 ) 𝒳 k Bk ∈ {−1,0,1}| 𝒳 k−1 |×| 𝒳 k | (i, j) ఆٛ ୯ମͷ͖ ͷݩʹॱংΛఆΊ͓ͯ͘ɻL୯ମ ΛλϓϧͱΈͳͨ͠ͷ ʹରͯ͠ɺۮஔ ͰঢॱʹͰ͖Δͱ͖ɺͦ͏Ͱͳ͍ͱ͖ΛׂΓͯͨͷΛ୯ମͷ͖ͱ͍͏ɻ 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
४උଓߦྻ ۩ମྫ
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 ΄Ͳɺ͕ۉҰͱͳ͍ͬͯΔɻ E(xk ) = x⊤ k Lk xk = ∥Bk xk ∥2 2 + ∥B⊤ k+1 xk ∥2 2
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
%JTDSFUF4JNQMJDJBM//4JNQMJDJBM'JMUFS • 4JNQMJDJBMGJMUFS ҎԼͷΑ͏ʹॻ͚Δ ЋɺЌύϥϝʔλ • )PEHF-BQMBDJBOͷ͖ͷॏͶ߹ΘͤͰؔΛදݱ DG,SZMPW෦ۭؒ๏
• )PEHF-BQMBDJBOΛLճ͔͚Δͱ Lϗοϓઌ·ͰͷใΛऔΓࠐΊΔ - L ϗοϓҎ্ͷใݟ͑ͳ͍ VOEFSSFBDIJOH - )PEHF-BQMBDJBOΛॏͶΔ΄Ͳ ใ͕֦ࢄ͍͖ࣦͯ͠ΘΕͯ͠·͏ PWFSTNPPUIJOH f : ℝ| 𝒳 k | → ℝ| 𝒳 k | l = 0 l = 1 l = 2
%JTDSFUF4JNQMJDJBM//࣍ݩͷ߹ • %JTDSFUF4JNQMJDJBM//ͷϨΠϠ ͔Β Ϛοϐϯά ɺ ߴ࣍ݩͷϓϩδΣΫγϣϯ ͱ࣍ݩͷϓϩδΣΫγϣϯ Λ༻͍ͯҎԼͷΑ͏ʹॻ͚Δ[Isufi+
2022] (l − 1) l xl k,d = BT k xl k−1 xl k,u = Bk+1 xl k+1
%JTDSFUF4JNQMJDJBM//ଟ࣍ݩͷ߹ • ߴ࣍ݩͷTJNQMJDJBMTJHOBMΛ ͱͯ͠ɺ%JTDSFUF4JNQMJDJBM//ͷϨΠϠ ͔Β Ϛοϐϯά ҎԼͷΑ͏ʹॻ͚Δ[Isufi+ 2022]
(l − 1) l
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
$04*.0ภඍํఔࣜʹΑΔߟ • %JTDSFUF4JNQMJDJBM//Λํʹ࿈ଓԽ͠ภඍํఔࣜΛಘΔ DG/FVSBM0%& • ఆ࣌ؒൃలํఔࣜͷղߦྻࢦؔͰهड़Ͱ͖Δ
$04*.0ภඍํఔࣜʹΑΔߟ • ఆ࣌ؒൃలํఔࣜͷղߦྻࢦؔͰهड़Ͱ͖Δ • ͨͩ͠ɺߦྻࢦؔҎԼͷΑ͏ʹఆٛ͞ΕΔ
