Harnessing the Power of Vicinity-Informed Analysis for Classification under Covariate Shift

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)BSOFTTJOHUIF1PXFSPG7JDJOJUZ *OGPSNFE"OBMZTJTGPS$MBTTJ fi DBUJPO VOEFS$PWBSJBUF4IJGU ୈճβοϐϯάηϛφʔ ෱஍Ұే ஜ೾େֶཧݚ"*1 IUUQTBSYJWPSHBCT +PJOUXPSLXJUI .JUTVIJSP'VKJLBXB 5TVLVCB3*,&/"*1 :PIFJ"LJNPUP 5TVLVCB3*,&/"*1 +VO 4BLVNB 5PLZP5FDI3*,&/"*1

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ࣗݾ঺հ w ໊લ෱஍Ұే 'VLVDIJ ,B[VUP w ॴଐஜ೾େֶγεςϜ৘ใܥॿڭ w ܦྺ w ஜ೾େֶγεςϜ৘ใ޻ֶઐ߈Պത࢜ޙظ՝ఔमྃ w ཧݚ"*1ಛผݚڀһ w ݱࡏஜ೾େֶγεςϜ৘ใܥॿڭ w ݱࡏཧݚ"*1٬һݚڀһ w ݚڀڵຯ w ػցֶशʹ͓͚ΔόΠΞεʢެฏੑɼసҠֶशɼҼՌਪ࿦ʣ w ਺ཧ౷ܭɼಛʹɼ൚ؔ਺ਪఆ

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ࠓ೔ͷ࿩͸సҠֶश

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సҠֶशͷશ͕ͯॻ͔Εͨຊʂ ങ͍·͠ΐ͏ʂ ዁౓λΠϜ

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໨࣍ wసҠֶश wڞมྔγϑτԼʹ͓͚Δཧ࿦ղੳ w݁Ռͷৄࡉ

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సҠֶश

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෼ྨ໰୊ ϥϕϧ෇͖σʔλ ֶशΞϧΰϦζϜ ෼ྨث h 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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෼ྨ໰୊ ֶशΞϧΰϦζϜ ෼ྨث h( )=Ҝࢠ ϥϕϧ෇͖σʔλ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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෼ྨ໰୊ ֶशΞϧΰϦζϜ ෼ྨث h( )=Ҝࢠ ͳΔ΂͘౰ͨΔΑ͏ h Λબ୒͍ͨ͠ ϥϕϧ෇͖σʔλ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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సҠֶश ֶशΞϧΰϦζϜ ෼ྨث h( )=Ҝࢠ ιʔεσʔλ ༧ଌ࣌ʹҟͳΔ ੑ࣭ͷσʔλ λʔήοτ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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సҠֶश ֶशΞϧΰϦζϜ ෼ྨث h( )=Ҝࢠ ιʔεσʔλ ༧ଌ࣌ʹҟͳΔ ੑ࣭ͷσʔλ λʔήοτσʔλ ༧ଌ࣌ͱಉ͡ੑ࣭ͷ σʔλΛগྔ؍ଌ ιʔεσʔλ͸ େྔʹ֬อՄೳ λʔήοτ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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సҠֶश ֶशΞϧΰϦζϜ ෼ྨث h( )=Ҝࢠ ιʔεσʔλ ιʔεσʔλΛ׆༻͠ ͯΑΓߴਫ਼౓ͷ ༧ଌΛ࣮ݱ λʔήοτσʔλ ༧ଌ࣌ͱಉ͡ੑ࣭ͷ σʔλΛগྔ؍ଌ ιʔεσʔλ͸ େྔʹ֬อՄೳ ༗༻ͳ৘ใΛநग़ʢసҠʣ λʔήοτ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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సҠֶशͷ੒ޭ wྫ0 ff i DF)PNFEBUBTFU wͭͷυϝΠϯ Ξʔτ ΫϦοϓΞʔτ ϓϩμΫτ ϦΞϧ wͷΧςΰϦ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ 1BQFSTXJUI$PEFIUUQTQBQFSTXJUIDPEFDPNTPUBEPNBJOBEBQUBUJPOPOP ff i DFIPNF ෼ྨਫ਼౓

