ELK ASIA PACIFIC JOURNAL OF COMPUTER SCIENCE AND INFORMATION SYSTEMS. ISSN: ; ISSN: (Online) Volume 1 Issue 2 (2015)

Transcription

1 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: ; I: (Onne) Voume Iue (05) wwwekourncom PERFORMACE EVAUATIO OF HOPFIED AOCATIVE MEMORY FOR COMPREED IMAGE Mnu Prtp ngh Dr Pndey Vk Pndey Deprtment of Computer cence, Inttute of Engneerng & Technoogy, Dr B R Ambedkr Unverty, Khndr, Agr (U P) Deprtment of Mthemtc & Computer cence Rn Durgwt Unverty, Jbpur (M P) Ind Deprtment of Mthemtc & Computer cence Rn Durgwt Unverty, Jbpur (M P) Ind Abtrct Th pper degned to nyze the performnce of Hopfed neur network for torge nd rec of compreed mge In th pper we re conderng the mge of dfferent ze Thee mge re frt compreed by ung wveet trnformton The compreed mge re then preproceed nd the feture vector of thee mge re obtned The trnng et cont wth the pttern nformton of the preproceed nd compreed mge Here ech nput pttern of ze 900 X Ech pttern of trnng et encoded nto Hopfed neur network ung hebbn nd peudo nvere ernng rue Once the pttern of trnng et re encoded then we mute the performnce of trned neur network for the preented noy pttern of the redy encoded pttern Thee noy tet pttern re o compreed nd preproceed mge The performnce for octve memory phenomen of Hopfed neur network nyzed The ny condered n term of uccefu nd correct recng of the pttern n the form of orgn compreed mge It found from muted reut tht the performnce of Hopfed neur network for recng of the pttern nd then the recontructon of mge decree the noe or dtorton n the orgn mge bove 0 % It o found tht the Hopfed neur network fed to rec the correct mge f the preented prototype nput pttern of the orgn mge contnng the noe more thn 50 % Keyword: Hopfed eur etwork, Aoctve memory, Compreed Imge torge & recng, pttern torge network Introducton Pttern torge & recng e pttern octon one of promnent method for the pttern recognton tk tht one woud ke to reze ung n rtfc neur network (A) octve memory feture Pttern torge genery ccomphed by feedbck network contng of proceng unt wth nonner bpor output functon

2 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) The Hopfed neur network mpe feedbck neur network () whch be to tore pttern ocy n the form of connecton trength between the proceng unt Th network cn o work for the pttern competon on the preentton of prt nformton or prototype nput pttern The tbe tte of the network repreent the memorzed or tored pttern nce the Hopfed neur network wth octve memory [-] w ntroduced, vrou modfcton [-0] re deveoped for the purpoe of torng nd retrevng memory pttern fxed-pont ttrctor The dynmc of thee network hve been tuded extenvey becue of ther potent ppcton [-4] The dynmc determne the retrev quty of the octve memore correpondng to redy tored pttern The pttern nformton n n unuperved mnner encoded um of correton weght mtrce n the connecton trength between the proceedng unt of feedbck neur network ung the ocy vbe nformton of the pre nd pot ynptc unt whch condered fn or prent weght mtrx The neur network ppcton ddre probem n pttern cfcton, predcton, fnnc ny, nd contro nd optmzton [5] In mot current ppcton, neur network re bet ued d to humn decon mker nted of ubttute for them The eur etwork hve been degned to mode the proce of memory rec n the humn brn [6] Aocton n humn brn refer to the phenomenon of one thought cung u to thnk of nother Correpondngy, octve memory the functon where the brn be to tore nd rec nformton, gven prt knowedge of the nformton content [7] Aoctve Memory dynmc ytem whch h number of tbe tte wth domn of ttrcton round them If the ytem trt t ny tte n the domn, t w converge to the ocy tbe tte, whch ced n ttrctor [8] One uch mode, decrbng the orgnzton of neuron n uch wy tht they functon Aoctve Memory or o ced Content Addrebe Memory, w propoed by J J Hopfed nd w nmed fter hm Hopfed Mode It fuy connected neur network mode n whch pttern cn be tored by dtrbutng mong neuron nd we cn retreve one of the prevouy preented pttern from n 4

