%!PS-Adobe-2.0 %%Creator: dvips(k) 2020.1 Copyright 2020 Radical Eye Software %%Title: arXiv:2209.11215v3 [cs.LG] 15 Apr 2023 %%CreationDate: Tue Apr 18 10:20:53 2023 %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: EUSM10 EUSM7 EUFM10 %%DocumentPaperSizes: Letter %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z arxiv3.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2023.04.18:1020 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: psfrag.pro 0 0 %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. 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div stampsize mul def pop 252 -32 moveto currentpoint /pdf@lly exch pdf@bottom add def /pdf@llx exch 2 sub def 0.5 setgray (arXiv:2209.11215v3 [cs.LG] 15 Apr 2023) show currentpoint /pdf@ury exch pdf@top add def /pdf@urx exch 2 add def /pdfmark where{pop}{userdict /pdfmark /cleartomark load put}ifelse [ /H /I /Border [0 0 1] /BS <</S/D/D[2 6]/W 1>> /Color [0 1 1] /Action << /Subtype /URI /URI (http://arxiv.org/abs/2209.11215v3)>> /Subtype /Link /Rect[pdf@llx pdf@lly pdf@urx pdf@ury] /ANN pdfmark grestore TeXDict begin HPSdict begin 1 0 bop 0 0 a SDict begin [/Producer (dvips + Distiller)/Title ()/Subject ()/Creator (LaTeX with hyperref)/Author ()/Keywords () /DOCINFO pdfmark end 0 0 a 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.1) cvn /DEST pdfmark end -8 -316 a Black 0 TeXcolorgray Black 0 TeXcolorgray 0 0 a SDict begin [/Count 2/Dest (section.1) cvn/Title (1 Introduction) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.1.1) cvn/Title (1.1 Our contributions) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.1.2) cvn/Title (1.2 Prior work) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 2/Dest (section.2) cvn/Title (2 Background on SGMs) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.2.1) cvn/Title (2.1 Background on denoising diffusion probabilistic modeling \(DDPM\)) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.2.2) cvn/Title (2.2 Background on the critically damped Langevin diffusion \(CLD\)) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 3/Dest (section.3) cvn/Title (3 Results) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.3.1) cvn/Title (3.1 Results for DDPM) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.3.2) cvn/Title (3.2 Consequences for arbitrary data distributions with bounded support) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.3.3) cvn/Title (3.3 Results for CLD) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (section.4) cvn/Title (4 Technical overview) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 5/Dest (section.5) cvn/Title (5 Proofs for DDPM) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.5.1) cvn/Title (5.1 Preliminaries on Girsanov's theorem and a first attempt at applying