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సҠֶशͷఆࣜԽɾ ཧ࿦ղੳͷ໨ඪ

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෼ྨ໰୊ͷֶश ֶशΞϧΰϦζϜ ෼ྨث h( )=Ҝࢠ h ʹΑΔ෼ྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ ϥϕϧ෇͖σʔλ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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෼ྨ໰୊ͷֶश ֶशΞϧΰϦζϜ ෼ྨث h ʹΑΔ෼ྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ ϥϕϧ෇͖σʔλ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ h(X) = ̂ Y (X, Y) ∼ P (X, Y) iid ∼ P = (X1 , Y1 ), ⋮ , (Xn , Yn ) ෼ྨޡࠩʢظ଴ޡࠩʣ errP (h) = 𝔼 P [1{h(X) ≠ Y}]

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ʢڭࢣ͋ΓʣసҠֶश ֶशΞϧΰϦζϜ ෼ྨث h( )=Ҝࢠ ιʔεσʔλ h ʹΑΔλʔήοτͰ ͷ෼ྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ λʔήοτσʔλ λʔήοτ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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ʢڭࢣ͋ΓʣసҠֶश ֶशΞϧΰϦζϜ ෼ྨث h(X) = ̂ Y ιʔεσʔλ P h ʹΑΔλʔήοτͰ ͷ෼ྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ λʔήοτσʔλ Q λʔήοτ Q (X, Y)P iid ∼ P = (X1 , Y1 ), ⋮ , (XnP , YnP ) (X, Y)Q iid ∼ Q = (XnP +1 , YnP +1 ), ⋮ , (XnP +nQ , YnP +nQ ) nP ≫ nQ ෼ྨޡࠩʢظ଴ޡࠩʣ errQ (h) = 𝔼 Q [1{h(X) ≠ Y}] (X, Y) ∼ Q

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ֶशཧ࿦ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ࢒༨ޡࠩͱαϯϓϧαΠζ ͷؔ܎ʢαϯϓϧෳࡶ౓ʣΛ໌Β͔ʹ͍ͨ͠

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ֶशཧ࿦ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ࢒༨ޡࠩͱαϯϓϧαΠζ ͷؔ܎ʢαϯϓϧෳࡶ౓ʣΛ໌Β͔ʹ͍ͨ͠ αϯϓϧαΠζେ αϯϓϧαΠζখ ΞϧΰϦζϜ͕ग़ྗͨ͠෼ྨثͷޡࠩ σʔλ͕୔ࢁ͋Δ΄Ͳখ͘͞ͳΔʢʁʣ ࢒༨ޡࠩ Լ͛ΒΕͳ͍ ޡࠩͷݶք ޡࠩେ ޡࠩখ

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ֶशཧ࿦ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ࢒༨ޡࠩͱαϯϓϧαΠζ ͷؔ܎ʢαϯϓϧෳࡶ౓ʣΛ໌Β͔ʹ͍ͨ͠ αϯϓϧαΠζେ αϯϓϧαΠζখ ޡࠩେ ޡࠩখ errP (h) ℰP (h) = errP (h) − inf h*:Մଌؔ਺ errP (h*) inf h*:Մଌؔ਺ errP (h*) 𝔼 [ℰP (h)] ≤ U(n) n

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Ұகੑ w࢒༨ޡ͕ࠩαϯϓϧαΠζແݶେͷ࣌ʹʹऩଋ wਖ਼֬ʹ͸ͲΜͳ෼෍ʹରͯ͠΋ˢ͕੒Γཱͭ͜ͱ αϯϓϧαΠζେ αϯϓϧαΠζখ Ұகੑ͋Γ Ұகੑͳ͠ ޡࠩେ ޡࠩখ n

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సҠֶशͷֶशཧ࿦ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ࢒༨ޡࠩͱιʔεͷαϯϓ ϧαΠζ ͱλʔήοτͷαϯϓϧαΠζ ͷؔ܎Λ໌Β ͔ʹ͍ͨ͠ nP nQ

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సҠֶशͷֶशཧ࿦ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ࢒༨ޡࠩͱιʔεͷαϯϓ ϧαΠζ ͱλʔήοτͷαϯϓϧαΠζ ͷؔ܎Λ໌Β ͔ʹ͍ͨ͠ nP nQ ͲΕ͚ͩιʔεͷσʔλΛ׆༻Ͱ͖͔ͨʁ

