Prob 6 - ans 5

Prob 6 - ans 5 - A rb itra ry-N o rm S ep a ra t ingP lan e...

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Unformatted text preview: A rb itra ry-N o rm S ep a ra t ingP lan e O .L .M anga sa r ian y A b stract A p lan esepa ra t ing tw opo in tse ts in n-d im en s iona lrea lspace iscon stru c ted su ch tha t it m in im izesth esum o fa rb itra ry-no rm d istan ceso fm isc la ss iedpo in tsto th ep lan e .In con tra st top rev iou sapp roach estha tu sed su rroga testo th em isc la ss ied-po in td istan ce-m in im iza t ion p rob lem ,th ep resen tapp roach isba sedonano rm-d ep end en texp lic itc lo sed fo rm fo rth ep ro- jec t iono fapo in tonap lan e .T h isp ro jec t ion isu sed to fo rm u la teth esepa ra t ing-p lan ep rob lem a sam in im iza t iono faconv ex fun c t iononaun itsph ere inano rm dua lto tha to fth ea rb itra ry no rm u sed .F o rth e-no rm on ly ,th ep rob lem canb eso lv ed inpo lynom ia lt im eby so lv ing n lin ea rp rog ram so rby so lv ingab ilin ea rp rog ram .F o ra llo th er p-no rm s , p ( ; ],a re la ted d ec is ionp rob lem to th em in im iza t ionp rob lem isNP-com p le te .F o ragen era l p-no rm ,th em in- im iza t ionp rob lem canb etran s fo rm edv iaan exac tp ena lty fo rm u la t ion tom in im iz ing th esum o faconv ex fun c t ionandab ilin ea r fun c t iononaconv ex se t .F o rth eon eand inn ityno rm s ,a n itesu ccess iv e lin ea r iza t iona lgo r ithm isp ropo sed fo rso lv ing th eexac tp ena lty fo rm u la t ion . In trodu c t ion O n eo fth e fundam en ta lp rob lem so fm ach in e lea rn ing istha to fd iscr im ina t ingb e tw een tw on ite po in tse ts in n-d im en s iona lrea lspace R n .W h en th econv exhu llso fth etw ose tsdono t in tersec t , as ing le lin ea rp rog ram cancon stru c tastr ic tsepa ra t ingp lan esu ch tha teacho fth etw oop en ha lfspacesgen era tedby th ep lan econ ta in son eo fth etw ose ts [,,].Su chap lan eco rrespond s toap ercep tronand cana lsob eob ta in edby th e itera t iv ep ercep tron lea rn inga lgo r ithm [,] w h ich canb e in terp re teda sth eM o tzk in-S choenb erg itera t iv esch em e fo rso lv ingcon s isten t lin ea r in equa lit ies .W h en th econv exhu llso fth etw ose ts in tersec tth e itera t iv esch em e fa ilsb ecau se th eund er ly ing lin ea r in equa lit iesa re in con s isten t ,w h ileth e lin ea rp rog ramm ingapp roachm u st b ep rov id edw ithan erro rcr iter ion tob em in im ized .W ep ropo sea sacr iter ionh ereth esum o f a rb itra ry-no rm d istan cesto th esepa ra t ingp lan eo fpo in ts ly ingon th ew rongs id eo fth ep lan e . U n fo rtuna te ly ,ifp rec ised istan cesa reu sed ,lin ea r ityo fth eob jec t iv e fun c t ion is lo st ,a sw illb e show n in th isw o rk .W ha tm anyo fth ep rev iou sapp roach eshav etend ed tou se ,a red istan ce su rroga testha tm a in ta in ed lin ea r ityo fth ep rob lem [,,,]bu td idno tm ea su red istan ces o fv io la t ingpo in tstoth esepa ra t ingp lan e .In con tra st ,ou rapp roachh ered ep end sonac lo sed fo rm fo rm u la fo rth ep ro jec t iono fapo in tin R n on toag iv enp lan eu s ingana rb itra ryno rm , w h ich...
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Prob 6 - ans 5 - A rb itra ry-N o rm S ep a ra t ingP lan e...

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