[DL輪読会]Weakly-Supervised Disentanglement Without Compromises

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June 30, 20

deep learning

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2020/06/19
Deep Learning JP:
http://deeplearning.jp/seminar-2/

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1.

8FBLMZ4VQFSWJTFE%JTFOUBOHMFNFOU DEEP LEARNING JP [DL Papers] 8JUIPVU$PNQSPNJTFT QSFTFOUFS,FJ"LV[BXB http://deeplearning.jp/

2.

ॻࢽ৘ใ w λΠτϧ8FBLMZ4VQFSWJTFE%JTFOUBOHMFNFOU8JUIPVU$PNQSPNJTFT w ஶऀ'SBODFTDP-PDBUFMMP #FO1PPMF (VOOBS3ÅUTDI #FSOIBSE4DIÖMLPQG  0MJWJFS#BDIFN .JDIBFM5TDIBOOFO w 'JSTUͷ-PDBUFMMP͞Μ͸l$IBMMFOHJOH$PNNPO"TTVNQUJPOTJOUIF 6OTVQFSWJTFE-FBSOJOHPG%JTFOUBOHMFE3FQSFTFOUBUJPOTz CFTUQBQFSBXBSE BU*$.- ͷਓ w ձٞ*$.- w Ұ‫طͱ͏͍Ͱݴ‬ଘ‫ʹڀݚ‬ൺ΂ͯσʔληοτʹඞཁͳ‫ࢣڭ‬ʢΞϊςʔγϣϯʣͷগͳ ͍ɼཧ࿦෼ੳ෇͖ͷEJTFOUBOHMFNFOUख๏ͷఏҊ

3.

എ‫ܠ‬%JTFOUBOHMFE3FQSFTFOUBUJPO w %JTFOUBOHMFE3FQSFTFOUBUJPO͸ɼ‫ػ‬ցਂ૚ֶशʹ͓͚Δॏཁͳ֓೦ w ΏΔ͍ఆٛ‫؍‬ଌσʔλΛߏ੒͢ΔҼࢠΛ͏·͘ଊ͍͑ͯΔද‫ݱ‬ w ࿦จʹΑͬͯ৭ʑͳఆ͕ٛ͋Δɽྫ &BTUXPPEBOE8JMMJBNT 4IVFUBM  w -PDBUFMMPFUBM -PDBUFMMPFUBM ঺հ࿦จ ͷఆٛ#JKFDUJWFCBTFE w ਅͷҼࢠz = [z1, . . . , zd]TͱɼϞσϧͷજࡏม਺z ̂ = [z1̂ , . . . , zd̂ ]TͷؒʹʢॱংΛআ͍ͯʣ ̂ ରͷରԠ͕ଘࡏ͢Δͱ͖ʹɼz͸%JTFOUBOHMFE3FQSFTFOUBUJPOͰ͋Δ w ̂ )Λຬͨ͢ ͋Δؔ਺fͱQFSNVUBUJPOπ͕ଘࡏ͠ɼ͢΂ͯͷiʹରͯ͠ɼzi = fi(zπ(i) w ஫ҙʢൃදऀͷཧղʣਅͷҼࢠz͸ɼ໬౓Λ࠷େʹ͢Δ͔ͭɼਓؒͷ௚‫ʹײ‬Ԋ͏Ҽࢠ Λzਅͷzͱ‫ݺ‬ΜͰ͍Δ͚ͩʢ࣮ࡍʹγϛϡϨʔλʹΞΫηεͰ͖Δ৔߹΋͋Δ͚Ͳʜʣ