Girsanov's theorem) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.5.2) cvn/Title (5.2 Checking the assumptions of Girsanov's theorem and the Girsanov discretization argument) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.5.3) cvn/Title (5.3 Proof of Theorem 2) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.5.4) cvn/Title (5.4 Auxiliary lemmas) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.5.5) cvn/Title (5.5 Proof of Corollary 5) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 5/Dest (section.6) cvn/Title (6 Proofs for CLD) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.6.1) cvn/Title (6.1 Background on the CLD process) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.6.2) cvn/Title (6.2 Girsanov discretization argument) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.6.3) cvn/Title (6.3 Proof of Theorem 6) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.6.4) cvn/Title (6.4 Auxiliary lemmas) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (subsection.6.5) cvn/Title (6.5 Lower bound against CLD) /OUT pdfmark end 0 0 a 0 0 a SDict begin [/Count 0/Dest (section.7) cvn/Title (7 Conclusion) /OUT pdfmark end 0 0 a 0 0 a SDict 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pdfmark end 3729 404 a FJ(\(5.5\))1518 670 y(=)1606 567 y Fw(N)f FB(\000)p FA(1)1617 591 y Fy(X)1617 770 y Fw(k)q FA(=0)1763 670 y Fx(E)1819 682 y Fw(Q)1871 663 y Fi( )1871 702 y Ft(T)1949 557 y Fy(Z)2032 578 y FA(\()p Fw(k)q FA(+1\))p Fw(h)1996 746 y(k)q(h)2248 670 y FI(k)p FH(s)2329 682 y Fw(T)j FB(\000)p Fw(k)q(h)2508 670 y FJ(\()p FH(X)2609 682 y Fw(k)q(h)2689 670 y FJ(\))19 b FI(\000)f(r)c FJ(ln)f FH(q)3026 682 y Fw(T)c FB(\000)p Fw(t)3155 670 y FJ(\()p FH(X)3256 682 y Fw(t)3286 670 y FJ(\))p FI(k)3360 636 y FA(2)3410 670 y FJ(d)p FH(t)14 b(:)-4 927 y FJ(W)-7 b(e)36 b(then)f(use)h(a)e(discretization)e (argumen)n(t)h(based)i(on)g(sto)r(c)n(hastic)f(calculus)e(to)k(further) f(b)r(ound)h(this)e(quan)n(tit)n(y)-7 b(.)59 b(The)0 1027 y(result)26 b(is)h(the)h(follo)n(wing)23 b(theorem.)-3 1193 y SDict begin H.S end -3 1193 a -3 1193 a SDict begin 12 H.A end -3 1193 a -3 1193 a SDict begin [/View [/XYZ H.V]/Dest (theorem.9) cvn /DEST pdfmark end -3 1193 a 0 TeXcolorgray FC(Theorem)j(9)i FJ(\(discretization)21 b(error)i(for)g(DDPM\))p FC(.)p 0 TeXcolorgray 39 w FG(Supp)l(ose)k (that)g(Assumptions)p 0.5 0 0 TeXcolorrgb 2665 1193 a SDict begin H.S end 2665 1193 a 0.5 0 0 TeXcolorrgb FG(1)p 0.5 0 0 TeXcolorrgb 2707 1140 a SDict begin H.R end 2707 1140 a 2707 1193 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (assumption.1) cvn H.B /ANN pdfmark end 2707 1193 a 0 TeXcolorgray FG(,)p 0.5 0 0 TeXcolorrgb 2759 1193 a SDict begin H.S end 2759 1193 a 0.5 0 0 TeXcolorrgb FG(2)p 0.5 0 0 TeXcolorrgb 2802 1140 a SDict begin H.R end 2802 1140 a 2802 1193 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (assumption.2) cvn H.B /ANN pdfmark end 2802 1193 a 0 TeXcolorgray FG(,)h(and)p 0.5 0 0 TeXcolorrgb 3013 1193 a SDict begin H.S end 3013 1193 a 0.5 0 0 TeXcolorrgb FG(3)p 0.5 0 0 TeXcolorrgb 3055 1140 a SDict begin H.R end 3055 1140 a 3055 1193 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (assumption.3) cvn H.B /ANN pdfmark end 3055 1193 a 0 TeXcolorgray 27 w FG(hold.)39 b(L)l(et)27 b FH(Q)3500 1163 y FB( )3500 1216 y Fw(T)3596 1193 y FG(and)h FH(P)3820 1153 y Fw(q)3850 1161 y Ft(T)3808 1217 y Fw(T)-6 1293 y FG(denote)34 b(the)g(me)l(asur)l(es)g(on)g(p)l (ath)h(sp)l(ac)l(e)g(c)l(orr)l(esp)l(onding)g(to)f(the)h(r)l(everse)f (pr)l(o)l(c)l(ess)h FJ(\()p 0.5 0 0 TeXcolorrgb 2673 1293 a SDict begin H.S end 2673 1293 a 0.5 0 0 TeXcolorrgb FJ(2.4)p 0.5 0 0 TeXcolorrgb 2779 1239 a SDict begin H.R end 2779 1239 a 2779 1293 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 2779 1293 a 0 TeXcolorgray FJ(\))f FG(and)h(the)f(SGM)h(algorithm)g(with)0 1392 y FH(L)57 1362 y FA(2)94 1392 y FG(-ac)l(cur)l(ate)29 b(sc)l(or)l(e)h(estimate)g(initialize)l(d)i(at)e FH(q)1508 1404 y Fw(T)1560 1392 y FG(.)39 b(Assume)29 b(that)g FH(L)23 b FI(\025)g FJ(1)29 b FG(and)h FH(h)23 b Fn(.)g FJ(1)p FH(=L)p FG(.)37 b(Then,)794 1575 y Fr(TV)q FJ(\()p FH(P)1004 1535 y Fw(q)1034 1543 y Ft(T)992 1599 y Fw(T)1085 1575 y FH(;)14 b(Q)1188 1541 y FB( )1188 1596 y Fw(T)1258 1575 y FJ(\))1290 1541 y FA(2)1350 1575 y FI(\024)23 b Fr(KL)p FJ(\()p FH(Q)1639 1541 y FB( )1639 1596 y Fw(T)1727 1575 y FI(k)18 b FH(P)1852 1535 y Fw(q)1882 1543 y Ft(T)1840 1599 y Fw(T)1933 1575 y FJ(\))23 b Fn(.)g FJ(\()p FH(")2147 1541 y FA(2)2147 1596 y(score)2314 1575 y FJ(+)18 b FH(L)2454 1541 y FA(2)2490 1575 y FH(dh)h FJ(+)f FH(L)2740 1541 y FA(2)2777 1575 y Fo(m)2841 1541 y FA(2)2841 1596 y(2)2878 1575 y FH(h)2926 1541 y FA(2)2963 1575 y FJ(\))c FH(T)25 b(:)0 1758 y FC(Pro)s(of.)f FJ(W)-7 b(e)28 b(start)f(b)n(y)h(pro)n (ving)439 1913 y Fw(N)6 b FB(\000)p FA(1)451 1938 y Fy(X)451 2117 y Fw(k)q FA(=0)597 2017 y Fx(E)652 2029 y Fw(Q)704 2009 y Fi( )704 2049 y Ft(T)783 1904 y Fy(Z)866 1924 y FA(\()p Fw(k)q FA(+1\))p Fw(h)829 2093 y(k)q(h)1082 2017 y FI(k)p FH(s)1163 2029 y Fw(T)j FB(\000)p Fw(k)q(h)1342 2017 y FJ(\()p FH(X)1443 2029 y Fw(k)q(h)1523 2017 y FJ(\))18 b FI(\000)g(r)c FJ(ln)g FH(q)1860 2029 y Fw(T)9 b FB(\000)p Fw(t)1989 2017 y FJ(\()p FH(X)2090 2029 y Fw(t)2119 2017 y FJ(\))p FI(k)2193 1983 y FA(2)2244 2017 y FJ(d)p FH(t)23 b Fn(.)g FJ(\()p FH(")2502 1983 y FA(2)2502 2037 y(score)2669 2017 y FJ(+)18 b FH(L)2809 1983 y FA(2)2846 2017 y FH(dh)g FJ(+)g FH(L)3095 1983 y FA(2)3132 2017 y Fo(m)3196 1983 y FA(2)3196 2037 y(2)3233 2017 y FH(h)3281 1983 y FA(2)3318 2017 y FJ(\))c FH(T)25 b(:)-3 2279 y