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Rethinking Controllable Variational AutoencodersHuajie Shao Yifei Yang Haohong Lin Longzhong Lin Yizhuo Chen Qinmin Yang Han ZhaoIEEE Conference on Computer Vision and Pattern Recognition
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[-500(W)80(e)-200(fundamentally)-199(e)15(xplain)-200(wh)5(y)-200(controlling)-199(the)-200(v)25(alue)-200(of)]TJ 8.468 -11.955 Td [(KL-di)25(v)15(er)18(gence)-306(can)-305(learn)-306(disentangled)-306(representations)]TJ 0 -11.955 Td [(from)-277(an)-278(information)-277(bottleneck)-278(perspecti)25(v)15(e.)-392(W)80(e)-278(sho)25(w)]TJ 0 -11.955 Td [(that)-375(KL-di)25(v)15(er)18(gence)-374(is)-375(an)-375(upper)-375(bound)-374(of)-375(the)-375(mutual)]TJ 0 -11.955 Td [(information)-250(between)-250(inputs)-250(and)-250(their)-250(encodings.)]TJ
0 g 0 G
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[(19)]TJ
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[(,)]TJ
0 1 0 rg 0 1 0 RG
[-158(36)]TJ
0 g 0 G
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ET
/S/Span<>BDC
BT
/F81 8.9664 Tf 62.924 81 Td [(L)]TJ/F39 5.9776 Tf 6.348 -0.996 Td [(v)-36(ae)]TJ/F79 8.9664 Tf 14.563 0.996 Td [(=)]TJ/F86 8.9664 Tf 9.727 0 Td [(E)]TJ/F39 5.9776 Tf 5.978 -1.395 Td [(q)]TJ/F20 4.9813 Tf 3.411 -1.4 Td [<1e>]TJ/F36 5.9776 Tf 4.691 1.4 Td [(\()]TJ/F84 5.9776 Tf 2.878 0 Td [(z)]TJ/F42 5.9776 Tf 3.697 0 Td [(j)]TJ/F84 5.9776 Tf 2.27 0 Td [(x)]TJ/F36 5.9776 Tf 4.369 0 Td [(\))]TJ/F79 8.9664 Tf 3.377 1.395 Td [([log)]TJ/F80 8.9664 Tf 15.998 0 Td [(p)]TJ/F39 5.9776 Tf 4.627 -1.063 Td [<12>]TJ/F79 8.9664 Tf 4.22 1.063 Td [(\()]TJ/F83 8.9664 Tf 3.584 0 Td [(x)]TJ/F81 8.9664 Tf 5.599 0 Td [(j)]TJ/F83 8.9664 Tf 2.56 0 Td [(z)]TJ/F79 8.9664 Tf 4.716 0 Td [(\)])]TJ/F81 8.9664 Tf 8.191 0 Td [<00>]TJ/F80 8.9664 Tf 9.216 0 Td [(D)]TJ/F39 5.9776 Tf 7.599 -0.996 Td [(K)-69(L)]TJ/F79 8.9664 Tf 12.101 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(q)]TJ/F39 5.9776 Tf 4.118 -1.063 Td [<1e>]TJ/F79 8.9664 Tf 4.934 1.063 Td [(\()]TJ/F83 8.9664 Tf 3.584 0 Td [(z)]TJ/F81 8.9664 Tf 4.715 0 Td [(j)]TJ/F83 8.9664 Tf 2.56 0 Td [(x)]TJ/F79 8.9664 Tf 5.6 0 Td [(\))]TJ/F81 8.9664 Tf 3.584 0 Td [(jj)]TJ/F80 8.9664 Tf 5.119 0 Td [(p)]TJ/F79 8.9664 Tf 4.626 0 Td [(\()]TJ/F83 8.9664 Tf 3.584 0 Td [(z)]TJ/F79 8.9664 Tf 4.716 0 Td [(\)\))]TJ/F80 8.9664 Tf 7.167 0 Td [(;)]TJ
ET
EMC
BT
/F67 8.9664 Tf 275.907 81 Td [(\(1\))]TJ
0 g 0 G
0 g 0 G
0 g 0 G
ET
1 0 0 1 310.043 634.359 cm
q
.25041 0 0 .25041 0 0 cm
q
1 0 0 1 -40 -129 cm
/Im1 Do
Q
Q
0 g 0 G
1 0 0 1 -310.043 -634.359 cm
BT
/F67 8.9664 Tf 329.205 616.725 Td [(Figure)-250(1.)-310(Diagram)-250(of)-250(the)-250(PI)-250(controller)-250(of)-250(ControlV)135(AE.)]TJ
0 g 0 G
0 g 0 G
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0 1 0 rg 0 1 0 RG
[(40)]TJ
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1 0 0 rg 1 0 0 RG
[-272(1)]TJ
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[(.)-378(It)-273(designs)-272(a)-273(PI)-272(controller)-273(to)-273(dynamical)1(ly)-273(tune)-273(the)]TJ 0 -11.955 Td [(weight)]TJ/F22 9.9626 Tf 29.901 0 Td [(\f)]TJ/F67 9.9626 Tf 8.944 0 Td [(in)-279(the)-280(follo)25(wing)]TJ/F22 9.9626 Tf 66.769 0 Td [(\f)]TJ/F67 9.9626 Tf 6.16 0 Td [(-V)135(AE)-279([)]TJ
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[(12)]TJ
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[(])-280(objecti)25(v)15(e)-279(to)-279(balance)]TJ -111.774 -11.956 Td [(the)-250(disentanglement)-250(learning)-250(and)-250(reconstruction.)]TJ
ET
/S/Span<>BDC
BT
/F81 8.9664 Tf 320.939 438.521 Td [(L)]TJ/F39 5.9776 Tf 6.348 -0.996 Td [(c)]TJ/F79 8.9664 Tf 6.4 0.996 Td [(=)]TJ/F86 8.9664 Tf 9.728 0 Td [(E)]TJ/F39 5.9776 Tf 5.977 -1.395 Td [(q)]TJ/F20 4.9813 Tf 3.412 -1.4 Td [<1e>]TJ/F36 5.9776 Tf 4.69 1.4 Td [(\()]TJ/F84 5.9776 Tf 2.879 0 Td [(z)]TJ/F42 5.9776 Tf 3.697 0 Td [(j)]TJ/F84 5.9776 Tf 2.269 0 Td [(x)]TJ/F36 5.9776 Tf 4.37 0 Td [(\))]TJ/F79 8.9664 Tf 3.376 1.395 Td [([log)]TJ/F80 8.9664 Tf 15.999 0 Td [(p)]TJ/F39 5.9776 Tf 4.626 -1.063 Td [<12>]TJ/F79 8.9664 Tf 4.22 1.063 Td [(\()]TJ/F83 8.9664 Tf 3.584 0 Td [(x)]TJ/F81 8.9664 Tf 5.6 0 Td [(j)]TJ/F83 8.9664 Tf 2.56 0 Td [(z)]TJ/F79 8.9664 Tf 4.715 0 Td [(\)])]TJ/F81 8.9664 Tf 8.192 0 Td [<00>]TJ/F80 8.9664 Tf 9.215 0 Td [(\f)-52(D)]TJ/F39 5.9776 Tf 13.26 -0.996 Td [(K)-69(L)]TJ/F79 8.9664 Tf 12.102 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(q)]TJ/F39 5.9776 Tf 4.117 -1.063 Td [<1e>]TJ/F79 8.9664 Tf 4.934 1.063 Td [(\()]TJ/F83 8.9664 Tf 3.584 0 Td [(z)]TJ/F81 8.9664 Tf 4.716 0 Td [(j)]TJ/F83 8.9664 Tf 2.56 0 Td [(x)]TJ/F79 8.9664 Tf 5.599 0 Td [(\))]TJ/F81 8.9664 Tf 5.826 0 Td [(k)]TJ/F80 8.9664 Tf 6.849 0 Td [(p)]TJ/F79 8.9664 Tf 4.626 0 Td [(\()]TJ/F83 8.9664 Tf 3.584 0 Td [(z)]TJ/F79 8.9664 Tf 4.715 0 Td [(\)\))]TJ/F80 8.9664 Tf 7.168 0 Td [(:)]TJ