3 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) exmpe whch mr to, or noy veron of t [7, 8] The network octe ech eement of pttern wth bnry neuron The neuron re updted ynchronouy nd n pre They re ntzed wth n nput pttern nd the network ctvton converge to the coet ernt pttern [9] Th dynmc behvor of the neuron trongy depend on the ynptc trength between neuron The pecfcton of the ynptc trength conventony referred to ernng [0] ernng empoy number of ernng gorthm perceptron, hebbn, peudo nvere, M etc [] Whe choong ernng gorthm, there re number of conderton The foowng conderton re ued n th pper: ) The mxmum cpcty of the network b) The network bty to dd pttern ncrementy to the network c) The network bty to correcty rec pttern tored n the network d) The network bty to octe noy pttern to t orgn pttern e) The network bty to octe new pttern t neret neghbor Pont d nd e woud o vry wth the number of pttern currenty tored n the network The mpet rue tht cn be ued to trn network the hebbn rue but t uffer from number of probem uch : ) The mxmum cpcty mted to ut 04, where the number of neuron n the network [] b) The rec effcency of the network deterorte the number of pttern tored n the network ncree [] c) The network bty to correct noy pttern o extremey mted nd deterorte wth pckng denty of the network d) ew pttern coud hrdy be octed to the tored pttern The next rue to be condered to overcome the ddvntge of the hebbn rue w the peudo nvere ernng rue The tndrd peudo nvere rue known to be better thn the hebbn rue n term of the cpcty (), rec effcency nd pttern correcton [4] In th pper we re conderng the mge of dfferent ze Thee mge re frt compreed by ung wveet trnformton The compreed mge re then preproceed nd the feture vector of thee mge re obtned 5

4 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) Therefore for ech mge pttern vector of ze 900 X contructed The trnng et cont wth the pttern nformton of the preproceed nd compreed mge Here ech nput pttern of ze 900 X Ech pttern of trnng et encoded nto Hopfed neur network ung hebbn nd peudo nvere ernng rue Once the pttern of trnng et re encoded then we mute the performnce of trned neur network for the preented noy pttern of the redy encoded pttern Thee noy tet pttern re o compreed nd preproceed mge The performnce for octve memory phenomen of Hopfed neur network nyzed The ny condered n term of uccefu nd correct recng of the pttern n the form of orgn compreed mge It found from muted reut tht the performnce of Hopfed neur network for recng of the pttern nd then the recontructon of mge decree the noe or dtorton n the orgn mge bove 0 % It o found tht the Hopfed neur network fed to rec the correct mge f the preented prototype nput pttern of the orgn mge contnng the noe more thn 50 % Th pper orgnzed foow: ecton II provde bref decrpton of the Hopfed network octve memory nd t torge nd updte dynmc ecton III eborte the Peudo nvere Rue, the octed probem nd meure to overcome them ecton IV conder the pttern formton for compreed mge ecton V contn the experment whoe reut hve been comped nd dcued n ecton VI Concuon then foow n ecton VII Hopfed etwork Aoctve Memory The propoed Hopfed mode cont of (900 = 0 X 0) neuron nd (900 X 900) connecton trength Ech neuron cn be n one of the two tte e ±, nd (9) bpor pttern hve to be memorzed n the Hopfed neur network of octve memory Hence, to tore (9) number of pttern n th pttern torge network, the weght mtrx w uuy determned by the Hebbn rue foow: w T x x or, w () () 6

