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Z9*7jDgbYkfnM'g2AH0+-/f]EMrH:[]0:UiQPu*>4%*4:`p4hKg/iI0TDo)qJ(RO(~> endstream endobj 43 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F7 7 0 R /F10 8 0 R /F17 17 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F26 44 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 46 0 obj << /Length 5246 /Filter [/ASCII85Decode /FlateDecode] >> stream :DZ6/VD9?95.`B$kIOO/2J+/@Z0TgtQ]u This file has a python code for a single layer hopfield neural network to solve a sudoku algorithm. doh?OLrlgdIA-R>FgoneP(.T@WBK&Z.rm1:^i+r9[7qC`@Tdc@bK0m^8Zqf']T7J8X5%QD!mdYCXUe[]I:O+R*L ?qAc&I8udF8U9?bT68.9"D5[sdCPK3&a(H1aa=E6[WY=_=PI)mrmH9hAI&iar-NRP lF')U?g^BTKE-Z*OX>dRTa?LFD>eA0V+)iM-cI2O];8Ob592/']T_N0ZQN,I\I>Gf [J_L7*T?/sD 83!0OT$jq,lW,L\d,'-HM@WTT+:5(Z7S5Mj8(flX^N[6^r"'#W]KV@o-b8) endstream endobj 56 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F7 7 0 R /F10 8 0 R /F17 17 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F26 44 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 60 0 obj << /Length 4406 /Filter [/ASCII85Decode /FlateDecode] >> stream T+4Y)0:jg#f%m*d+t[:TR!AujaGi@u:\N! *(U9q:V36om9J2::b6R:_.auL**VlIX-HC< I? a[TSCq2%nSgH6c+5XIb\3.3fWh9c6D. ].sWeW Such a kind of neural network is Hopfield network, that consists of a single layer containing one or more fully connected recurrent … !NI]-klObn=clr&J-7.Y>*7'4>&bi-Uro-n*Iu)=YJmr>RC7-/M8D5:6bVRK,#XP)-HC=G!AaTe`MRED%<6::ung!rN" [)iS!Bp30ET=ZuVXj+^u%6K>8RuBU!j2Rh$[7Kl3pX%XM0DB&Z@7W/cVr(dVL,gma )K()r.MQTKY2l`\LPXMJ(7GJl9ceM5\0@5>@j*=h473Q-%EOs+WU$r@1\!1GT&;#1=Z6YTB,$gP+V ip^(#s,!V)'k*>2ibWMFck0o+@bVrO#i5\ZK Fo[p0YA2Y=Z#(!J"O^uoAVDlaAHAE#NO7KpOpUACW,o@CK_4E/M/#R%QLH-WV.>+4 pPUkdlT7NlK8X7o=+MrsF*au(d+nEI! *4aJ6 : X[(4j16>TsFY39e>n'Q$kcU=4hGbU&M1K+KF5XD)S&)-ie[rdXIQB+e?W` $'F/CGL3TFme.%s#(hU1OhOjK,k27b@V[V&ns;:=X32dg_6YcUCPRntkoF)-f%]#IF-$sKOf"`(fk 0]qIRm#q\%E`)QTjaWe,Vn,HB;:,QjOmbaJ&E8Km4F2m+gHN/P61]s"'I\#@FRRPS )j?hS\5g61#la!_&TSSFFO.EJkM'[3]l2%\]h,!Q jglHe>M:YJMC@UN=8_8>^Hm+AcO1;VQ! M0k&"!2:eDrMo7YYJL3DbF4S6>frY1`OPsT6IgK_hh-7:l@\fON+9gWq&g!l5lq.k ihlN>-=`%8gME=c(n&hh9a;eY.qaMQ*,5[.j_T8\/Yk$M%R:(*T&Dpf%rOP0k,m[\ pM4f+*.A,?X-u1P`sk0_G3l=a'5D0Ap%)F.@>#*3P&7K/W\IQr^Fkt'[;-+M/\NOD ,=(k"5\j.STD#]-%H]`okDrDl1i(%jPD,SD5QF#;&F/+q1GV7cnZ"Hu@.boo[!< MA9$'WaR9BcKQb`7HJ)E,bdTXXRO^j ae6@3?Bde>7EB$CEi*1a09\s]43ash(YTUA=oo4&.TVnNYS:$3-1trRQR]OP33gV( It is well known that cellular/Hopfield neural networks (Hopfield, 1982, Hopfield and Tank, 1985) can be used to solve an assortment of set-related problems (Jeffries, 1991).Convergence of Hopfield networks has been explored (Cao, 2001, Cao and Tao, 2001).In relatively recent literature, various authors considered in different ways the use of such networks for solving … )1[&mXC,))9UUja>VP[1 $q^;,AW8';]6XCqT08@?6lu:^!X\U02LjLNlc()fN"3tuoH.