ET
EMC
BT
/F67 8.9664 Tf 534.657 438.521 Td [(\(2\))]TJ/F67 9.9626 Tf -225.795 -21.151 Td [(Dif)25(ferent)-239(fr)1(o)-1(m)]TJ/F22 9.9626 Tf 60.384 0 Td [(\f)]TJ/F67 9.9626 Tf 6.161 0 Td [(-V)135(AE)-238(that)-239(assigns)-239(a)-238<0278>15(ed)-239(weight)-238(to)-239(the)-238(KL)]TJ -66.545 -11.955 Td [(term,)-257(ControlV)135(AE)-255(adopts)-256(a)-255(positional)-255(PI)-256(controller)-255(to)-256(com-)]TJ 0 -11.955 Td [(pute)-308(the)-308(weight)]TJ/F22 9.9626 Tf 65.648 0 Td [(\f)]TJ/F67 9.9626 Tf 9.227 0 Td [(using)-308(the)-308(actual)-307(KL-di)25(v)15(er)18(gence)-308(as)-308(feed-)]TJ -74.875 -11.956 Td [(back)-250(during)-250(training)-250(as)-250(follo)25(ws:)]TJ
ET
/S/Span<>BDC
BT
/F80 8.9664 Tf 345.51 353.275 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))-285(=)]TJ/F80 8.9664 Tf 15.871 0 Td [(K)]TJ/F39 5.9776 Tf 7.804 -0.997 Td [(p)]TJ/F80 8.9664 Tf 4.338 0.997 Td [<1b>]TJ/F79 8.9664 Tf 5.591 0 Td [(\()]TJ/F81 8.9664 Tf 3.584 0 Td [<00>]TJ/F80 8.9664 Tf 7.168 0 Td [(e)]TJ/F79 8.9664 Tf 4.284 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\)\))]TJ/F81 8.9664 Tf 9.215 0 Td [<00>]TJ/F80 8.9664 Tf 9.215 0 Td [(K)]TJ/F39 5.9776 Tf 7.804 -0.997 Td [(i)]TJ 9.928 12.205 Td [(t)]TJ/F82 8.9664 Tf -5.23 -2.69 Td [(X)]TJ/F39 5.9776 Tf 0.332 -18.905 Td [(j)]TJ/F36 5.9776 Tf 3.404 0 Td [(=0)]TJ/F80 8.9664 Tf 11.111 10.387 Td [(e)]TJ/F79 8.9664 Tf 4.284 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(j)]TJ/F79 8.9664 Tf 4.281 0 Td [(\))-228(+)]TJ/F80 8.9664 Tf 14.847 0 Td [(\f)]TJ/F39 5.9776 Tf 5.193 -0.997 Td [(min)]TJ/F80 8.9664 Tf 14.301 0.997 Td [(;)]TJ
ET
EMC
BT
/F67 8.9664 Tf 534.657 353.275 Td [(\(3\))]TJ/F67 9.9626 Tf -225.795 -30.209 Td [(where)]TJ/F22 9.9626 Tf 28.132 0 Td [(e)]TJ/F19 9.9626 Tf 4.639 0 Td [(\()]TJ/F22 9.9626 Tf 3.875 0 Td [(t)]TJ/F19 9.9626 Tf 3.597 0 Td [(\))-520(=)]TJ/F22 9.9626 Tf 21.985 0 Td [(C)]TJ/F25 9.9626 Tf 11.013 0 Td [<00>]TJ/F22 9.9626 Tf 10.928 0 Td [(D)]TJ/F21 6.9738 Tf 8.248 -1.495 Td [(K)-70(L)]TJ/F19 9.9626 Tf 13.146 1.495 Td [(\()]TJ/F22 9.9626 Tf 3.875 0 Td [(q)]TJ/F21 6.9738 Tf 4.447 -1.495 Td [<1e>]TJ/F19 9.9626 Tf 5.289 1.495 Td [(\()]TJ/F75 9.9626 Tf 3.874 0 Td [(z)]TJ/F25 9.9626 Tf 5.092 0 Td [(j)]TJ/F75 9.9626 Tf 2.768 0 Td [(x)]TJ/F19 9.9626 Tf 6.046 0 Td [(\))]TJ/F25 9.9626 Tf 3.875 0 Td [(k)]TJ/F22 9.9626 Tf 4.981 0 Td [(p)]TJ/F19 9.9626 Tf 5.012 0 Td [(\()]TJ/F75 9.9626 Tf 3.875 0 Td [(z)]TJ/F19 9.9626 Tf 5.092 0 Td [(\)\)\()]TJ/F22 9.9626 Tf 11.623 0 Td [(t)]TJ/F19 9.9626 Tf 3.598 0 Td [(\))]TJ/F67 9.9626 Tf 3.874 0 Td [(,)-414(which)-380(is)-381(the)]TJ -178.884 -11.955 Td [(dif)25(ference)-233(between)-233(desired)-233(KL-di)25(v)15(er)18(gence)]TJ/F22 9.9626 Tf 171.104 0 Td [(C)]TJ/F67 9.9626 Tf 10.154 0 Td [(and)-233(the)-233(actual)]TJ -181.258 -11.955 Td [(one)-338(at)-339(training)-338(step)]TJ/F22 9.9626 Tf 82.102 0 Td [(t)]TJ/F67 9.9626 Tf 3.598 0 Td [(;)]TJ/F22 9.9626 Tf 6.579 0 Td [<1b>]TJ/F19 9.9626 Tf 6.051 0 Td [(\()]TJ/F22 9.9626 Tf 3.874 0 Td [(:)]TJ/F19 9.9626 Tf 2.767 0 Td [(\))]TJ/F67 9.9626 Tf 7.245 0 Td [(is)-338(a)-339(sigmoid)-338(function;)]TJ/F22 9.9626 Tf 93.072 0 Td [(\f)]TJ/F21 6.9738 Tf 5.636 -1.495 Td [(min)]TJ/F67 9.9626 Tf 18.681 1.495 Td [(is)]TJ -229.605 -11.956 Td [(an)-301(application-speci\002c)-302(constant,)-314(such)-302(as)-301(0;)]TJ/F22 9.9626 Tf 175.844 0 Td [(K)]TJ/F21 6.9738 Tf 8.462 -1.494 Td [(p)]TJ/F67 9.9626 Tf 7.608 1.494 Td [(and)]TJ/F22 9.9626 Tf 17.39 0 Td [(K)]TJ/F21 6.9738 Tf 8.461 -1.494 Td [(i)]TJ/F67 9.9626 Tf 6.321 1.494 Td [(are)]TJ -224.086 -11.955 Td [(positi)25(v)15(e)-293(coef)25(\002cients)-293(for)-293(the)-293(P)-293(term)-293(and)-294(I)-293(term)-293(respecti)25(v)15(ely)65(.)]TJ 0 -11.955 Td [(Ne)15(xt,)-250(we)-250(will)-250(describe)-250(the)-250(basic)-250(idea)-250(of)-250(a)-250(PI)-250(controller)55(.)]TJ/F66 10.9589 Tf 0 -20.653 Td [(2.3.)-250(PID)-250(Contr)18(ol)-250(Algorithm)]TJ/F67 9.9626 Tf 11.955 -18.175 Td [(The)-299(PID)-299(is)-300(a)-299(simple)-299(yet)-299(ef)25(fecti)25(v)15(e)-299(control)-300(algorithm)-299(that)]TJ -11.955 -11.955 Td [(can)-459(stabilize)-459(system)-459(output)-459(to)-459(a)-459(desired)-460(v)25(alue)-459(via)-459(feed-)]TJ 0 -11.955 Td [(back)-354(control)-353([)]TJ
