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`,+$aRS#"$-Fb!% Y 1j s & A & A A % 8 .' .$ &# #3 " A 4HA A * 3!4658792;:1#<+)"=,+ #% %->?,+(#,@,+$BA(),<"!"DC E NCPOQ%-,)#RS&MT#,)@!"U#VRW#"(-FX!%ZY[M$+\*] Y$S#V=DY dc&,+"eG$)Yf#!Rg,) E Z"8*# #"=,<F`,+RW#"$ FX!% Yh2i,+ E !"$),+#"=,k.H#l-,i +Fb++Y<,)@"j:m7n,)W!-*`#g'T!G$"(YoFbl#"K#"$#%->],+ E "=*6#Z#"8,lprq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l J 8 * /! 3 $ * E 2 , ; & "( = 4 A& @ * +/ # & 5 ; 2 # # " J . )/ 3 A / + #* < " # " =/!#5 + A @ - +/ /! " = >6 . ) A 25 # * * 8 - ) A ; $9 IA IA /!# & &# A / # 3 4/ +/ /!#5 * # ! A * 5 # *"9 = # !61 1 # & # "9 %9 / * 4/! # * * > )( , # ! # * * % 5 * * )/ * # * # 0 4 # * # +/ ) * 9 & 2 ; 0 0 B " *61 2 & ) & #7 = & 0 * * 9 # * < "9 2 & / / 3 2 9 * # /! *# + '7* 313 5 '* # * ( 2 5 # / + * "9 ,/! +/ ( & "( /! '( 8 %# = / < < = & * *# /!5< ! .( , 0 B ' 8 ' 1 *# 9 )( & / %/ 61 &# & 1 > & * /!#5 # # 2A A A A A ? A G ! A 3 A:EHGIG A A A A A EHGIGIGA A 3 A E GIG E A H A / + A +EHG JE JIJ 861 " "9 ;" ? HA . #7 * ( * >/! * $/ * / 8 ' - : -/! & ' . +/ ) 8 2 ## '* HA /!#5< '( & 1A /! # GIGHG +/ ) 531 1 * HA / 0 # /! $ % $/! * ## 0? A & , $? # >2 / 3 '( +/ A A)EHGHGIG3A41 A A EHGIGIG A # 5 3A / / &*A / 7* A 313 3A=EHGHG EA /! B / & #3 9 / 0 ! # 1 # = * 0* 9 9 &( * 9 /! * 5 6 A /! * < * 9 & # * / *# / A # A * 5 A % * & * # 5 ! # / B " B/! & @ +/ A / + # = * / 0 . ! 3. 7 3A A 8 + * /! * / & A * 4 =/! # A@/!# . * = J & /! #; +/ ( " <? 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rate, % 20 10 5 3 2 20 50 100 500 1000 Number of Labeled Points 5000 20000 50000 [&{c[&s [&{c[&s F 4eu^`[ m :. sGwhadw _`[&am [^6q[&badw4e[[^6^`re^B^`whad[&_r^s [^6[bapbuv y[^`_r {cw wbs ub{cw y[ {c[&sx rcbad_qrvxGwh^`[sa`r y[_+a}N: : Gw_6[P{ccb[m 80 Laplacian 19,935 points total Laplacian 2,000 points total Laplacian 600 points total best k−NN, k=1,3,5 70 Error rate, % 60 50 40 30 22 50 100 500 1000 5000 Number of Labeled Points 10000 18000 F 4eu^`[ m 8 |8 p : [& _)4e^`reux_ sGwhadw 6_ [&am ^`^6re^ `^ whad[&_ re^s 9[&^`[ba buv y[^`_ rB{cw wbs ub{cw y[ {c[&sx rc bad_qrvxGwh^`[sa`r y[ _+a : : w _`[&{ccb[m | 30 Laplacian 13168 points total Laplacian 2000 points total Laplacian 600 points total best k−NN, k=1,3,5 28 26 24 Error rate, % 22 20 18 16 14 12 10 8 20 50 100 500 1000 Number of Labeled Points F 4eu^`[ m . . swhaw_6[&am ^`^6re^ `^ whad[&_ re^ps 9[&^`[ba bYuv y[{c[&s x rebad_qrev xGw^6[sadr y[_+a : : Gwh_`[{cb[hm 5000 10000 [&^`_Rrh]{ w y[{[s wbs ub{ w 30 3−NN on the held out set 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