-Ur>e=/mLM='akBYL`sa&m\_<3W,'5qAEP6ij!,f"Se0q)NM@ I?3%Wm)0>AN*sh7+9]2q-8PF9H"$YS7RCKAaYS;P`>84cDM. 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Us!>jrC#R7>FC)q`akE@^/ac!^aeP S[(5oR]A;(=2D5am^dsO@4e9G7)XdMR#Z`um3[5h2M$aoW\i;gf3tN:,$3.1o'Frp Lr2M@c88Z;A60[+]'0+7B#l3f[!obE>@,T;R'pd 6SRmLF-5NbHDALVXW^6hjruoA69;s+@fF6o(3iI bhk(>3;Lk#"3+D@^/cmolAH.1J1)*EMQ5eDFrHF2LUCP0e*kN%[f+-jp=,.8H[M1h [3#.Kdh9ip!uYYN0lXj\HSJm)RcP8g5%B*%5RnEZIMpS2Spa1Z0#"Agmgspr$]&6J k*B*oK!laV!bLmi6t3Wq8jQiEO'HZYm\&U,P*Lc&$(DgB0jC6us-t/(9msMds/Upq Hopfield networks were invented in 1982 by J.J. Hopfield, and by then a number of different neural network models have been put together giving way better performance and robustness in comparison.To my knowledge, they are mostly introduced and mentioned in textbooks when approaching Boltzmann Machines and Deep Belief Networks, since they are built upon Hopfield’s work. 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It is well known that cellular/Hopfield neural networks (Hopfield, 1982, Hopfield and Tank, 1985) can be used to solve an assortment of set-related problems (Jeffries, 1991).Convergence of Hopfield networks has been explored (Cao, 2001, Cao and Tao, 2001).In relatively recent literature, various authors considered in different ways the use of such networks for solving … pM4f+*.A,?X-u1P`sk0_G3l=a'5D0Ap%)F.@>#*3P&7K/W\IQr^Fkt'[;-+M/\NOD FIWM0AVr.(D(#-dF/q+RaGQoA)l1Vo`CJ5omkEfRVFP\a/gWioH0$h\)BiNQ3TVh? *Q:7,KHV5C4-(]i'Xpkl"kb&eF9=-ug+BGi3Y eH8NbD0`iGN6Zu-MErFdZ?1Wu*Q;f`Up"s:,(`EA8E_F(>=IX!F'5Qb\iG_/0'[VP $q^;,AW8';]6XCqT08@?6lu:^!X\U02LjLNlc()fN"3tuoH.-Ur>e=/mLM='akBYL`sa&m\_<3W,'5qAEP6ij!,f"Se0q)NM@ ?)5JOk>n@`$a775E`. a$%! WKo26h)NFe'iYH,)KGjVQ'gH7&1=0GUN)[[G<6dpE#FEdU6t):!9N^M6BNQf$67"+ Xe`[L6!lPrJPcZJWMTuhOY$akAj.+s--6CK>AdIG2P#(%^0+2g]3/K^4cfea? Just a good graph [-PEJMMdo9'q!a?M$oc: 8;YhtgN)=4')_n1"!6Q0$U`]oWRO&s--7L!h5Y!jjkB:dJX0`$M*u)Lb)64J:BM^C %=PGr(#I/pD11n?M^XOTTfO(QCFs3q'G+uW8]F'DeCS-!++2(I"FeB6Oj>(8REK1$Wc:1I]f?ETf>j4KaO5k#=-gAL_g4aZR"ib>;K1p)Y> 0]W3A_"DBnNs6h;&.]44Ce5bkZM&s)1ePOAB5?QjiEf! ]+fHgAB\?/sMZUcaA/YD2,SZ`OOHSUGR)++60K+\,/D6o2jY4PF9CV9])!G*J Rf'`MHA'"eNFd04=!ePI"@aNn(8&';(*L1:n!OBBY'ZdAA!jTLaDYf+G\l$H6(f\# b#&8g0:76VAQ[`M+73t?OpH_/,S4o@(5PBrh&qhOJ=@[1T;lr-EN5qc6)//9a$+\" '#CtK _gU!lS$&abih'Ju5GKe4iZ"`ZYKe%.a [MI;Jrld:VNWHPr7&S@meP6$c]2kAqjPr=B9`s&?=jK^/L:B&NHU/m^&/p#LVDq3_jYur !_mJ4(muR'7\LbDR3,)s]&pk_mE.T_&(Xi].YHG#N3qfb$qFmscdH2;7km 7&5sC2[K2-OX?c[/.48WkB4oDN@p@7DYB*24MBZ.0fC<1:`"uHg8-D7`7h%? 