0 1 0 rg 0 1 0 RG
[(1)]TJ
0 g 0 G
[(,)]TJ
0 1 0 rg 0 1 0 RG
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[(].)-898(An)-446(o)15(v)15(er)20(-)]TJ 0 -11.956 Td [(shoot)-250(is)-249(temporary)-250(o)15(v)15(ercompensation,)-249(where)-250(the)-250(controlled)]TJ 0 -11.955 Td [(v)25(ariable)-461(oscillates)-461(around)-462(the)-461(set)-461(point.)-944(In)-461(our)-462(case,)-514(it)]TJ 0 -11.955 Td [(means)-286(that)-285(the)-286(actual)-285(KL-di)25(v)15(er)18(gence)-286(may)-285(signi\002cantly)-286(\(al-)]TJ 0 -11.955 Td [(beit)-216(temporarily\))-216(e)15(xceed)-217(the)-216(desired)-216(v)25(alue,)-223(when)-216(set)-216(point)-217(is)]TJ 0 -11.955 Td [(abruptly)-262(changed.)-347(This)-262(ef)25(fect)-262(w)10(ould)-262(cause)-262(some)-263(latent)-262(f)10(ac-)]TJ 0 -11.955 Td [(tors)-374(to)-375(come)-374(out)-375(earlier)-374(than)-374(e)15(xpected,)-406(and)-374(be)-375(entangled,)]TJ 0 -11.956 Td [(thereby)-438(producing)-437(poor)20(-quality)-438(disentanglement.)-872(T)80(o)-438(ad-)]TJ 0 -11.955 Td [(dress)-299(this)-298(problem,)-311(we)-298(de)25(v)15(elop)-299(a)-299(h)5(ybrid)-298(annealing)-299(method)]TJ 0 -11.955 Td [(that)-328(changes)-329(the)-328(set)-329(point)-328(more)-329(gradually)65(,)-348(as)-328(illustrated)-329(in)]TJ 0 -11.955 Td [(Fig.)]TJ
1 0 0 rg 1 0 0 RG
[-245(7)]TJ
0 g 0 G
[(\(b\))-246(in)-245(Appendix.)-308(It)-246(combines)-245(step)-245(function)-245(with)-246(ramp)]TJ 0 -11.955 Td [(function)-336(to)-335(smoothly)-336(increase)-336(the)-335(tar)18(get)-336(KL-di)25(v)15(er)18(gence)-336(in)]TJ 0 -11.955 Td [(order)-223(to)-224(pre)25(v)15(ent)-223(o)15(v)15(ershoot)-224(and)-223(thus)-224(better)-223(disentangle)-224(latent)]TJ 0 -11.956 Td [(f)10(actors)-250(one)-250(by)-250(one.)]TJ 11.955 -11.955 Td [(The)-252(combination)-252(of)-252(the)-252(abo)15(v)15(e)-252(three)-253(methods)-252(allo)25(ws)-252(Dy-)]TJ -11.955 -11.955 Td [(namicV)135(AE)-401(to)-400(e)25(v)20(olv)15(e)]TJ/F22 9.9626 Tf 89.168 0 Td [(\f)]TJ/F19 9.9626 Tf 6.161 0 Td [(\()]TJ/F22 9.9626 Tf 3.875 0 Td [(t)]TJ/F19 9.9626 Tf 3.597 0 Td [(\))]TJ/F67 9.9626 Tf 7.868 0 Td [(along)-401(a)-401(f)10(a)20(v)20(orable)-400(trajectory)-401(to)]TJ -110.669 -11.955 Td [(separate)-310(disentanglement)-311(learning)-310(and)-310(reconstruction)-311(opti-)]TJ 0 -11.955 Td [(mization.)-305(W)80(e)-235(summarize)-235(the)-235(proposed)-235(incremental)-235(PI)-235(algo-)]TJ 0 -11.956 Td [(rithm)-250(in)-250(Algorithm)]TJ
1 0 0 rg 1 0 0 RG
[-250(1)]TJ
0 g 0 G
[(.)]TJ/F66 10.9589 Tf 0 -19.243 Td [(4.1.)-250(Set)-250(P)20(oint)-250(Guidelines)]TJ/F67 9.9626 Tf 11.955 -17.932 Td [(In)-238(this)-237(section,)-240(we)-238(fundamentally)-238(analyze)-237(ho)25(w)-238(to)-238(choose)]TJ -11.955 -11.956 Td [(a)-427(set)-427(point)-427(for)-427(the)-427(tar)18(get)-427(KL-di)25(v)15(er)18(gence)]TJ/F22 9.9626 Tf 171.771 0 Td [(C)]TJ/F67 9.9626 Tf 12.086 0 Td [(of)-427(Dynamic-)]TJ -183.857 -11.955 Td [(V)135(AE)-376(\(same)-376(to)-377(ControlV)135(AE\).)-376(In)-376(DynamicV)135(AE,)-376(latent)-377(f)10(ac-)]TJ 0 -11.955 Td [(tors)-277(transmit)-277(through)-277(the)-277(information)-277(bottleneck)-277(for)-278(the)-277(re-)]TJ 0 -11.955 Td [(construction)-334(of)-334(input)-334(data.)-561(In)-334(information)-334(theory)65(,)-355(one)-334(bit,)]TJ
0 g 0 G
0 g 0 G
ET
q
1 0 0 1 308.862 719.601 cm
[]0 d 0 J 0.797 w 0 0 m 236.25 0 l S
Q
0 g 0 G
BT
/F66 9.9626 Tf 308.862 710.346 Td [(Algorithm)-250(1)]TJ/F67 9.9626 Tf 54.236 0 Td [(Incremental)-250(PI)-250(Control)-250(algorithm.)]TJ
0 g 0 G
ET
q
1 0 0 1 308.862 705.998 cm
[]0 d 0 J 0.398 w 0 0 m 236.25 0 l S
Q
0 g 0 G
0 g 0 G
BT
/F67 7.9701 Tf 313.421 695.786 Td [(1:)]TJ
0 g 0 G
/F66 8.9664 Tf 10.684 0 Td [(Input:)]TJ/F67 8.9664 Tf 28.596 0 Td [(desired)-328(KL)]TJ/F80 8.9664 Tf 43.731 0 Td [(C)]TJ/F67 8.9664 Tf 7.216 0 Td [(,)-348(coef)25(\002cients)]TJ/F80 8.9664 Tf 49.412 0 Td [(K)]TJ/F39 5.9776 Tf 7.804 -0.996 Td [(p)]TJ/F67 8.9664 Tf 4.338 0.996 Td [(,)]TJ/F80 8.9664 Tf 5.36 0 Td [(K)]TJ/F39 5.9776 Tf 7.804 -0.996 Td [(i)]TJ/F67 8.9664 Tf 3.161 0.996 Td [(,)]TJ/F80 8.9664 Tf 5.36 0 Td [(\f)]TJ/F39 5.9776 Tf 5.193 -0.996 Td [(min)]TJ/F67 8.9664 Tf 14.3 0.996 Td [(,)-348(iterations)]TJ/F80 8.9664 Tf -182.275 -10.959 Td [(N)]TJ/F67 8.9664 Tf 8.335 0 Td [(,)-250(windo)25(w)]TJ/F80 8.9664 Tf 35.39 0 Td [(T)]TJ
