%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: t.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips t %DVIPSParameters: dpi=400, compressed, comments removed %DVIPSSource: TeX output 1997.01.12:1337 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 0 N 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D<13035B131F137FEA03FFB5FC139FEAF81F1200B3B3A5EB7FC0007FB512C0A3 1A3579B427>I<1218121CEA3FE090B612C0A316801600485C0070C7120E5D00605CA25D 5D00E0495AC7FC4A5A4AC7FCA2140E5CA25C1478147014F05C1301A3495AA21307A3130F 5CA3131FA5133FA96DC8FC130E22387CB527>55 D57 D73 D82 D97 D101 D108 D<3907E03FC000FFEBFFF09038E1C0FC391FE7007CD807EE7FEA03 EC01F87FA25BA25BB3A3486CEB3F80B53887FFFEA327237DA22B>110 DI<3907E03FC039 FFE1FFF89038E7C0FE390FEF003FD807FCEB1F806C48EB0FC049EB07E049EB03F016F8A2 ED01FCA3150016FEA8ED01FCA316F8150316F06D130716E06DEB0FC06DEB1F8001EFEB3F 009038E3C0FC9038E1FFF09038E03FC091C8FCAA487EB512C0A327327EA22B>I<3807E0 7E38FFE1FF9038E38F80390FE71FC0EA07EEEA03ECEBF80FEC078049C7FCA35BB3A2EA07 F8B512E0A31A237EA21E>114 D<1360A613E0A31201A21203A21207120F383FFFFEB5FC A23803E000B01403A93801F006A2EA00F8140CEB7E18EB3FF0EB07E018327EB01E>116 D123 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 400dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 0 1 0 0 bop 31 689 a @beginspecial 0 @llx 3 @lly 514 @urx 110 @ury 4252 @rwi @setspecial %%BeginDocument: SplineLogoINI.eps %AI3_ColorUsage: Black&White %AI3_TemplateBox: 260.5 56 260.5 56 %AI3_TileBox: 0 0 567 816 %AI3_DocumentPreview: Header userdict /Adobe_packedarray 5 dict dup begin put /initialize % - initialize - { /packedarray where { pop } { Adobe_packedarray begin Adobe_packedarray { dup xcheck { bind } if userdict 3 1 roll put } forall end } ifelse } def /terminate % - terminate - { } def /packedarray % arguments count packedarray array { array astore readonly } def /setpacking % boolean setpacking - { pop } def /currentpacking % - setpacking boolean { false } def currentdict readonly pop end Adobe_packedarray /initialize get exec currentpacking true setpacking userdict /Adobe_cmykcolor 4 dict dup begin put /initialize % - initialize - { /setcmykcolor where { pop } { userdict /Adobe_cmykcolor_vars 2 dict dup begin put /_setrgbcolor /setrgbcolor load def /_currentrgbcolor /currentrgbcolor load def Adobe_cmykcolor begin Adobe_cmykcolor { dup xcheck { bind } if pop pop } forall end end Adobe_cmykcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_cmykcolor eq { end } if } def /setcmykcolor % cyan magenta yellow black setcmykcolor - { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat Adobe_cmykcolor_vars /_setrgbcolor get exec pop } def /currentcmykcolor % - currentcmykcolor cyan magenta yellow black { Adobe_cmykcolor_vars /_currentrgbcolor get exec 3 { 1 sub neg 3 1 roll } repeat 0 } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_cshow 3 dict dup begin put /initialize % - initialize - { /cshow where { pop } { userdict /Adobe_cshow_vars 1 dict dup begin put /_cshow % - _cshow proc {} def Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if userdict 3 1 roll put } forall end end } ifelse } def /terminate % - terminate - { } def /cshow % proc string cshow - { exch Adobe_cshow_vars exch /_cshow exch put { 0 0 Adobe_cshow_vars /_cshow get exec } forall } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_customcolor 5 dict dup begin put /initialize % - initialize - { /setcustomcolor where { pop } { Adobe_customcolor begin Adobe_customcolor { dup xcheck { bind } if pop pop } forall end Adobe_customcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_customcolor eq { end } if } def /findcmykcustomcolor % cyan magenta yellow black name findcmykcustomcolor object { 5 packedarray } def /setcustomcolor % object tint setcustomcolor - { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def /setoverprint % boolean setoverprint - { pop } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_IllustratorA_AI3 61 dict dup begin put % initialization /initialize % - initialize - { userdict /Adobe_IllustratorA_AI3_vars 46 dict dup begin put % paint operands /_lp /none def /_pf {} def /_ps {} def /_psf {} def /_pss {} def /_pjsf {} def /_pjss {} def /_pola 0 def /_doClip 0 def % paint operators /cf currentflat def % - cf flatness % typography operands /_tm matrix def /_renderStart [/e0 /r0 /a0 /o0 /i0 /i0 /i0 /i0] def /_renderEnd [null null null null /e1 /r1 /a1 /clip] def /_render -1 def /_rise 0 def /_ax 0 def % x character spacing (_ax, _ay, _cx, _cy follows awidthshow naming convention) /_ay 0 def % y character spacing /_cx 0 def % x word spacing /_cy 0 def % y word spacing /_leading [0 0] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def % typography operators /Tx {} def /Tj {} def % compound path operators /CRender {} def % printing /_AI3_savepage {} def % color operands /_gf null def /_cf 4 array def /_if null def /_of false def /_fc {} def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc {} def /_i null def Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3 { dup xcheck { bind } if pop pop } forall end end Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3_vars begin newpath } def /terminate % - terminate - { end end } def % definition operators /_ % - _ null null def /ddef % key value ddef - { Adobe_IllustratorA_AI3_vars 3 1 roll put } def /xput % key value literal xput - { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop % integer npop - { { pop } repeat } def % marking operators /sw % ax ay string sw x y { dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add } def /swj % cx cy fillchar ax ay string swj x y { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add 6 2 roll /_cnt 0 ddef {1 index eq {/_cnt _cnt 1 add ddef} if} forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss % ax ay string matrix ss - { 4 1 roll { % matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss % cx cy fillchar ax ay string matrix jss - { 4 1 roll { % cx cy fillchar matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke }ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def % path operators /sp % ax ay string sp - { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp % cx cy fillchar ax ay string jsp - { { % cx cy fillchar ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath }ifelse 2 copy rmoveto } exch cshow 5 npop } def % path construction operators /pl % x y pl x y { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c % x1 y1 x2 y2 x3 y3 c - { curveto } def /C /c load def /v % x2 y2 x3 y3 v - { currentpoint 6 2 roll curveto } def /V /v load def /y % x1 y1 x2 y2 y - { 2 copy curveto } def /Y /y load def /l % x y l - { lineto } def /L /l load def /m % x y m - { moveto } def } {%else /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def }ifelse % graphic state operators /d % array phase d - { setdash } def /cf {} def % - cf flatness /i % flatness i - { dup 0 eq { pop cf } if setflat } def /j % linejoin j - { setlinejoin } def /J % linecap J - { setlinecap } def /M % miterlimit M - { setmiterlimit } def /w % linewidth w - { setlinewidth } def % path painting operators /H % - H - {} def /h % - h - { closepath } def /N % - N - { _pola 0 eq { _doClip 1 eq {clip /_doClip 0 ddef} if newpath } { /CRender {N} ddef }ifelse } def /n % - n - {N} def /F % - F - { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf }ifelse } { /CRender {F} ddef }ifelse } def /f % - f - { closepath F } def /S % - S - { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps }ifelse } { /CRender {S} ddef }ifelse } def /s % - s - { closepath S } def /B % - B - { _pola 0 eq { _doClip 1 eq % F clears _doClip gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S }ifelse } { /CRender {B} ddef }ifelse } def /b % - b - { closepath B } def /W % - W - { /_doClip 1 ddef } def /* % - [string] * - { count 0 ne { dup type (stringtype) eq {pop} if } if _pola 0 eq {newpath} if } def % group operators /u % - u - {} def /U % - U - {} def /q % - q - { _pola 0 eq {gsave} if } def /Q % - Q - { _pola 0 eq {grestore} if } def /*u % - *u - { _pola 1 add /_pola exch ddef } def /*U % - *U - { _pola 1 sub /_pola exch ddef _pola 0 eq {CRender} if } def /D % polarized D - {pop} def /*w % - *w - {} def /*W % - *W - {} def % place operators /` % matrix llx lly urx ury string ` - { /_i save ddef 6 1 roll 4 npop concat userdict begin /showpage {} def false setoverprint pop } def /~ % - ~ - { end _i restore } def % color operators /O % flag O - { 0 ne /_of exch ddef /_lp /none ddef } def /R % flag R - { 0 ne /_os exch ddef /_lp /none ddef } def /g % gray g - { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G % gray G - { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k % cyan magenta yellow black k - { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K % cyan magenta yellow black K - { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x % cyan magenta yellow black name gray x - { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X % cyan magenta yellow black name gray X - { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def % locked object operator /A % value A - { pop } def currentdict readonly pop end setpacking % annotate page operator /annotatepage { } def %AI3-Grid.0 18 18 3 0.833333 0.833333 0.833333 3 Adobe_cmykcolor /initialize get exec Adobe_cshow /initialize get exec Adobe_customcolor /initialize get exec Adobe_IllustratorA_AI3 /initialize get exec 0 A 0 O 0 g 0 i 2 J 0 j 1 w 2 M []0 d %AI3_Note: 0 D 426.4 109.2 m 510.7 109 L 511.2 108.7 512.1 108.1 512 107.4 C 512 106.3 511.1 106 510.4 105.5 C 504 105.6 L 371.6 105.8 L 360.5 104.6 347.9 106.3 337.3 102 C 333.2 99.6 333.5 94.4 331.6 90.6 C 328.5 82.9 326.4 74.7 323.6 66.8 C 308.5 21.2 L 307 16.8 305.3 12.4 304 7.9 C 303.2 6.5 301.7 5.3 300.2 4.8 C 299.1 5.2 298.2 6.2 297.7 7.3 C 297.7 10.1 299.1 12.9 299.9 15.7 C 309.1 43.7 318.3 72.1 327.7 100.2 C 329.8 104.4 333.2 109 338.5 109 C 339.6 109.3 341 109 341.9 109.4 C 409.4 109.2 L 426.4 109.2 L f 215.8 107.2 m 216.6 104.2 214.2 101.8 213.8 98.9 C 201.6 61.4 L 196 45.3 191.1 29.2 185.5 13.2 C 183.4 10.2 181.5 6.6 177.6 5.5 C 171.2 4.7 L 11.5 4.6 L 2.8 4.7 L 1.9 5.1 2 6 2 6.8 C 3.5 7.8 L 69.6 7.8 L 137.6 7.9 L 149.7 9.4 163.2 7.5 175.1 11.4 C 181.4 14.8 181 23.2 184 29 C 192.7 54.6 200.8 80.4 209.6 106.1 C 210.1 107.8 211.6 108.2 213 109.2 C 214.4 109.4 215.1 108.1 215.8 107.2 C f 255.7 106.3 m 256.8 104.1 256.6 102 257.2 99.5 C 257.8 89.9 258.7 79.9 259.2 70.4 C 259.4 65.7 259.8 61.1 260 56.2 C 260.8 56.1 259.9 55.6 260.3 55.3 C 261 40.5 262 26 263.6 11.6 C 263.9 11.1 264.2 11.5 264.6 11.5 C 267 16.5 268.6 22.2 270.4 27.5 C 279.8 53.8 288.4 80.8 297.8 107 C 298.3 108.4 300 108.1 301.1 109 C 302.4 109 303.5 108 304 106.9 C 304.2 105 L 293 72.7 281.9 40 270.7 7.8 C 269.2 5.5 266.1 3.5 263 4.2 C 260.3 4.4 258.1 5.8 257 8.4 C 255.9 15.9 255.8 24.4 254.9 31.8 C 255.1 32.5 254.9 33.8 254.8 34.6 C 253.8 54.3 252.4 75.5 250.9 95.6 C 250.8 97.7 250.8 99.8 250 101.5 C 247 98.5 246.7 93.7 245.2 89.9 C 236.9 66.6 228.9 43.2 221 19.8 C 219.1 15.1 217.8 10.1 215.6 5.8 C 214.5 4.9 213.2 3.4 211.7 4.2 C 210.8 5.3 210 6.1 209.9 7.6 C 220.7 40.4 231.8 72.8 243.4 105.4 C 245.6 107.4 248.5 109.7 251.6 108.6 C 253.2 108.7 255.1 107.9 255.7 106.3 C f 158.5 80.9 m 158.8 79.1 155.8 79.6 156.2 77.4 C 156.5 77.1 157 77.4 157.3 77.2 C 157.7 76.6 157 76.1 156.7 76 C 155.8 76 L 155 73.4 153.9 71 153.1 68.4 C 152.8 68.2 152.4 68.3 152 68.3 C 152 70.9 153.8 73.1 154.2 75.7 C 153.6 76.1 152.8 76.4 153.5 77.2 C 156.1 77 154.6 80.8 157.3 80.9 C 158.5 80.9 L f 101.6 80.6 m 100.8 76.6 99.1 72.6 97.9 68.5 C 97.7 68.1 97.1 68.4 96.7 68.3 C 97.3 72.5 99.4 76.3 100.3 80.5 C 100.6 81 101.2 80.7 101.6 80.6 C f 129 80.8 m 129.1 79.9 128.8 78.2 127.4 79 C 127.6 79.6 127.6 81.2 129 80.8 C f 163.1 80.5 m 163.5 80 162.6 79.5 162.7 78.8 C 162.4 78.6 162 78.7 161.6 78.7 C 161.4 79.1 161.6 79.9 161.9 80.4 C 162.3 80.7 162.6 80.6 163.1 80.5 C f 166.1 80.5 m 165.7 79.9 166.2 78.4 164.9 78.7 C 164.4 78.7 L 164.4 79.5 164.7 81.2 166.1 80.5 C f 366.4 78.4 m 366.1 76.6 365.1 75 363.6 74.2 C 365 72.3 362.9 70.5 363.6 68.6 C 363.3 67.8 362.4 68.2 361.7 68.3 C 361.2 70.2 363.8 72.4 361.4 73.4 C 359.8 73.2 357.7 73.9 356.5 72.8 C 355.2 68.4 L 354.9 68 354.2 68.1 353.8 68.3 C 354.5 72.5 356.4 76.5 357.7 80.6 C 360.7 80.1 365.4 82.1 366.4 78.4 C f 380.3 80.5 m 380 79.3 379.5 77.8 379.2 76.4 C 380.5 76.8 382.3 77.8 383.9 76.9 C 385.5 75.6 384 73.7 383.8 72.2 C 382.8 70.9 383.3 67.9 381.2 68.2 C 381.4 70.6 383.2 73 382.9 75.6 C 382.2 76.5 380.9 75.9 380 75.7 C 377.2 74.1 377.7 70.3 376 68 C 375.2 68.2 L 376.2 72.2 377.9 76.2 379 80.4 C 379.3 80.7 379.8 80.7 380.3 80.5 C f 401.9 80.5 m 401.1 77.1 399.1 74 399.4 70.3 C 399.9 69.5 401.1 69.2 402.1 69.4 C 407.2 70.5 406.5 76.6 408.4 80.4 C 408.8 80.6 409.3 80.5 409.8 80.5 C 408.5 77 408.3 72.7 405.6 69.6 C 403.8 68.1 400.9 