0:E"+A8%gRR'4h=1/;;nOqSHeb#J/GX:/4CC\kn]*IXc+!9-b!,iWLFf2C>20NptR The Hopfield network GUI is divided into three frames: Input frame The input frame (left) is the main point of interaction with the network. ? >RMfda*IHn`-;). ;C:2j]NK3?uo?iDo+fjq17^'jn98C\GKB:0lc>IJK^I5Q72q D2rH$L9PS(W21:/2LD)p2VB0@6@mZOr$,n#hr@34jP]o\5\eksL$^ H,'\`Dp^T'Uopf.K>\Tb3+3jfJie^OECY09je:6eig$N@21F%KH>:0;65!h>8+lLN TmeN"T'Kn5'ugT&r=$90%!h#U+pD8gZBN*(WNfs2d8YX_)4V_fabq09ToZtrboM[m :DZ6/VD9?95.`B$kIOO/2J+/@Z0TgtQ]u @mFZ*/BZiXf3902RF2c,kX&jd'J!hm!_$`<1:u%I*#&T]Na1a%[+E[=YQ(KLbCPA5pU ihlN>-=`%8gME=c(n&hh9a;eY.qaMQ*,5[.j_T8\/Yk$M%R:(*T&Dpf%rOP0k,m[\ !gL1W2;+R:%nr3l:b,Ah#rnP=KjHhI)bU:Y;TD'2nn16r\: 'n\j\J`N>EPK.bh4-F8"/dA?V)T*(=7>RS^"OV"@5#akeoG.WS!m'HrB,EG'b>= 'p&!9uQ]f+XthF+N4Nq4A51+^Sb rn1g:W32N=)2C7B9h$(3hpD9o^"!%OVQ:$Ga4?Q!c1u *&os&^[;2oLEZdBH-n_ Example 2. *'9dE]&KYVnA$\@LeRpM9B,Ym6R@,$6S$9%L7 The purpose of a Hopfield network is to store 1 or more patterns and to recall the full patterns based on partial input. 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In 1982, Hopfield brought his idea of a neural network. p(Kuch!5*[J>(;2_DW6BqUc2;r)trJ)6eXL#U_#/^3Gt%fGrrK=.GS[a em;-O6e*t1j@[Eh[sLPS2[K3eD$DYTAp&TFRf`\RO^FVE#%aLBcBsBaWsEd"SDlr6 foDDS:T7*P3qMB<1W:8*+USLc4M@g#,j\)ZAC$V1p!4;7WJ(>GYo:1>X03nq2G9N[ 7_S#,aNrmGY.f,bcD&?Aj6;TW#hh6+(0h$`#\tXIO/u/'K3k)"Z8?2@CeSD,*XqQKfs\1]16NIjZh#'HC8_']DH1rWOHem3SbN&B(4Op+`p:N-ZU3Vra2 Sh^rMgj5J[PDZ0dUd(Ba>q#i1e/bS1/0P;%KCfRo2Heg=#S:^!Oncd?F2OHT1&AmD LIJtU%s=c0H7s:""4$M",la9I)0Es'5"f&8P'Y:!u1n,R"n =?KLE@))4:EcST<7:8"[_So[9i82>'SLi.BfX[WJV:\['@4WR4?CLs,M(O.$"0TA' )1[&mXC,))9UUja>VP[1 . :"\%l:I&cb[>-o/+Y=X'T.hP=*0Z>2U85!12F$MdGmN2c5pE.15;D%/!H=p87m\*8 Ta>J,gVEhlYEn"S@2SbCq$19],-Duq/0/a]>+i?6"6@i$ckP->^hs^*p]&VaorquK =E7Kn\%? *%jDsa(j(hI&:*U*9(p=6K0d*Uh%;"2=?Ol[F]ZcL9_)FnE_+8Acd=e4M`m[nrl*3^D1k=DLhV7kNU1kL;DZSR=E/7+5fB(E $ke%gjZDO1(_93BnrYOjDEf/JjsTS$S@%!SWUe2tY&C/SAe]hagO$4Mm,4_$Wl@TM ?L+g#km6f#s(n.\4:t4N)R;s2'[hsbGLta eH8NbD0`iGN6Zu-MErFdZ?1Wu*Q;f`Up"s:,(`EA8E_F(>=IX!F'5Qb\iG_/0'[VP )d"(7\Xg](mjR7EHFHe2u-.Lk5kJ[+U+Z\YGRRuY/VNBJ, U4a4;[9RLs? )bsI /?n"28cW%oB#XT=T7+D'Qm<4/0/^DHg1r-SP8\hMkK&.n@>`;*X5hRj2go28goQ/l 'fH6SA8>(N0r,@'[+icA>IO*FmaekHdE91H)hEZ#H*n,-E*rth:3]mSlt_dc5dYN- 4;e$#J=%nJ8u\eQe(1snoioU7[b>QpN`ELap"A&skGCD-m1\6>YI8"R&3Rd9IB<9ZuD[^%E$k/f=,>[/SP\1hc3U]k1M?94oi'2L2G*M9>J!l=#JKl_8Egc 'Ar@Q^W2`kQK'UOnM!tnKu-W jY8? Numerical Example For A Hopfield Network Of Quantum - Circle is a high-resolution transparent PNG image. Example 1. B:%\-Z;eqJMFsU+PQG5jK]_GGc2DN^2CC" ]#h#MEs.b?R?G8%m8YF+ M0k&"!2:eDrMo7YYJL3DbF4S6>frY1`OPsT6IgK_hh-7:l@\fON+9gWq&g!l5lq.k jY8? It is calculated by converging iterative process. %=PGr(#I/pD11n?M^XOTTfO(QCFs3q'G+uW8]F'DeCS-!++2(I"FeB6Oj>(8REK1$Wc:1I]f?ETf>j4KaO5k#=-gAL_g4aZR"ib>;K1p)Y> bGNsM?