0 g 0 G
/F67 7.9701 Tf -54.409 -10.959 Td [(2:)]TJ
0 g 0 G
/F66 8.9664 Tf 10.684 0 Td [(Output:)]TJ/F67 8.9664 Tf 33.668 0 Td [(weight)]TJ/F80 8.9664 Tf 26.648 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.327 0 Td [(\))]TJ/F67 8.9664 Tf 5.826 0 Td [(at)-250(training)-250(step)]TJ/F80 8.9664 Tf 55.537 0 Td [(t)]TJ
0 g 0 G
/F67 7.9701 Tf -144.935 -10.959 Td [(3:)]TJ
0 g 0 G
/F66 8.9664 Tf 10.684 0 Td [(Initialization)]TJ/F67 8.9664 Tf 49.323 0 Td [(:)]TJ/F80 8.9664 Tf 5.272 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\(0\))-285(=)-286(150)-250(\(100\))]TJ/F80 8.9664 Tf 61.118 0 Td [(;)-171(y)]TJ/F39 5.9776 Tf 8.634 -0.996 Td [(K)-69(L)]TJ/F79 8.9664 Tf 12.102 0.996 Td [(\(0\))-285(=)-286(0)]TJ
0 g 0 G
/F67 7.9701 Tf -152.794 -10.959 Td [(4:)]TJ
0 g 0 G
/F66 8.9664 Tf 10.684 0 Td [(f)25(or)]TJ/F80 8.9664 Tf 13.467 0 Td [(t)]TJ/F79 8.9664 Tf 5.888 0 Td [(=)-285(1)]TJ/F66 8.9664 Tf 16.576 0 Td [(to)]TJ/F80 8.9664 Tf 9.711 0 Td [(N)]TJ/F66 8.9664 Tf 10.577 0 Td [(do)]TJ
0 g 0 G
/F67 7.9701 Tf -66.903 -10.959 Td [(5:)]TJ
0 g 0 G
/F67 8.9664 Tf 20.646 0 Td [(Sample)-250(KL-di)25(v)15(er)18(gence,)]TJ/F80 8.9664 Tf 87.377 0 Td [(y)]TJ/F39 5.9776 Tf 4.539 -0.996 Td [(K)-69(L)]TJ/F79 8.9664 Tf 12.101 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ
0 g 0 G
/F67 7.9701 Tf -131.575 -10.958 Td [(6:)]TJ
0 g 0 G
/F80 8.9664 Tf 20.646 0 Td [(y)]TJ/F79 8.9664 Tf 4.861 0 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))-286(=)]TJ/F82 8.9664 Tf 15.871 6.724 Td [(P)]TJ/F39 5.9776 Tf 9.727 -2.463 Td [(t)]TJ 0 -7.001 Td [(t)]TJ/F42 5.9776 Tf 2.851 0 Td [<00>]TJ/F39 5.9776 Tf 5.756 0 Td [(T)]TJ/F80 8.9664 Tf 7.373 2.74 Td [<0b>]TJ/F39 5.9776 Tf 5.906 -0.997 Td [(i)]TJ/F80 8.9664 Tf 3.162 0.997 Td [(y)]TJ/F39 5.9776 Tf 4.538 -0.997 Td [(K)-69(L)]TJ/F79 8.9664 Tf 12.102 0.997 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(i)]TJ/F79 8.9664 Tf 3.153 0 Td [(\))]TJ
0 g 0 G
/F67 7.9701 Tf -106.441 -10.959 Td [(7:)]TJ
0 g 0 G
/F80 8.9664 Tf 20.647 0 Td [(e)]TJ/F79 8.9664 Tf 4.284 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.327 0 Td [(\))]TJ/F81 8.9664 Tf 6.144 0 Td [( )]TJ/F80 8.9664 Tf 11.775 0 Td [(C)]TJ/F81 8.9664 Tf 9.264 0 Td [<00>]TJ/F80 8.9664 Tf 9.215 0 Td [(y)]TJ/F79 8.9664 Tf 4.86 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ
0 g 0 G
/F67 7.9701 Tf -80.012 -10.959 Td [(8:)]TJ
0 g 0 G
/F80 8.9664 Tf 20.647 0 Td [(dP)]TJ/F79 8.9664 Tf 11.938 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 6.143 0 Td [( )]TJ/F80 8.9664 Tf 11.775 0 Td [(K)]TJ/F39 5.9776 Tf 7.804 -0.996 Td [(p)]TJ/F79 8.9664 Tf 4.339 0.996 Td [([)]TJ/F80 8.9664 Tf 2.56 0 Td [<1b>]TJ/F79 8.9664 Tf 5.591 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(e)]TJ/F79 8.9664 Tf 4.284 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\)\))]TJ/F81 8.9664 Tf 9.215 0 Td [<00>]TJ/F80 8.9664 Tf 9.215 0 Td [<1b>]TJ/F79 8.9664 Tf 5.591 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(e)]TJ/F79 8.9664 Tf 4.284 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F81 8.9664 Tf 5.376 0 Td [<00>]TJ/F79 8.9664 Tf 9.215 0 Td [(1\)\)])]TJ
0 g 0 G
/F67 7.9701 Tf -142.553 -10.959 Td [(9:)]TJ
0 g 0 G
/F80 8.9664 Tf 20.647 0 Td [(dI)]TJ/F79 8.9664 Tf 9.505 0 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 6.144 0 Td [( )]TJ/F80 8.9664 Tf 11.775 0 Td [(K)]TJ/F39 5.9776 Tf 7.804 -0.996 Td [(i)]TJ/F80 8.9664 Tf 3.162 0.996 Td [(e)]TJ/F79 8.9664 Tf 4.284 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.327 0 Td [(\))]TJ
0 g 0 G
/F67 7.9701 Tf -81.128 -10.959 Td [(10:)]TJ
0 g 0 G
/F66 8.9664 Tf 24.632 0 Td [(if)]TJ/F80 8.9664 Tf 9.961 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F81 8.9664 Tf 5.376 0 Td [<00>]TJ/F79 8.9664 Tf 9.215 0 Td [(1\))]TJ/F80 8.9664 Tf 10.751 0 Td [(<)-285(\f)]TJ/F39 5.9776 Tf 14.921 -0.996 Td [(min)]TJ/F66 8.9664 Tf 16.542 0.996 Td [(then)]TJ
0 g 0 G
/F67 7.9701 Tf -100.643 -10.959 Td [(11:)]TJ
0 g 0 G
/F80 8.9664 Tf 34.594 0 Td [(dI)]TJ/F79 8.9664 Tf 9.505 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 6.143 0 Td [( )]TJ/F79 8.9664 Tf 11.776 0 Td [(0)]TJ/F67 8.9664 Tf 6.849 0 Td [(//)-250(wind)-250(up)]TJ
0 g 0 G
/F67 7.9701 Tf -75.779 -10.959 Td [(12:)]TJ
0 g 0 G
/F66 8.9664 Tf 24.632 0 Td [(end)-250(if)]TJ
0 g 0 G
/F67 7.9701 Tf -24.632 -10.959 Td [(13:)]TJ
0 g 0 G