67.3 398.8 68.4 C 397.9 69 397.3 70 397.4 70.9 C 398.5 74.1 399.1 77.5 400.6 80.5 C 401.9 80.5 L f 423.7 80.5 m 424.2 79.8 423.4 79.3 423.4 78.7 C 423 78.5 422.5 78.6 422.2 78.6 C 421.9 79.5 422.4 81 423.7 80.5 C f 459.8 79.9 m 459.7 79.4 459.8 78.5 459 78.5 C 458.2 78.6 L 458 79.3 458.4 80 458.8 80.4 C 459.1 80.3 460.2 80.6 459.8 79.9 C f 469.9 80.2 m 470.1 79.5 469.7 78.8 469.3 78.2 C 468.2 78.4 L 468.2 79.1 468.5 80.8 469.9 80.2 C f 472.9 80.2 m 472.8 79.1 472.5 77.8 471.1 78.4 C 471 79.4 471.5 80.7 472.9 80.2 C f 124.3 79.7 m 124 79 124 78.2 123.7 77.5 C 124 77.2 125.3 77.3 124.7 76.4 C 124.5 75.5 123.8 76.4 123.4 76 C 122.5 74 121.7 71.9 121.2 69.8 C 121.3 69.1 122.9 69.5 122.2 68.5 C 121.7 67.9 120.7 68.2 120.1 68.3 C 119.1 68.9 120.1 70.1 120.1 71 C 120.7 72.6 121.4 74.1 121.7 75.7 C 121.2 76.3 120.6 75.9 120.8 76.9 C 123.6 76.4 121.7 80.7 124.3 79.7 C f 132.8 79.7 m 132.6 78.9 132.3 78.2 132.2 77.4 C 132.5 77.2 133 77.4 133.3 77.2 C 133.4 76.2 132.6 75.8 131.8 75.8 C 130.8 73.9 130.1 72 129.7 69.7 C 130.1 69.3 131.4 69.2 130.6 68.4 C 129.8 68 129.1 68 128.4 68.4 C 128.1 70.9 129.8 73.2 130.2 75.7 C 129.1 76.2 L 129.3 77 129.7 77.2 130.6 77.3 C 131.3 78.2 131 80.5 132.8 79.7 C f 146.6 77.5 m 147.2 77.1 148 77.4 147.7 76.3 C 147.4 75.6 146.6 76.2 146.2 75.8 C 145.3 73.9 144.7 71.9 144.1 69.8 C 144.4 69.2 145.6 69.6 145.2 68.6 C 144.8 67.9 143.9 67.9 143.3 68.2 C 142.6 69 L 143.1 71.3 143.9 73.5 144.6 75.7 C 144 76.1 143.2 76.4 143.9 77.2 C 146.4 76.6 144.7 80.5 147.2 79.8 C 146.6 77.5 L f 463.4 79.4 m 464.1 78.5 461.9 76.6 464 76.8 C 464.4 76.4 464 75.6 463.4 75.6 C 462.5 75.6 L 460.6 69.4 L 460.8 69 461.1 69 461.5 69 C 461.8 68.5 461.2 67.8 460.7 67.7 C 460 67.8 459.3 67.7 458.9 68.4 C 459.2 70.9 460.7 73 460.8 75.5 C 460.1 75.3 459.6 76.3 460.2 76.7 C 462.7 76.3 461.1 80 463.4 79.4 C f 478.1 79.2 m 477.5 77.2 L 477.8 76.8 479.2 77 478.6 76.1 C 478.5 75.2 477.4 75.9 477 75.4 C 476.3 73.4 475.5 71.5 475 69.5 C 475.1 69 475.6 69 476 69 C 476.1 68.6 476 67.7 475.3 67.7 C 474.6 67.7 473.8 67.5 473.4 68.3 C 473.6 70.7 474.7 72.9 475.4 75.1 C 475.1 75.7 473.9 75.7 474.7 76.7 C 477.3 76.2 475.4 80.6 478.1 79.2 C f 1 g 364.4 77.9 m 364.2 76.6 363.7 75.5 362.4 75 C 360.6 75 359.1 74.7 357.5 75.1 C 357.6 76.6 358.5 77.9 359 79.1 C 361 78.7 363.6 80.1 364.4 77.9 C f 0 g 173.3 76.8 m 173.4 75.2 171.5 75.9 170.6 75.4 C 168.7 73.5 168.6 70.7 167.5 68.4 C 167.1 68.3 166.7 68.1 166.3 68.5 C 169.1 77 L 169.3 77.4 169.9 77.3 170.2 77.2 C 170.5 76.4 L 171.4 76.6 173.1 78.3 173.3 76.8 C f 110.2 76.2 m 110 73.5 108.7 71 108 68.4 C 106.8 68.4 L 107.4 70.8 108.5 72.9 108.6 75.5 C 108.1 76.3 106.9 76.3 106.1 76 C 103 74.6 103.3 70.9 102 68.4 C 101.8 68.2 101.4 68.3 101 68.3 C 101.5 71.4 102.8 74 103.7 77.2 C 104.2 77.2 105 77.2 104.9 76.4 C 106.4 77.2 109.1 78.2 110.2 76.2 C f 118.6 76.8 m 119 76.3 119.2 75.5 118.9 74.9 C 118.6 74.8 118.3 74.7 118 74.8 C 117.6 75.4 117.4 75.8 116.8 76.2 C 115.4 76.3 114.1 76.2 113.5 74.9 C 114 73.3 116.1 73.6 117.4 72.8 C 118.3 72.2 118 71.4 117.7 70.6 C 117.2 69.2 115.7 68.5 114.5 68.2 C 113.2 67.9 112.2 68 111.1 68.6 C 110.8 69.4 110 70.2 110.8 71 C 112.9 71.8 111.3 68.8 113.4 69.4 C 114.7 69.1 116 69.6 116.5 70.8 C 116.3 72.7 113.9 72 112.7 72.8 C 111.8 73.3 112.3 74.3 112.3 75 C 112.9 76.5 114.4 77 115.7 77.4 C 116.8 77.5 117.7 77.3 118.6 76.8 C f 392.2 77.3 m 391.8 75.7 L 387.4 76 387.9 70.9 386.4 68.3 C 386.1 68 385.4 68 385.1 68.3 C 386 71.2 386.7 74.2 388 77 C 388.5 77 389.6 77.1 389.2 76.3 C 390 76.3 391 77.6 392.2 77.3 C f 136.6 77.3 m 136 75.2 135.4 72.7 134.6 70.6 C 134.5 69.8 135.3 69.4 135.8 69.2 C 140.1 69.4 139.9 74.2 141.1 77 C 141.5 77.2 141.9 77.4 142.3 77.3 C 141.6 74.4 140.5 71.4 139.6 68.4 C 138.5 68.4 L 138.5 69 L 137.2 68.4 135.2 67.5 133.7 68.5 C 132.4 71.6 135.1 74.2 135.4 77.2 C 135.7 77.4 136.2 77.2 136.6 77.3 C f 160.4 77.3 m 160.4 74.7 158.7 72.4 158.9 69.7 C 159.7 69 160.9 69.4 161.8 69.8 C 163.9 71.5 164.4 74.5 165.1 77 C 165.3 77.4 166.1 77.3 166.4 77.2 C 165.4 74.2 164.6 71.2 163.4 68.3 C 162.7 68.1 162.4 68.5 162.4 69.1 C 161.4 68.3 159.4 67.8 157.9 68.4 C 156.7 68.8 157.2 70.4 157.4 71.2 C 159.4 77.2 L 159.8 77.1 160 77.4 160.4 77.3 C f 418.1 76 m 417.8 73.2 416.7 70.7 415.9 68.2 C 415.5 68 414.9 67.9 414.6 68 C 414.9 70.6 416.5 72.6 416.4 75.4 C 415.8 76.3 414.6 75.7 413.9 75.7 C 411 74.3 411.1 70.6 409.8 68 C 408.7 68 L 409.1 71.1 410.7 73.9 411.5 76.9 C 412.7 76.9 L 412.7 76.4 L 414.2 77 417 78.1 418.1 76 C f 127.8 77.2 m 127 74.3 126.4 71 124.9 68.3 C 124.1 68.4 L 124.9 71.4 125.7 74.4 126.8 77.2 C 127.8 77.2 L f 369.8 77 m 369.3 74.6 367.8 72.2 368 69.7 C 368.6 68.9 369.9 69.2 370.6 69.5 C 373.6 71 372.9 74.8 374.6 77.2 C 375.6 77.2 L 374.6 74.2 373.7 71.2 372.7 68.2 C 371.8 68.2 L 371.7 68.5 371.9 68.8 371.5 69 C 370.2 68 368.6 67.9 367.2 68.2 C 365.9 68.7 366.3 70.6 366.7 71.5 C 367.4 73.4 367.8 75.5 368.8 77.2 C 369.8 77 L f 439.7 75.6 m 440.1 74.3 439.5 73.1 439.2 72 C 433 72 L 432.1 71.6 432.5 70.6 432.6 70 C 433.4 68.7 435 69 436.1 69.5 C 437.2 69.7 437.1 71.4 438.6 70.8 C 438.1 69.3 436.8 68.7 435.6 67.9 C 434.4 67.7 432.9 67.7 431.9 68.4 C 430.9 69.6 430.7 70.9 431.3 72.4 C 432 74.7 434 76.9 436.6 77.2 C 438 77.5 439.1 76.7 439.7 75.6 C f 447.4 76.9 m 447.3 74.4 443.6 76 443.3 73.2 C 441.7 68.2 L 441.5 67.7 440.9 68 440.5 67.9 C 440.7 71 442.4 73.8 443.2 76.8 C 443.6 77.1 443.8 76.8 444.4 76.9 C 444.4 76.2 L 445.3 76.1 446.3 77.5 447.4 76.9 C f 454.4 75.8 m 454.4 74.5 L 452.8 73.7 453.2 75.9 451.8 75.8 C 450.8 75.7 449.5 75.9 449 74.6 C 449.6 72.7 452.7 73.6 453.5 71.6 C 453.5 70.3 452.6 69.1 451.6 68.4 C 450.1 67.9 448.2 67.2 446.8 68.2 C 446.1 68.8 445.4 69.8 446 70.7 C 446.4 70.9 446.8 70.8 447.2 70.8 C 447.3 70.1 447.6 69.4 448.3 69.1 C 449.4 68.8 450.7 69.1 451.6 69.8 C 451.5 70.3 452 70.4 451.9 70.9 C 450.7 72.7 445.9 71.5 447.8 75 C 448.6 76.2 449.9 76.9 451.3 77.2 C 452.5 77 453.9 77.1 454.4 75.8 C f 419.9 68.2 m 419.6 67.8 419 67.9 418.7 68 C 419.1 71.1 420.6 74 421.6 77 C 422.6 76.9 L 419.9 68.2 L f 425.8 76.9 m 425.9 75.1 425.4 72.4 426.1 70.4 C 428 72.4 429.3 74.8 430.8 76.9 C 432.1 77 L 431.8 75.8 L 426 68 L 424.4 68.2 L 424.2 71 424.4 74.4 424.2 76.9 C 424.6 77.1 425.3 77.1 425.8 76.9 C f 472.4 75.6 m 470.6 69.2 L 471.3 69.1 471.6 68.4 471 67.9 C 470.4 67.8 470 67.6 469.4 67.9 C 468.7 70 467.4 67.3 466.1 67.7 C 465 67.5 464.1 67.8 463.6 68.6 C 463.1 69.9 463.8 70.9 464.4 71.9 C 466.1 74.1 470.9 71.8 470.8 75.2 C 469.8 