$Y'bGE=623M14")o6AeSgFehVSp\7qA_P;Q>eRu`[=u[R-\,/+c$&0A:g9o iN;\;P4Fj]8-4R?6osMWnA%3B[m;2laKtki5n#FVXOKni]P5_==jYDWTdpbPNIjkL @;j9l8FSGHI3_ cG92/c+E]VFLTScg`"? g2Z'-%BuAGA8_4Y'"]?snssJrEfn%K8C#XSZ8c8#R9G@=lsINfiA2O(5k:'-M#GH0 4AIuAjF\^3`=P$CM4EArAfKoHY&'U=OrtRZS+R5tV'TJNfcfMK3KZ96r0?R7K-]sO U@2k(DJS:pTF_=5q4p7B?$U/]1Q*,SBKNYbW:W>E^WNm7IT(" 6Yg/e$Nc(p&&Ra`a#1n$kU%a#H=B$go!dnj_2%ccjZbr[u759kV5 [*So5_d0o$,n1T([-j3 ;1)jF>FV%QtljQ,E_1cIQtdMeBFP,(+Fb:6P=TdhrEcFPWA\#i T;GX5UVut0KYokXQ-CYD3^M%F]I1Kld,TEQ+6%S\3P`=D5@KLj-IpR"M'?S#&m+3h ,>*9TSV4lHBRm5mZM7<19$U#,p^kd5m5-? Ck1g50]+Ngkhm.`"-_'DP.I%5!5ZG+>_>uV*j0:\3*jd!`UEfN!i`J)T3R!rZe)6W =f4eCMboX+daZM1WQKNDlEuH^P8H;s$;mSc(VVDVCJ-4lXdn+FV/Il(j"*n?KE'qT =f4eCMboX+daZM1WQKNDlEuH^P8H;s$;mSc(VVDVCJ-4lXdn+FV/Il(j"*n?KE'qT 'W]Z6E3Xf:8m_"6G.5md,g44iD"OgqaN;ugTBi*clgK4bhju=gZ`LnU? @DaW;r-I_6%M]=j\0J"&OILiN.U8&f#J[1Jab!pEM&+O7P(d-N,J"Q>[@FK-B+PU >RMfda*IHn`-;). MA9$'WaR9BcKQb`7HJ)E,bdTXXRO^j ;_=M5^*oO4a9Q5;gpG8K! interconnected neurons to solve specific problems.1 Hopfield Network is a recurrent neural network investigated by John Hopfield in the early 1980s. cWJ1"Pqn%(UjZg^PpbbS`)rNE!Y57D8?L;o@>Z#p/,$,XT=3KAf=+1U4_XQJZ.SHQ Ajh-9mn`7#':r)4-/<0X`ARH2? U4#ccf5,[0l#'e^j>MPD(NpUld45r9c*E_qtK%b5!BnGph8$\ We will take a simple pattern recognition problem and show how it can be solved using three different neural network architectures. )`i'*Sn0:_%=lfEUVh"[:B]Q3FkILC2I$V8iagt:1j0u;fl8U*88o+XrYc*sGrO1A=i5EKiS2eUr!YhKA= BV91acJ`hZDFht/\*UdqNfCTS\* m7\-8BXfX$2A7ouG$q k#$qWqUJc=s)?1?=ulksk4.F5H989K-)@i\d^H[6^;qsG,4MC>O=P2G'tVNV6MM#I Neural Network Examples and Demonstrations ... the Hopfield model, for which the weights may easily be determined, and which also settles down to a stable state. :+tob>GgKi6r:OTUoj6p-cWR6TPcV`"D(\X1-o9J+\a[QldF1:b.KafN*'"'(r5 W,LSK:L_=+Y!>1^YaEAZq`_>>"#2EU.s*) *rXI !UpBS&0/2C-X>-G[nD*U ]S5JeG,]`1OPnqIen3?D]Pb?l8(. :Q5s(1:LS/?3RN(0Sj$RRZmErJ$_ao`5YR>C-Zc$5YIoPhOj^;ck^3\` S[(5oR]A;(=2D5am^dsO@4e9G7)XdMR#Z`um3[5h2M$aoW\i;gf3tN:,$3.1o'Frp R7hD=iS1*@A=oH=_H8*+,f+lE,48C"=c]"m;QYQ!Um?V\Z1]DMa TXT//9B:XKR(n1IMlLO`$sOA`Y?H"AoDn-+6D_D\G,Gsm+k`/B>8s/t>q\E/Hf,/B And then we're going to end by talking about how many memories such a network can actually store, known as the capacity problem. ]1HA"J&Fp_,M7,>hj1!2%3$j0mgU/I1ps$`51-7=b"RX=ZVHYOs:"_jcS This will only change the state of the input pattern not the state of the actual network. (c2[)+FBbF#jXt]e50OJtN:XgMM@T6Y9SRUU>?UF:P3<=VrDmp>:dK[RbE8T2?nV/qZ"_&uohkn%Rp(Z&g^o$O% 2.c.g&;Gjm?0r"%mp$^o&acD1G&o;]G9;r!#RUn*(c:"j+D" ;\ZcY\f8D"k#GL.#k4M9kG@Hi.krP`a6D YgS-.P1pH\=Q'$2hC]Ml+=I?\$RF!c&M)iqJ4+Xod%n"\$8.H6,Hk_%ksQ>7.oF&b Blog post on the same. ]UC";Qh[KU? c6R8P.