4.
[beta] 
എ‫͠ͳࢣڭܠ‬ͷ%JTFOUBOHMFNFOU͸ෆՄೳ
<-PDBUFMMPFUBM>
w

ཧ࿦తͳઆ໌ਅͷʢEJTFOUBOHMF͞ΕͨʣҼࢠͷ෼෍P(z)ʹରͯ͠ҎԼΛຬͨ͢f͕ແ਺ʹଘࡏ
J p(z) = p( f(z))ʢೋͭͷ֬཰ม਺z, f(z)ͷ෼෍͕Ұகʣ
∂fi(u)
JJ ͢΂ͯͷ࣍‫ݩ‬i, jʹ͍ͭͯ
for a . e . u ∈ supp(z)
∂uj

w

ཁ͸ɼP( f(z))ͱ͍͏๬·͘͠ͳ͍ʢFOUBOHMF͍ͯ͠Δʣ෼෍͕ແ਺ଘࡏͯ͠ɼͦͷp( f(z))ͱp(z)͸ࣝผෆՄೳֶशʹ
ΑͬͯͲͪΒ͕ಘΒΕΔ͔อূͰ͖ͳ͍

w

ྫ ‫׬‬શͳ‫Ͱ͠ͳࢣڭ‬%JTFOUBOHMFNFOUΛୡ੒͠Α͏ͱ͢Δβ7"& 'BDUPS7"&ͳͲ͕͋Δ

w

BHHSFHBUFEQPTUFSJPSq(z) = q(z | x)pdata(x)dxͷ֤࣍‫͕ݩ‬ಠཱʹͳΔΑ͏ͳ੍໿Λ༩͍͑ͯΔ
∫

w

Ծʹq(z)ͷ֤࣍‫͕ݩ‬ಠཱͰ΋ɼͦΕ͕p(z)ʢͷ֤࣍‫ݩ‬ʣͱCJKFDUJWFͳؔ܎Λ͔࣋ͭɼp( f(z))ͱCJKFDUJWFͳؔ܎Λ࣋
͔ͭʹҰக͢Δ͔อূͰ͖ͳ͍ɽͭ·Γɼ֤࣍‫ݩ‬ͷಠཱੑΛอূͰ͖ͯ΋ͦΕ͕ਓؒͷ௚‫ͱײ‬੔߹͢Δอূ͸ͳ͍ɽ

w

஫ҙp(z)͸ਅͷҼࢠͷ෼෍ɽϞσϧʢ7"&ʣͷࣄલ෼෍Ͱ͸ͳ͍
5.

എ‫͠ͳࢣڭܠ‬ͷ%JTFOUBOHMFNFOU͸ෆՄೳ <-PDBUFMMPFUBM> w w ࣮‫ݧ‬తͳઆ໌‫׬‬શͳ‫Ͱ͠ͳࢣڭ‬͸Ϟσϧબ୒͕೉͍͠ w ֶशࡁΈϞσϧͷද‫Ͳ͕ݱ‬ͷఔ౓ʮEJTFOUBOHMFNFOU͞Ε͍ͯΔ͔ʯ͸ॳ‫ظ‬஋ͳͲʹ‫͘ڧ‬ґଘʢࠨਤʣ w ‫͠ͳࢣڭ‬ͷࢦඪʢ&-#0΍࠶ߏ੒‫ ࠩޡ‬,-߲ͳͲʣͱʮEJTFOUBOHMFNFOUͷࢦඪʯʹ૬ؔ͸ͳ͍ʢӈਤʣ w β7"&ͳͲͷ‫ط‬ଘ‫ڀݚ‬΋Ϟσϧબ୒ʹ͸ҼࢠͷϥϕϧΛ࢖͍ͬͯΔ ༨ஊl3FQSPEVDJOHUIFTFFYQFSJNFOUTSFRVJSFTBQQSPYJNBUFMZ(16ZFBST /7*%*"1 z

6.