5 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) nd, or, w w where,, () ( ) nd w 0 (4) {,,, ;,, nd ; wth et of pttern to be memorzed nd the number of proceng unt} The network ntzed : 0 0 for to (5) The ctvton vue nd output of every unt n Hopfed mode cn repreent : y w t nd t gn y ;,,, ; (6) (7) where gn repectvey y for y 0 nd y 0 Aoctve memory nvove the retrev of memorzed pttern n repone to the preentton of ome prototype nput pttern the rbtrry nt tte of the network Thee nt tte hve certn degree of mrty wth the memorzed pttern nd w be ttrcted towrd them wth the evuton of the neur network Hence, n order to memorze 9 cnned mge of n 900-unt bpor Hopfed neur network, there houd be one tbe tte correpondng to ech tored pttern Thu t the end, the memory pttern houd be fxed-pont ttrctor of the network nd mut tfy the fxed-pont condton : y or, y w w (8) where y 0 (9) Therefore, the foowng ctvton dynmc equton mut tfy to ccomph the pttern torge: f ( w ) ; (0) where,,,, ; et the pttern et be P { x, x,, x } where [ x (,,, ), x x (, (,,,,, ) () wth,,, 900 nd,,,9] ow, the nt weght hve been condered w 0 (ner to zero) for ), 7

6 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) 8 ' nd ' From the ynptc dynmc vector we hve the foowng equton for encodng the pttern nformton : X X W W T od new () nd new od W W () mry for the th pttern, we hve: T X X W W (4) Thu, fter the ernng for the pttern, the fn prent weght mtrx cn be repreented : W (5) ow, to repreent W n the convenent repreentton form, et u ume foowng notton:, ,,, ,,,, (6) o tht, from equton (66) & (67), we get:

7 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) W 0 Th qure mtrx condered the prent weght mtrce becue t repreent the prt outon or ub-optm outon for the pttern recng correpondng to the preented prototype nput pttern vector Hopfed uggeted tht the mxmum mt for the torge 0 5 n network wth neuron, f m error n recng owed ter, th w theoretcy ccuted p 0 4 by ung repc method [5] Wermn [6] howed tht the mxmum number of memore m tht cn be tored n network of n neuron nd reced excty e tht cn where c potve contnt greter thn one It h been dentfed tht the torge cpcty trongy depend on ernng cheme Reercher hve propoed dfferent ernng cheme, nted of the Hebbn rue to ncree the torge cpcty of the Hopfed neur network [7, 8] Grdner howed tht the utmte cpcty w be p functon of the ze of the bn of ttrcton [9] 0 0 (7) Pttern rec nvove ettng the nt tte of the network equ to n nput vector ξ The tte of the ndvdu unt re then updted repetedy unt the over tte of the network tbe Updtng of unt my be ynchronou or ynchronou [0] In the ynchronou updte the unt of the network re updted mutneouy nd the tte of the network frozen unt updte mde for the unt Whe n the ynchronou updte, unt eected t rndom nd t tte updted ung the current tte of the network Th updte v rndom choce of unt contnued unt no further chnge n the tte tke pce for the unt e the network reche tbe tte Ech tbe tte of the network correpond to tored pttern tht h mnmum hmmng dtnce from the nput pttern [] Ech tbe tte of the network octed n energy E nd hence tht tte ct pont ttrctor And durng updte the network move from n nt hgh energy tte to the neret ttrctor A tbe tte whch re mr to ny of ξ of 9

8 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) trnng et re ced Fundment Memore Aprt from them there re other tbe tte, ncudng nvere of fundment memore The number of uch fundment memore nd the nture of ddton tbe tte depend upon the ernng gorthm tht empoyed Peudo nvere Rue In Hopfed, we cn ue the peudo nvere ernng rue to encode the pttern nformton f pttern vector re even non orthogon It provde the more effcent method for ernng n the feedbck neur network mode The peudo nvere weght mtrx gven by W = Ξ Ξ - (8) where Ξ the mtrx whoe row re ξn nd Ξ - t peudo nvere The mtrx wth the property tht Ξ - Ξ = I [] The peudo nvere rue nether oc nor ncrement compred to the hebbn rue Th men tht the updte of prtcur connecton doe not depend on the nformton vbe on ether de of the connecton nd o pttern cnnot be ncrementy dded to the network Thee probem cn be oved by modfyng the rue n uch wy tht ome chrctertc of hebbn ernng re o ncorported uch tht octy nd ncrementy enured The hebbn rue gven : W = / ξ * ξ for (9) = = 0, for =,, where, the number of unt/neuron n the network ξ for = to re the vector / mge to be tored, where ech component of ξ bnry e ech ξ = ± for = to ow the peudo nvere of the weght mtrx cn be ccuted Wpnv = W t * (W * W t ) - (0) Where, W t the trnpoe of the weght mtrx W nd (W * W t ) - the nvere of the product of W nd t trnpoe Th method w overcome the octy nd ncrementy probem octed wth the peudo nvere rue In ddton t h the beneft of the peudo nvere rue n term of the torge cpcty nd rec effcency over the hebbn rue Pttern rec refer to the dentfcton nd retrev of the correpondng mge when n mge preented nput to the 0