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సҠֶशͷֶशཧ࿦ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ࢒༨ޡࠩͱιʔεͷαϯϓ ϧαΠζ ͱλʔήοτͷαϯϓϧαΠζ ͷؔ܎Λ໌Β ͔ʹ͍ͨ͠ nP nQ ιʔεαϯϓϧαΠζେ ιʔεαϯϓϧαΠζখ λʔήοτޡࠩେ λʔήοτޡࠩখ nP errQ (h) ℰQ (h) = errQ (h) − inf h*:Մଌؔ਺ errQ (h*) inf h*:Մଌؔ਺ errQ (h*) 𝔼 [ℰQ (h)] ≤ U(nP , nQ )

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ιʔεαϯϓϧαΠζʹର͢ΔҰகੑ w࢒༨ޡ͕ࠩιʔεαϯϓϧαΠζແݶେͷ࣌ʹʹऩଋ wਖ਼֬ʹ͸ͲΜͳ෼෍ʹରͯ͠΋ˢ͕੒Γཱͭ͜ͱ Ұகੑ͋Γ Ұகੑͳ͠ ιʔεαϯϓϧαΠζେ ιʔεαϯϓϧαΠζখ λʔήοτޡࠩେ λʔήοτޡࠩখ nP

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ιʔεαϯϓϧαΠζʹର͢ΔҰகੑ w࢒༨ޡ͕ࠩιʔεαϯϓϧαΠζແݶେͷ࣌ʹʹऩଋ wਖ਼֬ʹ͸ͲΜͳ෼෍ʹରͯ͠΋ˢ͕੒Γཱͭ͜ͱ Ұகੑ͋Γ Ұகੑͳ͠ ιʔεαϯϓϧαΠζେ ιʔεαϯϓϧαΠζখ λʔήοτޡࠩେ λʔήοτޡࠩখ nP ιʔεαϯϓϧΛ࢖ֶͬͯश͕Ͱ͖͍ͯΔ ˠసҠͷ੒ޭ

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෼෍γϑτ ֶशΞϧΰϦζϜ ෼ྨث f( )=Ҝࢠ ιʔεσʔλ λʔήοτσʔλ ιʔεσʔλͱ༧ଌ࣌ͷσʔλ͕ શ͘ҟͳΔͱ༧ଌ͸Ͱ͖ͳ͍ ιʔεͱλʔήοτ͸Կ͔͠ΒͷҙຯͰࣅ͍ͯΔඞཁ͕͋Γ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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$PWBSJBUF4IJGU ιʔε λʔήοτ ෼ྨنଇ͸ಉҰ ೖྗσʔλ͸ҟͳΔ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ

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$PWBSJBUF4IJGU ιʔε λʔήοτ ෼ྨنଇ͸ಉҰ ೖྗσʔλ͸ҟͳΔ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ ෼ྨنଇ͕ಉ͡ ˠιʔε͚ͩͰ΋෼ྨ͕੒ޭ͢Δ ˠҰகੑʹసҠͷ੒ޭ

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$PWBSJBUF4IJGU ιʔε λʔήοτ PX QX PY|X QY|X PX ≠ QX PY|X (Y = 1|X) = QY|X (Y = 1|X) = η(X) $PWBSJBUFTIJGUԾఆ η(X) = 1 2

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طଘͷཧ࿦త݁Ռ

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෼෍ؒڑ཭Λ࢖ͬͨ൚ԽޡࠩʹΑΔ্ք #FO%BWJEFUBM 1BSLFUBM "NJOJBOFUBM ʜ wཧ࿦ղੳͷ໨ඪ 𝔼 [ℰQ (h)] ≤ U(nP , nQ ) λʔήοτ෼෍Ͱଌͬͨ࢒ࠩޡࠩ

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෼෍ؒڑ཭Λ࢖ͬͨ൚ԽޡࠩʹΑΔ্ք #FO%BWJEFUBM 1BSLFUBM "NJOJBOFUBM ʜ w൚ԽޡࠩղੳΛ௨ͨ͠࢒ࠩޡࠩͷ্ք 𝔼 [ℰQ (h)] ≤ errP,nP (h) + d(PX , QX ) + n−c P ιʔεͷܦݧޡࠩerrP,nP (h) = 1 nP nP ∑ i=1 1{h(Xi ) ≠ Yi } ෼෍ؒڑ཭ ιʔεͷܦݧޡࠩ ෼෍͕ࣅ͍ͯΔ΄ͲసҠֶश্͕ख͍͘͘ ݟͨ໨͕ࣅ͍ͯΔ

