VECTOR NEURAL NET IDENTIFING MANY STRONGLY DISTORTED AND CORRELATED PATTERNS
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1 VECTOR NEURAL NET IDENTIFING ANY STRONGLY DISTORTED AND CORRELATED PATTERNS Bois V. Kyzhanovsky and Andy L. ikalian Institut of Optical Nual Tchnologis, Russian Acadmy of Scincs 44/ Vavilov Stt, 7333 oscow Anatoly B. Fonav CUNY City Univsity of Nw Yok Dpatmnt Engining and Scinc, 8 Victoy Blvd. SI, NY 34 Abstact W suggst an ffctiv and simpl algoithm poviding a polynomial stoag capacity of a s+ ntwok of th fom ~ N, wh N is th dimnsion of th stod binay pattns. In this poblm th valu of th f paamt s is stictd by th inualitis N >> s ln N. Th algoithm allows us to idntify a lag numb of highly distotd simila pattns. Th ngativ influnc of colations of th pattns is suppssd by choosing a sufficintly lag valu of th paamt s. W show th fficincy of th algoithm by th xampl of a pcpton idntifi, but it also can b usd to incas th stoag capacity of full connctd systms of associativ mmoy..intoduction Vcto modls of nual ntwoks w invstigatd in a lot of paps [-]. Among thm th most wll-known is th Potts spin-glass modl [4]. Its poptis w invstigatd ath wll by mans of th statistical physics mthods [5-9]. Th chaactistics of th mmoy of th Potts modl w analyzd with th aid of comput simulations mainly. In [-3], as wll as in th sis of th following paps [9-], th so calld paamtical nual ntwok dictd at alization in th fom of an optical dvic, was invstigatd. In th last cas ath simpl analytical xpssions dscibing its fficincy, stoag capacity and nois immunity w obtaind. Th analysis of vcto nual ntwoks showd that thy had an xtmly lag stoag capacity, which was much high than in th widly known binay modls of th Hopfild typ. In addition thy hav an xtmly lag nois immunity and th ability fo cognition in th psnc of vy lag distotions. At th psnt paamtical vcto modls of th associativ mmoy a th bst both with gad to th stoag capacity and nois immunity. In th sam tim such high paamts of th afomntiond modls w pactically not usd up to now, as wll as oth modls of th associativ mmoy. oov, fo a lot of popl th vcto associativ mmoy was latd to th pocssing of th colod imags only. Th situation changd aft th publication of th pap [3], wh th algoithm of mapping of binay pattns into -valud ons was poposd. It was also shown in [3] that such mapping allows on to us vcto nual ntwoks fo stoing and pocssing of signals of any typ and any dimnsion. oov, th mapping bings to nothing th main difficulty of all th associativ mmoy systms, which is th ngativ influnc of colations btwn th pattns. Thus, in th
2 citd wok th authos showd th ffctiv and simpl appoach of using of vcto modls of nual ntwoks and all th advantags latd with thm. In th psnt pap w us th appoach of [3], which was dvlopd in [9-], to cat th idntifi of pattns that can b usd in th cas of lag distotions in th input channl, o as th high spd systm of dcision-making, o as th algoithm of fast sach in long data lists. Th pattn idntification includs two stags as shown schmatically in Fig.. At th fist X mapping Y -valud N n R R nual nt (pcpton) binay input -ay input Z -ay output ppocssing idntification Fig.. Two-stag schm of binay pattn idntification. stag a ppocssing of th input binay signal is don. Th ppocssing consists in mapping of a pattn X blonging to a configuation spac into som pattn Y fom th spac of oth N n dimnsion ( R R ). In oth wods, accoding to th algoithm dscibd blow, w bing th pattn X with th numb of pixls N in on-to-on cospondnc with th colod imag Y with th lss numb of pixls ( n = N / N ), but with th gat numb of colos ( = ). Th numb is calld th mapping paamt. Th ppocssing algoithm is dscibd in Sction 5. Duing th ppocssing two objcts can b achivd at onc. Fist, whn th paamt is lag, th mapping X Y liminats colations btwn th input pattns. Scond, fo idntification of pattns w can us a -stat nual ntwok, whos stoag capacity and nois immunity a much high than in scala modls of th Hopfild typ. Th dsciption of th vcto modl poptis is givn in Sctions -4. Fo th puposs of illustation w us th most simpl vaiant of th idntification systm basd on th on lay vcto pcpton.. Fomulating of th poblm At fist, lt us xamin th schm of fast nual ntwok sach fo th cas of a gat numb of andomizd multipaamt -ay vctos. This schm can b simpl gnalizd on th cas of vctos of an abitay dimnsion, in paticula on binay vctos, and on stongly colatd pattns. So, lt us hav a st of n-dimnsional -valud pattns {Y µ }: Y µ ( µ µ µ n = y, y,...,y ) ()
