Reducing Large Diagonals in Kernel Matrices through Semidefinite ...

1 downloads 0 Views 154KB Size Report
A method for reducing large diagonals based on the use of Semi-Definite Programming (SDP) is pro- posed. The kernel matrix generated by SDP is guaranteed.
Reducing Large Diagonals in Kernel Matrices through Semidefinite Programming Hsiao-Mei Lu

Sumeet Gupta

Department of Bioengineering University of Illinois at Chicago

Department of Bioengineering University of Illinois at Chicago

Department of Bioengineering University of Illinois at Chicago

[email protected]

[email protected]

[email protected]

ABSTRACT



     !"#$  %&'(*)+-, .0/1 2435 %67 98 :;.036=#> 84#*,91 / ? A@(%2#  CBD 8A'(%,EF$>G)0 > H%%-I J 8 KL/ %/M0 M8 $M %9 L  N! 85@(%-# OP*)Q $'(%  >ICRS%8T)A!"4*)/  >E>&) > U *) V8E/E!D.0 1;W%  YXK>M > :;.0WGXC%%M$*)&-,G.0WGX7 Q>/ M-*) U22  '(? )%  I J 8?0  ,P%O02% -M 2%!"# G!Z8G9B[@(%%\ D %/M> >0I

Keywords

.0/2\35 %K67 98 ].0% F)0%

 KX^>M >]_\> W >]2`G%\6a%8T)0

1. INTRODUCTION

`G%^%8T)0G#A P!?LM$%1;!F1b$& >=>1  8?8 $4> '(G!c*'(M&2%!d# P = % $ 7 M)0   G >e98T)I J 8V/2a' %f#1 M8 N 5gB@T*BDh%Oi j-k llk lmn;I5Re 98 %!2/ F)0 > - M@a!58&%8T)04 U=- b,a@0*BD=48P@(%cI R[ *?!"/ % a#58U /N2 -? -hh8 >8% )   0!"*/0?  DU8 $5 *5  % E2 M *)+  8&  I J 8@(9K  9@o0'T )%h+%pY1  -KB?,& L/ 9^/ M8EU *$5 

%C-,P8N/N! + %&0T)/ 9P2%bBq#2*)V2 -& +8Y!d/   IqRg  M) > ,-]Cf)-hE%*)q  %/ ,q 01  )%D8G#2)A2 -D%O0    ,-I J 8 g01b)02)  !28?@(%E8D)0 %  b,h!28? /5 4) 8Nr%O0   s,Y!H/ >PT,h@(%0#@(N8?@(%%%8T) #!D8LY2/$E M8P!dE    * t    0!d#  g   hi uTk v0kHl$j*n;I w *B'(%*]DBD8%[)M$o%-*)xaV'( %] 8a@(%%4#$ O )yy)#0T)/ %// ,t8A 1; *)z>q)0 F> ]U {I I]g8V) >G-A! 8Y# O Y/ 98o>%&8 V8E|21s) > C%I `G%D#  !"} %M y8 L  *)y@%]?I >0I J 8~ %2 )0 %~8/)2€)  9*)ƒ‚> WG {I„67  >…))0%+†W-f!Y‡ -> % > :;6=Nˆ‰Š()o>%M$7= $ ' %*]KBD8%A* M8 2-& 2)hVxsBM)0‘ N7s !‘P E8D> '(%*Ž-/ IQ’x8%'(%^8?) ;BNM)‘#b !L‘Y-  /0? A8U*Ž-/% ]-8U 2- D%DYGU8%g G2 %  *)aT/h2%*I J t 98 '(o2%%a*)0  9 [ …8 0  !&/ M8[ M8 $M %9 L  & 78G@(%Z#$ OH]2.0 M8Z“ @(0!g”9•D–—˜i l$u$n 02*)E%8T)h *)Eh8D  * A!2!d/ %   //PA8E@%^#$ OV +M)%Gf*)/ Y8E)0,T1   GM>Y!C8P# OIG™U%M ,-]8 U# >EBD  h>/M-#8 $h8#%Bš# O+BD # 2  '( )0%