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෼෍ؒڑ཭Λ࢖ͬͨ൚ԽޡࠩʹΑΔ্ք #FO%BWJEFUBM 1BSLFUBM "NJOJBOFUBM ʜ ʹͰ͖ͳ͍ ͜ΕΒͷ্քͰ͸αϯϓϧαΠζʹର͢ΔҰகੑΛࣔͤͳ͍ w൚ԽޡࠩղੳΛ௨ͨ͠࢒ࠩޡࠩͷ্ք 𝔼 [ℰQ (h)] ≤ errP,nP (h) + d(PX , QX ) + n−c P

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֬཰ൺΛ࢖্ͬͨք ,QPUVGF .BFUBM 'FOHFUBM w֬཰ൺ wֶशΞϧΰϦζϜ ρ(x) = dQX dPX (x) h = arg minh 1 nP ∑nP i=1 ρ(Xi )ℓ(h, (Xi , Yi )) ιʔε λʔήοτ PX QX ௿͍ॏΈ ߴ͍ॏΈ λʔήοτͬΆ͍σʔλΛ ߴ͘ධՁ͢Δ

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֬཰ൺΛ࢖্ͬͨք ,QPUVGF .BFUBM 'FOHFUBM w֬཰ൺ wֶशΞϧΰϦζϜ ρ(x) = dQX dPX (x) h = arg minh 1 nP ∑nP i=1 ρ(Xi )ℓ(h, (Xi , Yi )) 𝔼 [ℰQ (h)] ≤ C ( ln(nP ) nP ) c ҰகੑΛ͍ࣔͤͯΔʁ

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֬཰ൺΛ࢖্ͬͨք ,QPUVGF .BFUBM 'FOHFUBM w֬཰ൺ wֶशΞϧΰϦζϜ ρ(x) = dQX dPX (x) h = arg minh 1 nP ∑nP i=1 ρ(Xi )ℓ(h, (Xi , Yi )) 𝔼 [ℰQ (h)] ≤ C1 ( ln(nP ) nP ) c1 + C2 n−c2 Q ͷਪఆʹҰகੑΛ ્֐͢Δ߲͕ݱΕΔ ρ ֶशʹ֬཰ൺΛ࢖͍ͬͯΔ ࣮ࡍʹ͸ಘΒΕͳ͍ ͜ΕΒͷ্քͰ͸αϯϓϧαΠζʹର͢ΔҰகੑΛࣔͤͳ͍

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ڑ཭ۭؒϕʔε෼෍ؒྨࣅ౓ʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ཭্ۭؒͷ௒ٿΛ΋ͱʹͨ͠ྨࣅ౓ 1BUIBLFUBM wڑ཭ۭؒ w൒ܘ ͷ௒ٿ ( 𝒳 , ρ) r Bρ (x, r) = {x′ ∈ 𝒳 : ρ(x, x′ ) ≤ r} ΔPMW (P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ͷ࣌ ҰகੑΛ࣋ͭΞϧΰϦζϜΛߏங ΔPMW (P, Q; r) = O(r−τ) (τ < ∞) 𝔼 [ℰQ (h)] ≤ Cn−c P (c > 0) ࣮ࡍ͸ 1BUIBLFUBM ͸ճؼઃఆͰ͋Δ͕ɼ্هྨࣅ౓͸෼ྨʹద༻Մೳʢຊ࿦จʣ

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ڑ཭ϕʔε෼෍ؒྨࣅ౓ʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ཭্ۭؒͷ௒ٿΛ΋ͱʹͨ͠ྨࣅ౓ ΔPMW (P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ׂΓࢉ͕ى͜ΔՄೳੑ ιʔε PX QX λʔήοτ ॏͳ͍ͬͯΔʢઈର࿈ଓʣ ˠׂ͸ى͜Βͳ͍