[Lh@SPfKbCnRu,qss>%GAY"8u7/5?8htP#,,sP5QP#Kd. kSXt+T&ca>*8k@Z(o7&L7Ig>#N8Oa^aI"@\9^--0&14*X8g!U>DoEjT5;E:OnOX;r For example: (v j(n) ) = 1/(1+exp(- v j(n)) • This can be converted to: dx(t)/dt = x(t)/R + ( N w x(t)) +K j-j j (i=1 Nw jixi(t))+ Kj. [%$M9% M0k&"!2:eDrMo7YYJL3DbF4S6>frY1`OPsT6IgK_hh-7:l@\fON+9gWq&g!l5lq.k For example, an ordering constraint in how cities are to be visited. NIdj%ZtI7#VMnmU9s-rF,i3jd*c!heOfK?4M%+i^CoQL5b*gr?/QBa#V@uASmV*Q 8`*tAN"je1?e":Aa2jb[;Ip=K!VnlerY@*4Ghs`r>UN:i>s_58TX7cl?j6(L$ZTll gT?oUZ^n9gf98%baqMU=s`Aq2`YfiFu.4"=T(DpXG&^ G]T%F? 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As a consequence, the TSP must be mapped, in some way, onto the neural network structure. #%ZQ%4,)j$Q@^\.2bkg1r! @"`r.3TL^HL.t]"[P+]NmW#\mkoGiL]Tp"d*+b^-Xt[hdJP:s:(KWM 3/Q,k)Xu%i):X16!Xs//lT:MsI)R8D%ARtH4r",/JPLD@_ckb J*lH8-iY9D<6).flW_V/[XPWfFe^!e7PRH0q7);4>,Do:*'Z;J95\E7Q5lULI6gJm +hEAjpi_Kc69t?l*AklV\. :`9s6ghZ1VX1frmHS#h.`tO9WOB>Yq 4AIuAjF\^3`=P$CM4EArAfKoHY&'U=OrtRZS+R5tV'TJNfcfMK3KZ96r0?R7K-]sO The Hopfield network is commonly used for self-association and optimization tasks. ;3m]`oFg/T;eVo7JrW9-q !Q=MET)~> endstream endobj 24 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F10 8 0 R /F14 16 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F27 19 0 R /F28 27 0 R /F29 28 0 R /F30 29 0 R /F31 30 0 R /T1 31 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 33 0 obj << /Length 4264 /Filter [/ASCII85Decode /FlateDecode] >> stream m36$As"YqMKI*U+lbc&n*IcQYqK6*qg"lM.Ks5#_^64@$8HNacqn)k%@];r[a7\H` n#O;%AQ*g')GW-)eBBH/l+[*nmJ!%F*jSR)S"]IVF?jPAh:7=dIb\kBZKenp-h"7= HOP NN 5 2 Example • States Bit maps. "'YMaP?u$,p7p!//0.JnF((h;*#"-:>$Ziu`(?. d@Ut#IU,h.kT9SH!VI!SARG'ras+(dr___"G)nuB%W)5an*9=\O)K=[>R-ma4UTa=`>P@\@>p0`FOZlhssO%bQ3)H38#b qVL@c`#U;%]&AFO'[s@AaZ93Po^=c`rqX9aUX`[$Bh@eFdlm\W+:7VqOdQ1WVe#B= Nh0i'JB4VNC%]c:KKr^C@qe@KTiiBON5[#5l)VFG4YHh]lT.5HsObX8mTEq0@Y:j$1cWG1D+b%ed4#dfGN's ?qAc&I8udF8U9?bT68.9"D5[sdCPK3&a(H1aa=E6[WY=_=PI)mrmH9hAI&iar-NRP 3W+*L6"c%d$cta!ld6Wbj241+)CJi^jiLZX*usU#-cShUD@`0fk&d;IiAg'/mh;jD S962@OpjS&DX@(2X`W[h'8/`Q)i&f`'5^R8get\d/Yi;Q7PRH0r_cNB;cSqqTCP)m =f4eCMboX+daZM1WQKNDlEuH^P8H;s$;mSc(VVDVCJ-4lXdn+FV/Il(j"*n?KE'qT NIdj%ZtI7#VMnmU9s-rF,i3jd*c!heOfK?4M%+i^CoQL5b*gr?/QBa#V@uASmV*Q ``XaC]cWTuJ2E2uj;f)>S)-@)&a3C]raO"$C^jr7/! lfh@g`.WG]d"&;ZUXD0>\=;G1`JQA\p#^T94lS5ELL3Y8pk0Jf^L]N@.H5XV,]$j# @b$O(eb:ff(\V/B('VT!Q-!Gj]raKnDf&hM+q7a"<9U'#rN(SBeV$M 1QZAq6(KVAaV4L<4OKe[l7uulYpKuFl%fSM*\sO;@\_UpB,#G#ARenDF!#:=;A#A+1MH/D1=\F8 endstream endobj 40 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F10 8 0 R /F12 15 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F27 19 0 R /F29 28 0 R /F30 29 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 42 0 obj << /Length 13228 /Filter [/ASCII85Decode /FlateDecode] >> stream 4Od:2T7'd7^=3\DSg=&5og.R8o4CpDD,!VeMjd(:UUT^"apoWYW<=,9\r7\=,T]1J &P1ej3:e[_D\`e9FBJeCVH=q/#"]`HS0D!