FJ(Then,)j(w)n(e)f(giv)n(e)f(the)i(appro)n(ximation)22 b(argumen)n(t)k(to)h(pro)n(v)n(e)f(the)i(inequalit)n(y)c(\()p 0.5 0 0 TeXcolorrgb 2483 2279 a SDict begin H.S end 2483 2279 a 0.5 0 0 TeXcolorrgb FJ(5.5)p 0.5 0 0 TeXcolorrgb 2590 2225 a SDict begin H.R end 2590 2225 a 2590 2279 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.5) cvn H.B /ANN pdfmark end 2590 2279 a 0 TeXcolorgray FJ(\).)125 2378 y FC(Bound)31 b(on)h(the)f(discretization)h(error.)37 b FJ(F)-7 b(or)27 b FH(t)c FI(2)g FJ([)p FH(k)s(h;)14 b FJ(\()p FH(k)21 b FJ(+)e(1\))p FH(h)p FJ(],)27 b(w)n(e)g(can)g (decomp)r(ose)717 2561 y Fx(E)772 2573 y Fw(Q)824 2553 y Fi( )824 2593 y Ft(T)889 2561 y FJ([)p FI(k)p FH(s)993 2573 y Fw(T)9 b FB(\000)p Fw(k)q(h)1172 2561 y FJ(\()p FH(X)1273 2573 y Fw(k)q(h)1353 2561 y FJ(\))19 b FI(\000)f(r)c FJ(ln)f FH(q)1690 2573 y Fw(T)c FB(\000)p Fw(t)1819 2561 y FJ(\()p FH(X)1920 2573 y Fw(t)1950 2561 y FJ(\))p FI(k)2024 2527 y FA(2)2061 2561 y FJ(])892 2698 y Fn(.)23 b Fx(E)1035 2710 y Fw(Q)1087 2690 y Fi( )1087 2730 y Ft(T)1152 2698 y FJ([)p FI(k)p FH(s)1256 2710 y Fw(T)9 b FB(\000)p Fw(k)q(h)1435 2698 y FJ(\()p FH(X)1536 2710 y Fw(k)q(h)1616 2698 y FJ(\))19 b FI(\000)f(r)c FJ(ln)f FH(q)1953 2710 y Fw(T)c FB(\000)p Fw(k)q(h)2133 2698 y FJ(\()p FH(X)2234 2710 y Fw(k)q(h)2313 2698 y FJ(\))p FI(k)2387 2664 y FA(2)2424 2698 y FJ(])1136 2835 y(+)18 b Fx(E)1275 2847 y Fw(Q)1327 2827 y Fi( )1327 2867 y Ft(T)1392 2835 y FJ([)p FI(kr)c FJ(ln)e FH(q)1659 2847 y Fw(T)d FB(\000)p Fw(k)q(h)1839 2835 y FJ(\()p FH(X)1940 2847 y Fw(k)q(h)2020 2835 y FJ(\))19 b FI(\000)f(r)c FJ(ln)f FH(q)2357 2847 y Fw(T)c FB(\000)p Fw(t)2486 2835 y FJ(\()p FH(X)2587 2847 y Fw(k)q(h)2667 2835 y FJ(\))p FI(k)2741 2801 y FA(2)2778 2835 y FJ(])1136 2972 y(+)18 b Fx(E)1275 2984 y Fw(Q)1327 2964 y Fi( )1327 3004 y Ft(T)1392 2972 y FJ([)p FI(kr)c FJ(ln)e FH(q)1659 2984 y Fw(T)d FB(\000)p Fw(t)1789 2972 y FJ(\()p FH(X)1890 2984 y Fw(k)q(h)1970 2972 y FJ(\))18 b FI(\000)g(r)c FJ(ln)f FH(q)2306 2984 y Fw(T)c FB(\000)p Fw(t)2436 2972 y FJ(\()p FH(X)2537 2984 y Fw(t)2566 2972 y FJ(\))p FI(k)2640 2938 y FA(2)2677 2972 y FJ(])892 3150 y Fn(.)23 b FH(")1019 3115 y FA(2)1019 3170 y(score)1185 3150 y FJ(+)18 b Fx(E)1324 3162 y Fw(Q)1376 3142 y Fi( )1376 3182 y Ft(T)1441 3058 y Fy(h)1480 3054 y(\015)1480 3104 y(\015)1480 3154 y(\015)1526 3150 y FI(r)c FJ(ln)1702 3094 y FH(q)1739 3106 y Fw(T)9 b FB(\000)p Fw(k)q(h)p 1702 3131 217 4 v 1727 3207 a FH(q)1764 3219 y Fw(T)g FB(\000)p Fw(t)1929 3150 y FJ(\()p FH(X)2030 3162 y Fw(k)q(h)2110 3150 y FJ(\))2142 3054 y Fy(\015)2142 3104 y(\015)2142 3154 y(\015)2188 3075 y FA(2)2225 3058 y Fy(i)2283 3150 y FJ(+)18 b FH(L)2423 3115 y FA(2)2473 3150 y Fx(E)2529 3162 y Fw(Q)2581 3142 y Fi( )2581 3182 y Ft(T)2646 3150 y FJ([)p FI(k)p FH(X)2780 3162 y Fw(k)q(h)2877 3150 y FI(\000)g FH(X)3029 3162 y Fw(t)3058 3150 y FI(k)3100 3115 y FA(2)3137 3150 y FJ(])c FH(:)3729 3150 y SDict begin H.S end 3729 3150 a 3729 3150 a SDict begin 15 H.A end 3729 3150 a 3729 3150 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.6) cvn /DEST pdfmark end 3729 3150 a FJ(\(5.6\))125 3388 y(W)-7 b(e)21 b(m)n(ust)e(b)r(ound)i(the)h(c)n(hange)d(in)h(the)h(score)e (function)i(along)d(the)j(forw)n(ard)e(pro)r(cess.)34 b(If)21 b FH(S)27 b FJ(:)c Fx(R)3080 3358 y Fw(d)3148 3388 y FI(!)g Fx(R)3308 3358 y Fw(d)3374 3388 y FJ(is)c(the)i(mapping)0 3487 y FH(S)5 b FJ(\()p FH(x)p FJ(\))191 3484 y(:)208 3487 y(=)23 b(exp\()p FI(\000)p FJ(\()p FH(t)17 b FI(\000)h FH(k)s(h)p FJ(\)\))c FH(x)p FJ(,)27 b(then)h FH(q)1177 3499 y Fw(T)9 b FB(\000)p Fw(k)q(h)1380 3487 y FJ(=)23 b FH(S)1519 3499 y FA(#)1577 3487 y FH(q)1614 3499 y Fw(T)9 b FB(\000)p Fw(t)1761 3487 y FI(\003)17 b Fr(no)n(rmal)o FJ(\(0)p FH(;)d FJ(1)j FI(\000)g FJ(exp\()p FI(\000)p FJ(2)d(\()p FH(t)j FI(\000)g FH(k)s(h)p FJ(\)\)\).)38 b(W)-7 b(e)27 b(can)g(then)h(use)f([)p 0 0 0.5 TeXcolorrgb 3649 3487 a SDict begin H.S end 3649 3487 a FJ(LL)-7 b(T22)3890 3487 y SDict begin 12 H.L end 3890 3487 a 3890 3487 a SDict begin [/Subtype /Link/Dest (cite.0@leelutan2022generative) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3890 3487 a 0 TeXcolorgray -2 w FJ(,)0 3587 y(Lemma)23 b(C.12])h(\(or)g(the)h(more)e(general)g(Lemma)p 0.5 0 0 TeXcolorrgb 1561 3587 a SDict begin H.S end 1561 3587 a 0.5 0 0 TeXcolorrgb FJ(16)p 0.5 0 0 TeXcolorrgb 1644 3534 a SDict begin H.R end 1644 3534 a 1644 3587 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.16) cvn H.B /ANN pdfmark end 1644 3587 a 0 TeXcolorgray 25 w FJ(that)j(w)n(e)e(pro)n(v)n(e)f (in)h(Section)p 0.5 0 0 TeXcolorrgb 2569 3587 a SDict begin H.S end 2569 3587 a 0.5 0 0 TeXcolorrgb FJ(6.4)p 0.5 0 0 TeXcolorrgb 2675 3534 a SDict begin H.R end 2675 3534 a 2675 3587 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (subsection.6.4) cvn H.B /ANN pdfmark end 2675 3587 a 0 TeXcolorgray FJ(\))i(with)e FH(\013)g FJ(=)e(exp\()p FH(t)13 b FI(\000)g FH(k)s(h)p FJ(\))23 b(=)g(1)13 b(+)g FH(O)r FJ(\()p FH(h)p FJ(\))0 3687 y(and)27 b FH(\033)211 3657 y FA(2)272 3687 y FJ(=)c(1)18 b FI(\000)g FJ(exp\()p FI(\000)p FJ(2)c(\()p FH(t)k FI(\000)g FH(k)s(h)p FJ(\)\))23 b(=)f FH(O)r FJ(\()p FH(h)p FJ(\))29 b(to)f(obtain)558 3816 y Fy(\015)558 3866 y(\015)558 3915 y(\015)604 3911 y FI(r)14 b FJ(ln)780 3855 y FH(q)817 3867 y Fw(T)9 b FB(\000)p Fw(k)q(h)p 780 3892 217 4 v 805 3968 a FH(q)842 3980 y Fw(T)g FB(\000)p Fw(t)1006 3911 y FJ(\()p FH(X)1107 3923 y Fw(k)q(h)1187 3911 y FJ(\))1219 3816 y Fy(\015)1219 3866 y(\015)1219 3915 y(\015)1266 3836 y FA(2)1326 3911 y Fn(.)23 b FH(L)1471 3877 y FA(2)1507 3911 y FH(dh)c FJ(+)f FH(L)1757 3877 y FA(2)1794 3911 y FH(h)1842 3877 y FA(2)1892 3911 y FI(k)p FH(X)2003 3923 y Fw(k)q(h)2082 3911 y FI(k)2124 3877 y FA(2)2180 3911 y FJ(+)g(\(1)g(+)g FH(L)2495 3877 y FA(2)2532 3911 y FJ(\))c FH(h)2626 3877 y FA(2)2677 3911 y FI(kr)g FJ(ln)e FH(q)2921 3923 y Fw(T)d FB(\000)p Fw(t)3051 3911 y FJ(\()p FH(X)3152 3923 y Fw(k)q(h)3231 3911 y FJ(\))p FI(k)3305 3877 y FA(2)1326 4092 y Fn(.)23 b FH(L)1471 4058 y FA(2)1507 4092 y FH(dh)c FJ(+)f FH(L)1757 4058 y FA(2)1794 4092 y FH(h)1842 4058 y FA(2)1892 4092 y FI(k)p FH(X)2003 4104 y Fw(k)q(h)2082 4092 y FI(k)2124 4058 y FA(2)2180 4092 y FJ(+)g FH(L)2320 4058 y FA(2)2356 4092 y FH(h)2404 4058 y FA(2)2455 4092 y FI(kr)c FJ(ln)f FH(q)2700 4104 y Fw(T)c FB(\000)p Fw(t)2829 4092 y FJ(\()p FH(X)2930 4104 y Fw(k)q(h)3010 4092 y FJ(\))p FI(k)3084 4058 y FA(2)-3 4275 y FJ(where)27 b(the)h(last)e(line)g(uses)h FH(L)c FI(\025)g FJ(1.)125 4375 y(F)-7 b(or)27 b(the)g(last)g(term,)572 4557 y FI(kr)14 b FJ(ln)f FH(q)817 4569 y Fw(T)c FB(\000)p Fw(t)946 4557 y FJ(\()p FH(X)1047 4569 y Fw(k)q(h)1127 4557 y FJ(\))p FI(k)1201 4523 y FA(2)1261 4557 y Fn(.)23 b FI(kr)14 b FJ(ln)e FH(q)1593 4569 y Fw(T)d FB(\000)p Fw(t)1723 4557 y FJ(\()p FH(X)1824 4569 y Fw(t)1853 4557 y FJ(\))p FI(k)1927 4523 y FA(2)1982 4557 y FJ(+)18 b FI(kr)c FJ(ln)f FH(q)2310 4569 y Fw(T)c FB(\000)p Fw(t)2439 4557 y FJ(\()p FH(X)2540 4569 y Fw(k)q(h)2620 4557 y FJ(\))19 b FI(\000)f(r)c FJ(ln)f FH(q)2957 4569 y Fw(T)c FB(\000)p Fw(t)3086 4557 y FJ(\()p FH(X)3187 4569 y Fw(t)3217 4557 y FJ(\))p FI(k)3291 4523 y FA(2)1261 4692 y Fn(.)23 b FI(kr)14 b FJ(ln)e FH(q)1593 4704 y Fw(T)d FB(\000)p Fw(t)1723 4692 y FJ(\()p FH(X)1824 4704 y Fw(t)1853 4692 y FJ(\))p FI(k)1927 4658 y FA(2)1982 4692 y FJ(+)18 b FH(L)2122 4658 y FA(2)2173 4692 y FI(k)p FH(X)2284 4704 y Fw(k)q(h)2382 4692 y FI(\000)g FH(X)2534 4704 y Fw(t)2563 4692 y FI(k)2605 4658 y FA(2)2655 4692 y FH(;)-3 4875 y FJ(where)27 b(the)h(second)f(term)g(ab)r(o)n(v)n(e)f(is)g(absorb)r (ed)h(in)n(to)f(the)i(third)f(term)g(of)g(the)h(decomp)r(osition)c(\()p 0.5 0 0 TeXcolorrgb 3053 4875 a SDict begin H.S end 3053 4875 a 0.5 0 0 TeXcolorrgb FJ(5.6)p 0.5 0 0 TeXcolorrgb 3159 4821 a SDict begin H.R end 3159 4821 a 3159 4875 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.6) cvn H.B /ANN pdfmark end 3159 4875 a 0 TeXcolorgray FJ(\))q(.)37 b(Hence,)767 5058 y Fx(E)823 5070 y Fw(Q)875 5050 y Fi( )875 5090 y Ft(T)940 5058 y FJ([)p FI(k)p FH(s)1044 5070 y Fw(T)9 b FB(\000)p Fw(k)q(h)1223 5058 y FJ(\()p FH(X)1324 5070 y Fw(k)q(h)1404 5058 y FJ(\))19 b FI(\000)f(r)c FJ(ln)f FH(q)1741 5070 y Fw(T)c FB(\000)p Fw(t)1870 5058 y FJ(\()p FH(X)1971 5070 y Fw(t)2000 5058 y FJ(\))p FI(k)2074 5023 y FA(2)2111 5058 y FJ(])943 5195 y 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FJ([)p FI(k)p FH(X)7 b FI(k)1982 5366 y FA(2)2017 5400 y FJ(])19 b(+)f FI(f)p FJ(1)f FI(\000)h FJ(exp\()p FI(\000)p FJ(2)p FH(t)p FJ(\))p FI(g)c FH(d)22 b FI(\024)h FH(d)c FI(_)f Fo(m)3062 5366 y FA(2)3062 5421 y(2)3113 5400 y FH(:)p 0 TeXcolorgray 1908 5649 a FJ(14)p 0 TeXcolorgray eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.15) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 0 a SDict begin H.S end 0 0 a 0 0 a SDict begin 12 H.A end 0 0 a 0 0 a SDict begin [/View [/XYZ H.V]/Dest (Item.7) cvn /DEST pdfmark end 0 0 a 0 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Fn(.)23 b FH(\016)1578 1736 y Fy(Z)1661 1757 y Fw(t)1624 1925 y(s)1704 1849 y Fx(E)14 b FJ([)p FI(k)1848 1828 y FJ(\026)1824 1849 y FH(X)1893 1861 y Fw(r)1929 1849 y FI(k)1971 1815 y FA(2)2008 1849 y FJ(])g(d)p FH(r)21 b FJ(+)d FH(\016)s(d)24 b Fn(.)e FH(\016)2466 1815 y FA(2)2517 1849 y FJ(\()p FH(d)d FJ(+)f Fo(m)2758 1815 y FA(2)2758 1870 y(2)2795 1849 y FJ(\))h(+)f FH(\016)s(d)1436 2030 y Fn(.)23 b FH(\016)1564 1996 y FA(2)1601 2030 y Fo(m)1665 1996 y FA(2)1665 2050 y(2)1721 2030 y FJ(+)18 b FH(\016)s(d)c(;)-3 2209 y FJ(where)27 b(w)n(e)g(used)h(Lemma)p 0.5 0 0 TeXcolorrgb 845 2209 a SDict begin H.S end 845 2209 a 0.5 0 0 TeXcolorrgb FJ(10)p 0.5 0 0 TeXcolorrgb 928 2155 a SDict begin H.R end 928 2155 a 928 2209 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.10) cvn H.B /ANN pdfmark end 928 2209 a 0 TeXcolorgray FJ(.)p 3839 2209 4 57 v 3843 2156 50 4 v 3843 2209 V 3893 2209 4 57 v 125 2368 a(W)-7 b(e)28 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FJ(\()p FH(t)p FJ(\))29 b FG(for)i(every)g FH(t)23 b FI(\024)f FH(T)1214 2971 y Fw(n)1259 2959 y FG(.)-13 3122 y(Then,)29 b(for)e(every)h FH(")23 b(>)f FJ(0)p FG(,)28 b FH(f)858 3134 y Fw(n)926 3122 y FI(!)23 b FH(f)35 b FG(uniformly)28 b(over)g FJ([0)p FH(;)14 b(T)22 b FI(\000)12 b FH(")p FJ(])p FG(.)38 b(In)26 b(p)l(articular,)j FH(f)2589 3134 y Fw(n)2634 3122 y FJ(\()p FI(\001)12 b(^)g FH(T)25 b FI(\000)12 b FH(")p FJ(\))22 b FI(!)i FH(f)9 b FJ(\()p FI(\001)j(^)g FH(T)24 b FI(\000)12 b FH(")p FJ(\))26 b FG(uniformly)-7 3222 y(over)31 b FJ([0)p FH(;)14 b(T)e FJ(])p FG(.)0 3385 y SDict begin H.S end 0 3385 a 0 3385 a SDict begin 12 H.A end 0 3385 a 0 3385 a SDict begin [/View [/XYZ H.V]/Dest (theorem.13) cvn /DEST pdfmark end 0 3385 a 0 TeXcolorgray FC(Lemma)30 b(13.)p 0 TeXcolorgray 40 w FG(Consider)h FH(f)h FJ(:)23 b([0)p FH(;)14 b(T)e FJ(])22 b FI(!)h Fx(R)1342 3355 y Fw(d)1416 3385 y FG(c)l(ontinuous,)30 b(and)g FH(f)2062 3397 y Fw(")2121 3385 y FJ(:)23 b([0)p FH(;)14 b(T)e FJ(])21 b FI(!)j Fx(R)2535 3355 y Fw(d)2609 3385 y FG(s.t.)30 b FH(f)2792 3397 y Fw(")2827 3385 y FJ(\()p FH(t)p FJ(\))24 b(=)f FH(f)9 b FJ(\()p FH(t)18 b FI(^)h FJ(\()p FH(T)30 b FI(\000)18 b FH(")p FJ(\)\))30 b FG(for)g FH(")23 b(>)g FJ(0)p FG(.)