എ‫ܠ‬ऑ‫ʹࢣڭ‬ΑΔ%JTFOUBOHMFNFOU w -PDBUFMMPFUBM͸ɼ%JTFOUBOHMFNFOUʹ͓͍ͯɼᶃ‫׬‬શͳ‫͠ͳࢣڭ‬͸ෆՄೳᶄ‫͋ࢣڭ‬ Γ͸ߴίετͳ͜ͱ͔Βɼऑ‫ࢣڭ‬ͷར༻Λਪ঑ͨ͠ w ‫ͦࢣڭ‬ΕͧΕͷ‫؍‬ଌ͕ͲͷཁૉΛ͔࣋ͭͱ͍͏ϥϕϧ w ऑ‫׬ࢣڭ‬શͳ‫ࢣڭ‬ΑΓίετͷ௿͍ϥϕϧɽ৭ʑͳ‫ܗ‬ଶ͕͋Γ͏Δ w ྫ 4IVFUBMʹΑΔ෼ྨ

7.

ऑ‫ࢣڭ‬ͷྫ)PTPZB w ਤ%͋Δάϧʔϓ͕‫ڞ‬௨ͷҼࢠΛ࣋ͭͱΘ͔ͬͯΔͱ͖ʢऑ‫ࢣڭ‬ʣʹɼ‫ڞ‬௨ͷҼࢠʹରԠ͢Δ જࡏม਺[ΛɼͦͷҼࢠΛ࣋ͭ‫؍‬ଌͷάϧʔϓ͔Βਪ࿦͠ɼͦͷଞͷҼࢠ͸֤ը૾͔Βਪ࿦͢Δ w ஫ҙೋͭͷҼࢠʢάϧʔϓͰ‫ڞ‬༗ඇ‫ڞ‬༗ʣΛ%JTFOUBOHMF͢Δख๏Ͱ͋Γɼβ7"&ͷΑ͏ʹ શͯͷ࣍‫֤͕ݩ‬ʑҼࢠʹରԠ͢Δͷͱ͸ҟͳΔ

8.

঺հ࿦จͷ֓ཁ w ϖΞσʔλ(x1, x2)Λ༻͍ͨEJTFOUBOHMFEද‫ֶݱ‬श w σʔλʹର͢ΔԾఆɼσʔλऩूίετ͕‫ط‬ଘ‫ڀݚ‬ΑΓখ͍͞ w (x1, x2)ʹର͢Δ੍໿͸ɼ‫ڞ‬௨͢Δk ≥ 1‫ݸ‬ͷཁૉΛ࣋ͭ͜ͱɽ۩ମతʹԿΛ‫ڞ‬༗ ͍ͯ͠ΔͳͲͷϥϕϧ͸͍Βͳ͍ w ্‫ه‬ͷϖΞσʔλ͕EJTFOUBOHMFEද‫ݱ‬ͷֶशʹे෼ͳ͜ͱΛཧ࿦తʹࣔ͠ɼ·ͨ ࣮༻తͳख๏ΛఏҊ͢Δ w ఏҊख๏Ͱ֫ಘͨ͠EJTFOUBOHMFEද‫ݱ‬ͷ‫ޙ‬ଓλεΫʹ͓͚Δ༗ޮੑΛ࣮‫ࣔ͢Ͱݧ‬ w w ‫ޙ‬ଓλεΫઢ‫ܗ‬Ϋϥε෼ྨɼ‫ڞ‬มྔγϑτͷ‫Ͱݩ‬ͷ൚ԽੑೳͳͲ ಛʹ‫ڞ‬มྔγϑτͱEJTFOUBOHMFNFOUͷؔ܎Λ࣮‫ݧ‬తʹࣔͨ͠ͷ͸͓ͦΒ͘ॳ

9.