/F80 8.9664 Tf 24.632 0 Td [(d\f)]TJ/F79 8.9664 Tf 10.435 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 6.144 0 Td [( )]TJ/F80 8.9664 Tf 11.775 0 Td [(dP)]TJ/F79 8.9664 Tf 11.938 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))-228(+)]TJ/F80 8.9664 Tf 14.847 0 Td [(dI)]TJ/F79 8.9664 Tf 9.505 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.327 0 Td [(\))]TJ
0 g 0 G
/F67 7.9701 Tf -110.011 -10.959 Td [(14:)]TJ
0 g 0 G
/F80 8.9664 Tf 24.632 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 6.144 0 Td [( )]TJ/F80 8.9664 Tf 11.775 0 Td [(d\f)]TJ/F79 8.9664 Tf 10.436 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.327 0 Td [(\))-228(+)]TJ/F80 8.9664 Tf 14.847 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F81 8.9664 Tf 5.376 0 Td [<00>]TJ/F79 8.9664 Tf 9.215 0 Td [(1\))]TJ
0 g 0 G
/F67 7.9701 Tf -111.153 -10.959 Td [(15:)]TJ
0 g 0 G
/F66 8.9664 Tf 24.632 0 Td [(if)]TJ/F80 8.9664 Tf 7.72 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F80 8.9664 Tf 6.144 0 Td [(<)-285(\f)]TJ/F39 5.9776 Tf 14.92 -0.996 Td [(min)]TJ/F66 8.9664 Tf 16.543 0.996 Td [(then)]TJ
0 g 0 G
/F67 7.9701 Tf -82.531 -10.958 Td [(16:)]TJ
0 g 0 G
/F80 8.9664 Tf 34.594 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 6.143 0 Td [( )]TJ/F80 8.9664 Tf 11.776 0 Td [(\f)]TJ/F39 5.9776 Tf 5.193 -0.997 Td [(min)]TJ
0 g 0 G
/F67 7.9701 Tf -70.279 -9.962 Td [(17:)]TJ
0 g 0 G
/F66 8.9664 Tf 24.632 0 Td [(end)-250(if)]TJ
0 g 0 G
/F67 7.9701 Tf -24.632 -10.959 Td [(18:)]TJ
0 g 0 G
/F66 8.9664 Tf 24.632 0 Td [(Retur)15(n)]TJ/F80 8.9664 Tf 29.499 0 Td [(\f)]TJ/F79 8.9664 Tf 5.661 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.327 0 Td [(\))]TJ
0 g 0 G
/F67 7.9701 Tf -66.703 -10.959 Td [(19:)]TJ
0 g 0 G
/F66 8.9664 Tf 14.669 0 Td [(end)-250(f)25(or)]TJ
0 g 0 G
ET
q
1 0 0 1 308.862 483.406 cm
[]0 d 0 J 0.398 w 0 0 m 236.25 0 l S
Q
BT
/F67 9.9626 Tf 308.862 454.666 Td [(also)-220(cal)1(led)-220(the)-220(information)-219(entrop)10(y)-220(of)-219(a)-220(binary)-219(random)-220(v)25(ari-)]TJ 0 -11.955 Td [(able,)-336(is)-318(often)-319(used)-318(to)-319(encode)-318(data)-319(or)-318(transmit)-319(information.)]TJ 0 -11.956 Td [(Hence,)-375(we)-350(adopt)-350(one)-349(bit)-350(theory)-350(to)-350(analyze)-350(the)-350(mutual)-350(in-)]TJ 0 -11.955 Td [(formation)]TJ/F25 9.9626 Tf 42.658 0 Td [(I)]TJ/F19 9.9626 Tf 6.16 0 Td [(\()]TJ/F75 9.9626 Tf 3.874 0 Td [(x)]TJ/F22 9.9626 Tf 6.047 0 Td [(;)]TJ/F75 9.9626 Tf 4.428 0 Td [(z)]TJ/F19 9.9626 Tf 5.092 0 Td [(\))]TJ/F67 9.9626 Tf 7.24 0 Td [(between)-338(the)-338(input)-337(data)]TJ/F75 9.9626 Tf 95.911 0 Td [(x)]TJ/F67 9.9626 Tf 9.412 0 Td [(and)-338(the)-338(latent)]TJ -180.822 -11.955 Td [(v)25(ariable)]TJ/F75 9.9626 Tf 34.971 0 Td [(z)]TJ/F67 9.9626 Tf 5.092 0 Td [(,)-330(which)-314(can)-315(be)-314(used)-314(as)-314(a)-314(lo)25(wer)-315(bound)-314(of)-314(the)-314(de-)]TJ -40.063 -11.955 Td [(sired)-250(KL-di)25(v)15(er)18(gence)-250(for)-250(DynamicV)135(AE.)]TJ 11.955 -11.955 Td [(Let)]TJ/F22 9.9626 Tf 17.656 0 Td [(M)]TJ/F67 9.9626 Tf 15.127 0 Td [(denote)-439(the)-439(capacity)-440(of)-439(v)25(ariational)-439(information)]TJ -44.738 -11.956 Td [(bottleneck)-253(for)-253(the)-253(complete)-252(reconstruction)-253(of)]TJ/F22 9.9626 Tf 182.226 0 Td [(N)]TJ/F21 6.9738 Tf 8.005 -1.494 Td [(x)]TJ/F67 9.9626 Tf 7.535 1.494 Td [(data)-253(sam-)]TJ -197.766 -11.955 Td [(ples.)-310(Then)-250(we)-250(ha)20(v)15(e)]TJ
ET
/S/Span<>BDC
BT
/F79 8.9664 Tf 372.602 340.944 Td [(2)]TJ/F39 5.9776 Tf 4.608 4.307 Td [(M)]TJ/F81 8.9664 Tf 10.646 -4.307 Td [<15>]TJ/F80 8.9664 Tf 9.727 0 Td [(N)]TJ/F39 5.9776 Tf 7.367 -0.997 Td [(x)]TJ/F79 8.9664 Tf 7.23 0.997 Td [(=)]TJ/F81 8.9664 Tf 5.631 0 Td [(\))]TJ/F80 8.9664 Tf 11.776 0 Td [(M)]TJ/F81 8.9664 Tf 12.431 0 Td [<15>]TJ/F79 8.9664 Tf 9.727 0 Td [(log)]TJ/F36 5.9776 Tf 11.903 -2.242 Td [(2)]TJ/F80 8.9664 Tf 5.687 2.242 Td [(N)]TJ/F39 5.9776 Tf 7.367 -0.997 Td [(x)]TJ
ET
EMC
BT
/F67 8.9664 Tf 530.174 340.944 Td [(\(11\))]TJ/F67 9.9626 Tf -221.312 -19.077 Td [(Therefore,)-356(we)-335(can)-334(deri)25(v)15(e)-335(a)-335(lo)25(wer)-334(bound)-335(of)-335(a)-334(set)-335(point)-335(for)]TJ 0 -11.956 Td [(the)-250(tar)18(get)-250(KL-di)25(v)15(er)18(gence)]TJ/F22 9.9626 Tf 101.877 0 Td [(C)]TJ/F67 9.9626 Tf 7.833 0 Td [(,)-250(satisfying)]TJ
ET
/S/Span<>BDC
BT
/F80 8.9664 Tf 391.343 291.831 Td [(C)]TJ/F81 8.9664 Tf 9.775 0 Td [<15>]TJ/F80 8.9664 Tf 9.728 0 Td [(M)]TJ/F81 8.9664 Tf 12.431 0 Td [<15>]TJ/F79 8.9664 Tf 9.727 0 Td [(log)]TJ/F36 5.9776 Tf 11.903 -2.242 Td [(2)]TJ/F80 8.9664 Tf 5.687 2.242 Td [(N)]TJ/F39 5.9776 Tf 7.367 -0.997 Td [(x)]TJ
ET
EMC
BT