75.9 468.9 76 467.9 75.6 C 466.8 75.3 466.7 73.6 465.4 74.3 C 465.3 75.5 466.6 76.1 467.5 76.7 C 469.1 77.2 471.8 77.4 472.4 75.6 C f 458.6 76.9 m 458.8 75.7 458.1 74.6 457.7 73.6 C 456 68 L 455.7 67.8 455.2 67.9 454.8 67.9 C 455.4 70.9 456.5 74.1 457.7 76.9 C 458.6 76.9 L f 1 g 438.4 74.8 m 439.1 73.2 437.1 73 436.3 73.2 C 435.3 73.3 434.3 73.2 433.4 73.6 C 433.9 74.8 435 75.4 436.2 75.7 C 437.1 75.8 438.1 75.7 438.4 74.8 C f 0 g 395.8 73.6 m 396.1 72.7 395.3 71.8 394.3 72 C 391.4 72.1 L 391.1 74.9 394.4 73.3 395.8 73.6 C f 1 g 469.2 70.3 m 468 69.6 466.8 68.7 465.4 69.2 C 464.7 69.4 465 70.3 465.1 70.8 C 466.2 72.2 468.2 71.6 469.7 72.2 C 470.1 71.7 469.5 70.9 469.2 70.3 C f 0 g 108.5 53 m 108.3 52.6 108.5 51.8 107.8 51.8 C 106.9 52.1 106.3 50.9 106.1 50.3 C 106 50.1 105.8 49.8 106.1 49.7 C 107.3 49.4 L 107.4 48.3 106.2 48.4 105.5 48.2 C 103 40.9 L 102.8 40.5 102.3 40.8 102 40.7 C 102.1 43.2 103.5 45.6 104 48.1 C 103.5 48.4 102.7 48.5 103.2 49.3 C 105.9 49.3 104.6 53.2 107.5 53.2 C 108.5 53 L f 93.4 52.8 m 93.2 52.2 93.3 51.1 92.4 51.1 C 91.7 51.2 L 91.5 52.2 92.2 53.8 93.4 52.8 C f 155.5 53 m 155.9 52.4 155.4 51.9 155.3 51.4 C 154.8 51.1 154.5 51.3 154 51.2 C 153.8 51.8 154.2 52.1 154.3 52.6 C 154.4 53 155 53.1 155.5 53 C f 159.2 52.9 m 158.8 50.6 157.6 48.7 157.3 46.3 C 159.6 46.6 161.1 50.2 163.8 49.6 C 159.6 46.2 L 160.2 44.4 160.7 42.7 161.3 40.9 C 158 39.4 159.7 43.9 158 45 C 155.6 44.7 156.3 42.1 155.2 40.7 C 154.3 40.8 L 155.2 44.8 156.9 48.7 157.9 52.7 C 158.2 52.9 158.8 53.2 159.2 52.9 C f 47 52.8 m 48.1 49.6 48.8 46.3 50.2 43.3 C 51.4 46.4 52.5 49.6 53.4 52.8 C 54.8 52.8 L 50.9 40.7 L 49.2 40.7 L 48.2 43.8 47.3 47 46 50 C 45.5 49.6 L 42.6 40.6 L 41.3 40.6 L 42.4 44.7 43.8 48.6 45.1 52.7 C 45.7 53.2 46.3 52.6 47 52.8 C f 356.5 51.1 m 356.3 49.5 355.7 48.1 354.2 47.2 C 355.3 46.6 355.3 45 355.1 44 C 353.6 39.8 348.9 40.6 345.1 40.6 C 345 40.8 344.8 40.6 344.6 40.7 C 345.9 44.7 346.9 48.7 348.4 52.7 C 348.7 52.9 L 351.2 52.4 355.3 54.3 356.5 51.1 C f 379.2 52.8 m 379 51.5 378.6 50.1 378.1 48.8 C 379.5 49.1 381.2 50 382.8 49.3 C 384 48.9 383.5 47.2 383.3 46.4 C 381.4 40.6 L 380.8 40.6 380 40.3 380.2 41.2 C 380.7 43.5 382.5 45.7 381.8 48.1 C 381.1 48.5 380 48.4 379.1 48.1 C 376.5 46.5 376.5 43.3 375.5 40.8 C 375.2 40.3 374.7 40.6 374.3 40.6 C 375 44.8 377.1 48.5 378 52.7 C 378.4 52.7 378.8 53.1 379.2 52.8 C f 150.6 52.1 m 151 51.1 149.2 49.3 151.3 49.3 C 151.3 48.3 150.3 48.6 149.6 48.4 C 148.6 46.4 148.2 44.2 147.7 42 C 148.7 41.6 L 148.9 40.6 148 40.5 147.2 40.4 C 146.6 40.4 146.4 40.9 146 41.4 C 148.2 48.1 L 147.1 48.6 L 147.2 49.5 147.8 49.6 148.6 49.7 C 149.4 50.4 148.7 52.6 150.6 52.1 C f 1 g 354.7 50.6 m 354.9 49.6 354.3 48.6 353.6 48.1 C 352.2 47.3 350.1 47.7 348.6 47.8 C 348.4 48.9 349.4 50 349.6 51.2 C 351.3 51.2 353.3 51.9 354.7 50.6 C f 0 g 122.5 49.7 m 123.1 47.3 120.2 48.7 119.4 47.3 C 118 45.3 117.7 42.8 116.6 40.7 C 115.8 40.8 L 116.3 43.8 117.6 46.5 118.4 49.3 C 118.8 49.4 119.1 49.6 119.5 49.6 C 119.8 48.6 L 120.5 49.3 121.6 49.7 122.5 49.7 C f 145.1 48.4 m 143.3 42.4 L 143.4 41.7 144.6 41.4 143.6 40.7 C 142.8 40.2 141.7 40.9 141.5 41.6 C 140.2 40.9 138.7 39.9 137 40.7 C 135.7 41.1 136.5 42.7 136.4 43.6 C 137.6 46.1 140.6 45.7 142.9 46.2 C 143.3 46.7 143.7 47.2 143.5 47.9 C 142.9 48.8 141.4 48.7 140.5 48.4 C 139.3 48.2 139.6 46.3 138.1 46.9 C 138 48 138.8 48.5 139.6 49.1 C 141.2 49.9 144 50.2 145.1 48.4 C f 62.6 48.4 m 62.9 47.1 62.8 45.7 62.3 44.6 C 56 44.5 L 55.3 44.1 55.6 43.3 55.6 42.6 C 56.2 41.6 57.6 41.5 58.7 41.8 C 60 41.9 60 43.9 61.6 43.4 C 61.6 42.2 60.1 41.5 59.2 40.8 C 57.9 40.2 56.1 40.4 55 41 C 53.7 41.9 54.3 43.5 54.5 44.8 C 55.3 47.1 56.9 49.5 59.6 49.7 C 60.8 49.6 62.2 49.5 62.6 48.4 C f 80 49.6 m 79.8 47.2 76.1 48.4 75.8 45.7 C 74.3 40.8 L 74 40.6 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49.5 72.8 49.6 C 73.1 49.3 L 70.3 40.9 L 70.1 40.5 69.4 40.6 69.1 40.8 C 69.1 41.3 L 67.9 40.8 66.2 40 64.7 40.7 C 63.4 41.2 64 42.9 64.2 43.8 C 65 45.7 65.1 48 66.4 49.6 C 67.2 49.6 L f 89.4 40.7 m 88.4 40.7 L 89 43.6 90.1 46.8 91.2 49.6 C 92.3 49.4 L 89.4 40.7 L f 154.6 49.2 m 151.7 40.7 L 150.7 40.7 L 151.3 43.8 152.4 46.8 153.5 49.6 C 153.8 49.5 154.5 49.8 154.6 49.2 C f 402.4 48.5 m 403.5 49.2 405 50 406.6 49.3 C 407.7 48.9 407.1 47.8 407.3 46.9 C 405.1 40.6 L 404.8 40.4 404.4 40.4 404 40.4 C 404 43 406 45.4 405.6 48 C 404.9 48.7 403.8 48.3 403.2 47.9 C 400.9 46.2 401.1 43 400 40.7 C 399.7 40.4 399.3 40.4 398.9 40.4 C 398.6 43 400.7 45.2 400.6 47.8 C 400.3 48.9 398.8 48.2 398.2 48 C 395.9 46.2 395.4 43.3 394.7 40.7 C 394.4 40.4 393.9 40.4 393.6 40.6 C 393.9 43.5 395.3 46.3 396.1 49.2 C 396.6 49.5 396.8 49.2 397.3 49.3 C 397.3 49.1 397.3 48.9 397.6 48.7 C 398.9 49.5 401.3 50.1 402.4 48.5 C f 388.1 49.2 m 386 42.4 L 386.6 41.1 388.2 41.6 389.2 42 C 391.5 43.8 391.4 46.9 392.5 49.3 C 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142.4 43.7 141.8 42.5 140.6 42.1 C 139.6 41.9 138.7 41.4 137.9 42.1 C 137.6 42.5 137.3 43.1 137.8 43.6 C 139 44.7 140.8 44.8 142.6 44.9 C f gsave annotatepage grestore showpage Adobe_IllustratorA_AI3 /terminate get exec Adobe_customcolor /terminate get exec Adobe_cshow /terminate get exec Adobe_cmykcolor /terminate get exec Adobe_packedarray /terminate get exec %%EndDocument @endspecial 836 852 a FE(In)n(ternal)27 b(Rep)r(ort)g(97{01)259 1039 y FD(The)j(LBG-U)g(metho)r(d)f(for)h(v)n(ector)f(quan)n(tization)f (-)h(an)145 1139 y(impro)n(v)n(emen)n(t)e(o)n(v)n(er)i(LBG)h(inspired)f (from)g(neural)g(net)n(w)n(orks)1182 1947 y FC(by)1018 2056 y FB(Bernd)22 b(F)-5 b(ritzk)n(e)31 3381 y FA(Ruhr-Univ)n (ersit\177)-28 b(at)17 b(Bo)r(c)n(h)n(um)31 3448 y(Institut)g(f)r(\177) -29 b(ur)17 b(Neuroinformatik)31 3514 y(44780)g(Bo)r(c)n(h)n(um)976 3648 y @beginspecial 0 @llx 0 @lly 110 @urx 108 @ury 793 @rwi @setspecial %%BeginDocument: unibolo.eps %Version: 2.0 %Copyright: 1988, NeXT, Inc. /__NXdef{1 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cvx copy dup 2 4 -1 roll put __NXProcessRectArgs }{{__NXRectPath stroke} __NXProcessRectArgs}ifelse grestore }__NXbdef /xyshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index 2 mul 1 add get add exch 3 index 3 index 2 mul get add exch moveto pop }for pop pop }__NXbdef /xshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show exch 3 index 3 index get add exch moveto pop }for pop pop }__NXbdef /yshow{ 0 1 3 index length 1 sub{ currentpoint 4 index 3 index 1 getinterval show 