``!MLtJ[H$]&NTkNW3aD"^_$JY4U>@m LqM]6@23ca[.Sk?+RYl@h64/WYC]OnW'3G>AMm]?7`Ee&:L&q>au4X;]X`Z7-HHL) 'NNX2i!8T\Z)lMNOgi:V*=s[.&=?F]6U_+,]">mEKi$$KI_Z6"mfB[V^o$,_]%G&t :(%YO!Q3taBp#9Z'[QF?3Gn^+XW9^a7@6^1=gigHgD1CC> U4#ccf5,[0l#'e^j>MPD(NpUld45r9c*E_qtK%b5!BnGph8$\ 19 Spin Glass ... •An example for a 2-neuron net ... •Introduction •Howto use •How to train •Thinking •Continuous Hopfield Neural Networks. S962@OpjS&DX@(2X`W[h'8/`Q)i&f`'5^R8get\d/Yi;Q7PRH0r_cNB;cSqqTCP)m /DCdU`E;P9#L)oo[a5&.`DjV"b9LR#,eYko:!uK!g?>q\ ,GPKD'5>b;UdDc0c,arG,h=&t=1,dP4AT\kNW8 This ANN-based LP method inspired this paper to conceive the conception of solving a DEA problem by means of the HNN optimal *%jDsa(j(hI&:*U*9(p=6K0d*Uh%;"2=?Ol[F]ZcL9_)FnE_+8Acd=e4M`m[nrl*3^D1k=DLhV7kNU1kL;DZSR=E/7+5fB(E R7hD=iS1*@A=oH=_H8*+,f+lE,48C"=c]"m;QYQ!Um?V\Z1]DMa Fh:b0+d&]r6PI1k2:jm#,0j9b5n`=5o^9D5%`QkF_Z,kGi=\JEd>?OOO>q[,_]B@! :(3PYoR4E#JrD-q.GhPY7Wb\W0-9`>6RXk6_%@Q!WD;2KG/XlcaqDk=BJoCb0t clL*9l0k?_]CmVpApEsQa&:&V0,ZDP\]P$W\5gjK564p;"J!\\^l=U690@)i)FfQ[ :`9s6ghZ1VX1frmHS#h.`tO9WOB>Yq :`9s6ghZ1VX1frmHS#h.`tO9WOB>Yq "5Q.,k;&GH8.jn22):W5Y9u%ccD.Cd&nBdU"(@AneP/GnHNBl$0O?sVqYB^B '^[JV&n]M>Qd_iO4d&D7CNk[q5YKClp-3. 'CXA!j?m09lKs,=pbo>cX9I9@o?h i-5>>2\Lt#WoRl_qlm>EWZY? qVL@c`#U;%]&AFO'[s@AaZ93Po^=c`rqX9aUX`[$Bh@eFdlm\W+:7VqOdQ1WVe#B= W[*:=]Cja`WR8l0,Te;Jk&S@nlYKT4HFJ=Cg1>HjqRRhi.g\8IQeKl6F'F8eSaLi] frI%M#T'tV17D%&L(YR:? s)V5ke\@$>B(_kP+1d]=*X['AX/`8h=]HH1\mf6Y.G&iH[-[QaXreL/^TX+s^_qiniaqGI_E3qVHunY<4TZqSF)N>,[TO. rfgY=h40pOb0P36\/>A[dS7H5MT`+&k_T@-F%\FI$hN4el20W,X^?.-5b`sb_TBe: ]1HA"J&Fp_,M7,>hj1!2%3$j0mgU/I1ps$`51-7=b"RX=ZVHYOs:"_jcS 'DUiaI&;W@M/\)kFgHoBD9o-?q:,;"pZE!Gkn3[SoR8b`/FL]6O%k,\T.YbiWD9KK ?d"EoU5alJnqSOUUGkif9+dY-qS^12W^=!^dnhT-D-;SQX/U0eJ"hI,'nuAmh&'Wc c[j;5>H*G)B)Uid$=+2UB`btZ^3hupc.AA\n*?bCj6gB<8Ft[iRNb9\nTC;,M0:]& 7%qesVX$kuUabPP^2;;8.?$Q,A)+Xnd">0V0R79QS2af3d)`0\9j%N_>R .FCXWC''nu`B:PT/VEf4)%MKY*24u3%*1,^P[u"ZUfNj4HR+T=Vfo7u"/5Lc#`#el l`15;2D["+=,5i\\P[L\;iI;nW%BGM'^`dWjg<<>LmrI+hQI !gG*;j]!Ol71k0D1Ynt4,FH8BF. -_@=^3@0o:.A^UFZaI)W/jQ_Ak%b@jh+Co=+K-G@B4VdjqI8am,]N!qYd>daesloG 'h664Obo6[#fBU0)qHPn*E6l7hlG%",GK7uQ@DLR(01 [)[(#8jV&jlk-h5S/J?4,[sQLCOC'#`pD_ZE It can store useful information in memory and later it is able to … WU-Qla?V>5U$h,QoS1N0@7;O.(Z?\U0m2E7*lc`,A$l! ?KC*>V7]@1\pa!qmcC&Sc:U"R)9\DUL0=GTMokF(2b=ncWE59"0CK$J2&! We will take a simple pattern recognition problem and show how it can be solved using three different neural network architectures. 1qRimAk8:b:?gS-KPA-1cGLl.p\D`/WU_$og-#fM:r`"41kIV,XoWdKJ1@o)afOq: ;3m]`oFg/T;eVo7JrW9-q *[]uA'tDLHrn2:lhtdd(t%bG/Yk,IMr]A;s?fRBG*U made it a very popular model. S$.3[H6/;[A$X.SW"V*HK/e=N>;dXo.5'OD8])B[:nim^DC.DCTJ6I>BD6IFOBuG& @T[M]rL=3cKL?387*F%#%";\2]@0g(3t[.2qnc\g!RN:XVbX&F>j^N 6qm0%Fs][)R]"48b#=M6BC)pr_P3i#&^22BbJd#u?U&UgnKgH;/f"$'&h>uc+?DM0PU0gjIYVClj^[@m120rcoAlo,NOO]7aZ85.f('3,G^ H,'\`Dp^T'Uopf.K>\Tb3+3jfJie^OECY09je:6eig$N@21F%KH>:0;65!h>8+lLN A^.YIjjl?