9 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) network A oon n mge fed nput to the network, the network trt updtng tef In the current pper, we ue ynchronou updte of the network unt to fnd ther new tte Th updte v rndom choce of unt contnued unt no further chnge n the tte tke pce for the unt Tht, the tte t tme (t+) the me the tte t tme t for the unt (t+) = (t) for () uch tte referred to the tbe tte In tbe tte the output of the network w be tbe (trned) pttern tht h mnmum hmmng dtnce from the nput pttern [] The network d to hve converged nd reced the pttern f the output mtche the pttern preented nty nput For pttern octon, the pttern tored n n octve memory ct ttrctor nd the rget hmmng dtnce wthn whch mot tte fow to the pttern defned the rdu of the bn of ttrcton [] Ech tte of the Hopfed etwork octed wth n energy vue, whoe vue ether reduce or remn the me the tte of the network chnge [] The energy functon of the network gven by V W () Hopfed h hown tht for ymmetrc weght wth no ef feedbck connecton nd bpor output functon, the dynmc of the network ung ynchronou updte wy ed towrd energy mnm t equbrum The network tte correpondng to thee energy mnm re termed tbe tte [4] nd the network ue ech of thee tbe tte for torng ndvdu pttern 4 Pttern formton ung Imge Preproceng technque The pttern formton n eent tep for performng the octve feture n Hopfed neur network mode Hence to contruct the pttern nformton to encode the pttern, the preproceng tep re requred Preproceng, n the form of mge enhncement, of the mge requred to convert the mge nto utbe pttern for torge n the Hopfed etwork The term mge enhncement refer to mkng the mge cerer for ey further operton The mge condered for the tudy re the mge of the mpreon of dfferent ndvdu The

10 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) mge re not of perfect quty to be condered for torge n network Hence enhncement method re requred to reve the fne det of the mge whch my remn uncovered due to nuffcent nk or mperfect mpreon The enhncement method woud ncree the contrt between mge component nd connect the broken or ncompete mge component The mge were frt cnned gry mge nd then trnform n wveet to retn the fne det n the mge The mge w then ubected to bnrzton Bnrzton refer to converon of gryce mge to bck nd whte mge Typcy bnrzton convert n mge of up to 56 gry eve to bck nd whte mge hown n fgure () nd (b) Fg : () Orgn Greyce Imge Fg : (b) Bnry Imge 0 After ttnng bnrzton, the need w to convert the bnry mge to bpor mge, nce Hopfed network work bet wth bpor dt A bpor mge one where ech pxe h vue ether + or - Hence, pxe vue re verfed nd thoe wth vue 0 re converted to -, thu convertng the bnry mge to bpor mge Fny the mge converted to bpor vector

11 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) A the mge vector re tored nto comprehenve mtrx whch h the foowng tructure P [ ] 9900 () The equton repreentng the trnng et Th trnng et ued to encode the pttern nformton of the nne preproceed mge n to Hopfed neur network nd modfed by terng the vue of k rndomy choen pxe Ao, ume vector ynew to tore the new tte of the network Conder vrbe count ntzed to vue 5 Impementton det nd experment degn The pttern n the form of bpor vector creted n ecton 4 were then tored n the Hopfed network v the foowng gorthm eprtey for hebbn nd peudo nvere rue n eprte weght mtrce Agorthm: Pttern torge nd Recng The gorthm for pttern rec n Hopfed eur etwork torng pttern foow: The gorthm woud be mpemented both for Hebbn nd Peudo nvere rue nd reut woud be recorded Aume pttern x, of ze, redy tored n the network Intze weght to tore pttern (Ue Hebbn nd Peudo nvere Rue) per the equton 4 nd 0 repectvey for Pttern torge Whe ctvton of the net re converged perform tep to 8 For ech nput vector x, repet tep to 7 et nt ctvton of the net equ to the extern nput vector x, y = x (= to n) 4 Perform tep 5 to 7 for ech unt y 5 Compute the net nput Y x Y, * W