/F67 8.9664 Tf 530.174 291.831 Td [(\(12\))]TJ/F67 9.9626 Tf -221.312 -19.077 Td [(W)80(e)-261(will)-261(further)-262(v)15(erify)-261(this)-261(result)-261(empirically)-261(by)-262(conducting)]TJ 0 -11.955 Td [(a)-250(set)-250(of)-250(e)15(xperiments)-250(in)-250(Section)]TJ
1 0 0 rg 1 0 0 RG
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/F79 8.9664 Tf 170.818 342.723 Td [(1)-228(+)]TJ/F80 8.9664 Tf 15.871 0 Td [(a)]TJ 6.077 6.081 Td [(x)]TJ/F36 5.9776 Tf 5.224 -0.996 Td [(2)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F85 8.9664 Tf -103.03 -19.116 Td [(,)]TJ/F80 8.9664 Tf 9.533 0 Td [(f)]TJ/F36 5.9776 Tf 4.473 -0.996 Td [(2)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(x)]TJ/F36 5.9776 Tf 5.223 -0.996 Td [(1)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F80 8.9664 Tf 3.584 0 Td [(;)-171(x)]TJ/F36 5.9776 Tf 9.319 -0.996 Td [(2)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.327 0 Td [(\))]TJ/F80 8.9664 Tf 3.584 0 Td [(;)-171(x)]TJ/F36 5.9776 Tf 9.319 -0.996 Td [(3)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\)\))]TJ/F80 8.9664 Tf -136.546 -15.492 Td [(x)]TJ/F36 5.9776 Tf 5.223 -0.996 Td [(3)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 5.376 0 Td [(+)-228(1\))-286(=)]TJ/F80 8.9664 Tf 29.694 0 Td [(x)]TJ/F36 5.9776 Tf 5.223 -0.996 Td [(2)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F85 8.9664 Tf 6.143 0 Td [(,)]TJ/F80 8.9664 Tf 9.534 0 Td [(f)]TJ/F36 5.9776 Tf 4.473 -0.996 Td [(3)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(x)]TJ/F36 5.9776 Tf 5.223 -0.996 Td [(1)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F80 8.9664 Tf 3.583 0 Td [(;)-171(x)]TJ/F36 5.9776 Tf 9.32 -0.996 Td [(2)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F80 8.9664 Tf 3.584 0 Td [(;)-171(x)]TJ/F36 5.9776 Tf 9.319 -0.996 Td [(3)]TJ/F79 8.9664 Tf 4.151 0.996 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\)\))]TJ
ET
EMC
BT
/F67 8.9664 Tf 271.424 348.621 Td [(\(16\))]TJ/F67 9.9626 Tf -209.357 -52.043 Td [(In)-308(order)-308(to)-308(analyze)-307(the)-308(stability)-308(of)-308(the)-308(abo)15(v)15(e)-308(non-linear)]TJ -11.955 -11.955 Td [(state)-387(space)-388(model,)-421(one)-388(commonly)-387(used)-388(method)-387(is)-387(to)-388(lin-)]TJ 0 -11.955 Td [(earize)-284(it)-284(at)-283(an)-284(equilibrium)-284(point)-284([)]TJ
0 1 0 rg 0 1 0 RG
[(14)]TJ
0 g 0 G
[(].)-411(In)-284(this)-284(paper)40(,)-292(we)-284(use)]TJ 0 -11.956 Td [(the)-250(follo)25(wing)-250(equilibrium)-250(point:)]TJ
ET
/S/Span<>BDC
BT
/F80 8.9664 Tf 98.767 243.214 Td [(x)]TJ/F42 5.9776 Tf 5.223 4.307 Td [<03>]TJ/F79 8.9664 Tf 6.877 -4.307 Td [(=)-285(\()]TJ/F80 8.9664 Tf 13.311 0 Td [(x)]TJ/F42 5.9776 Tf 5.224 4.307 Td [<03>]TJ/F36 5.9776 Tf 0 -5.885 Td [(1)]TJ/F80 8.9664 Tf 4.317 1.578 Td [(;)-171(x)]TJ/F42 5.9776 Tf 9.319 4.307 Td [<03>]TJ/F36 5.9776 Tf 0 -5.885 Td [(2)]TJ/F80 8.9664 Tf 4.317 1.578 Td [(;)-171(x)]TJ/F42 5.9776 Tf 9.319 4.307 Td [<03>]TJ/F36 5.9776 Tf 0 -5.885 Td [(3)]TJ/F79 8.9664 Tf 4.317 1.578 Td [(\))-286(=)-285(\()]TJ/F80 8.9664 Tf 19.455 0 Td [(g)]TJ/F42 5.9776 Tf 4.716 4.307 Td [<00>]TJ/F36 5.9776 Tf 5.756 0 Td [(1)]TJ/F79 8.9664 Tf 4.151 -4.307 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(C)]TJ/F79 8.9664 Tf 7.216 0 Td [(\))]TJ/F80 8.9664 Tf 3.583 0 Td [(;)-171(C)-14(;)-172(C)]TJ/F79 8.9664 Tf 22.112 0 Td [(\))]TJ/F80 8.9664 Tf 3.583 0 Td [(;)]TJ
ET
EMC
BT
/F67 8.9664 Tf 271.424 243.214 Td [(\(17\))]TJ/F67 9.9626 Tf -221.312 -18.495 Td [(where)]TJ/F22 9.9626 Tf 28.082 0 Td [(g)]TJ/F24 6.9738 Tf 5.109 3.615 Td [<00>]TJ/F18 6.9738 Tf 6.227 0 Td [(1)]TJ/F19 9.9626 Tf 4.469 -3.615 Td [(\()]TJ/F25 9.9626 Tf 3.875 0 Td [<01>]TJ/F19 9.9626 Tf 2.767 0 Td [(\))]TJ/F67 9.9626 Tf 7.618 0 Td [(denotes)-376(the)-375(in)40(v)15(erse)-376(function)-376(and)]TJ/F22 9.9626 Tf 137.141 0 Td [(x)]TJ/F24 6.9738 Tf 5.694 3.615 Td [<03>]TJ/F18 6.9738 Tf 0 -6.088 Td [(2)]TJ/F19 9.9626 Tf 9.667 2.473 Td [(=)]TJ/F22 9.9626 Tf 12.836 0 Td [(x)]TJ/F24 6.9738 Tf 5.694 3.615 Td [<03>]TJ/F18 6.9738 Tf 0 -6.088 Td [(3)]TJ/F67 9.9626 Tf 4.58 2.473 Td [(.)]TJ -233.759 -11.956 Td [(Ne)15(xt,)-222(we)-215(apply)-215(the)-215(\002rst-order)-215(T)80(aylor)-215(e)15(xpansion)-215(to)-215(the)-215(abo)15(v)15(e)]TJ 0 -11.955 Td [(Eq.)-250(\()]TJ
1 0 0 rg 1 0 0 RG
[(16)]TJ
0 g 0 G
[(\),)-250(yielding)]TJ
ET
/S/Span<>BDC
BT
/F80 8.9664 Tf 130.614 183.31 Td [(X)]TJ/F79 8.9664 Tf 8.314 0 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 5.376 0 Td [(+)-228(1\))-286(=)]TJ/F80 8.9664 Tf 29.694 0 Td [(AX)]TJ/F79 8.9664 Tf 15.224 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F80 8.9664 Tf 3.583 0 Td [(;)]TJ