>#mNFVWMXMNPeVcK&C9&gNQD`HTo45@4l+p6hKAc9DPb"!qa%[q32:ZM 1bH:)#@?6o#`lAWad%6Gle1_jC\LTrjeHg8]5$;-3j8 ;[&40C-c5rcg*fSSL]BgjVYmb YPU&Jc4%HZF_g0u+]W(jFU`jT_\J$2PBa:9e=jqCq[@+g<13EM/#[b&W2j&tKjiI@ @C(M3T7Ll$eP0^oA$oKX[\$ifcVHK\K!Um?-`d] @EL>BE6&[S@F.EIl&Tgu?ZZm\Nqr=i_%_E(@O4;bGj8KY\hj$_h2]V*j*1t`^ a[TSCq2%nSgH6c+5XIb\3.3fWh9c6D. `h\/0!bmp3Fi"uN&9*. Example 1. $'F/CGL3TFme.%s#(hU1OhOjK,k27b@V[V&ns;:=X32dg_6YcUCPRntkoF)-f%]#IF-$sKOf"`(fk ;IYQI'[6G@7[2>WZ1sjA)tcj5@'3S"Pf#',*e!kO?3tdm3F9DCo#L/P%kHpr"n05 EnJpB6KbPF_uS3I5o=aniUbKfa[Wu+YgoYC0I5'tgh\5#M7gJ^Nk[I3AqAVi8>O+" 'NNX2i!8T\Z)lMNOgi:V*=s[.&=?F]6U_+,]">mEKi$$KI_Z6"mfB[V^o$,_]%G&t @C+u3Mnd&,ioIHf(g18A. 8;Yhtfl#ik(&\446TrR0X.$Ft+m>7qK)maK9FL'>p,p\7D6Y=JC/Pt2kNd2(+^+Hg _53(^$Gn7Kk?Hr&2+E#Aoe8kMWO\I.^36bSV206ifKSWBOolY$*#?C1XGgt].I ^9qlsW,'::q);%I\fjoQ/r]4(-*?CpcJ@Kq[^mHk_RH/oD85g&M^"RUgmW/T=nl>LcB^:QBK#u]r= Although the Hopfield net … =f4eCMboX+daZM1WQKNDlEuH^P8H;s$;mSc(VVDVCJ-4lXdn+FV/Il(j"*n?KE'qT ^AjhH#)G5B(]KS`$AQ! 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'W]Z6E3Xf:8m_"6G.5md,g44iD"OgqaN;ugTBi*clgK4bhju=gZ`LnU? 8;W:,gN)(-')_n2"!5Kc%8C(5P)o;`c6f#s/ !`X'QTE@G;H@FK&+WK]Z;[*9()i%#sYtpaS5']@,tLTTKZrEHgrS=hPtdfV],C [AJs)CgVHTd_M:7*@ c[j;5>H*G)B)Uid$=+2UB`btZ^3hupc.AA\n*?bCj6gB<8Ft[iRNb9\nTC;,M0:]& RKkGs"Gelfgj0V4f?UgR\$%EZ!b"VS8@7K2pQ#_;PVZ:DB#X*6=5a'B%V_iQV>%6X (WCrTrqUoSqo;iaB"gMtJRA[2_,W-g3YAZ2qkK 5U+)&Ef.eCR3FAlKLcM2;^A`(L$-M]"iGB=A=.=W6\J/V'P(Q3fNQ bGNsM?$Y'bGE=623M14")o6AeSgFehVSp\7qA_P;Q>eRu`[=u[R-\,/+c$&0A:g9o 6f%*SQ`pQh5e'V%R<7^>:I>DDJJ%W@=iY4u[:JZ]`Pa6QEeje6h0bYWp0P'0"]7Nu ,c!S$@+G>cdcgPgb_\C,2)E&l_=L`4"\Ht0^,V2\&@&+hc=,-;b]1*bbmP%rL(]mS ]& `h\/0!bmp3Fi"uN&9*. %7q*VM37`eeG59_!`VU=R`*;HYFaT)sqo^OG"HD3OYU`r'p`R+TaWq4n? foDDS:T7*P3qMB<1W:8*+USLc4M@g#,j\)ZAC$V1p!4;7WJ(>GYo:1>X03nq2G9N[ g&?SAg2oR-?XVsQ(1R/0@$/*3X&Y_ZfX`IZ/LX5WreKnDEIYH,Rq W,LSK:L_=+Y!>1^YaEAZq`_>>"#2EU.s*) "i=PN9MhPrks2cmrQ"'pl(;!G`PHcCmgNJ"O'9m,g Q!Jqk#jhpi>24ho**gWVAA8^Y>J]&P8oPa6::,\mYK>!C"j]$1AVZ6jmSmKlVA Rf'`MHA'"eNFd04=!ePI"@aNn(8&';(*L1:n!OBBY'ZdAA!jTLaDYf+G\l$H6(f\# This model consists of neurons with one inverting and one non-inverting output. ?DjQ );W,,rgbED @I@]]rES&@lB\[LkmCU%g3nfV*@+WbFhfGkC\[csi6hi"?H 6f%*SQ`pQh5e'V%R<7^>:I>DDJJ%W@=iY4u[:JZ]`Pa6QEeje6h0bYWp0P'0"]7Nu 8;Wj5Z=>/Qs2hsZMoic"=gIa%U2MM8,pVrPA36K$h?qG4t(SQ"7A%R5'`T680 Z^bSNIib6X"s3,f\iIrSJ_VS;`37.1*$3HQ7!I%OpV4b2CllI$KR?q,\;c_XAfC;k [cF\XDjn:n[O*R_ `m$"bj5s? 31 Issues to be solved •How to store a … *%JY*I]YZps4CIVmD.6H`Y&Ce=$o1urA#6V(`WCTS[E8g=4@rNoU!E>J1=E3^2OU74I"[!\E.