 ]^f!48f02% &*Ž-/ *)q-,=@(9N#$1  OI J 8 E /AB?h'(*)-,+8f -T)/ % !U %   *\@(%%Z#i l*u$n{I J 8  M8=)4A#@%1  #$ OhBD8  M8f C8™gM›#$ OY!H84T)

)#@%1 K# OI J 8 G@(%K# O=  8o2E/)oBD 8  LM )M).036œ!"YM  >0IXK  , /0M  %/ U8$'&8*BDo848&%8T)f *o $'(G2%!"1 # @(%&8& $/M ,>%M$##  PBD 8 $>=)0 F> oi l$u*n;I w *B'(%*]C8A  a=)8 M8  C!T8^!d/ % P *)?*)/ K8C)0,0   QM> 8*'(U,-%?h2G/)%L-T)I Œs8 Q2%*]BNC2ND%B+*)/ % P%8T)G! )0 F>1  ?*)Af )%  U0>M >ai lˆn{I J 8#  !"*/0P!?8 P%8T)+$#8G8>%M*) @(%%C  L K2  '(h)%  h )78U8h   b,7*/0&!"  !C2 Tg o8G!d*$/&  %h# g/ M8 >)HI ’+o *)t8 A@(%4BD 8z.T3G6!df8V %F

*$  !KbB#)M%U!^>h9O0 ='/%g>%M$*)a!" 8?  %($M*,Y%O02% -Q!dN * %/ )h# L1 /I J 8G%O02% -?8*BD48G%| % '(D!K8 ?%B 98T)!dD8*)/ % a!5) >Z%%-I 2. SUBPOLYNOMIAL KERNEL METHODS FOR LARGE DIAGONALS IN KERNEL MATRICES

.T/2f'( %7# M8 a$+/2%'T )z0$O0 # 8Z *&2  )9*)h^h*O0 #$C %-M  ! 8?L/ %/?   ž*$ Y0   ?/>>%L*)E-,h3C01  @+i jTk^lmn{I’x8%=/*)a!"G    * \]Q.03G6ag#78 /4 h -#8 >8%)   \!d*$/Y P8 $

‡ŒLŸU`GWWGˆŠT†5ŠM)7R4?6 .0ŒL™U`GWW¡’+@T8a7WGME6a  >E =‡  0!d#  ]HuˆˆŠ

>&j¢

M%D&> 'f9?!Z  $,#F2*)M  >Y)$MPBD 8 t0 #?8-,29I J 87 #D8-,02%A!"/ ) -,h84.036 *$ >&> 8 58?D85#O0  ž 8Y $M >7#$> +29bBN% 8Y ,V &!8 M  >)$M0I J 8G '$ A!"g# >P8G)MY - Y8 >8%g)0 % !d*$/G  P D84 *$U)0    2/ )  L/ 9*)t 878 >8…)0 % N!d/   P %2 )A *g)0   =2/ ) 4 f8 /D  I ™U '(fPM  >h%D!  /N'( %     GBD 8 @T*BDE C2%        hl  Gl -]T%Be2 -   U >)=EF2Z-,#8&.T3G6  % M)0 >EY8P)0    !d/ % 

Z:5Z: !



!$#%!&

!

 :  ^ YY8 g!K2  '(P)0%

 @(%&i Š$n;† ! & ! ! :   < h)0 > Z-I ŒbAM %   ?8?2A%'(*)8 $N8U.036aNBD 8A8  @T  ) !?@(%^# O )f&2%!" Bgi l$u*n{I J 7)* BD 8a8U !^>%g) >Q%T].T 98\“ @(! ”9•?–—˜Di l$u*nZ02*)7h *?ML!d# fh*)0/  8E)0,0   &M>!8E%TG!N8Y# OI J 89, /)aY & !1s *)7/2 ,0 ! \@(9c])% ! *)f :   9 L< … >\2: .0^:  *< 0K:  9 2< 1 < H .0K:  *< 0K:  9 2< 1 H M= :{O ˆ NQPRN[l$*<  & ! ! :cm-< !" 8A> '(@(% :  2 7 < S.0^:  *< 0K:  2< 19I w *B?1 '(9*]48 A98T)tBD *)y9 q#$ OtBD8  M9 8x#,e >%a2=2  '(o )%  I R 0- *)0/7B?f8 2*)-,e.0 M8Z“ @(0!”9•h–—˜xi l$u*ngo'(A8  /  *)af8G/!K8&    *\@& %H#Vi l*Š$n† :;j< T Y: 5G)0 > H%%-]T  U8g/2 ,T1  @(9H *f2G  )%*)a  C+I 3. REDUCING LARGE DIAGONALS THROUGH SEMIDEFINITE PROGRAMMING