঺հ࿦จͷ֓ཁ w ϖΞσʔλ(x1, x2)Λ༻͍ͨEJTFOUBOHMFEද‫ֶݱ‬श w σʔλʹର͢ΔԾఆɼσʔλऩूίετ͕‫ط‬ଘ‫ڀݚ‬ΑΓখ͍͞ w (x1, x2)ʹର͢Δ੍໿͸ɼ‫ڞ‬௨͢Δk ≥ 1‫ݸ‬ͷཁૉΛ࣋ͭ͜ͱɽ۩ମతʹԿΛ‫ڞ‬༗ ͍ͯ͠ΔͳͲͷϥϕϧ͸͍Βͳ͍ w ্‫ه‬ͷϖΞσʔλ͕EJTFOUBOHMFEද‫ݱ‬ͷֶशʹे෼ͳ͜ͱΛཧ࿦తʹࣔ͠ɼ·ͨ ࣮༻తͳख๏ΛఏҊ͢Δ w ఏҊख๏Ͱ֫ಘͨ͠EJTFOUBOHMFEද‫ݱ‬ͷ‫ޙ‬ଓλεΫʹ͓͚Δ༗ޮੑΛ࣮‫ࣔ͢Ͱݧ‬ w w ‫ޙ‬ଓλεΫઢ‫ܗ‬Ϋϥε෼ྨɼ‫ڞ‬มྔγϑτͷ‫Ͱݩ‬ͷ൚ԽੑೳͳͲ ಛʹ‫ڞ‬มྔγϑτͱEJTFOUBOHMFNFOUͷؔ܎Λ࣮‫ݧ‬తʹࣔͨ͠ͷ͸͓ͦΒ͘ॳ

10.

σʔλʹର͢ΔԾఆ w σʔλͷਅͷੜ੒աఔΛҎԼ ͷΑ͏ʹԾఆ͍ͯ͠Δ p(z) = d Πi=1p(zi), ⋆ p(z̃) = d Πi=1p(z̃i) ⋆ S ∼ p(S), x1 = g (z), x2 = g ( f(z, z̃, S)) w f͸ɼi ∉ SͷཁૉΛz̃iʹஔ ‫͢׵‬ΔΑ͏ͳૢ࡞ʢਤΛࢀর

11.

঺հ࿦จͷ֓ཁ w ϖΞσʔλ(x1, x2)Λ༻͍ͨEJTFOUBOHMFEද‫ֶݱ‬श w σʔλʹର͢ΔԾఆɼσʔλऩूίετ͕‫ط‬ଘ‫ڀݚ‬ΑΓখ͍͞ w (x1, x2)ʹର͢Δ੍໿͸ɼ‫ڞ‬௨͢Δk ≥ 1‫ݸ‬ͷཁૉΛ࣋ͭ͜ͱɽ۩ମతʹԿΛ‫ڞ‬༗ ͍ͯ͠ΔͳͲͷϥϕϧ͸͍Βͳ͍ w ্‫ه‬ͷϖΞσʔλ͕EJTFOUBOHMFEද‫ݱ‬ͷֶशʹे෼ͳ͜ͱΛཧ࿦తʹࣔ͠ɼ·ͨ ࣮༻తͳख๏ΛఏҊ͢Δ w ఏҊख๏Ͱ֫ಘͨ͠EJTFOUBOHMFEද‫ݱ‬ͷ‫ޙ‬ଓλεΫʹ͓͚Δ༗ޮੑΛ࣮‫ࣔ͢Ͱݧ‬ w w ‫ޙ‬ଓλεΫઢ‫ܗ‬Ϋϥε෼ྨɼ‫ڞ‬มྔγϑτͷ‫Ͱݩ‬ͷ൚ԽੑೳͳͲ ಛʹ‫ڞ‬มྔγϑτͱEJTFOUBOHMFNFOUͷؔ܎Λ࣮‫ݧ‬తʹࣔͨ͠ͷ͸͓ͦΒ͘ॳ

12.

෼ੳఆཧʢཁ໿ʣ ∫∫∫ w ԾఆϖΞσʔλ(x1, x2) ∼ p(x1, x2) = p(x1, x2, z, z̃, S)dzdz̃dS͕ແ਺ʹಘΒΕΔ w Ծఆ‫ڞ‬༗͢ΔҼࢠͷ਺ | S | = k͕༩͑ΒΕ͍ͯΔ w ԾఆS, S′ ∼ p(S)ʹରͯ͠ɼP(S ∩ S′ = {i}) > 0, ∀i ∈ [d] w ݁Ռੜ੒ϞσϧΛֶशͯ͠ࣄલ෼෍p(zî )ɼσίʔμʔq(x1 | z)̂ ɼp(S)̂ ͕ಘΒΕΔͱ͢Δɽ BHHSFHBUFEQPTUFSJPSq(z)̂ = q(z ̂ | x1)p(x1)dx͸CJKFDUJWJUZͷҙຯͰ%JTFOUBOHMF͞Ε͍ͯΔ ∫

13.