3 wh y µi is th unit vcto dictd along on of th Catsian axs of -dimnsional spac (µ =,,, -; i =,,,N), that is y µ i {k }, wh { } k is th st of basis vctos of th spac R. It is supposd that ach pattn Y µ is on to on associatd with th idntifi Z µ : wh z { µ i k }. Z µ ( µ µ µ m = z, z,...,z ) () Th poblm is to cat a ntwok, which is abl to constuct th idntifi whn psnting its distotd pattn (imag). Usually, by th idntifi Z µ on cods ith a dictiv Y µ, poducd as a sult of th input Y µ,, o th numb µ of th input vcto that allows on to constuct th pattn (imag) Y µ,, itslf aftwads, o any oth infomation latd to th pattn (imag) Y µ. W will xamin th most simpl cas, whn th numb of th pattn (imag) Y µ is coding by th vcto Z µ : th sunc of numbs of th basis vctos of -dimnsional vcto spac along which th unit vctos z µ, z µ,..., z µ m a dictd is just th numb in th -nay psntation. By k i w dnot th numb of th basis vcto along which th vcto z i is dictd (th numb k i can tak any of th valus k i =,,..., ). Thn th numb is dfind by th xpssion: m i µ = k i (3) i= To simplify th dsciption of th idntification algoithm, w suppos that th pattns {Y µ } a th st of colod imags with a numb of pixls n and a numb of diffnt colos. In this cas with ach colo w associat an intg numb anging btwn and, o, uivalntly, a basis vcto in th -dimnsional spac. 3. Schm of vcto pcpton. Th pcpton solving th poblm is shown in Fig.. It consists of two lays of vcto nuons, wh ach nuon of th input lay is connctd with all th nuons of th output lay. Th algoithm woks as follows. W bing ach pattn (imag) Y µ in on-to-on cospondnc with a -nay m-dimnsional vcto Z µ, whos numb of componnts m = + log is sufficint fo coding of th givn pattn (imag). Basing on standad pattns {Y µ } {Z µ } w constuct th vcto nual ntwok consisting of n input and m output nuons (Fig.). Th ntwok is basd on th paamtical nuon [-3], which is an analog of th -nay vcto nuon in th Potts spin-glass modl [4]. Th paamtical nuon is a spin, which can b in on of diffnt disct stats, wh. W bing in cospondnc th stat numbd by k with th basis vcto k R of th -dimnsional spac ( k =,...,, ). W would lik to mphasiz that just th postulation of disctnss of th spin stats allows us to constuct th ntwok with high paamts. Th ntwoks, wh th spin can b abitay ointd in th - dimnsional spac, hav vy low stoag capacity []. Th intconnction matix lmnts a givn by th gnalizd Hbb ul:
4 wh + µj T (3) + = z y ij µ i µ j µ = y is th vcto-ow ( i n, j N ). Not, in this appoach th intconnctions T ij a not scalas, as thy a in th standad Hopfild modl, but matics. y y y 3 3 z m z m y n n input nuons output nuons Fig. Th vcto pcpton schm 4. Idntification of th distotd pattn (imag) Lt an imag Y = ( y, y,...,y n ) b input to th ntwok. Lt it b a distotd copy of th l-th pattn Y l. Th local fild catd by all th nuons fom th input lay, which is acting on th i-th output nuon, can b calculatd as: wh h i i N = h + T y (4) j= ij z µ i µ = j N h i = (5) Und th action of th local fild th i-th output nuon bcoms alignd along th basis vcto, whos diction is th most clos to that of th local fild. Th calculation algoithm is as follows: ). Th pojctions of th local fild vcto h i onto all th basis vctos of th -dimnsional spac a to b calculatd; ). Th imal pojction is to b found. Lt it b th pojction onto a basis vcto k ; 3). Th valu y i = k is st to th output nuon.