3 index 3 index get add moveto pop }for pop pop }__NXbdef /arct{arcto pop pop pop pop}__NXbdef /setbbox{pop pop pop pop}__NXbdef /ucache{}__NXbdef /ucachestatus{mark 0 0 0 0 0}__NXbdef /setucacheparams{cleartomark}__NXbdef /uappend{systemdict begin cvx exec end}__NXbdef /ueofill{gsave newpath uappend eofill grestore}__NXbdef /ufill{gsave newpath uappend fill grestore}__NXbdef /ustroke{ gsave newpath dup length 6 eq {exch uappend concat}{uappend}ifelse 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pop}def /currentwindowtype{pop 0}def /setalpha{pop}def /currentalpha{1.0}def /hidecursor{}def /obscurecursor{}def /revealcursor{}def /setcursor{4 {pop}repeat}bind def /showcursor{}def /NextStepEncoding where not{ /NextStepEncoding StandardEncoding 256 array copy def 0 [129/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/Ccedilla/Egrave /Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/Ugrave/Uacute /Ucircumflex/Udieresis/Yacute/Thorn/mu/multiply/divide/copyright 176/registered 181/brokenbar 190/logicalnot 192/onesuperior 201/twosuperior 204/threesuperior 209/plusminus/onequarter/onehalf/threequarters/agrave /aacute/acircumflex/atilde/adieresis/aring/ccedilla/egrave/eacute /ecircumflex/edieresis/igrave 226/iacute 228/icircumflex/idieresis/eth /ntilde 236/ograve/oacute/ocircumflex/otilde/odieresis 242/ugrave/uacute /ucircumflex 246/udieresis/yacute 252/thorn/ydieresis] {dup type /nametype eq {NextStepEncoding 2 index 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y(mo)q(di\014ed)g(co)q(deb)q(o)q(ok)h (un)o(til)g(con)o(v)o(ergence,)e(another)h(v)o(ector)e(is)i(mo)o(v)o (ed,)g(and)h(so)e(on.)38 b(Since)22 b(a)g(strictly)249 1605 y(monotonous)16 b(impro)o(v)o(emen)o(t)e(of)i(the)h(LBG-generated) f(co)q(deb)q(o)q(oks)h(is)g(enforced,)f(it)h(can)f(b)q(e)h(pro)o(v)o (ed)f(that)249 1665 y(LBG-U)j(terminates)e(in)i(a)f(\014nite)h(n)o(um)o (b)q(er)f(of)f(steps.)27 b(Exp)q(erimen)o(ts)18 b(with)h(arti\014cial)f (data)g(demonstrate)249 1726 y(signi\014can)o(t)i(impro)o(v)o(emen)o (ts)f(in)i(terms)e(of)h(RMSE)g(o)o(v)o(er)e(LBG)j(com)o(bined)f(with)h (only)f(mo)q(destly)h(higher)249 1787 y(computational)16 b(costs.)110 1882 y Ft(Keyw)n(ords:)42 b FA(co)r(deb)r(o)r(ok)23 b(construction,)j(data)f(compression,)i(gro)n(wing)f(neural)g(net)n(w)n (orks,)i(LBG,)g(v)n(ector)110 1949 y(quan)n(tization)110 2130 y FD(1)90 b(In)n(tro)r(duction)110 2251 y FA(V)-5 b(ector)20 b(quan)n(tization)f(\(V)n(Q\))i([4])g(is)h(an)f(imp)r(ortan) 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Fr(n)p FA(-dimensional)g Fs(c)m(o)m(deb)m(o)m(ok)23 b FA(\(or)17 b(reference\))e(v)n(ectors)i Ft(w)1544 2658 y Fp(1)1569 2650 y Fr(;)i(:)9 b(:)g(:)h(;)19 b Ft(w)1756 2658 y Fo(M)1805 2650 y FA(.)193 2716 y(The)i(co)r(deb)r(o)r(ok)d Fz(C)24 b FA(is)e(assumed)e(to)g(b)r(e)g(kno)n(wn)h(to)g(b)r(oth)e(sender)h (and)g(receiv)n(er.)32 b(T)-5 b(o)22 b(transmit)d(an)n(y)j(data)110 2783 y(v)n(ector)12 b Fq(\030)r FA(,)j(the)d(sender)g(determines)f(the) h(nearest)f(co)r(deb)r(o)r(ok)f(v)n(ector)i Ft(w)1588 2791 y Fo(j)1625 2783 y FA(according)f(to)h(some)h(distance)e(measure) 110 2849 y Fr(d)p FA(\()p Fq(\030)q Fr(;)e Ft(w)261 2857 y Fo(i)279 2849 y FA(\))21 b(and)g(transmits)f(the)h(index)g Fr(j)j FA(of)e(this)f(v)n(ector.)33 b(The)21 b(particular)e(distance)h (measure)g(w)n(e'll)j(assume)110 2916 y(throughout)15 b(this)j(pap)r(er)f(is)h(the)g(common)f(squared)g(Euclidean)g(distance) 802 3079 y Fr(d)p FA(\()p Ft(a)o Fr(;)9 b Ft(b)q FA(\))15 b(=)1065 3010 y Fo(n)1038 3026 y Fn(X)1043 3144 y Fo(i)p Fp(=1)1118 3079 y FA(\()p Fr(a)1169 3087 y Fo(i)1200 3079 y Fz(\000)d Fr(b)1279 3087 y Fo(i)1297 3079 y FA(\))1319 3056 y Fp(2)1454 3079 y FA(for)j Ft(a)p Fr(;)9 b Ft(b)16 b Fz(2)g FA(R)1736 3059 y Fo(n)1765 3079 y Fr(:)622 b FA(\(1\))193 3225 y(F)-5 b(or)19 b(notational)e(con)n(v)n(enience)g(w)n (e)i(further)e(de\014ne)g(an)i(index)f(function)g Fr(I)t FA(\()p Fq(\030)r FA(\))g(whic)n(h)h(deliv)n(ers)g(for)f(a)g(giv)n(en) 110 3292 y(data)c(v)n(ector)g Fq(\030)i FA(the)e(index)h(of)f(the)g (nearest)f(co)r(deb)r(o)r(ok)e(v)n(ector)j(\(in)h(case)e(there)h(is)h (one)f(single)g(nearest)f(co)r(deb)r(o)r(ok)110 3358 y(v)n(ector\))f(or)g(the)g(nearest)f(co)r(deb)r(o)r(ok)e(v)n(ector)j (with)h(minim)n(um)h(index)e(\(in)h(case)f(that)f(there)g(are)h(sev)n (eral)h(co)r(deb)r(o)r(ok)110 3425 y(v)n(ectors)k(with)h(minim)n(um)i (distance\):)618 3557 y Fr(I)t FA(\()p Fq(\030)r FA(\))15 b(=)h(min)9 b Fz(f)p Fr(i)15 b Fz(j)h FA(\()p Fr(d)p FA(\()p Ft(w)1106 3565 y Fo(i)1124 3557 y Fr(;)9 b Fq(\030)s FA(\))15 b Fz(\024)g Fr(d)p FA(\()p Ft(w)1370 3565 y Fo(j)1394 3557 y Fr(;)9 b Fq(\030)r FA(\)\))15 b Fz(8)g Fr(j)j Fz(2)d(f)p FA(1)p Fr(;)9 b(:)g(:)g(:)i(;)e(M)d Fz(gg)j Fr(:)438 b FA(\(2\))p 110 3606 945 3 v 171 3641 a Fm(\003)195 3657 y Fl(to)13 b(app)q(ear)h(in)g Fk(Neur)n(al)g(Pr)n(o) n(c)n(essing)g(L)n(etters)p Fl(,)g(V)l(ol.)20 b(5,)13 b(No.)20 b(1,)14 b(1997,)e(cop)o(yrigh)o(t)h(has)g(b)q(een)h (transferred)g(to)g(Klu)o(w)o(er)f(Academic)110 3709 y(Publishers)173 3748 y Fm(y)195 3764 y Fl(email)i(fritzk)o (e@neuroinformatik.ruhr-uni-b)q(o)q(c)o(h)o(um.de,)i(T)l(el.)k(+49-234) 13 b(7007845,)f(F)-5 b(AX.)16 b(+49-234)d(7094210)1278 4000 y FA(1)p eop %%Page: 2 3 2 2 bop 31 3 a FA(2)1067 b Fj(IR-INI)18 b(97{01,)1493 1 y FA(c)1477 3 y Fz(\015)h Fj(1997)36 b(Klu)n(w)n(er)18 b(Academic)f(Publishers)p 382 1023 4 882 v 386 146 1658 4 v 410 235 a Fz(C)h FA(:=)e(\(randomly)h(initialized)i(from)e(data)h (set)f Fz(D)r FA(\))p 386 300 100 4 v 485 737 4 441 v 489 300 1554 4 v 513 386 a(determine)g(the)g(set)g Fz(R)1000 394 y Fo(i)1037 386 y FA(for)h(all)h Ft(w)1245 394 y Fo(i)1279 386 y Fz(2)c(C)p 489 447 V 845 573 a Ft(w)891 581 y Fo(i)925 573 y FA(:=)1040 535 y(1)p 1005 560 97 3 v 1005 611 a Fz(jR)1067 619 y Fo(i)1086 611 y Fz(j)1131 520 y Fn(X)1117 639 y Fo(\030)q Fi(2R)1206 645 y Fh(i)1233 573 y Fq(\030)58 b FA(\(for)17 b(all)i Ft(w)1548 581 y Fo(i)1581 573 y Fz(2)d(C)s FA(\))p 489 737 1554 4 v 410 823 a(UNTIL)h(no)h Ft(w)729 831 y Fo(i)765 823 y FA(c)n(hanges)f(an)n(ymore)p 386 886 1658 4 v 410 971 a(RETURN)f Fz(C)p 386 1023 V 2039 1023 4 882 v 868 1200 a FA(Figure)h(1:)25 b(The)18 b(LBG)g(metho)r(d.)115 1362 y(The)h(decision)h(to)g(tak)n(e)g(the)g(smallest)g(index)g(in)h(case)e (of)h(a)h(set)e(of)h(nearest)f(v)n(ectors)g(is)i(arbitrary)-5 b(.)30 b(Other)31 1428 y(tie-breaking)18 b(rules,)g(e.g.)24 b(taking)18 b(the)f(largest)g(index,)i(w)n(ould)f(do)g(as)g(w)n(ell.) 