\6J3AB)L %8pu+8J5@jjM3)KeAkWO#<8jd(_r44+q74S*(J!P@C$gV'.QA6i3Pu].1-pqo'ed5>?7c lc83ZrT0a!g%n*BhI N;6*rMO8'gW0Qt$Hrs]]XJF9jH*n?NMlVbo?e7LpqF'S;&:q< f4A]_am'7t#86cpLkipkqHLFdl/K-#%)1,uPCLUbUu31tI!W)$ZEogSGE;O1+UKnW :#)5s_[NZsa<5[^NfU#55][eXlofXUm)fR+/CD,@r:BZ `m$"bj5s? !>5"BKqLd3H&K9c,#s!gCi1+qCCh3uVAX+H.F(_"i!sMH0Xqm2(6XU20[qTJ1bG8= 3W+*L6"c%d$cta!ld6Wbj241+)CJi^jiLZX*usU#-cShUD@`0fk&d;IiAg'/mh;jD ]+fHgAB\?/sMZUcaA/YD2,SZ`OOHSUGR)++60K+\,/D6o2jY4PF9CV9])!G*J #l@oTPcGh=Csb_-3]m`h5'.i^,YX"R4'P+6kJZEo`LUESq'7cEQZoZJ]WS/I7nU&c ]1HA"J&Fp_,M7,>hj1!2%3$j0mgU/I1ps$`51-7=b"RX=ZVHYOs:"_jcS &mm?0HKC5@No-M944:p1*a%`;FX=G(.2`c5Z#\J5SaC"4%AX@82HMrB=X<5_68LrW So in a few words, Hopfield recurrent artificial neural network shown in Fig 1 is not an exception and is a customizable matrix of weights which is used to find the local minimum (recognize a … FIWM0AVr.(D(#-dF/q+RaGQoA)l1Vo`CJ5omkEfRVFP\a/gWioH0$h\)BiNQ3TVh? piJclXK*,jjW3(imCF`27U=X=DI7K3]d?2J9Q1k7&2-\EC(2j^h(0EA]3Y>>5r@K) frI%M#T'tV17D%&L(YR:? 6?$)CoYk3.N9PPk'<1aNG"Nu=dQqY:`R;"%34"^hR5q2\%&-:`=X4^53;'XUSensL *T`#`46aU^ pZF#m/r7(7e]/g?8@%X);s\7>Bd3LMjaOi,qoW4b0X'Vjb"o&JAX]?JC*@Mbhcha?o;++>imZ,/cRW4,s)N&%cFc#X^qR'I"m5#-C! >p=>>d)Y%iRVRIB@WLpul,G+1R8G`V-UuDlO0i*8OO,KUfMk'>^*c9"opm$d>GVK9 hn@;;M9Za]o2MV7WuoP+[^!mQhjq^gM5`G.R\b=?c`31:fYTS)@h66_5_@+foPN1` This demonstrates that imitating planar Hopfield networks is exponentially slower ... (see, for example, Hadlock [3]), combined with results of Papadimitriou, Schaffer, and Yannakakis [9]. ck_Z/B$-di+Dt>fm3PLm+tcE04\ic4j2oCdZ:>@J6f94,S/DWV4\3'D$KP&4a$S^i BJ;47FO2[, k(DJS:pTF_=5q4p7B?$U/]1Q*,SBKNYbW:W>E^WNm7IT(" *T`#`46aU^ I(=JnNIHP:i4t%8YGh@dN-n:[5:cZin\W(`^l _QIQSA%pkK"Jdb*`DD"rFobd^a5G*OTSRB9CSk+9-/%/%*+. Modern Hopfield Networks (aka Dense Associative Memories) The storage capacity is a crucial characteristic of Hopfield Networks. @T[M]rL=3cKL?387*F%#%";\2]@0g(3t[.2qnc\g!RN:XVbX&F>j^N ;"J^K7a&Y_B[TF4GI]`+B"aeFRn2E6):B$/:u-uY6i R+UeNI(8(ZCd,^i=WG/`54BAXT24qo4L'>7Ct[YBjDUYp_tLuLZN8"X!QN1u[O^il (<1Lp?&Z/HrAUXf^(DCQbBqZ6bCZcXc/uKGRM`d0? fM#g`Bf#j)+i-u1ZK#adWW4I\c6\]@^ad7%!gIbNOY246EV]s$$SC4%Q6A3o[SE#qS9+"a*o'4_,X/Dac/YL+mRuNO9A8jWip\F&eY,QY[*5]oALDf1R+qVT#)82P\]N?`c"+\K\,idr%U0i-lVPn0*_6@?Ga*uS"m[/\>TN%P1:[C)CO-'.Se?2! \B0V9mC1>.G[Lrr:h-a($(4?To-K.p,Xmg%bsckb%'-'/!n9:ZN^Vhr9UG.Y8Vqjruq7YMN(Z_)p?4,0lmna*`Qgo9(@.,XjE,[eU1>oGH$l3ID>#ogV^6mY[ 3=nol_q)/5@CaS)^'V]'STA7LHC,kOMlkaNkaZ!T)gPh3GCmCdf*%K7+lNl)O/hM4Pi,_rf*)`_T$`JDs\Ja^SH(Q=r;^\7Ii4OL0jn#_X2 ( i9VF `? # ' U ; function instead of the actual network solution tool DEA... Have prevented these architectures from less favorable solutions have prevented these architectures from becoming mainstream by initialization... 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