ET
EMC
BT
/F67 8.9664 Tf 271.424 183.31 Td [(\(18\))]TJ/F67 9.9626 Tf -221.312 -18.496 Td [(where)]TJ
ET
/S/Span<>BDC
BT
/F80 8.9664 Tf 80.824 147.316 Td [(X)]TJ/F79 8.9664 Tf 8.314 0 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))-285(=)-286([)]TJ/F80 8.9664 Tf 18.431 0 Td [(x)]TJ/F36 5.9776 Tf 5.223 -0.997 Td [(1)]TJ/F79 8.9664 Tf 4.151 0.997 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 5.631 0 Td [<00>]TJ/F80 8.9664 Tf 9.216 0 Td [(x)]TJ/F42 5.9776 Tf 5.223 4.307 Td [<03>]TJ/F36 5.9776 Tf 0 -5.885 Td [(1)]TJ/F80 8.9664 Tf 4.317 1.578 Td [(;)-171(x)]TJ/F36 5.9776 Tf 9.319 -0.997 Td [(2)]TJ/F79 8.9664 Tf 4.151 0.997 Td [(\()]TJ/F80 8.9664 Tf 3.584 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 5.632 0 Td [<00>]TJ/F80 8.9664 Tf 9.215 0 Td [(x)]TJ/F42 5.9776 Tf 5.223 4.307 Td [<03>]TJ/F36 5.9776 Tf 0 -5.885 Td [(2)]TJ/F80 8.9664 Tf 4.317 1.578 Td [(;)-171(x)]TJ/F36 5.9776 Tf 9.32 -0.997 Td [(3)]TJ/F79 8.9664 Tf 4.151 0.997 Td [(\()]TJ/F80 8.9664 Tf 3.583 0 Td [(t)]TJ/F79 8.9664 Tf 3.328 0 Td [(\))]TJ/F81 8.9664 Tf 5.632 0 Td [<00>]TJ/F80 8.9664 Tf 9.215 0 Td [(x)]TJ/F42 5.9776 Tf 5.224 4.307 Td [<03>]TJ/F36 5.9776 Tf 0 -5.885 Td [(3)]TJ/F79 8.9664 Tf 4.317 1.578 Td [(])]TJ/F39 5.9776 Tf 2.56 4.307 Td [(T)]TJ/F80 8.9664 Tf 5.837 -4.307 Td [(;)]TJ
ET
EMC
BT
/F67 8.9664 Tf 271.424 147.316 Td [(\(19\))]TJ/F67 9.9626 Tf -221.312 -18.495 Td [(and)]TJ/F22 9.9626 Tf 18.297 0 Td [(A)]TJ/F67 9.9626 Tf 11.383 0 Td [(is)-393(the)-392(Jacobian)-393(matrix)-392(at)-393(equilibrium)-392(point)]TJ/F22 9.9626 Tf 181.24 0 Td [(x)]TJ/F24 6.9738 Tf 5.694 3.615 Td [<03>]TJ/F67 9.9626 Tf 4.58 -3.615 Td [(,)-428(as)]TJ -221.194 -11.955 Td [(de\002ned)-280(in)-280(Eq.)-281(\()]TJ
1 0 0 rg 1 0 0 RG
[(22)]TJ
0 g 0 G
[(\))-280(in)-280(Appendix)]TJ
1 0 0 rg 1 0 0 RG
[-280(C)]TJ
0 g 0 G
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0 g 0 G
0 g 0 G
0 g 0 G
/F66 9.9626 Tf 258.75 629.037 Td [(Theor)18(em)-302(5.1.)]TJ
0 g 0 G
/F70 9.9626 Tf 61.841 0 Td [(Let)]TJ/F22 9.9626 Tf 15.744 0 Td [(a)-375(>)]TJ/F19 9.9626 Tf 20.483 0 Td [(0)]TJ/F70 9.9626 Tf 7.994 0 Td [(and)-302(assume)]TJ/F22 9.9626 Tf 50.298 0 Td [(g)]TJ/F24 6.9738 Tf 5.11 3.616 Td [(0)]TJ/F19 9.9626 Tf 2.795 -3.616 Td [(\()]TJ/F22 9.9626 Tf 3.874 0 Td [(x)]TJ/F19 9.9626 Tf 5.694 0 Td [(\))]TJ/F22 9.9626 Tf 7.608 0 Td [(<)]TJ/F19 9.9626 Tf 11.483 0 Td [(0)]TJ/F22 9.9626 Tf 4.981 0 Td [(;)]TJ/F25 9.9626 Tf 4.428 0 Td [(8)]TJ/F22 9.9626 Tf 5.535 0 Td [(x)-375(>)]TJ/F19 9.9626 Tf 20.91 0 Td [(0)]TJ/F70 9.9626 Tf 4.981 0 Td [(.)]TJ -233.759 -11.955 Td [(Then)-357(DynamicV)60(AE)-356(is)-357(stable)-356(at)-357(the)-357(equilibrium)-356(point)]TJ/F22 9.9626 Tf 219.325 0 Td [(C)]TJ/F70 9.9626 Tf 11.386 0 Td [(if)]TJ -230.711 -11.955 Td [(and)-264(only)-263(if)-264(the)-263(par)15(ameter)10(s)-264(of)-263(the)-264(PI)-263(contr)45(oller)111(,)]TJ/F22 9.9626 Tf 188.72 0 Td [(K)]TJ/F21 6.9738 Tf 8.462 -1.494 Td [(i)]TJ/F70 9.9626 Tf 5.942 1.494 Td [(and)]TJ/F22 9.9626 Tf 17.569 0 Td [(K)]TJ/F21 6.9738 Tf 8.461 -1.494 Td [(p)]TJ/F70 9.9626 Tf 4.605 1.494 Td [(,)]TJ -233.759 -11.955 Td [(satisfy)-250(the)-250(following)-250(conditions)]TJ
ET
/S/Span< 0, K_p > 0 \\\\ \\end {aligned} \\right . )>>BDC
BT
/F44 7.9701 Tf 315.74 663.288 Td [(8)]TJ 0 -7.173 Td [(>)]TJ 0 -2.391 Td [(>)]TJ 0 -2.391 Td [(>)]TJ 0 -2.391 Td [(>)]TJ 0 -2.391 Td [(<)]TJ 0 -14.346 Td [(>)]TJ 0 -2.391 Td [(>)]TJ 0 -2.391 Td [(>)]TJ 0 -2.392 Td [(>)]TJ 0 -2.391 Td [(:)]TJ/F38 7.9701 Tf 7.527 29.106 Td [(K)]TJ/F39 5.9776 Tf 7.142 -0.996 Td [(p)]TJ/F35 7.9701 Tf 6.22 0.996 Td [(+)]TJ/F38 7.9701 Tf 8.469 0 Td [(K)]TJ/F39 5.9776 Tf 7.141 -1.215 Td [(i)]TJ/F38 7.9701 Tf 5.514 1.215 Td [(<)]TJ/F41 7.9701 Tf 8.939 0 Td [<00>]TJ/F35 7.9701 Tf 7.782 5.55 Td [(4\(1)-236(+)]TJ/F38 7.9701 Tf 22.112 0 Td [(a)]TJ/F35 7.9701 Tf 4.498 0 Td [(\))]TJ
ET
q
1 0 0 1 374.474 653.739 cm
[]0 d 0 J 0.359 w 0 0 m 29.903 0 l S
Q
BT