ŒsA8 N % fBN02h%8T)A/ >Eh) |%-D1 0( 98E*)0/ ?84$>)0 >  E8D@(%%# OI ŸU/0?!dY/$ = 4*)7a8G/G!C.T )%  XK1 >M >A:;.0WX%M ž =!K *401 >M >0]%gI >0IDilˆn{IC.0WXq)*5BD 8E8?  ž*  !N T'(%O7!"/ 9 G$'(%U8Y %T'(%Oo h!N% F)0%

  #$  ]/9U!58Y I J 8&>%MK!d ! Y.0WGXe 4> '(74!"*BD†   dc Ufe I *I gE: e tM]Opuv>xw] o S>xw yjM0]Og zg U  > y  $: 6; ˆ=*l] P2 '… ){gE: e A8Y2*)+ $ !Q8U)$MhY*$=tb0 #‘Eh2*))0 >YE!dD8 %L >7 $*x I } 8h2*))0 >]8h*)/ *)   1 s,-]2)9

)=U8& %UT)/ %42%bB=L >A2 - D**)HI ŒsV8 &L/)0,-]HBNEBD 8+72%  !",+a T'(9Oo L&!d/ %  !K8&.TWGXe8DBD Z P/?*7!F 8G >   @%T#$ OhU%Be# OPBD8g)0 > ?>% $>I J 84 '$ E!d8N!"h/ # ^G/M % %%M o/-?!"„* M8o) >Q%T4!K8G@(% #$ O7A)/ %h8h) >C)  ]2BD8 E@(% > 89B¡#$ O#2  '(P )%  I J 8P/-D!Q8 /M % A *A2U -*)f-,/ >Y)0 |%-?n ~L% % '( !"/ % K P8D.TWGXQI w %]BN  )%^85!d*BD >P E1 L?!d7I _H%|  C 2A87 >  4@(%?# OM *)e!F 8 M  > )$M0IR4/#8 |-l]    7 ) 'T )*) -|P /%?!5*O0 #$ ,A*Ž-/  žI_\9

C€B‚j6 : CR2   C;G< !„! !„! 25?)0 F> -#$ OgBD8%:jƒC  BC IQ_\%fcI…Gl Gl U {> M :Lla†‡Pk†    %%-IQ} L*]\%;g 2Ya) >5# O7BD 8+ , l]Š g$D82  4 2 )0 >EY8‹{6 8f- M@V†

‡ŒLŸU`GWWGˆŠT†5ŠM)7R4?6 .0ŒL™U`GWW¡’+@T8a7WGME6a  >E =‡  0!d#  ]HuˆˆŠ

>&jv

ˆ      



III



ˆ





l



g 





!

’+g%*,-*)E8g%8T)!K.0WGXq@%2T)0

*$ a)0%1  9 2)[ x8V%'0 /a% % z [ ] ~b/ 9 xBD 8¡ .T3G6š!"C 

*$ \I J 8?%'/ !28?%8T)&BN % )0/ %)Aa8U!"*BD >AsBN)MY9I

6c87 - M@ 





  

4.1 Datasets

III

  "!$#&%(')%%*',+.-/'102'435! 0768/9:9? 



l

 



ˆ



III





ˆ

$: 6, l] APGRD|2,9 OP >T/ %% )gM,0]-%O0% '( g !"^mˆ4/^)Yuu4#- %Y /^N/)P!" ‰jˆˆ#>%E:d!d*$/M%]28huˆˆˆBD 8=8 8 >8%Lh  #^ - b,  %h8E /&# %) !"G %F