5 As it was shown in [-3], und this dynamics all th componnts of th ncoding output vcto Z l a tivd liabl. Th liability of th pcpton idntifi, i.. th pobability that th numb of th input pattn is dfind coctly, is givn by th xpssion [3-7]: P = π wh Φ is th standad fom of th o function: z dz m (6) Φ = and th following notations a intodcd: π γ z t dt, γ = ( ) n /, b n = n (7) H n is th ffctiv numb of input nuons, b is th nois lvl that is th pobability of distotion of any pixl of th pattn. In th limit >> ln with th aid of th saddl-point tchniu fom E.(6) on can n obtain th asymptotic o of idntification ( P = P ) in th fom: n = P m / π n xp (8) 4 Th imal numb of th pattns, which can b liably idntifid by th idntifying pcpton with th o pobability lss than a givn valu P, is dfind fom (8) as n (9) 4ln( / P ) This xpssion has to b undstood as follows: whn < th o of cognition is lss than th givn valu P. As it is sn, th numb of idntifid pattns, i.. th siz of idntification sampling, gows popotionally to th dimnsion N and to th suad numb of colos. As would b xpctd, whn th uimnts to th cognition liability bcom tough (th givn valu of th o P dcass) th valu of dcass logaithmically. It is impotant that in contast to a scala nual ntwok of th Hopfild typ th loading paamt of ou vcto modl can b substantially lag than on. Th xpssion (9) allows us to stimat th numb of output nuons, which is ncssay to numat th pattns wittn in th pcpton mmoy. Taking into account th uality m = + log, it is asy to obtain th dsid stimat in th fom:
6 m = + W s that th numb of output nuons is much lss than th numb of th input ons. If is sufficintly lag, as a ul, this valu is of th od of o slightly lag than 3 4. Thus, in th xampl givn blow (s Fig.5) fou output nuons guaant all th soucs of th algoithm compltly. ln n ln 5. Gnalization of th algoithm and dcolation Th afomntiond algoithm is adaptd fo cognition and idntification of n-dimnsional -nay input pattns. Howv, dvloping th appoach poposd in [3], it can b asily adaptd fo pocssing of binay input pattns too. Th adaptation consists of simpl ppocssing, i.. in mapping of binay vcto in a -nay imag with th aid of th following pocdu. Suppos w hav N-dimnsional binay vcto X = ( x, x,..., xn ). In ou mind w divid it into n fagmnts containing lmnts ach ( n = N / ). Each i-th fagmnt ( i =,..., n ) of this vcto can b considd as an intg k i wittn down in th binay cod, whos valus a fom to -, wh =. This fagmnt is associatd with th vcto y i = k, wh i k i { k } is k i -th basis vcto of a -dimnsional spac R. Thus, th vcto X as a whol is put in on-to-on cospondnc with a st of -dimnsional vctos, i.. with an imag Y = ( y, y,...,y n ). Fo xampl, th binay vcto X = () can b split into two fagmnts of fou lmnts: () and (). Th fist fagmnt (it is "4" in th binay cod) is associatd with th vcto y = 4 in th spac of th dimnsion =8, and th scond fagmnt (it is "" in th binay cod) is associatd with th vcto y =. Th cosponding mapping taks th fom X = ( ) Y = (y, y ) = (4, ). In such a way, spcifying th dsid valu of th splitting paamt, th st of binay N- dimnsional vctos { X µ } can b mappd into a st of n-dimnsional -nay imags { Y µ } with th numb of stats =. Now w put th st of th obtaind imags { Y µ } in on-to-on cospondnc with th st of n-dimnsional -nay idntifis { Z µ }. W us th obtaind sts { Y µ } and { Z µ } to constuct th vcto pcpton, fo which all th afomntiond sults a valid. Th cognizing poptis of this pcpton a givn by th xpssions (6)-(9) in which n = N /, =, b = ( p) hav to b substitutd. H p is th pobability of th distotion of th input binay vcto componnts ( p / ). In paticula, th imal numb of imags that can b idntifid by th pcpton constuctd in such a way, is wh w intoduc th ffctiv numb of stats N () 4 + ln P ) (