/F38 7.9701 Tf 375.841 645.776 Td [(ag)]TJ/F42 5.9776 Tf 8.814 2.269 Td [(0)]TJ/F35 7.9701 Tf 2.684 -2.269 Td [(\()]TJ/F38 7.9701 Tf 3.294 0 Td [(x)]TJ/F42 5.9776 Tf 4.766 2.745 Td [<03>]TJ/F36 5.9776 Tf 0 -5.287 Td [(1)]TJ/F35 7.9701 Tf 4.318 2.542 Td [(\))]TJ -76.45 -13.69 Td [(0)]TJ/F38 7.9701 Tf 4.235 0 Td [(:)]TJ/F35 7.9701 Tf 2.352 0 Td [(5)]TJ/F38 7.9701 Tf 4.234 0 Td [(K)]TJ/F36 5.9776 Tf 7.705 3.311 Td [(2)]TJ/F39 5.9776 Tf -0.563 -5.303 Td [(p)]TJ/F38 7.9701 Tf 4.714 1.992 Td [(ag)]TJ/F42 5.9776 Tf 8.814 3.311 Td [(0)]TJ/F35 7.9701 Tf 2.684 -3.311 Td [(\()]TJ/F38 7.9701 Tf 3.293 0 Td [(x)]TJ/F42 5.9776 Tf 4.767 3.311 Td [<03>]TJ/F36 5.9776 Tf 0 -5.303 Td [(1)]TJ/F35 7.9701 Tf 4.317 1.992 Td [(\))]TJ/F36 5.9776 Tf 3.293 3.311 Td [(2)]TJ/F35 7.9701 Tf 6.033 -3.311 Td [(+)-236(2[)]TJ/F38 7.9701 Tf 15.055 0 Td [(K)]TJ/F39 5.9776 Tf 7.142 -0.996 Td [(p)]TJ/F41 7.9701 Tf 6.22 0.996 Td [<00>]TJ/F35 7.9701 Tf 8.468 0 Td [(8)]TJ/F38 7.9701 Tf 4.235 0 Td [(K)]TJ/F39 5.9776 Tf 7.141 -1.215 Td [(i)]TJ/F35 7.9701 Tf 3.162 1.215 Td [(\(1)-236(+)]TJ/F38 7.9701 Tf 17.877 0 Td [(a)]TJ/F35 7.9701 Tf 4.498 0 Td [(\)])]TJ/F38 7.9701 Tf 5.646 0 Td [(g)]TJ/F42 5.9776 Tf 4.316 3.311 Td [(0)]TJ/F35 7.9701 Tf 2.684 -3.311 Td [(\()]TJ/F38 7.9701 Tf 3.293 0 Td [(x)]TJ/F42 5.9776 Tf 4.767 3.311 Td [<03>]TJ/F36 5.9776 Tf 0 -5.303 Td [(1)]TJ/F35 7.9701 Tf 4.317 1.992 Td [(\))]TJ/F41 7.9701 Tf 5.176 0 Td [<00>]TJ/F35 7.9701 Tf 8.468 0 Td [(8\(1)-236(+)]TJ/F38 7.9701 Tf 22.112 0 Td [(a)]TJ/F35 7.9701 Tf 4.498 0 Td [(\))]TJ/F38 7.9701 Tf 5.645 0 Td [(<)]TJ/F35 7.9701 Tf 8.939 0 Td [(0)]TJ/F38 7.9701 Tf -209.537 -13.765 Td [(K)]TJ/F39 5.9776 Tf 7.142 -1.215 Td [(i)]TJ/F38 7.9701 Tf 5.514 1.215 Td [(>)]TJ/F35 7.9701 Tf 8.939 0 Td [(0)]TJ/F38 7.9701 Tf 4.234 0 Td [(;)-177(K)]TJ/F39 5.9776 Tf 10.905 -0.996 Td [(p)]TJ/F38 7.9701 Tf 6.691 0.996 Td [(>)]TJ/F35 7.9701 Tf 8.939 0 Td [(0)]TJ
ET
EMC
BT
/F67 7.9701 Tf 531.834 609.035 Td [(\(20\))]TJ/F70 9.9626 Tf -211.017 -22.703 Td [(W)92(e)-250(pr)45(o)10(vide)-250(the)-250(detailed)-250(pr)45(oof)-250(in)-250(Appendix)]TJ
1 0 0 rg 1 0 0 RG
[-250(C)]TJ
0 g 0 G
[(.)]TJ
0 g 0 G
/F66 9.9626 Tf -11.955 -22.704 Td [(Remark)-354(5.1.)]TJ
0 g 0 G
/F70 9.9626 Tf 58.998 0 Td [(The)-354(assumption)-353(of)]TJ/F22 9.9626 Tf 78.657 0 Td [(g)]TJ/F24 6.9738 Tf 5.109 3.615 Td [(0)]TJ/F19 9.9626 Tf 2.795 -3.615 Td [(\()]TJ/F22 9.9626 Tf 3.874 0 Td [(x)]TJ/F19 9.9626 Tf 5.694 0 Td [(\))]TJ/F22 9.9626 Tf 8.556 0 Td [(<)]TJ/F19 9.9626 Tf 12.43 0 Td [(0)]TJ/F22 9.9626 Tf 4.982 0 Td [(;)]TJ/F25 9.9626 Tf 4.427 0 Td [(8)]TJ/F22 9.9626 Tf 5.535 0 Td [(x)]TJ/F70 9.9626 Tf 9.218 0 Td [(basically)]TJ -200.275 -11.956 Td [(asks)-247(that)-247(the)-246(KL)-247(term)-247(in)-247(the)-247(objecti)1(ve)-247(to)-247(be)-247(a)-247(monotonously)]TJ 0 -11.955 Td [(decr)37(easing)-247(function)-248(of)-247(the)-248(coef)18(\002cient)]TJ/F22 9.9626 Tf 149.022 0 Td [(\f)]TJ/F19 9.9626 Tf 6.161 0 Td [(\()]TJ/F22 9.9626 Tf 3.874 0 Td [(t)]TJ/F19 9.9626 Tf 3.598 0 Td [(\))]TJ/F70 9.9626 Tf 3.874 0 Td [(,)-248(and)-247(we)-248(also)-247(fur)20(-)]TJ -166.529 -11.955 Td [(ther)-280(empiri)1(cally)-280(corr)45(obor)15(ate)-280(its)-279(validity)-280(on)-279(two)-280(benc)15(hmark)]TJ 0 -11.955 Td [(datasets)-265(as)-265(shown)-265(in)-265(Appendix)]TJ
1 0 0 rg 1 0 0 RG
[-265(C.1)]TJ
0 g 0 G
[(.)-355(In)-265(addition,)-269(we)-265(c)15(hoose)]TJ/F22 9.9626 Tf 0 -11.955 Td [(K)]TJ/F21 6.9738 Tf 8.461 -1.495 Td [(p)]TJ/F70 9.9626 Tf 6.929 1.495 Td [(and)]TJ/F22 9.9626 Tf 17.268 0 Td [(K)]TJ/F21 6.9738 Tf 8.462 -1.495 Td [(i)]TJ/F70 9.9626 Tf 5.641 1.495 Td [(that)-233(meet)-233(the)-234(abo)10(ve)-233(conditions)]TJ/F67 9.9626 Tf 123.31 0 Td [(\()]TJ
1 0 0 rg 1 0 0 RG
[(20)]TJ
0 g 0 G
[(\))]TJ/F70 9.9626 Tf 18.921 0 Td [(to)-233(verify)-234(the)]TJ -188.992 -11.955 Td [(stability)-250(of)-250(DynamicV)60(AE)-250(in)-250(Appendix)]TJ
1 0 0 rg 1 0 0 RG
[-250(C.1)]TJ
0 g 0 G
[(.)]TJ/F66 11.9552 Tf 0 -26.689 Td [(6.)-250(Experiments)]TJ/F67 9.9626 Tf 11.955 -19.856 Td [(W)80(e)-631(e)25(v)25(aluate)-631(the)-631(performance)-631(of)-631(DynamicV)135(AE)-631(and)]TJ -11.955 -11.955 Td [(compare)-657(it)-656(ag)5(ainst)-657(e)15(xisting)-656(baselines,)-758(including)-657(Con-)]TJ 0 -11.955 Td [(trolV)135(AE)-225([)]TJ
0 1 0 rg 0 1 0 RG
[(40)]TJ
0 g 0 G
[(],)]TJ/F22 9.9626 Tf 56.59 0 Td [(\f)]TJ/F67 9.9626 Tf 6.161 0 Td [(-V)135(AE)]TJ/F21 6.9738 Tf 22.446 -1.494 Td [(H)]TJ/F67 9.9626 Tf 9.8 1.494 Td [([)]TJ
0 1 0 rg 0 1 0 RG
[(12)]TJ
0 g 0 G
[(],)]TJ/F22 9.9626 Tf 21.381 0 Td [(\f)]TJ/F67 9.9626 Tf 6.161 0 Td [(-V)135(AE)]TJ/F21 6.9738 Tf 22.446 -1.494 Td [(B)]TJ/F67 9.9626 Tf 9.1 1.494 Td [([)]TJ
0 1 0 rg 0 1 0 RG
[(5)]TJ
0 g 0 G
[(],)-230(F)15(actorV)135(AE)-225([)]TJ
0 1 0 rg 0 1 0 RG
[(17)]TJ
0 g 0 G
[(],)]TJ -154.085 -11.955 Td [(Lagrange)-248(Multiplier)-248(\(LM\))-248([)]TJ
0 1 0 rg 0 1 0 RG
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