*$ …i l%n{IhŸU&Y/L)0 L >/ 8V2%sBNV *1 %%/G)o#Q /G> '(%o8Y%O00 V'/! 8?>IKŒsEM)0%CGM h8?>%?)0 F> - Y8?@(%#$ OH] ?BNC))0*)E)0 % 2*)E 7i l$u*n;† l=ˆ ˆˆˆa!"*/0GBD 8+'/PˆfB%))*)o7* 98V /  )GM) ,Y 98A>/#!5l*ˆˆ&/N!\8g!"*/0 BNU8o%#2hM )0 '/U e:{=ˆ l*8Y.0WX5IU.0  &Z8&|21s)0 F> Z-491 # #/ 98>*)E #84T)

*)E@%]T84 > @(%1 %Q!d/ % V * 2Y/*)= o8Y) %   !d/ % t:Ll*I J 8G #\T/h2%N!^- M@0D8/)A28 &8U>%MP@(9Q#$ O7> 'T >#8&2LG  /01 M 9,-IQŒbG0M %   Z *&2C)%% *)   * ,G80/>8 8U- *)0/G!K %L1;' )$ \I .0  8G/2 ,0 H@(%%H M8 >UZ- 4 f8  > ?@(95# O]Q8   b,+!g7  Y!g2 T x8V#2)t!d*$/   % 7 % ,[ %*)HI›Œs 78 $M)…e/V8*B / MV8V%B @(%% a!d 8o2 %   +  *$ z)# \I Œby -ML*]g8 .0WX T) *)h@(95 ,h 98>?8D)0 F> -^ E8  > T@(%(#$ O]*85   b,G!"K%'(%,|21s) >  -?# ?/ 98>*)I 4. EXPERIMENTAL RESULTS AND DISCUSSION

J 8N)Mg9^!H™U/7”9•Q–— INi ‰$n  L^!>N%O0  '/!N!F u¢a %/E ,08L  h/@(% F :{Rg_Z_Z%^8 $CB%?

 %*)Y/58/>8 8=%1;0-  >eLI J 8 = ~b % '(V f)  %  1 D8DbB&s,2%C!Q/@ I J 8gM &!H8g*'(%M> |21s) > Q-D8 g!) >K%T4!^8  4@(%# OA!K8%G)$MY gnˆ ˆ0Kl J=:c J PŠY2 -I

‡ŒLŸU`GWWGˆŠT†5ŠM)7R4?6 .0ŒL™U`GWW¡’+@T8a7WGME6a  >E =‡  0!d#  ]HuˆˆŠ

>&j\J

J 8h>%M ž*$ q2%!"# Y!Nq.T3G6  U -*) -,a8hM)0%1;|+ M%%›)oM%%-  *) BD 8f8U@(%& I J 8 #2'/%?! U ) g A8 Lg ‡ }z@%#: : K <  9OH5:  V_^V Y a!U8f|21 ) > %%-K'%C8^!H8?) > %%-]( 1 2 ) >A#8h@(%%Q#  U!58Y *]H8&/1 2 ,0 c]H)=8h.0WGXQI J 8&M$ G&8*BDV,f!d 8h#  BD8  98 >%M*)V>-T) %/ G!dG8E% bB@(%I J 8GM$ U 2) >E8&.0WGXe%81 T)P$Yh/ M8V$>%G!"&RgZ Ah2hE  )=BD 8V8h! 8G/2,0 H%8T)I w *BN%'(%*]8UM g!dU28 %8T)DP2/048g!d™U/\I J 8 5/ C *2D%O0 *)5!d*BDIN.0  D4!8 |21s)0 F> 0-ZNQ8#l!d5Rg Ah2h ](K8*BD   CR   }Q >/Al]8P/2 ,0 FQ29M + %/ , 8 ?8%|% %4!^  % >E8'/?!dD8G%%-I J 89%!d]8 h/ E eV/LMT N*)/ % !g8 ) |%% ?2%sBN%E8D) >2 )Y|21s) >  - a8P/2 ,0 H@(%%\#$ O#!"U8 g)ME%h:c }Q >/0&u / )0G!^8&)0 F> K%%-I J 8 U 4BD8T,A8GM$1 #  ^# h# Y8D.0WGXV@(%#$  K!dC8N )$M%*IhR4U!dh™U/H]Z  %E5-$E>G8  l]-8*BD |  C |  }Q >/Pl]8D/2 ,0 %8T) *7D8 *'(U8GY %| %4)0 % 2*)72'I 