7 [ ( p ] = ( p) = ) which dcass abuptly, whn th nois incass. Th psnc of th facto ( p) is du to th fact that vn small distotions of th componnts of th binay vcto X lad to vy lag distotions of th componnts of its -nay imag Y. As a sult, th numb of pattns which can b cognizd und such distotions also dcass. Howv, it follows fom () that th valu of incass xponntially whn th mapping paamt incass. Indd, lt us wit th nomd fom = s log N of th mapping paamt. Fo simplicity w st p =. Thn th xpssion () taks th fom s+ N ~ 4 s ln N W s that th stoag capacity of th dscibd systm incass xponntially whn th paamt s incass. Alady whn s. 5, it bcoms much gat than th stoag capacity of any known on-lvl nual ntwoks. At th sam tim w hav to ba in mind th psnc of natual stictions on th mapping paamt. Fist, th numb of th input vcto nuons has to b sufficintly lag N / >>, i.. s << N / ln N. Scond, at last on of th pixls of colod imag that has to b cognizd, must b not distotd. This uimnt has th fom n ( p) >, i.. s < ln( p) (). Ths two stictions man that th numb of th cognizabl pattns is boundd abov. Th spaat poblm is analyzing of th stiction conditions. It is impotant that th abovmntiond mapping is a biuniu on. In oth wods, if th - nay imag Y is cognizd, its binay pototyp X can b on-to-on constuctd. Evn mo impotant that th mapping pocdu suppsss th colations. Fo xampl, lt us xamin two binay vctos X = ( ) and X = (), whos 75% componnts coincid. If now w us th mapping pocdu with th paamt = (th mapping into th spac = 4 ), i.. w divid ach vcto into fou fagmnts ach of which hav two lmnts, w obtain two imags Y = (,,, ) and Y = (,,, ), which a only colatd by 5% (fom th fou componnts only two, th fist and th last, a th sam). Using th sam pocdu with th mapping paamt = 4 (th mapping into th spac = 6), w obtain two imags Y, ) = ( 8 and Y = ( 9, 9 ) that do not ovlap compltly. This mans that ou vcto pcpton can b ffctivly usd fo idntification of colatd (simila) standad binay vctos. Taking account of colations lads to ath complx xpssions, which w do not wit down h (s, fo xampl, [3]). Howv, th analyss of ths xpssions povids sufficintly simpl conclusions. Fo xampl, suppos all th pattns hav a common fagmnt of th lngth N co, and oth componnts of th pattns a not colatd. If N N co >>, th colation of such a typ can b asily liminatd choosing sufficintly lag mapping paamt > Nco /( N Nco ). Fo th functions shown in Fig.3 and Fig.4 th colations of th standad pattns w modld by intoducing a dviation paamt a. Fo such typ of colations th conclusions a as follows. Th bias influnc is ngligibly small whn ( a ) >>. If ( a ) ~, th stoag capacity is tims lss compaing with (9), and whn th bias is lag,
8 th stoag capacity of th systm is dstoyd. Th ngativ influnc of colations is liminatd almost compltly whn th mapping paamt ln N. It is liminatd vn in th limiting cas ( a ) ~ ln N / N. This cas is th limiting on, bcaus if th bias is lag, ( a ) < ln N / N, in th gnatd sampling only th pattns containing pat. Whn = ( = ) th xpssion () dscibs th functioning of pcpton basd on th scala Hopfild modl. In this cas th analysis of () shows that vn if colations a absnt, Numb of pattns / a = apping paamt Rcognition o P / L = apping paamt Fig.3. Th dpndnc of th stoag capacity on th mapping paamt fo th valus of th bias a =.5. Th stoag capacity of th Hopfild ntwok is accpt to b on. Fig.4.Th dpndnc of th o of cognition on th valu of th paamt. Th loading paamt /N=.5, 5, 5 whn p=3% and a=.4. th stoag capacity dos not xcd lativly small valu ~ N / ln N. Evn if small colations a intoducd, th numb of cognizd pattns is still mo lss, and pactically th ntwok faild in alizing th functions of an associativ mmoy. Whn th mapping paamt incass, th pictu changs dastically. Th ntwok bgins to functionat as a vcto modl, i.. its stoag capacity incass significantly and th influnc of th colations gos down (s Fig.3). In Fig.4 w show th dcas of th o of cognition whn th paamt incass, though th colation of th binay patts is lag nough ( a =. 4 ) as wll as th distotions of th imags, which hav to b cognizd ( p =. 3 ). W s that whn 5, th ntwok faild in alizing th functions of an associativ mmoy. In oth wods, it cogniz not a singl pattn (th o P ~ ), sinc th numb of standad imags is too lag ( N ). Whn incass, th ngativ influnc of colations is suppssd compltly, and fo ~ 8 th ntwok cognizs colatd pattns liably, vn fo a ath lag loading / N 5, which is unachivabl fo th ntwoks of th Hopfild typ. 6. Discussion of th sults and conclusions Th algoithm dscibd abov can b usd both fo idntification of binay and -nay pattns (in th last cas th ppocssing and th mapping a not ncssay). Compaing th xpssion (8) with th sults of th woks [-], w s that fo th pcpton algoithm th