෼ੳͬ͘͟Γͨ͠ূ໌ͷࢦ਑ w ·ͣ͸Sʢx1, x2ʹ‫ڞ‬௨͢ΔҼࢠʣΛ‫ݻ‬ఆֶͯ͠शͨ͠৔߹Λߟ͑Δɽ ̂ ͱzi∉T ̂ ͕ಘΒΕΔ w ͜ͷஈ֊ͰɼҼࢠzi∈Sͱzi∉SʹରԠ͢Δzi∈T w ཁ͸ɼ‫ڞ‬௨͢ΔҼࢠͦΕҎ֎ͷҼࢠͷ%JTFOUBOHMFNFOU͕Ͱ͖Δ w छͷEJTFOUBOHMFNFOUͱ͍͏ҙຯͰ)PTPZBͱࣅ͍ͯΔɽσʔλʹର͢ΔԾఆ΋ɼϖΞ σʔλ(x1, x2) ∼ p(x1, x2 | S)͕ಘΒΕΔɼͱ͍͏΋ͷͰଟ෼ಉ͡ w ࣍ʹɼSΛϥϯμϜʹαϯϓϦϯάͨ͠৔߹Λߟ͑Δ w ‫ڞ‬௨͢ΔҼࢠ͕ϥϯμϜʹ‫ݱ‬ΕΔͷͰɼશͯͷҼࢠ͕EJTFOUBOHMFNFOU͞ΕΔ w ʢ͜͜ͷূ໌͸গ͠௥͍͑ͯͳ͍෦෼͕͋Δɽʣ

14.

ఏҊख๏ w ࣮૷ͷํ਑ w ਅͷࣄ‫ޙ‬෼෍Ͱ͸ p (zi | x1) = p (zi | x2) p (zi | x1) ≠ p (zi | x2) ∀i ∈ S ͕੒ΓཱͭͷͰɼ͜ͷ੍໿ΛϞσ ∀i ∈ S̄ ϧͷࣄ‫ޙ‬෼෍ʹ΋༩͍͑ͨ w ͔͠͠ɼ‫ڞ‬௨͢Δཁૉͷ਺k͕ʢఆ਺Ͱʣ༩͑ΒΕͳ͍ͱɼS͕ਪఆͰ͖ͳ͍ɽ ࣮༻্͸ɼk͸Θ͔Βͳ͍͔΋͠Εͳ͍͠ɼఆ਺Ͱ͸ͳ͍ʢϖΞ͝ͱʹҧ͏͔΋ ͠Εͳ͍ʣ͔΋͠Εͳ͍ w ͦ͜ͰɼkʢͱSʣ΋ώϡʔϦεςΟοΫͰϖΞ͝ͱʹਪఆ͢Δ

15.

ఏҊख๏ max 𝔼(x1,x2)𝔼q̃ϕ(ẑ | x1) log (pθ (x1 | z)̂ ) ϕ,θ +𝔼q̃ϕ(ẑ | x2) log (pθ (x2 | z)̂ ) −βDKL (q̃ϕ (ẑ | | x1) | p(z)̂ ) XIFSF q̃ϕ (zî | x1) = a (qϕ (zî | x1), qϕ (zî | x2)) q̃ϕ (zî | x1) = qϕ (zî | x1) −βDKL (q̃ϕ (ẑ | | x2) | p(z)̂ ) w ϖΞ(x1, x2)ʹରͯ͠ɼͦΕͧΕ࠶ߏ੒ͱ,-߲Λద༻ʢ7"&෩ʣ w ̂ qϕ (zî | x1), qϕ (zî | x2)͕ಉ͡ʹͳΔΑ͏ʹฏ‫਺ؔۉ‬aΛ‫ͤ·ט‬Δ ਪ࿦෼෍͸ɼ‫ڞ‬௨͢Δཁૉi ∈ Sʹؔͯ͠ w w ฏ‫਺ؔۉ‬͸ɼ࿨ͷ෼෍Λߟ͑Δ͔ʢ)PTPZBʣੵͷ෼෍Λߟ͑Δ͔ʢ#PVDIBDPVSUFUBMʣ ‫ڞ‬௨͢Δཁૉ͔Ͳ͏͔͸ɼ,-߲ͷେ͖͞δi = DKL (qϕ (zî | x1) ∥qϕ (zî | x2))͔ΒώϡʔϦεςΟοΫʹ൑அ ∀i ∈ S ̂ FMTF

16.

঺հ࿦จͷ֓ཁ w ϖΞσʔλ(x1, x2)Λ༻͍ͨEJTFOUBOHMFEද‫ֶݱ‬श w σʔλʹର͢ΔԾఆɼσʔλऩूίετ͕‫ط‬ଘ‫ڀݚ‬ΑΓখ͍͞ w (x1, x2)ʹର͢Δ੍໿͸ɼ‫ڞ‬௨͢Δk ≥ 1‫ݸ‬ͷཁૉΛ࣋ͭ͜ͱɽ۩ମతʹԿΛ‫ڞ‬༗ ͍ͯ͠ΔͳͲͷϥϕϧ͸͍Βͳ͍ w ্‫ه‬ͷϖΞσʔλ͕EJTFOUBOHMFEද‫ݱ‬ͷֶशʹे෼ͳ͜ͱΛཧ࿦తʹࣔ͠ɼ·ͨ ࣮༻తͳख๏ΛఏҊ͢Δ w ఏҊख๏Ͱ֫ಘͨ͠EJTFOUBOHMFEද‫ݱ‬ͷ‫ޙ‬ଓλεΫʹ͓͚Δ༗ޮੑΛ࣮‫ࣔ͢Ͱݧ‬ w w ‫ޙ‬ଓλεΫઢ‫ܗ‬Ϋϥε෼ྨɼ‫ڞ‬มྔγϑτͷ‫Ͱݩ‬ͷ൚ԽੑೳͳͲ ಛʹ‫ڞ‬มྔγϑτͱEJTFOUBOHMFNFOUͷؔ܎Λ࣮‫ݧ‬తʹࣔͨ͠ͷ͸͓ͦΒ͘ॳ

17.

࣮‫ݧ‬ w w σʔληοτE4QSJUFT $BST% 4NBMM/03# .1*% w Ҽࢠ͕શͯΞϊςʔγϣϯ͞Ε͍ͯΔʢධՁʹͷΈ࢖͏ʣ w ‫؍‬ଌ͸Ҽࢠ͔Βܾఆతͳࣸ૾͕ͱΕΔʢҼࢠҎ֎ͷϊΠζ͕ͳ͍ʣ ϕʔεϥΠϯ w ‫͠ͳࢣڭ‬β7"& "OOFBMFE7"& 'BDUPS7"& β5$7"& %*17"& w ऑ‫ࢣڭ‬ʢBWFSBHFؔ਺͕ҧ͏ೋͭʣ w (7"& )PTPZB .-7"& #PVDIBDPVSUFUBM

18.

࣮‫͠ͳࢣڭݧ‬ख๏ͱͷൺֱ w Ϟσϧ ͕ఏҊख๏ɼϞσϧબ୒ͳ͠ͷ৔߹ฏ‫ۉ‬తʹߴ͍είΞ w ͜ΕΒͷਤ͸Ϟσϧબ୒ͳ͕ͩ͠ɼ΋͠‫͠ͳࢣڭ‬ͷख๏͕Ϟσϧબ୒ͷͨΊʹҼ ࢠϥϕϧ৘ใΛ࢖ͬͯ΋ɼ$PNFUJUJWFͳੑೳ

19.

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