C   



Љ= J(j$m-j ˆ= ¢-l$j‰m ˆn ¢Tl$j‰Š Šm= ¢m(ˆ‰ ˆn ‰v(jˆ0lˆ= ‰‰(¢jj ]l Š(uˆv l jv0l*v‰

ˆ= ‰vjˆTl$¢ l Š(uˆvˆ ˆ= ‰‰¢jjŠ l jvTl*v‰ Š=m jˆ(*¢ Jˆ ˆ= ¢Šjj-l ˆ= ¢Šjj-llŠ(¢j l*jv‰

C Y 

 

 ) 

C 

 

 

l ˆn ˆˆˆŠ(u ˆn ˆˆˆŠv ˆn ˆˆˆm-¢

ˆ= ˆˆˆŠu l ˆ= ˆˆˆŠu ˆ= ˆˆˆmˆ

mŠ= ˆ‰¢m l%m(nŠ l‰ˆ Šˆ(¢nu \j J l%m(nv ˆˆ‰ ˆ= ˆˆˆŠv ˆ= ˆˆˆŠu l ˆ= ˆˆˆm‰

j$ms ¢u¢Tl j=Š j*j J‰ l%m(=v ˆˆ‰ ŠˆŠnv &v J







ˆ= ˆˆˆm-¢ ˆ= ˆˆˆm(ˆ ˆ= ˆˆˆm(‰ l







    !"#$&% '(%*)+$,*#.-0/1$23)*46578$,93*#.:  70); : -0/$.4)2*)*4o%O02% -I#Œb))   a8#)M7R4Z Ah2h] 80G)$MY%DBD 87) |%-D G%'(4B%G>%1 9M*)I J 8 $ ] l=ˆ Mˆˆˆ!d*$/^BD 8'/CˆUBN%4))*)  * M8y /V )eM )0 ,e 98t>/#!hjˆ0]Gl*jˆ0]  )Aujˆh/?!Z8%g!d/%NBN%U8#%Nh2gM) '/G t:{nˆ *l$%^BN BD88f 7'?   %**)HI J 8a*)0  9 >VM$ !F 8.TWGX^1l&T) *)V@%G$E >8- , *BN9G8  8g!58&/2 ,0  QI J 8   %/0h#,f 98> !2BD/?8?M%%A x!"C8g.0WGXK1l? *]-BD8%?8 '/  SB?? O)aPlP)0/ >&89O29 %T*I

5. CONCLUSIONS





Š0lˆvn l‰ l$j*J= jj¢ l$*j J= jj¢ Šˆmms ¢ˆ m(nŠ ˆ‰(¢$m lmŠ= l*‰ˆ j=m ¢u¢-ljŠn jj\J‰ 



J 8 g2%g8 g -T)/ *)=E9B .TWGX^1;*)a98T)f!" 8K*)/ % P!>N) > --Z P@(%(#$  % >9M*)!" 0'-  (@(%!d/ % I J 8C2%!"1 # g!\802)f%8T)BND   *,#L)f

‡ŒLŸU`GWWGˆŠT†5ŠM)7R4?6 .0ŒL™U`GWW¡’+@T8a7WGME6a  >E =‡  0!d#  ]HuˆˆŠ

>P‰ˆ

L / [!.03G6 J   l† 4 RgZAh2hI

b*,)*4 pN70,705*$*  )370   W ` PR. @ W  j E U  W  p -4 I @ W  j E+i W  i W % i W s  W ` ndD   W s E+i W j

Suggest Documents