-13 3484 y(Then)30 b FH(f)244 3496 y Fw(")303 3484 y FI(!)23 b FH(f)38 b FG(uniformly)31 b(over)f FJ([0)p FH(;)14 b(T)e FJ(])28 b FG(as)j FH(")22 b FI(!)h FJ(0)p FG(.)0 3621 y SDict begin H.S end 0 3621 a 0 3621 a SDict begin 12 H.A end 0 3621 a 0 3621 a SDict begin [/View [/XYZ H.V]/Dest (subsection.5.5) cvn /DEST pdfmark end 0 3621 a 95 x Fz(5.5)112 b(Pro)s(of)37 b(of)h(Corollary)p 0.5 0 0 TeXcolorrgb 1190 3716 a SDict begin H.S end 1190 3716 a 0.5 0 0 TeXcolorrgb Fz(5)p 0.5 0 0 TeXcolorrgb 1246 3652 a SDict begin H.R end 1246 3652 a 1246 3716 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.5) cvn H.B /ANN pdfmark end 1246 3716 a 0 TeXcolorgray 0 3869 a FC(Pro)s(of.)24 b FJ([Pro)r(of)30 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3143 y FJ(\))2253 3047 y Fy(\015)2253 3097 y(\015)2253 3147 y(\015)2299 3068 y FA(2)2337 3051 y Fy(i)2399 3143 y Fn(.)23 b FH(\016)2540 3030 y Fy(Z)2623 3050 y Fw(t)2587 3218 y(s)2666 3143 y Fx(E)15 b FJ([)p FI(k)2810 3122 y FJ(\026)2787 3143 y FH(X)2856 3155 y Fw(r)2892 3143 y FI(k)2934 3109 y FA(2)2989 3143 y FJ(+)j FI(k)3127 3122 y FJ(\026)3114 3143 y FH(V)3162 3155 y Fw(r)3199 3143 y FI(k)3241 3109 y FA(2)3278 3143 y FJ(])c(d)p FH(r)21 b FJ(+)d FH(\016)s(d)823 3323 y Fn(.)k FH(\016)950 3289 y FA(2)1001 3323 y FJ(\()p FH(d)d FJ(+)f Fo(m)1242 3289 y FA(2)1242 3344 y(2)1279 3323 y FJ(\))h(+)f FH(\016)s(d)c(;)-3 3504 y FJ(where)27 b(w)n(e)g(used)h(Lemma)p 0.5 0 0 TeXcolorrgb 845 3504 a SDict begin H.S end 845 3504 a 0.5 0 0 TeXcolorrgb FJ(17)p 0.5 0 0 TeXcolorrgb 928 3450 a SDict begin H.R end 928 3450 a 928 3504 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.17) cvn H.B /ANN pdfmark end 928 3504 a 0 TeXcolorgray 28 w FJ(again.)p 3839 3504 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b FB(\000)p FA(1)778 3805 y Fy(X)778 3984 y Fw(k)q FA(=0)924 3771 y Fy(Z)1007 3791 y FA(\()p Fw(k)q FA(+1\))p Fw(h)970 3959 y(k)q(h)1223 3884 y FI(k)p Fp(s)1308 3896 y Fw(T)j FB(\000)p Fw(k)q(h)1488 3884 y FJ(\()p FH(X)1589 3896 y Fw(k)q(h)1669 3884 y FH(;)14 b(V)1754 3896 y Fw(k)q(h)1834 3884 y FJ(\))19 b FI(\000)f(r)2037 3896 y Fw(v)2090 3884 y FJ(ln)13 b Fp(q)2221 3904 y Fw(T)c FB(\000)p Fw(t)2350 3884 y FJ(\()p FH(X)2451 3896 y Fw(t)2481 3884 y FH(;)14 b(V)2566 3896 y Fw(t)2595 3884 y FJ(\))p FI(k)2669 3850 y FA(2)2720 3884 y FJ(d)p FH(t)2796 3792 y Fy(\021)519 4177 y FI(\024)23 b Fx(E)28 b FJ(exp)803 4085 y Fy(\020)853 4177 y FH(C)6 b(T)12 b(h)1027 4143 y FA(2)1122 4177 y FJ(sup)1077 4251 y Fw(s)p FB(2)p FA([0)p Fw(;T)d FA(])1305 4177 y FJ(\()p FI(k)1403 4156 y FJ(\026)1379 4177 y FH(X)1448 4189 y Fw(s)1483 4177 y FI(k)1525 4143 y FA(2)1580 4177 y FJ(+)19 b FI(k)1719 4156 y FJ(\026)1706 4177 y FH(V)1754 4189 y Fw(s)1789 4177 y FI(k)1831 4143 y FA(2)1868 4177 y FJ(\))f(+)g FH(C)6 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