$04*.0ϞσϧͷఆࣜԽ • ఆ࣌ؒൃలํఔࣜͷղߦྻࢦؔͰهड़Ͱ͖Δ • Ҏ্ΑΓɺ$04*.0ͷϨΠϠʔΛҎԼͷΑ͏ʹߏ͢Δ - ࣮ࡍʹɺߦྻࢦؔΛϥϯΫۙࣅͯ͠ܭࢉ͢Δ
$04*.0ܗঢ়ͷϊΠζʹର͢Δݎ࿚ੑ • ଓߦྻʹઁಈΛ༩͑ͨͱ͖ͷ༧ଌ݁ՌͷͣΕ ͷ্ք͕ಘΒΕ͍ͯΔ δXk • ઁಈ͕ेখ͍͞ͱ͖ɺ༧ଌ݁ՌͷͣΕઁಈʹରͯ͠ઢܗʹൺྫ
$04*.00WFS4NPPUIJOHʹର͢Δߟ ࢄͷ߹ • %JTDSFUF4JNQMJDJBM//ͷ߹%JSJDIMFUΤωϧΪʔͷ্ք͕ )PEHF-BQMBDJBOͷݻ༗ͷଟ߲ࣜͱͳΔ - ҙͷॏΈʹରͯ͠PWFSTNPPUIJOHΛ੍͢ΔͨΊʹ)PEHF-BQMBDJBO ͷݻ༗Λେ͖ͤ͘ͶͳΒͣɺ୯ମෳମΛमਖ਼͠ͳ͚ΕͳΒͳ͍
$04*.00WFS4NPPUIJOHʹର͢Δߟ ࿈ଓͷ߹ • ࿈ଓͷ߹%JSJDIMFUΤωϧΪʔͷ্քΛθϩʹऩଋ͠ͳ͍Α͏ʹ ͢Δ͜ͱ͕Մೳ 1SPQ
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
࣮ݧ࣮Ͱͷݕূ • ͞·͟·ͳ࣮Ͱ4PU"ͦ͘͠Εʹ४͡ΔੑೳΛ࣮ݱ [Einizade+ 20025]
࣮ݧܗঢ়ͷϊΠζʹର͢Δݎ࿚ੑ • ୯ମෳମͷϊΠζ 4/3@ ͱ୯ମෳମͷϊΠζ 4/3@ Λ ͞·͟·ʹมԽͤ͞ਫ਼ΛධՁ -
͞·͟·ͳ࣍ݩͷใΛෳ߹ͤͯ͞༻͍ͯ͠ΔͨΊɺ 4/3@࠷খ/43@࠷େ·ͨ4/3@࠷େ/43@࠷খͷͱ͖ʹੑೳͷྼԽ͕গͳ͍ [Einizade+ 20025]
࣮ݧ0WFS4NPPUIJOHͷ੍ޚ • U ࢦؔͷݞʹͷ͍ͬͯΔύϥϝʔλ Λখ͘͢͞Δ ʙ ͜ͱʹΑͬͯ EJTDSFUFTJNQMJDJBM//ΑΓPWFSTNPPUIJOHΛ੍͢Δ͜ͱ͕Մೳ
[Einizade+ 20025]
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
࣍ • Ϟνϕʔγϣϯ • ४උ • %JTDSFUF4JNQMJDJBM// • $04*.0
• ࣮ݧ • ·ͱΊ
·ͱΊ • ୯ମෳମ ܗঢ়ͷτϙϩδʔΛࢄతʹද͢σʔλߏ ʹର͢ΔΈࠐΈԋࢉ Λ࿈ଓԽͨ͠$04*.0 DPOUJOVPVTTJNQMJDJBMOFVSBMOFUXPSL ΛఏҊ •
࣮Ͱͷ࣮ݧʹ͓͍ͯ4PU"ͦ͘͠Εʹ͍ۙੑೳ • $04*.0ʹ͓͍ͯܗঢ়ͷϊΠζʹର͢Δޡࠩͷ্քΛಋग़͠ɺ࣮ݧతʹ֬ೝ • $04*.0PWFSTNPPUIJOH (//ʹΑΔܗঢ়ೝࣝͰͱͳΓ͍͢ใͷ ࢄҳ ͷఔΛ੍ޚͰ͖Δ͜ͱΛূ໌͠ɺ࣮ݧతʹ࣮ূ ୯ମෳମͷྫ[Einizade+ 20025]