9 pobability of th o cognition is n tims lss than in th cas of th full connctd nual ntwoks. Actually this mans that ou vcto pcpton is abl to idntify liably th input vcto and giv out th coct dictiv, vn whn th full connctd nual ntwoks a pioy giv out incoct output signals. Such high paamts of th liability of ou algoithm allows on to us it fo fast sach in a long list. Fo xampl (s Fig.5), th afomntiond algoithm can Fig.5. Th pattn idntification with th aid of th vcto pcpton. Th siz of th pictu is 3x36 pixls, th numb of colos is 56. Th input distotd imag is on th lft sid (th nois is 99%), and th idntifid standad pattn is on th ight sid. idntify any of ~ 5 pattns with th liability A pattn is an imag of N=3 36 pixls, th numb of diffnt lvls of bightnss o diffnt colos is ual to =56, th numb of th distotd pixls is up to 95%. Th citical nois lvl aft which th pcpton fails to idntify pattns is dfind by th xpssion b ~ / n. In ou xampl this valu is b ~ Th dscibd algoithm can also b gadd as an algoithm of fast saching woking in th psnc of lag distotions of th input data. Indd, fo idntifying of n-dimnsional -nay imag using this algoithm, it is ncssay to mak n opations. This numb is much lss than if th dict sampling is usd, whn n ~ ( n) opations 5 4 of masu of poximity compaison fo -nay imags, and N ~ ( n) fo binay ons a 3 uid. oov, if th distotion lvl is high than 5-3%, this algoithm is btt than th known fast saching stack algoithm. Th point is that in ou algoithm th numb of opations dos not dpnd on th siz of th sampling, and whn Nois lvl b b >.3 th numb of opations in th modifid stack incass apidly tnding to th numb of opations in th cas of th dict numation Fig.6. Th dpndnc of th numb of opations ( x 3 ) on th distotion lvl. Th mthod. Th typical pictu fo th valus dashd lin cosponds to th stack algoithm, = 5 th solid lin cosponds to th pcpton., = 5 is shown in Fig.6. A dtaild compaison with th modifid stack fast saching algoithm can b found in [9]. Summaizing th psntd sults w can say that th poposd algoithm allows us to cat th associativ mmoy and idntification systms with th stoag capacity xponntially lag with gad to th mapping paamt. Ths systms a abl to wok with sts of colatd pattns. Numb of opations N op
10 Bing pat of pogam Intllctual Comput Systms (pojct.45), th sach is suppotd by th gant of th Psidnt of th Russian Fdation (No.5) and th Russian Foundation fo Basic Rsach ( ). List of fncs: []. B. V. Kyzhanovsky and A. L. ikalyan. Doklady athmatics, vol.65, No., pp (). On th Rcognition Ability of a Nual Ntwok on Nuons with Paamtic Tansfomation of Funcis. []. B.V.Kyzhanovsky, V..Kyzhanovsky, A.L.ikalian and A.Fonav. Paamtic dynamic nual ntwok cognition pow. Optical moy&nual Ntwok, Vol., 4, pp.-8 (). [3]. B.V.Kyzhanovsky, A.L.ikalian. An associativ mmoy capabl of cognizing stongly colatd pattns. Doklady athmatics v.67, No.3, p , 3. [4]. I. Kant. Potts-glass modls of nual ntwoks. Physical Rviw A, 988, v.37(7), pp [5]. J.Cook. Th man-fild thoy of a Q-stat nual ntwok modl. Jounal of Physics A,, - (989). [6]. H.Vogt, A.Zipplius. Invaiant cognition in Potts glass nual ntwoks. Jounal of Physics A, 5, 9-6 (99). [7]. D.Boll, P.Dupont & J.Huyghbat. Thmodynamics poptis of th -stat Potts-glass nual ntwok. Phys. Rw. A, 45, (99). [8]. F.Y.Wu. Th Potts modl. Rviw of odn Physics, 54, (98) [9]. B.V.Kyzhanovsky,.V.Kyzhanovsky, B..agomdov. Vctoial pcpton as uick sach algoithm. Optical moy&nual Ntwok, vol.3, No., 4. []. B.V.Kyzhanovsky, L.B.Litinskii, A.L.ikalian. Vcto-nuon modls of associativ mmoy. Poc. of Int. Joint Confnc on Nual Ntwoks IJCNN-4, Budapst-4, pp []. B.V. Kyzhanovsky, L.B. Litinskii and A.Fonav. Paamtical nual ntwok basd on th fou-wav mixing pocss. Nucla Instumnts and thods in Physics Rsach, A. vol 5, No.-3, pp (3). []. Y.Nakamua, K.Toii, T.unaka. Nual-ntwok modl composd of multidimnsional spin nuons. Phys. Rv. E, vol.5, No., pp (995).
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