12 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) 6 Determne the ctvton (output gn): Y, fy 0 fy 0 Brodct the vue of Y to other unt 7 Tet for convergence per equton 4 The foowng prmeter re ued to encode the mpe pttern of trnng et Tbe : Prmeter ued for Hebbn nd Peudo nvere ernng rue Prmeter Int tte of neuron Threhod vue of neuron Vue Rndomy Generted Vue Ether nd 000 The vue of threhod θ umed to be zero Ech unt rndomy choen for updte The mxmum number of pttern uccefuy reced by the bove procedure ponter to the mxmum torge cpcty of the Hopfed etwork, whch further dcued n the reut Further the rec effcency for noy pttern o determned up to wht percentge of error n the pttern cceptbe by the network nd convergence to the orgn pttern occur 6 Reut nd Dcuon The Hopfed network h the bty to recognze uncer pcture correcty Th men tht the network cn rec ctu pttern when the noy or prt cue of tht pttern re preented to the network It known nd h been hown [8] tht Hopfed network converge to the orgn pttern f up to 40% dtorted veron of tored pttern preented The pttern re tored n the network n the form of ttrctor on the energy urfce The network cn then be preented wth ether porton of one of the mge (prt cue) or n mge degrded wth noe (noy cue) nd through mutpe terton t w ttempt to recontruct one of the tored mge Fg : (c) Rec mge 4

13 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) A the pttern fter preproceng depcted n ecton 4 were converted nto bpor vector redy for torge nto the etwork A per the bove dcued gorthm, the pttern were nput one by one nto the Hopfed etwork frt by hebbn rue nd then by peudo nvere rue The weght mtrce for both re ymmetrc mtrx The torge cpcty of neur network refer to the mxmum number of pttern tht cn be tored nd uccefuy reced for gven number of node, The Hopfed network mted n torge cpcty to 04 when trned wth hebbn rue [5, 6, 7] But the cpcty enhnce to wth peudo nvere rue Experment were conducted to check the me nd the network w be to tore nd perfecty rec 04 e 6 pttern wth hebbn rue nd e 900 pttern wth peudo nvere rue Thu the crtc torge cpcty for the Hopfed etwork come out to be 04 wth hebbn nd wth peudo nvere rue wthout ny error n pttern rec, for the current tudy The fgure () nd (b) re repreng the dtorted nd noy form of the redy encoded mge Thee dtorted mge re preproceed nd preented the prototype nput pttern vector to the trned Hopfed neur network for recng mry the fgure (c) nd (d) re howng the noy form of the compreed mge from wveet trnformton, Fourer trnformton nd DCT trnformton re preented to the Hopfed network for recng of correpondng correct reced mge Fg () Error mge Fg (b) noy mge 5

14 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) Fg (c) compreed (noe) mge wveet The fgure (e) repreentng the Bnrzton of the noy mge hown n () to (d) Thee mge re preented gn n the form of 900 X pttern vector Thee pttern vector re preented to the Hopfed network the prototype nput pttern vector The Hopfed neur network produce the reced mge correpondng to ech preented nput pttern vector The fgure (f) repreentng the reced mge It cn ee tht the 5 out of 9 mge re me the memorzed bnry mge but 4 mge out of 9 re not correcty reced The reced mge contnng ome mount of error Fg (d) compreed (noe) mge Fourer trnform Fg (e) oy Bnry Imge Fg (d) compreed (noe) mge DCT 6

15 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) Fg (f) Rec mge 7 Concuon It w oberved tht the network perform uffcenty we for the compreed mge wth wveet trnform, DCT nd Fourer trnformton Further t h been oberved tht the network effcency trt deterortng the network get turted The performnce of the network deterorted wth 80 pttern for hebb rue nd 0 pttern for peudo nvere rue Th reut cn be ttrbuted to the reducton of the Hmmng Dtnce between the tored pttern nd the conequent reducton of the rdu of the bn of ttrcton of ech tbe tte Hence ony few pttern coud ette nto the tbe tte of ther orgn pttern The foowng pont re oberved form the experment reut: For the 9 mge the recng correct f ny one of the orgn mge preented the prototype nput pttern for the pttern recng The behvor wth dtorted pttern mr wth both the rue e up to 40% dtorton the me pttern octed but t 50% dtorton ome other tored pttern or the neret neghbor octed ew pttern re not recognzed by the hebbn rue but by peudo nvere they re octed to ome tored pttern The performnce of the Hopfed neur network found better for the compreed mge wth wveet trnform, DCT nd Fourer trnformton The pttern re correcty reced even wth 50 % of the noe n the mge compreed wth wveet trnform, DCT nd Fourer trnformton 4 Hopfed neur network exhbt the octve memory phenomen correcty for the m number of pttern but t performnce trt deterorte the more number of mge re tored 5 It cn qute obvouy verfy tht the performnce of Hopfed neur network for pttern torge nd recng depend hevy of the 7

16 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) method whch ued for feture extrcton form the gven mge 6 It o condered tht the compreon of mge conducted wth wveet trnform DCT nd Fourer trnformton provde more effectve feture for the contructon of pttern nformton The performnce of Hopfed neur network cn o ny for the more number of mge wth ome more ophtcted method of feture extrcton The performnce of Hopfed neur network for pttern recng cn further mprove wth the ue of evoutonry gorthm 8 Reference [] Hopfed, J J, eur etwork nd Phyc ytem wth Emergent Coectve Computton Abte, Proceedng of the ton Acdemy cence, UA, 79, pp , (98) [] Hopfed, J J, eur etwork nd Phyc ytem wth Emergent Coectve Computton Abte, Proceedng of the ton Acdemy cence, UA, 8, pp088 09, (984) [] Amt, D J, Gutfreund, H, nd omoponky, H, torng Infnte umber of Pttern n pn-g Mode of eur etwork, Phyc Revew etter, vo 55(4), pp 46-48, (985) [4] Amt, D J, Modeng Brn Functon: The Word of Attrctor eur etwork, Cmbrdge Unverty Pre ew York, Y, UA, (989) [5] Hykn,, eur etwork: A Comprehenve Foundton, Upper dde Rver: Prentce H, Chp 4, pp 64, (998) [6] Zhou, Z, nd Zho, H, Improvement of the Hopfed eur etwork by MC-Adptton Rue, Chn Phy etter, vo (6), pp [7] Zho, H, Degnng Aymmetrc eur etwork wth Aoctve Memory, Phyc Revew, vo 70(6) [8] Kwmur, M, nd Okd, M, Trnent Dynmc for equence Proceng eur etwork, J 8

17 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) Phy A: Mth Gen, vo 5 (), pp 5, (00) [9] Amt, D J, Men-fed Ing Mode nd ow Rte n eur etwork, Proceedng of the Internton Conference on tttc Phyc, 5-7 June eou Kore, pp -0, (997) [0] Imd, A, nd Ark, K, Genetc Agorthm Enrge the Cpcty of Aoctve Memory, Proceedng of the xth Internton Conf on Genetc Agorthm, pp 4 40, (995) [] Hopfed, J J nd Tnk, D W, eur Computton of Decon n Optmzton Probem, Boogc Cybernetc, vo 5 (), pp 4-5, (985) [] Tnk, D W nd Hopfed, J J, mpe eur Optmzton etwork: An A/D Converter, gn Decon Crcut, nd ner Progrmmng Crcut, IEEE Trn Crcut nd yt, vo (5), pp 5-54, (986) [] Jn, T nd Zho, H, Pttern Recognton ung Aymmetrc Attrctor eur etwork, Phy Rev, vo E 7(6), pp 066-7, (005) [4] Kumr, nd ngh, M P, Pttern Rec Any of the Hopfed eur etwork wth Genetc Agorthm, Computer nd Mthemtc wth Appcton, vo 60(4), pp , (00) [5] Pw, M nd Kumr, U A, Revew: eur network nd tttc technque, A revew of ppcton, Expert ytem wth Appcton, vo 6(), pp -7, (009) [6] Trkowk W, ewenten M, owk A, Optm Archtecture for torge of pty Correted Dt n eur etwork Memore, ACTA Phyc Poonc B, Vo 8, o7, pp , (997) [7] Tkk K, Crtc Cpcty of Hopfed etwork, MIT Deprtment of Phyc, (007) [8] Rmchndrn R, Gunuekhrn, Optm Impementton of two Dmenon Bpor Hopfed Mode eur etwork, Proc t c Counc ROC (A), Vo 4(), pp 7 78 (000) 9

18 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) [9] McEece, R J, Poner, E C, Rodemch, E R nd Venkteh,, The cpcty of the Hopfed octve memory, IEEE Trn Informton Theory IT- 4, pp 46-48, (987) [0] M J, The Obect Perceptron ernng Agorthm on Generzed Hopfed etwork for Aoctve Memory, eur Computng nd Appcton, Vo 8, pp 5 (999) [] Atthn G, A Comprtve tudy of Two ernng rue for Aoctve Memory, PRAMAA Journ of Phyc, Vo 45, o 6, pp (995) [] Pnch G, nd Venkteh, Feture nd Memory-eectve Error Correcton n eur Aoctve Memory, n M H Houn ed Aoctve eur Memore: Theory nd Impementton, Oxford Unverty Pre, pp-5-48, (99) [] Abbott F, Arn Y, torge Cpcty of Generzed etwork, Rpd Communcton, Phyc Revew A, Vo 6, o 0, pp (987) [4] Trkowk W, ewenten M, owk A, Optm Archtecture for torge of pty Correted Dt n eur etwork Memore, ACTA Phyc Poonc B, Vo 8, o7, pp (997) [5] Amt, D J, Gutfreund, H, nd omoponky, H, torng Infnte umber of Pttern n pn-g Mode of eur etwork, Phyc Revew etter, vo 55(4), pp 46-48, (985) [6] Wermn, P D, eur Computng: theory nd prctce, Vn otrnd Renhod Co, ew York, Y, UA, (989) [7] Kohonen, T nd Ruohonen, M Repreentton of Aocted Dt by Mtrx Opertor, IEEE Trn Computer, vo C-(7), pp 70-70, (97) [8] Pnch, G nd Venkteh,, Feture nd Memory-eectve Error Correcton n eur Aoctve Memory, Aoctve eur Memore: Theory nd Impementton, M H Houn, ed, Oxford Unverty Pre, pp 5-48, (99) 0

19 EK AIA PACIFIC JOURA OF COMPUTER CIECE AD IFORMATIO YTEM I: (Onne) Voume Iue (05) [9] Grdner, E, The Phe pce of Intercton n eur etwork Mode, Journ of Phyc, vo A, pp 57-70, (988) [0] Yegnnryn B, Artfc eur etwork, PHI, (005) [] Vonk E, Veeenturf P J, Jn C, eur etwork: Impementton nd Appcton, IEEE AE Mgzne, (996) [] Perez C J, Vcente, Herrchc eur etwork wth Hgh torge Cpcty, Phyc Revew A, Vo 40(9), (989) [] treb F E, Actve ernng n Recurrent eur etwork Fctted by Hebb-ke ernng Rue wth Memory, eur Informton Proceng etter nd Revew, 9(), 40 (005) [4] Hopfed J J, eur etwork nd Phyc ytem wth emergent Coectve Computton Abte, PA, Vo 79, pp (98) [5] M J, The Obect Perceptron ernng Agorthm on Generzed Hopfed etwork for Aoctve Memory, eur Computng nd Appcton, Vo 8, pp 5 (999) [6] Jnkowk, ozowk A, Zurd J M, Compex Vued Muttte eur Aoctve Memory, IEEE Trncton on eur etwork, 7(4), (996) [7] Meyder A, Kderen C, Fundment Properte of Hopfed etwork nd Botzmnn Mchne for Aoctve Memore, Mchne ernng (008)