115 1495 y(With)26 b(the)f(help)h(of)g(the)f(just-de\014ned)g(index)h (function)f(w)n(e)h(can)g(express)f(the)g(op)r(eration)f(of)i(a)g(v)n (ector)31 1561 y(quan)n(tizer)18 b(as)1046 1694 y Fr(Q)p FA(\()p Fq(\030)q FA(\))d(=)g Ft(w)1282 1704 y Fo(I)s Fp(\()p Fo(\030)q Fp(\))1364 1694 y Fr(:)944 b FA(\(3\))31 1783 y(Only)16 b(the)e Fs(index)i Fr(I)t FA(\()p Fq(\030)r FA(\))f(of)g(the)f Fs(winning)h FA(co)r(deb)r(o)r(ok)d(v)n(ector)i(has) h(to)f(b)r(e)g(transmitted)f(to)h(the)g(receiv)n(er.)23 b(Dep)r(end-)31 1849 y(ing)e(on)e(the)h(size)f(of)h(the)f(co)r(deb)r(o) r(ok)e(and)j(the)f(length)g(of)h(the)f(data)g(v)n(ectors)g(this)h(ma)n (y)h(result)e(in)h(a)g(dramatic)31 1915 y(reduction)d(of)h(the)f(amoun) n(t)g(of)h(data)g(to)f(b)r(e)g(transmitted.)115 1982 y(The)23 b(lo)n(w)h(data)f(rate,)h(ho)n(w)n(ev)n(er,)i(also)e(has)f (its)h(price.)40 b(The)24 b(receiv)n(er)e(do)r(es)h(not)g(get)g(the)f (original)i(data)31 2048 y(v)n(ector)c Fq(\030)r FA(,)i(but)e(the)g(co) r(deb)r(o)r(ok)e(v)n(ector)h Ft(w)932 2058 y Fo(I)s Fp(\()p Fo(\030)q Fp(\))1035 2048 y FA(whic)n(h)i(w)n(as)g(nearest)d(to)i Fq(\030)r FA(,)i(instead.)31 b(Summed)21 b(o)n(v)n(er)g(all)g(data)31 2115 y(v)n(ectors)d(there)e(is)j(a)f Fs(distortion)j(err)m(or)898 2218 y Fr(E)s FA(\()p Fz(D)r Fr(;)9 b Fz(C)s FA(\))15 b(=)1164 2165 y Fn(X)1160 2284 y Fo(\030)q Fi(2D)1256 2218 y FA(d\()p Fq(\030)q Fr(;)9 b Ft(w)1409 2228 y Fo(I)s Fp(\()p Fo(\030)q Fp(\))1490 2218 y FA(\))p Fr(:)796 b FA(\(4\))31 2378 y(A)19 b(crucial)f(step)g(in)h(applications)f(of)g (V)n(Q)h(is)h(to)e(\014nd)g(a)h(co)r(deb)r(o)r(ok)d Fz(C)22 b FA(whic)n(h)d(minimizes)g(the)f(distortion)f(error)31 2445 y(for)h(a)g(giv)n(en)h(data)e(set)g Fz(D)r FA(.)25 b(A)17 b(new)h(metho)r(d)f(to)g(\014nd)h(go)r(o)r(d)e(co)r(deb)r(o)r (oks)f(is)k(the)e(main)h(p)r(oin)n(t)g(of)g(this)g(pap)r(er.)31 2624 y FD(2)90 b(The)30 b(LBG)g(Metho)r(d)31 2746 y FA(A)17 b(large)f(n)n(um)n(b)r(er)h(of)f(co)r(deb)r(o)r(ok)e(construction)g (metho)r(ds)h(exist)i(suc)n(h)f(as)h(v)m(arious)f(\015a)n(v)n(ors)h(of) g Fr(k)r FA(-means)f([7])i(and)31 2812 y(di\013eren)n(t)j(neural)e(net) n(w)n(ork)i(mo)r(dels)f(\(e.g.,)h(the)f(self-organizing)f(feature)g (map)h([5],)j(the)d(neural)g(gas)g(metho)r(d)31 2878 y([8)q(])h(or)f(the)g(gro)n(wing)g(neural)g(gas)g(metho)r(d)f([2]\).)32 b(The)20 b(most)g(w)n(ell-kno)n(wn)i(approac)n(h,)e(ho)n(w)n(ev)n(er,)i (is)f(probably)31 2945 y(the)16 b(LBG)h(algorithm)f([6])h(\(named)e (after)g(its)i(authors)d(Linde,)j(Buzo)f(and)g(Gra)n(y\).)24 b(Since)16 b(LBG)h(is)g(the)f(starting)31 3011 y(p)r(oin)n(t)i(of)g (our)g(new)f(metho)r(d)g(w)n(e)h(describ)r(e)e(it)j(in)f(the)g(follo)n (wing.)115 3078 y(F)-5 b(or)19 b(notational)f(con)n(v)n(enience)h(w)n (e)h(denote)e(the)h(set)g(of)g(data)g(v)n(ectors)f(for)i(whic)n(h)f(a)h (particular)e(co)r(deb)r(o)r(ok)31 3144 y(v)n(ector)g Ft(w)244 3152 y Fo(i)281 3144 y FA(is)g(nearest)e(with)923 3210 y Fz(R)970 3218 y Fo(i)1003 3210 y FA(=)g Fz(f)p Fq(\030)h Fz(2)e(D)10 b(j)p Fr(I)t FA(\()p Fq(\030)s FA(\))15 b(=)g Fr(i)p Fz(g)9 b Fr(:)821 b FA(\(5\))115 3299 y(With)17 b(this)h(notation)e(the)h(LBG)h(algorithm)f(can)h(b)r(e) f(expressed)f(b)n(y)i(the)f(Nassi-Sc)n(hneidermann)f(diagram)31 3365 y(sho)n(wn)i(in)h(\014gure)e(1.)115 3432 y(A)e(single)g (computation)e(of)i(the)g(sets)f Fz(R)948 3440 y Fo(i)982 3432 y FA(together)f(with)i(the)g(follo)n(wing)h(up)r(date)d(of)i(the)g (co)r(deb)r(o)r(ok)d(v)n(ectors)31 3498 y Ft(w)77 3506 y Fo(i)110 3498 y FA(is)i(often)f(denoted)f(as)i Fs(Lloyd)i(iter)m (ation)p FA(.)26 b(Th)n(us,)15 b(the)e(LBG)h(algorithm)f(consists)g(of) h(sev)n(eral)f(Llo)n(yd)i(iterations)31 3565 y(go)n(v)n(erned)j(b)n(y)h (a)f(stopping)f(criterion.)115 3631 y(LBG)h(is)h(guaran)n(teed)d(to)i (decrease)e(the)i(distortion)f(error)g Fr(E)s FA(\()p Fz(D)r Fr(;)9 b Fz(C)s FA(\))18 b(in)h(eac)n(h)f(Llo)n(yd)g(iteration)f (or)h(to)g(lea)n(v)n(e)31 3697 y(it)g(unc)n(hanged,)d(in)j(whic)n(h)f (case)f(it)h(has)g(con)n(v)n(erged.)23 b(The)16 b(\014nal)h(co)r(deb)r (o)r(ok)d(corresp)r(onds)g(to)i(a)h(lo)r(cal)g(minim)n(um)31 3764 y(of)i(the)e(distortion)f(error)h(function)g([6].)p eop %%Page: 3 4 3 3 bop 110 3 a Fj(IR-INI)18 b(97{01,)477 1 y FA(c)462 3 y Fz(\015)g Fj(1997)f(Klu)n(w)n(er)i(Academic)e(Publishers)1084 b FA(3)110 197 y FD(3)90 b(The)30 b(Utilit)n(y)f(Measure)110 318 y FA(F)-5 b(or)21 b(the)e(follo)n(wing)i(considerations)c(w)n(e)k (assume)e(that)g(a)h(co)r(deb)r(o)r(ok)d Fz(C)24 b FA(has)c(b)r(een)f (constructed)e(b)n(y)k(LBG.)g(In)110 385 y(general)16 b(this)g(co)r(deb)r(o)r(ok)d(will)18 b(not)e(b)r(e)g(globally)h (optimal.)24 b(Ho)n(w)17 b(could)f(one)g(impro)n(v)n(e)h(this)f(co)r (deb)r(o)r(ok)e(without)110 451 y(creating)e(a)i(completely)g(new)f (one?)23 b(One)13 b(p)r(ossible)g(approac)n(h)g(is)h(to)g(iden)n(tify)g (co)r(deb)r(o)r(ok)d(v)n(ectors)i(whic)n(h)h(do)f(not)110 518 y(con)n(tribute)18 b(m)n(uc)n(h)j(to)f(the)f(reduction)g(of)h (distortion)f(error)f(and)i(mo)n(v)n(e)h(them)f(somewhere)e(else.)31 b(Ho)n(w)20 b(could)110 584 y(w)n(e)c(\014nd)e(out)g(ho)n(w)h(m)n(uc)n (h)h(a)f(particular)e(co)r(deb)r(o)r(ok)f(v)n(ector)i Ft(w)1395 592 y Fo(i)1429 584 y FA(actually)g(con)n(tributes)f(to)i (error)e(reduction?)22 b(W)-5 b(e)110 650 y(migh)n(t)18 b(compare)e(the)h(o)n(v)n(erall)i(distortion)d(error)g(for)i(the)f (curren)n(t)f(co)r(deb)r(o)r(ok)e Fz(C)22 b FA(with)17 b(the)g(error)g(whic)n(h)g(w)n(ould)110 717 y(o)r(ccur)g(if)h Ft(w)356 725 y Fo(i)393 717 y FA(w)n(as)g(remo)n(v)n(ed)h(from)e(the)h (co)r(deb)r(o)r(ok.)193 783 y(The)f(di\013erence)e(of)i(these)f(terms)g (indicates)f(ho)n(w)j(imp)r(ortan)n(t)d Ft(w)1558 791 y Fo(i)1594 783 y FA(is)i(for)f(error)g(reduction.)22 b(W)-5 b(e,)18 b(therefore,)110 850 y(de\014ne)f(the)g Fs(utility)24 b Fz(U)g FA(of)18 b(a)g(co)r(deb)r(o)r(ok)d(v)n(ector)i (as)h(follo)n(ws:)904 983 y Fz(U)5 b FA(\()p Ft(w)1011 991 y Fo(i)1030 983 y FA(\))15 b(=)g Fr(E)s FA(\()p Fz(D)r Fr(;)9 b(C)e Fz(n)s Ft(w)1383 991 y Fo(i)1401 983 y FA(\))12 b Fz(\000)h Fr(E)s FA(\()p Fz(D)r Fr(;)c Fz(C)s FA(\))722 b(\(6\))193 1080 y(It)24 b(is)h(ob)n(vious)f(that)f(the)g(remo)n(v)m (al)h(of)g Ft(w)1081 1088 y Fo(i)1124 1080 y FA(from)g Fz(C)k FA(only)c(a\013ects)f(the)g(distortion)g(error)f(for)i(those)f (data)110 1146 y(v)n(ectors)d(for)h(whic)n(h)h Ft(w)597 1154 y Fo(i)637 1146 y FA(w)n(as)f(the)g(nearest)e(co)r(deb)r(o)r(ok)f (v)n(ector,)k(i.e.,)h(the)e(v)n(ectors)f(in)i Fz(R)1987 1154 y Fo(i)2005 1146 y FA(.)35 