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ڑ཭ϕʔε෼෍ؒྨࣅ౓ʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ཭্ۭؒͷ௒ٿΛ΋ͱʹͨ͠ྨࣅ౓ ΔPMW (P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ׂΓࢉ͕ى͜ΔՄೳੑ ιʔε PX QX λʔήοτ ͣΕ͍ͯΔʢඇઈର࿈ଓʣ ˠׂ͕ى͜Δʂ

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ڑ཭ϕʔε෼෍ؒྨࣅ౓ʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ཭্ۭؒͷ௒ٿΛ΋ͱʹͨ͠ྨࣅ౓ ΔPMW (P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ׂΓࢉ͕ى͜ΔՄೳੑ ιʔε PX QX λʔήοτ ͣΕ͍ͯΔʢඇઈର࿈ଓʣ ˠׂ͕ى͜Δʂ ඇઈର࿈ଓͷঢ়ଶͰ͸αϯϓϧαΠζʹର͢ΔҰகੑΛࣔͤͳ͍

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ݱ࣮ੈքͰͷඇઈର࿈ଓੑ wྫ0 ff i DF)PNFEBUBTFU wͭͷυϝΠϯ Ξʔτ ΫϦοϓΞʔτ ϓϩμΫτ ϦΞϧ wͷΧςΰϦ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ ҟͳΔυϝΠϯͰ͸ग़ݱ͠ͳ͍ը૾͕͋Δˠඇઈର࿈ଓ

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طଘݚڀͷ·ͱΊͱຊ࿦จͷߩݙ ߩݙ wඇઈର࿈ଓͰ͋ͬͨͱͯ͠΋ιʔεʹର͢ΔҰகੑΛࣔͤ Δཧ࿦Λߏங wڑ཭ۭؒϕʔεͷཧ࿦Λ౷Ұతʹٞ࿦Ͱ͖Δํ๏࿦Λߏங ͠ɼఏҊ͢Δཧ࿦ͷΑΓૣ͍ऩଋ଎౓ͷୡ੒Λࣔ͢ ෼෍ؒڑ཭ ֬཰ൺ ڑ཭ۭؒϕʔε ຊݚڀ ιʔεҰகੑ ✔ ✔ ඇઈର࿈ଓ ✔ ✔

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ຊݚڀͷ݁Ռ

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΍ͬͨ͜ͱ w৽͍͠௒ٿΛ΋ͱʹͨ͠ྨࣅ౓ΛఏҊ Δ 𝒱 (P, Q; r) = ∫ 𝒳 inf x′ ∈ 𝒱 (x) 1 PX (B(x′ , r)) QX (dx) ۙ๣ू߹ 𝒱 (x) ͷ࣌ ҰகੑΛ࣋ͭΞϧΰϦζϜΛߏங Δ 𝒱 (P, Q; r) = O(r−τ) (τ < ∞) *O fi NVNΛऔΔ͜ͱͰׂΓࢉΛ ͋Δఔ౓ճආՄೳ

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//ΞϧΰϦζϜ k wιʔεʴλʔήοταϯϓϧΛ׆༻ͨ͠ //෼ྨث k (X, Y)P (X, Y)Q ιʔεαϯϓϧ λʔήοταϯϓϧ (X, Y) ݁߹ ςετೖྗX (X(1) , Y(1) ), . . . , (X(k) , Y(k) ) ͱڑ཭͕͍ۙ ݸΛநग़ X k ̂ ηk (X) = 1 k k ∑ i=1 Y(i) ̂ hk (X) = 1 { ̂ ηk (X) ≥ 1 2}

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λʔήοτ ͷ೉͠͞ Q wλʔήοταϯϓϧͷΈͰͷ෼ྨ໰୊ͷ೉͠͞ͷԾఆ w4NPPUIOFTT /PJTFDPOEJUJPO w4NPPUIOFTT ͷ)ÖMEFS࿈ଓੑ w/PJTFDPOEJUJPO 5TZCBLPWϊΠζ৚݅ η |η(x) − η(x′ )| ≤ Cα ρα(x, x′ ) QX (0 < |η(X)− 1 2 | ≤ t) ≤ Cβ tβ X ϥϕϧ͕ ϥϕϧ͕ η(X) 1 2 1 ϊΠζͷେ͖͞ ʢؒҧͬͨϥϕϧ͕ಘΒΕΔ֬཰ʣ େ͖͍ϊΠζك ۙ͘ͷϥϕϧ͸ಉ͡

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ۙ๣ू߹ w఺ ͷϥϕϧΛ༧ଌ͢Δͱ͖ϥϕϧ͕มΘΒͳ͍ۙ๣఺ ͷϥϕϧΛ༧ଌͨ݁͠ՌΛ࢖ͬͯྑ͍ X X′ 𝒱 (x) = { x′ ∈ 𝒳 : 2Cα ρα(x, x′ ) < η(x) − 1 2 } X 𝒱 (X) ڥքΛ௒͑ͳ͍͙Β͍ͷ େ͖͞ͷٿ

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సҠࢦ਺ɾࣗݾࢦ਺ wڑ཭ۭؒϕʔεྨࣅ౓ w Λ࢖ͬͨ ͷಛ௃ ͱ ͷಛ௃ Δ(P, Q; r) Δ (P, Q) τ Q ψ 𝔼 [ℰQ (h)] ≤ U(nP , nQ ) λʔήοτ෼෍Ͱଌͬͨ࢒ࠩޡࠩ wཧ࿦ղੳͷ໨ඪ 𝔼 [ℰQ (h)] ≤ C (nc(τ) P + nc(ψ) Q ) −1 ͷ߲ͱ ͷ߲ͷ଍͠ࢉ nP nQ Λେ͖͘͢Ε͹ʹऩଋ ˠҰகੑ nP

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సҠࢦ਺ɾࣗݾࢦ਺ wڑ཭ۭؒϕʔεྨࣅ౓ w Λ࢖ͬͨ ͷಛ௃ ͱ ͷಛ௃ సҠࢦ਺ ࣗݾࢦ਺ Δ(P, Q; r) Δ (P, Q) τ Q ψ Δ τ sup r∈(0,D 𝒳 ( r D 𝒳 ) τ Δ(P, Q; r) ≤ C Δ ψ sup r∈(0,D 𝒳 ( r D 𝒳 ) ψ Δ(Q, Q; r) ≤ C Δ(P, Q; r) = O(r−τ) Δ(Q, Q; r) = O(r−ψ)

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ओ݁Ռ ʢఆཧʣ ͸ ࿈ଓੑɼ ϊΠζ৚͕݅੒Γཱͪɼ ࣗݾࢦ ਺ Λ࣋ͭɽ ͸ సҠࢦ਺ Λ࣋ͭɽ //෼ྨث͸Ҏ Լͷ্քΛ࣋ͭɽ Q α β Δ 𝒱 ψ (P, Q) Δ 𝒱 τ k C (n 1 + β 2 + β +max{1,τ/α} P + n 1 + β 2 + β +max{1,ψ/α} Q ) −1

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ओ݁Ռ w௨ৗઃఆͷ࠷దϨʔτ͸ ʢ ͸࣍ݩʣ "VEJCFSU FUBM w࣮ࡍ ͸࣍ݩͱࣅͨΑ͏ͳੑ࣭Λ࣋ͭ n− 1 + β 2 + β + d/α d ψ ʢఆཧʣ ͸ ࿈ଓੑɼ ϊΠζ৚͕݅੒Γཱͪɼ ࣗݾࢦ ਺ Λ࣋ͭɽ ͸ సҠࢦ਺ Λ࣋ͭɽ //෼ྨث͸Ҏ Լͷ্քΛ࣋ͭɽ Q α β Δ 𝒱 ψ (P, Q) Δ 𝒱 τ k C (n 1 + β 2 + β +max{1,τ/α} P + n 1 + β 2 + β +max{1,ψ/α} Q ) −1 సҠࢦ਺ ࣗݾࢦ਺

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సҠࢦ਺ɾࣗݾࢦ਺ʹΑΔطଘ݁Ռͷ࠶ղऍ wطଘͷ݁Ռ͸ҟͳΔ Λ࢖͍ͬͯΔͱղऍͰ͖Δ 1BUIBLFUBM ,QPUVGFFUBM Δ ΔPMW (P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ΔDM (Q, Q; r) = sup x∈ 𝒳 Q 1 QX (B(x, r)) ΔBCN (Q, Q; r) = 𝒩 ( 𝒳 Q , ρ, r) ΔKM (Q, Q; r) = sup x∈ 𝒳 Q QX (B(x, r)) PX (B(x, r)) ඃෳ਺

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సҠࢦ਺ɾࣗݾࢦ਺ʹΑΔطଘ݁Ռͷ࠶ղऍ ʢఆཧʣ ͸ ࿈ଓੑɼ ϊΠζ৚͕݅੒Γཱͭɽ ʹ͍ͭ ͯҎԼͷ͍ͣΕ͔͕੒Γཱͭɽ ͕ ࣗݾࢦ਺ ɼ ͕ సҠࢦ਺ Λ࣋ͭ ͕ PS ࣗݾࢦ਺ ɼ ͕ సҠࢦ਺ Λ͔࣋ͭͭ ͜ͷ࣌ //෼ྨث͸ओఆཧͱಉ্͡քΛ࣋ͭɽͭ·Γɼ Q α β (P, Q) Q ΔPMW ψ (P, Q) ΔPMW τ Q ΔDM ΔBCN ψ (P, Q) ΔKM τ − ψ τ ≥ ψ k C (n 1 + β 2 + β +max{1,τ/α} P + n 1 + β 2 + β +max{1,ψ/α} Q ) −1 Λൺֱ͢Ε͹্քͷྑ͠ѱ͕͠ൺֱͰ͖Δ Δ

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ͷൺֱ Δ ʢఆཧʣ೚ҙͷ ʹ͍ͭͯ ͕࣋ͭ࠷খͷ సҠࢦ਺ɾࣗݾࢦ਺ w ఏҊ͍ͯ͠Δ ͷసҠࢦ਺ɾࣗݾࢦ਺͕Ұ൪খ͍͞ w ˠҰ൪ૣ͍ऩଋΛ্ࣔ͢ք͕ಘΒΕΔ (P, Q) τΔ 𝒱 ≤ τΔPMW ≤ τΔKM + min{ψΔDM , ψΔDM } ψΔ 𝒱 ≤ τΔPMW ≤ min{ψΔDM , ψΔDM } τΔ , ψΔ (P, Q) Δ Δ 𝒱

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࣮ݧ ͷਓ޻σʔλͷ࣮ݧΛ࣮ࢪ wӈਤͷ෼෍ɾճؼؔ਺ w ධՁࢦඪ wαΠζͷςετσʔληοτ Ͱܭࢉͨ͠࢒༨ޡࠩ 𝒳 = ℝ nP ∈ {28,29, . . . ,218}, nQ = 10 ੨ιʔε෼෍ͷີ౓ؔ਺ ᒵλʔήοτ෼෍ͷີ౓ؔ਺ ੺αϙʔτ͕ҟͳΔྖҬ ྘ճؼؔ਺ BMQIB CBUB UBV QTJ 1.8 PS BMQIB ♾ 0VS PS BMQIB PS ඇઈର࿈ଓΑΓ

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݁Ռ w1.8΋PVS΋ཧ࿦ό΢ϯυͱ ܏͖͕ಉ͡ wό΢ϯυ͸λΠτ w1.8͸ޡ͕ࠩݮΒͳ͍ wҰகੑ͕ͳ͍ w0VS͸ޡ͕ࠩݮ͍ͬͯΔ wҰகੑΛࣔ͢ α = 0.5,τ = 2.0 α = 0.25,τ = 2.0 ιʔεαϯϓϧαΠζ ιʔεαϯϓϧαΠζ

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·ͱΊ w$PWBSJBUFTIJGUԼͰιʔεαϯϓϧαΠζʹର͢ΔҰகੑ ΛࣔͤΔཧ࿦Λߏங w͜ͷঢ়گԼͰͷసҠͷ੒ޭΛࣔ͢ wಛʹۙ๣৘ใΛ׆༻͠ඇઈର࿈ଓͳঢ়گͰ΋ҰகੑΛࣔ͢ ͜ͱ͕Մೳ .JUTVIJSP'VKJLBXB :PIFJ"LJNPUP +VO4BLVNB BOE ,B[VUP'VLVDIJ)BSOFTTJOHUIF1PXFSPG7JDJOJUZ *OGPSNFE"OBMZTJTGPS$MBTTJ fi DBUJPOVOEFS$PWBSJBUF 4IJGUIUUQTBSYJWPSHBCT