b(Eac)n(h)20 b(data)h(v)n(ector)110 1212 y Fq(\030)i Fz(2)f(R)267 1220 y Fo(i)307 1212 y FA(is)g(mapp)r(ed)e(to)i(the)f(resp)r(ectiv)n(e)e(second-nearest)g(co)r (deb)r(o)r(ok)f(v)n(ector.)35 b(If)22 b(w)n(e)g(denote)e(the)g(set)h (of)h(all)110 1279 y(co)r(deb)r(o)r(ok)15 b(indices)j(with)1090 1345 y Fz(J)25 b FA(=)16 b Fz(f)p FA(1)p Fr(;)9 b(:)g(:)g(:)h(;)f(M)d Fz(g)p Fr(;)910 b FA(\(7\))110 1442 y(the)18 b(index)g(of)g(the)f(co)r (deb)r(o)r(ok)e(v)n(ector)i(second)g(nearest)f(to)i Fq(\030)i FA(is)f(giv)n(en)f(b)n(y)453 1575 y Fr(I)481 1552 y Fi(0)497 1575 y FA(\()p Fq(\030)q FA(\))d(=)g(min)10 b Fz(f)p Fr(i)15 b Fz(j)h Fr(i)f Fz(2)g(J)22 b(n)12 b Fr(I)t FA(\()p Fq(\030)r FA(\))28 b Fz(^)f FA(\()p Fr(d)p FA(\()p Ft(w)1336 1583 y Fo(i)1354 1575 y Fr(;)9 b Fq(\030)s FA(\))15 b Fz(\024)g Fr(d)p FA(\()p Ft(w)1600 1583 y Fo(j)1624 1575 y Fr(;)9 b Fq(\030)r FA(\)\))15 b Fz(8)g Fr(j)j Fz(2)d(J)22 b(n)12 b Fr(I)t FA(\()p Fq(\030)r FA(\))p Fz(g)d Fr(:)272 b FA(\(8\))193 1672 y(All)19 b(data)f(v)n(ectors)g(not)g(in)h Fz(R)811 1680 y Fo(i)848 1672 y FA(are)f(mapp)r(ed)f(to)h(the)g(same)g (co)r(deb)r(o)r(ok)e(v)n(ectors)h(as)i(b)r(efore)d(the)i(remo)n(v)m(al) h(of)110 1739 y Ft(w)156 1747 y Fo(i)175 1739 y FA(.)25 b(W)-5 b(e)19 b(can,)f(therefore,)e(rewrite)h(\(6\))h(as)g(the)g(follo) n(wing)h(sum)f(o)n(v)n(er)h(the)e(additional)h(error)f(v)m(alues)h(o)r (ccurring)110 1805 y(for)g(v)n(ectors)f(in)i Fz(R)495 1813 y Fo(i)513 1805 y FA(:)848 1938 y Fz(U)5 b FA(\()p Ft(w)956 1946 y Fo(i)974 1938 y FA(\))15 b(=)1083 1885 y Fn(X)1069 2004 y Fo(\030)q Fi(2R)1158 2010 y Fh(i)1185 1938 y FA(d\()p Fq(\030)q Fr(;)k Ft(w)1347 1948 y Fo(I)1370 1937 y Fg(0)1385 1948 y Fp(\()p Fo(\030)q Fp(\))1444 1938 y FA(\))12 b Fz(\000)g FA(d\()p Fq(\030)q Fr(;)19 b Ft(w)1695 1946 y Fo(i)1713 1938 y FA(\))667 b(\(9\))193 2095 y(A)17 b(v)n(ector)e(with)i(zero)e(utilit)n(y)j(could)e(b)r(e)g (remo)n(v)n(ed)g(without)f(increasing)h(the)f(o)n(v)n(erall)j(error)d Fr(E)s FA(\()p Fz(D)r Fr(;)9 b Fz(C)s FA(\))16 b(and)g(is)110 2162 y({)i(in)g(accordance)d(with)j(the)f(c)n(hosen)g(terminology)g({)g (useless.)24 b(Bey)n(ond)18 b(that,)f(ho)n(w)n(ev)n(er,)h(the)f(utilit) n(y)i(measure)110 2228 y(mak)n(es)g(it)f(p)r(ossible)f(to)h (distinguish)f(more)h(and)f(less)h(useful)g(co)r(deb)r(o)r(ok)d(v)n (ectors.)110 2410 y FD(4)90 b(The)30 b(LBG-U)f(Metho)r(d)110 2532 y FA(Ho)n(w)16 b(can)f(w)n(e)i(use)e(the)g(utilit)n(y)i(measure)d Fz(U)22 b FA(to)15 b(generate)e(b)r(etter)h(co)r(deb)r(o)r(oks)e(than)j (standard)f(LBG?)j(A)e(simple)110 2598 y(idea)g(is)g(to)g(p)r(erform)e (LBG)i(un)n(til)h(con)n(v)n(ergence,)e(store)f(the)h(resulting)g(co)r (deb)r(o)r(ok)e Fz(C)19 b FA(and)14 b(then)g(mo)n(v)n(e)i(the)e(v)n (ector)110 2664 y Ft(w)156 2672 y Fo(a)205 2664 y FA(with)22 b(minim)n(um)i(utilit)n(y)e(to)g(a)g(di\013eren)n(t)f(lo)r(cation.)35 b(W)-5 b(e)22 b(run)g(LBG)g(on)g(this)g(mo)r(di\014ed)e(co)r(deb)r(o)r (ok)f(un)n(til)110 2731 y(con)n(v)n(ergence)d(and)h(c)n(hec)n(k)h (whether)e(the)h(resulting)g(co)r(deb)r(o)r(ok)d Fz(C)1484 2711 y Fi(0)1518 2731 y FA(has)k(a)g(lo)n(w)n(er)g(distortion)e(error)h (than)f Fz(C)s FA(.)26 b(W)-5 b(e)110 2797 y(rep)r(eat)16 b(this)i(as)g(long)g(as)g(the)f(qualit)n(y)i(of)f(the)g(co)r(deb)r(o)r (oks)c(impro)n(v)n(es.)193 2864 y(Where,)k(ho)n(w)n(ev)n(er,)h(should)f (w)n(e)g(mo)n(v)n(e)i Ft(w)1061 2872 y Fo(a)1087 2864 y FA(?)26 b(A)18 b(rather)f(ob)n(vious)h(c)n(hoice)g(is)h(to)e(mo)n(v)n (e)j(it)e(to)g(the)g(p)r(osition)f(of)110 2930 y(that)f(co)r(deb)r(o)r (ok)e(v)n(ector)i Ft(w)685 2938 y Fo(b)707 2930 y FA(,)i(whic)n(h)f (curren)n(tly)f(\\generates")e(most)j(of)g(the)f(distortion)f(error.)23 b(In)17 b(the)g(vicinit)n(y)110 2997 y(of)j Ft(w)221 3005 y Fo(b)263 2997 y FA(the)f(v)n(ector)g Ft(w)574 3005 y Fo(a)620 2997 y FA(migh)n(t)h(b)r(e)f(more)g(useful)h(\(in)f (the)g(de\014ned)f(sense\))h(than)f(at)h(its)h(curren)n(t)e(p)r (osition.)28 b(T)-5 b(o)110 3063 y(express)17 b(this)g(more)g(formally) h(let)f(us)g(de\014ne)f(the)h(distortion)f(error)g(asso)r(ciated)f (with)j(a)f(particular)f(co)r(deb)r(o)r(ok)110 3129 y(v)n(ector)h Ft(w)322 3137 y Fo(i)359 3129 y FA(with)1017 3262 y Fr(E)s FA(\()p Ft(w)1128 3270 y Fo(i)1147 3262 y FA(\))e(=)1255 3210 y Fn(X)1242 3329 y Fo(\030)q Fi(2R)1331 3335 y Fh(i)1358 3262 y FA(d\()p Fq(\030)q Fr(;)9 b Ft(w)1511 3270 y Fo(i)1529 3262 y FA(\))p Fr(:)808 b FA(\(10\))193 3416 y(After)15 b(LBG)i(has)f(con)n(v)n(erged)g(for)g(the)f(curren)n(t)g(co)r(deb)r(o)r (ok)f(w)n(e)i(determine)f(the)h(v)n(ector)f Ft(w)2075 3424 y Fo(a)2119 3416 y FA(with)h(minim)n(um)110 3483 y(utilit)n(y)1001 3549 y Ft(w)1047 3557 y Fo(a)1089 3549 y FA(=)f(arg)j(min)1343 3563 y Ff(w)p Fi(2C)1453 3549 y Fz(U)5 b FA(\()p Ft(w)o FA(\))792 b(\(11\))110 3646 y(and)18 b(the)f(v)n(ector)g Ft(w)525 3654 y Fo(b)566 3646 y FA(with)h(maxim)n(um)h(error)988 3764 y Ft(w)1034 3772 y Fo(b)1072 3764 y FA(=)c(arg)j(max)1336 3777 y Ff(w)p Fi(2C)1446 3764 y Fr(E)s FA(\()p Ft(w)p FA(\))p Fr(:)779 b FA(\(12\))p eop %%Page: 4 5 4 4 bop 31 3 a FA(4)1067 b Fj(IR-INI)18 b(97{01,)1493 1 y FA(c)1477 3 y Fz(\015)h Fj(1997)36 b(Klu)n(w)n(er)18 b(Academic)f(Publishers)115 197 y FA(A)n(t)e(\014rst)g(sigh)n(t)h(one)f (migh)n(t)h(consider)e(to)h(simply)i(mo)n(v)n(e)g Ft(w)1331 205 y Fo(a)1373 197 y FA(directly)f(to)f(the)g(p)r(osition)f(of)i Ft(w)2046 205 y Fo(b)2068 197 y FA(.)24 b(In)16 b(this)g(case,)31 264 y(ho)n(w)n(ev)n(er,)21 b(b)r(oth)d(co)r(deb)r(o)r(ok)f(v)n(ectors)h (had)h(the)g(same)g(distance)f(to)h(all)h(co)r(deb)r(o)r(ok)d(v)n (ectors)h(in)i Fz(R)2090 272 y Fo(b)2132 264 y FA(\(also)f(to)g(all)31 330 y(others\))14 b(whic)n(h)i(w)n(ould)g(ha)n(v)n(e)g(the)f(follo)n (wing)h(undesired)e(consequences)f(for)i(an)n(y)i(Llo)n(yd)f(iteration) e(\(see)g(section)31 396 y(2\))j(p)r(erformed)d(on)i(the)g(mo)r (di\014ed)f(co)r(deb)r(o)r(ok:)21 b(All)c(data)e(v)n(ectors)h(in)h (former)e Fr(R)1701 404 y Fo(b)1741 396 y FA(w)n(ould)h(b)r(e)g(mapp)r (ed)f(to)h(one)g(of)31 463 y(the)j(t)n(w)n(o)h(v)n(ectors)e(ha)n(ving)i (iden)n(tical)f(p)r(ositions.)27 b(Due)19 b(to)g(our)g(de\014nition)f (of)i Fr(I)t FA(\()p Fr(\030)s FA(\))f(this)g(w)n(ould)h(b)r(e)e(the)h (v)n(ector)31 529 y(with)c(smallest)g(index.)23 b(Since)15 b(it)g(is)g(already)f(in)i(the)e(cen)n(ter)f(of)i(gra)n(vit)n(y)g(of)g (this)g(data)e(set)i(the)f(v)n(ector)g(w)n(ould)h(not)31 596 y(b)r(e)k(mo)n(v)n(ed)g(b)n(y)h(the)e(Llo)n(yd)i(iteration.)25 b(The)19 b(other)e(v)n(ector)h(w)n(ould)i(ha)n(v)n(e)f(an)g(empt)n(y)g (asso)r(ciated)d(data)i(set)h(and)31 662 y(w)n(ould)h(b)r(e)d(a)i (so-called)f(dead)g(unit)h(not)f(con)n(tributing)f(at)h(all)h(to)f (error)g(reduction.)24 b(W)-5 b(e)19 b(w)n(ould)g(ha)n(v)n(e)g(jump)r (ed)31 728 y(directly)f(to)g(another)e(p)r(o)r(or)g(lo)r(cal)i(minim)n (um)h(of)f(the)g(distortion)e(error)h(function.)115 795 y(F)-5 b(ortunately)g(,)18 b(it)h(is)h(rather)d(simple)j(to)e(ensure)g (that)g(the)g(t)n(w)n(o)i(v)n(ectors)e(divide)h(the)f(data)g(set)h Fr(R)2153 803 y Fo(b)2194 795 y FA(in)n(to)h(t)n(w)n(o)31 862 y(subsets.)29 b(W)-5 b(e)20 b(only)h(ha)n(v)n(e)f(to)f(add)h(a)g (little)g(o\013set)f(v)n(ector)g(to)h(one)f(of)h(them)f(whic)n(h)h(has) g(the)f(only)h(purp)r(ose)e(of)31 928 y(making)23 b(the)e(v)n(ectors)h (di\013eren)n(t)f(from)h(eac)n(h)g(other.)36 b(A)22 b(little)g(care,)h (ho)n(w)n(ev)n(er,)h(has)e(to)f(b)r(e)h(tak)n(en)g(to)f(ensure)31 994 y(that)e(the)g(o\013set)g(v)n(ector)f(is)j(not)e(so)g(large)g(that) g(it)g(mo)n(v)n(es)i(the)e(v)n(ector)f(it)i(is)g(applied)g(to)f(a)n(w)n (a)n(y)i(from)e(the)g(data)31 1061 y(v)n(ectors)f(in)h Fr(R)327 1069 y Fo(i)345 1061 y FA(.)27 b(W)-5 b(e)18 b(can)g(deriv)n(e)h(a)f(suitable)g(length)f(for)i(the)e(o\013set)h(v)n (ector)g(from)g(already)g(computed)e(v)m(alues)31 1127 y(as)j(follo)n(ws.)27 b(The)18 b(error)f Fr(E)s FA(\()p Ft(w)667 1135 y Fo(b)689 1127 y FA(\))h(can)g(b)r(e)g(used)g(to)g (compute)f(the)h(standard)e(deviation)1896 1080 y Fn(p)p 1952 1080 283 3 v 1952 1127 a Fr(E)s FA(\()p Ft(w)2063 1135 y Fo(b)2085 1127 y FA(\))p Fr(=)p Fz(jR)2197 1135 y Fo(b)2219 1127 y Fz(j)j FA(of)g(the)31 1194 y(v)n(ectors)f(in)g Fz(R)331 1202 y Fo(b)353 1194 y FA(.)25 b(The)18 b(length)f(of)h(the)g (o\013set)f(v)n(ector)g(should)h(b)r(e)f(m)n(uc)n(h)i(smaller)f(than)f (this.)24 b(T)-5 b(o)19 b(eliminate)f(the)31 1260 y(need)h(for)g(using) g(the)g(set)g(cardinalit)n(y)g Fz(jR)919 1268 y Fo(b)942 1260 y Fz(j)h FA(w)n(e)g(replace)e(the)h(standard)e(deviation)i(with) 1948 1213 y Fn(p)p 2004 1213 231 3 v 2004 1260 a Fr(E)s FA(\()p Ft(w)2115 1268 y Fo(b)2137 1260 y FA(\))p Fr(=)m(N)26 b FA(whic)n(h)31 1326 y(is)19 b(guaran)n(teed)d(to)h(b)r(e)h(not)f (larger)g(\()p Fr(N)k Fz(\025)16 b(jR)978 1334 y Fo(b)1000 1326 y Fz(j)p FA(\).)25 b(Then)18 b(w)n(e)g(c)n(ho)r(ose)f(a)h(small)h (constan)n(t)d Fr(";)9 b FA(0)15 b Fr(<)h(")f Fz(\034)h FA(1,)i(and)g(set)31 1393 y(the)g(length)f(of)h(the)f(o\013set)h(v)n (ector)f(to)1003 1460 y Fr(l)f FA(=)g Fr(")1121 1410 y Fn(p)p 1176 1410 V 50 x Fr(E)s FA(\()p Ft(w)1288 1468 y Fo(b)1310 1460 y FA(\))p Fr(=)m(N)5 b(:)874 b FA(\(13\))31 1560 y(T)-5 b(o)20 b(get)e(a)g(direction)g(for)g(the)g(o\013set)g(v)n (ector)g(w)n(e)h(c)n(ho)r(ose)f(a)h(random)e(v)n(ector)h Ft(u)i FA(from)e(the)g Fr(n)p FA(-dimensional)h(unit)31 1627 y(sphere:)656 1760 y Ft(u)d FA(=)f(\(random)i(from)g Fr(n)p FA(-dimensional)i(unit)f(sphere\))523 b(\(14\))31 1861 y(Finally)-5 b(,)21 b(w)n(e)d(p)r(erform)e(the)i(mo)n(v)n(e)886 1928 y Ft(w)932 1936 y Fo(a)974 1928 y FA(:=)e Ft(w)1094 1936 y Fo(b)1128 1928 y FA(+)c Fr(")1209 1878 y Fn(p)p 1265 1878 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b(the)f(last)h(section)e(w)n(e)i(ha)n(v)n(e)h (discussed)d(the)h(close)g(relation)g(of)h(LBG-U)g(to)g(some)f(neural)g (net)n(w)n(ork)h(mo)r(dels)110 861 y(and)g(w)n(e)g(ha)n(v)n(e)g(prop)r (osed)d(an)j(on-line)g(v)m(arian)n(t)g(of)f(the)g(utilit)n(y)i(measure) e(whic)n(h)h(is)g(curren)n(tly)f(b)r(eing)g(in)n(v)n(estigated)110 928 y(in)19 b(the)e(con)n(text)g(of)h(these)f(mo)r(dels.)110 1083 y Fa(Ac)n(kno)n(wledgmen)n(ts)110 1185 y FA(W)-5 b(e)22 b(lik)n(e)h(to)e(thank)g(Mic)n(hael)g(P\177)-28 b(otzsc)n(h)20 b(and)h(Jan)h(C.)g(V)-5 b(orbr)r(\177)-29 b(uggen)18 b(for)j(sev)n(eral)g(useful)g(commen)n(ts)g(and)g(sug-)110 1251 y(gestions,)16 b(and)g(Christoph)f(v)n(on)h(der)g(Malsburg)f(for)h (the)g(stim)n(ulating)g(researc)n(h)e(en)n(vironmen)n(t)i(he)g(pro)n (vides)g(us)110 1318 y(with.)110 1501 y FD(References)110 1622 y FA([1])28 b(B.)17 b(F)-5 b(ritzk)n(e.)21 b(Gro)n(wing)16 b(cell)g(structures)d({)i(a)h(self-organizing)e(net)n(w)n(ork)i(for)f (unsup)r(ervised)e(and)j(sup)r(ervised)196 1688 y(learning.)24 b Fs(Neur)m(al)d(Networks)p FA(,)f(7\(9\):1441{1460,)15 b(1994.)110 1799 y([2])28 b(B.)22 b(F)-5 b(ritzk)n(e.)32 b(A)21 b(gro)n(wing)f(neural)f(gas)i(net)n(w)n(ork)f(learns)g(top)r (ologies.)30 b(In)20 b(G.)i(T)-5 b(esauro,)20 b(D.)i(S.)f(T)-5 b(ouretzky)g(,)196 1865 y(and)15 b(T.)i(K.)f(Leen,)f(editors,)g Fs(A)m(dvanc)m(es)20 b(in)e(Neur)m(al)g(Information)i(Pr)m(o)m(c)m (essing)f(Systems)f(7)p FA(,)f(pages)d(625{632.)196 1932 y(MIT)k(Press,)g(Cam)n(bridge)f(MA,)h(1995.)110 2043 y([3])28 b(B.)c(F)-5 b(ritzk)n(e.)38 b(Gro)n(wing)23 b(grid)g(-)g(a)g(self-organizing)e(net)n(w)n(ork)h(with)h(constan)n(t)e (neigh)n(b)r(orho)r(o)r(d)e(range)i(and)196 2109 y(adaptation)16 b(strength.)22 b Fs(Neur)m(al)f(Pr)m(o)m(c)m(essing)h(L)m(etters)p FA(,)e(2\(5\):9{13,)c(1995.)110 2220 y([4])28 b(R.)19 b(M.)f(Gra)n(y)-5 b(.)26 b(V)-5 b(ector)17 b(quan)n(tization.)23 b Fs(IEEE)c(ASSP)g(Magazine)p FA(,)h(pages)d(4{29,)h(1984.)110 2330 y([5])28 b(T.)19 b(Kohonen.)24 b(Self-organized)17 b(formation)g(of)h(top)r(ologically)f(correct)f(feature)g(maps.)25 b Fs(Biolo)m(gic)m(al)e(Cyb)m(er-)196 2397 y(netics)p FA(,)d(43:59{69,)d(1982.)110 2507 y([6])28 b(Y.)18 b(Linde,)f(A.)g (Buzo,)g(and)g(R.)g(M.)h(Gra)n(y)-5 b(.)24 b(An)17 b(algorithm)f(for)h (v)n(ector)f(quan)n(tizer)g(design.)22 b Fs(IEEE)c(T)l(r)m(ansac-)196 2574 y(tions)j(on)f(Communic)m(ation)p FA(,)g(COM-28:84{95,)e(1980.)110 2685 y([7])28 b(J.)21 b(MacQueen.)27 b(Some)19 b(metho)r(ds)f(for)h (classi\014cation)f(and)h(analysis)h(of)f(m)n(ultiv)m(ariate)g(observ)m (ations.)27 b(v)n(ol-)196 2751 y(ume)21 b(1)g(of)f Fs(Pr)m(o)m(c)m(e)m (e)m(dings)26 b(of)c(the)h(Fifth)g(Berkeley)i(Symp)m(osium)e(on)f (Mathematic)m(al)k(statistics)d(and)g(pr)m(ob)m(a-)196 2817 y(bility)p FA(,)d(pages)d(281{297,)f(Berk)n(eley)-5 b(,)20 b(1967.)d(Univ)n(ersit)n(y)i(of)f(California)g(Press.)110 2928 y([8])28 b(T.)17 b(M.)f(Martinetz)d(and)i(K.)i(J.)f(Sc)n(h)n (ulten.)k(A)c(\\neural-gas")d(net)n(w)n(ork)j(learns)e(top)r(ologies.) 19 b(In)d(T.)g(Kohonen,)196 2995 y(K.)24 b(M\177)-28 b(akisara,)23 b(O.)h(Sim)n(ula,)h(and)d(J.)i(Kangas,)f(editors,)g Fs(A)o(rti\014cial)i(Neur)m(al)h(Networks)p FA(,)g(pages)21 b(397{402.)196 3061 y(North-Holland,)c(Amsterdam,)f(1991.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF