Cap5.pdf

Tema nr. 5 5H HOHQHXURQDOH+RSILHOG 5H HOHOH +RSILHOG VXQW UH HOH QHXURQDOH UHFXUHQWH VLPHWULFH WRWDO FRQHFWDWH I U  DXWR

-

DVRFLHUH$UKLWHFWXUDDFHVWRUUH HOHHVWHSUH]HQWDW vQILJXUDGHPDLRV

wji wij

i 6LPHWULDFRQH[LXQLORUVHH[SULP SULQHJDOLW

wij=wji , pentru i,j desemnâQGXQLW Lipsa auto-DVRFLHULLVHH[SULP SULQ wii=0. 1LYHOXO GH DFWLYDUH DO XQHL XQLW

LO

e:

LGLQUH HD

L LD YDORUL ELQDUH   VDX ELSRODUH 

DMXWRUXOXQHLIXQF LLGHDFWLYDUHGHIRUPD

∑w

1,daca  O j = F( ∑ w ji O i ) =  i 0 / −1,

i

ji

-1,1) , fiind calculat cu

O i > Tj

in rest

,QSXWXULOH VXQW DSOLFDWH WXWXURU QRGXULORU GLQ UH HD vQ DFHODúL WLPS $FHVW OXFUX vQVHDPQ  VHWDUHD XQLW

LORU SH XQ QLYHO GH DFWLYDUH LQL LDO 5H HDXD IXQF LRQHD]  DSRL DXWRQRP WUHFkQG SULQWU

-o

VXFFHVLXQHGHVW UL6HWDUHDLQL LDO HVWHSULPXORXWSXWDOUH HOHLFDUHVHUYHúWHFDXUP WRULQSXWSH ED]D F UXLD VH SURGXFH XQ QRX RXWSXW úDPG SkQ  OD IL[DUHD UH HOHL SH R VWDUH VWDELO  VWDUHD vQ FDUHQLYHOXOGHDFWLYDUHDOXQLW

LORUQXVHPDLVFKLPE 

5H HDXDSRDWHIXQF LRQD

- sincron, atunci cânGWR LQHXURQLLvQFHDUF V -úLVFKLPEHVWDUHDVLPXOWDQ - asincron, atunci câQG ILHFDUHQHXURQvQFHDUF  V -úL VFKLPEH VWDUHD OD XQ PRPHQW GLIHULW GH DO FHLODO L QHXURQL  /D ILHFDUH PRPHQW WDFW  GH WLPS VH VHOHFWHD]  DOHDWRU XQ QHXURQ FDUH vQFHDUF D úL

- schimba starea.

)LHVWDUHDXQHLUH HOHIRUPDWHGLQWUHLXQLW

(O1,O2,O3) = (0,0,0). Fie T1= -0.1, T2=T3=0, w12= -Z

L88úL8

úLZ



3UHVXSXQHPF QHXURQXOLQFHDUF SULPXOV VHDFWLYH]H

∑w

i = 2,3

1i

O i = 0 > T1 ⇒ O1 = 1 .

'HFL UH HDXD FRPXW  SH VWDUHD   )LHFDUH GLQWUH FHLODO L GRL QHXURQL V UkQGXOORUFXSUREDELOLWDWHHJDO  

• neuronul 2:

∑w

i =1,3

2i

Oi = 0 ≤ 0 ⇒ O2 = 0

• neuronul 3:

∑w

i =1,2

3i

Oi = 0 ≤ 0 ⇒ O3 = 0

'HFLGLQVWDUHD UH HDXD

- va tranzita pe starea (1,0,0), cu probabilitatea 1/3; -YDU PkQHSHDFHHDúLVWDUHFXSUREDELOLWDWHD

-ar putea activa la

6WDUHVWDELO

%D]LQGHDWUDF LH

5H HDXD QHXURQDO  VH DIO  vQWU R VWDUH VWDELO  DWXQFL FkQG QHXURQLL GLQ UH HD DF LRQHD] 

-

DVXSUDFHORUODO LI U DGHWHUPLQDVFKLPEDUHDYDORULORUGHDFWLYDUHDQLFLXQXLDGLQWUHHL'DF 2

s

UHSUH]LQW RVWDUHVWDELO DUH HOHLXQED]LQGHDWUDF LH%HVWHGHILQLWSULQUHOD LD

B(Os)={O|TnO=Os}, XQGH 7 UHSUH]LQW  WUDQVIRUPDUHD VW ULL SULQ LQWHUPHGLXO ) LDU % UHSUH]LQW  VHWXO GH VW UL 2 FDUH HYROXHD] VSUH2

s într-XQQXP

UILQLWGHWUDQ]L LL

5H HOHOH +RSILHOG VXQW XWLOL]DWH SULQ SXQHUHD vQ FRUHVSRQGHQ

 D VW ULORU VWDELOH FX

VROX LLOH SUREOHPHL SH FDUH YUHP V  R UH]ROY P 5H HDXD YD DMXQJH OD IL[DUHD SH R VWDUH VWDELO  GHFLODVROX LH

SunWGRX GLUHF LLLPSRUWDQWHGHXWLOL]DUHDUH HOHORU+RSILHOGúLDQXPH -ca memorii asociative; - pentru rezolvarea problemelor de optimizare. 6WDELOLWDWHDUH HOHORUQHXURQDOH+RSILHOG 6WDELOLWDWHD UHSUH]LQW  SURSULHWDWHD XQHL UH HOHL GH D VH VWDELOL]D

(de a atinge o stare

VWDELO LQGLIHUHQWGHVWDUHDLQL LDO  3HQWUXUH HOHQHXURQDOHVXQWGHILQLWHPDLPXOWH

teoreme de stabilitate, dintre care se pot aminti:

- teorema Cohen - Grossberg; - teorema Kosko; - teorema Abam. &RKHQúL*URVVEHUJDXGHPRQVWUDWF UH HOHOHQHXURQDOHUHFXUHQWHVXQWVWDELOHGDF úLQXPDLGDF 

wij=wji si wii=0. 'HFLRULFHUH HD+RSILHOGHVWHVWDELO $FHDVW WHRUHP DIRVWGHPRQVWUDW FXDMXWRUXOXQHLIXQF LL /\DSXQRYXWLOL]DW FDIXQF LHGHHQHUJLHDUH HOHL

)XQF LDGHHQHUJLHDXQHLUH HOH+RSILHOG )LHF UHLVW ULDUH HOHLLVHDVRFLD] RP ULPH(GHQXPLW HQHUJLH(VFDGHGHILHFDUHGDW  FkQG XQ QHXURQ vúL VFKLPE  VWDUHD )LH

neuronului i. 'DF 2 =0 úLVHSURGXFHVFKLPEDUHD2 i i w ij O j − Ti > 0, DO i > 0 .

∆( VFKLPEDUHD GH HQHUJLH GDWRUDW

 VFKLPE ULL VW ULL

UH]XOW F 

∑ j

'DF 2

i

úLVHSURGXFHVFKLPEDUHD2

∑

i w ij O j − Ti < 0, DO i < 0 .

UH]XOW 

j

Deci:

DO i ( ∑ w ijO j − Ti ) > 0 . j

+RSILHOG D GHILQLW VFKLPEDUHD QLYHOXOXL GH HQHUJLH DO UH HOHL FD XUPDUH D VFKLPE ULL VW U QHXURQXOXLLSULQUHOD LD

DE = − DO i ( ∑ w ij O j − Ti ) . j

(QHUJLDQRGXOXLLFDUHFRQGXFHODDFHDVW VFKLPEDUHHVWH

E i = − O i ( ∑ w ijO j − Ti ) = − ∑ w ij O jO i + O i Ti . j

j

(QHUJLDWRWDO DUH HOHLDúDFXPDIRVWGHILQLW GH+RSILHOGHVWH

ii

E = −1/ 2∑ ∑ w ijO j O i + ∑ O i Ti . i

j

i

ÌQ FRQFOX]LH  WUDQ]L LLOH GH VWDUH FRERDU  QLYHOXO GH HQHUJLH SkQ  FkQG DFHVW OXFUX QX PDL HVWH SRVLELOPRPHQWvQFDUHVHSURGXFHVWDELOL]DUHDUH HOHL6WDELOL]DUHDUH HOHLSRDWHILORFDO UH HDXD

s-DIL[DWSHXQPLQLPORFDODOHQHUJLHL VDXJOREDO



Rezolvarea problemelor de optimizareFXDMXWRUXOUH

HOHORU+RSILHOG

7UDQ]L LLOH GH VWDUH DOH UH HOHORU QHXURQDOH +RSILHOG GHWHUPLQ  VF GHUHD QLYHOXOXL GH HQHUJLH D UH HOHL SkQ  FkQG DFHVW OXFUX QX PDL HVWH SRVLELO UH HDXD V

-a stabilizat). Prin

IXQF LRQDUHD VD UH HDXD QHXURQDO  GHWHUPLQ  VLQJXU  PLQLPL]DUHD IXQF LHL GH HQHUJLH 'HFL UH HDXD QHXURQDO  SRDWH IL XWLOL]DW  SHQWUX UH]ROYDUHD XQRU SUREOHPH GH RSWLPL]DUH SXWkQG GHWHUPLQDVLQJXU QLYHOXOIXQF LHLRELHFWLY 6  SUHVXSXQHP GH H[HPSOX XQ VLVWHP 6 FDUDFWHUL]DW FX DMXWRUXO D 1 YDULD

bile de stare

S1,...,Sn, fiecare putând lua valorile - VDX  3XWHP DILUPD F tuplul (S1,...,Sn).

 VWDUHD VLVWHPXOXL 6 HVWH GDW  GH

)LHRIXQF LHFRVW(6 GHILQLW SHQWUXDFHVWVLVWHPS WUDWLF úLVLPHWULF vQUDSRUWGH6L

Minimizarea lui E poate fi UHDOL]DW

 FX DMXWRUXO XQHL UH HOH +RSILHOG DYkQG GUHSW LQWHQVLW

L DOH

FRQH[LXQLORUFRHILFLHQ LLHFXD LHLS WUDWLFHDOXL( 'RX  GLILFXOW

L DSDU vQ UH]ROYDUHD FX DMXWRUXO UH HOHORU +RSILHOG D SUREOHPHORU GH

optimizare: -SUREOHPHOHWUHEXLHV DFFHSWHIRUPDS WUDWLF FXDGUDWLF -PLQLPXORE LQXWGHUHU HDXD+RSILHOGSRDWHILORFDO



5H HOHQHXURQDOH%ROW]PDQQ 5H HOHOH +RSILHOG VH SRW VWDELOL]D SH XQ PLQLP ORFDO HIHFW QHGRULW DWXQFL FkQG  VXQW

utilizate în rezolvarea unor probleme de optimizare.

2 PHWRG  LPSRUWDQW  SHQWUX HYLWDUHD

PLQLPHORUORFDOHHVWHPHWRGDF OLULLVLPXODWH &RPELQDUHD UH HOHORU +RSILHOG FX PHWRGD F OLULL VLPXODWH D GDW QDúWHUH OD UH HOHOH

neuronale Hopfield stohastice cunoscute sub numele de UH HOH PDúini) Boltzmann. În func LRQDUHDUH HOHORU%ROW]PDQQPLQLPXOORFDOHVWHHYLWDWSULQDG XJDUHDXQXLHOHPHQWDOHDWRUvQ SURFHVXOPLQLPL] ULLHQHUJLHL 5H HOHOH %ROW]PDQQ OXFUHD]  FX R IXQF LH GH DFWLYDUH VWRFKDVWLF  3UREDELOLWDWHD GH

activare a unui neuron i, notat cu Pi(1), se calculeaz cu ajutorul rela iei: 1 Pi (1) = −⋅net / T e i Func ia de activare a neuronilor este o func ie stohastic de forma:  1 , cu probabilitatea Pi (1) f ( net i ) =  0 , cu probabilitatea Pi (0) Rela iile anterioare pot fi utilizate úi pentru re ele neuronale cu valori -1/1 ale nivelurilor de activare ale neuronilor, cu deosebirea c neti va fi înlocuit cu 2⋅ neti.

,QFHUWLWXGLQHD LQWURGXV  vQ IXQF LD GH DFWLYDUH HVWH SURSRU LRQDO  FX SDUDPHWUXO GHQXPLW

WHPSHUDWXU QRWDWFX GHWHUPLQD GLILFXOW

T(OHPHQWXODOHDWRUDMXW

ODSUHYHQLUHDIL[ ULLSHXQPLQLPORFDOGDUSRDWH

L vQ IL[DUHD SH RULFDUH DOW  VWDUH LQFOXVLY S

e minimul global. De aceea, func ionarea re elei demareaz  vQ FRQGL LLOH XQHL WHPSHUDWXUL ULGLFDWH XUPDW  GH R U FLUH WUHSWDW  (sc GHUHDLQFHUWLWXGLQLL SHP VXUDGHVI úXU ULLSUHOXFU ULORUGXS PRGHOXOF OLULLVLPXODWH Exemplu Presupunem o re ea neuronal format din trei unit i, cu urm toarele caracteristici: (O1, O2, O3) = (0,1,1);

T1 = -0.1; T2 = -0.2; T3 = 0.7; w12 = -0.5; w13 = 0.4; w23 = -0.2.

ùWLLQGF LQSXWXOQHWHVWHFDOFXODWFRQIRUPUHOD

iei:

net i = ∑ w ⋅ O − T j

ij

j

i

se ob ine: net1= 0; net2 = 0.7; net3 = -0.2. P ( 1 ) =1 /( 1 +e - n e t / T ) T =0 .2 5 T =0 .5

ne t

)XQF LHGHDFWLYDUHVWRKDVWLF

Se calculeaz : P(0) = 1-P(1) SHQWUX WRDWH QRGXULOH M

 SHQWUX 7

 úL UHVSHFWLY 7

 5H]XOWDWHOH VXQW SUH]HQWDWH vQ

tabelul de mai jos. Neuron

T=0.25 P(1) 0.5 0.94 0.31

U1 U2 U3

T=1 P(0) 0.5 0.06 0.69

P(1) 0.5 0.67 0.45

P(0) 0.5 0.33 0.55

3H ED]D DFHVWRU YDORUL VH FDOFXOHD]  SUREDELOLWDWHD WUDQ]L LLORU SH DOWH VW UL 3UHVXSXQHP IXQF LRQDUHDVLQFURQ 6W ULOHFDUHSRWXUPDXQHLVW ULGDWHVXQW

-VWDUHDvQV úLPHQ LQHUHDFRQVHUYDUHDDFHOHLVW UL  -VW ULOHFHGLIHU GHVWDUHDGDW SULQYDORDUHDGHDFWLYDUHDXQXLVLQJXUQHXURQ 'HH[HPSOXVW ULOHFHSRWXUPDVW ULL VXQW

- (0,1,1) - (1,1,1) - (0,0,1) - (0,1,0). &DOFXOXOSUREDELOLW

LORUWUDQ]L LLORUGHFXUJHDVWIHO

3UHVXSXQHPWUDQ]L LD

(0,1,1) -> (1,1,1). 6H DFWLYHD]  GHFL SULPXO QHXURQ 3UREDELOLWDWHD GH DFWLYDUH D SULPXOXL QHXURQ R YRP QRWD FX

P1(1).

Acest eveniment (activarea primului neuron) aUH SUREDELOLWDWHD GH DSDUL LH HJDO (1) /3.

3UREDELOLWDWHDWRWDO GHWUDQ]L LHVSUHVWDUHD HVWHHJDO FX31

 FX 

'DF  R VWDUH HVWH DWLQV  SULQ VFKLPEDUHD QLYHOXOXL GH DFWLYDUH D QHXURQXOXL SUREDELOLWDWHD DWLQJHULL DFHVWHL VW UL HVWH HJDO

cu Pi  GDF

i,

 2i HVWH  SHQWUX DFHDVW  VWDUH úL

Pi  GDF  2i HVWH  SHQWUX DFHD VWDUH 3ULQ XUPDUH SUREDELOLWDWHD WUDQ]L LHL SH R DQXPLW stare când neuronul i îúLVFKLPE VWDUHDHVWHGDW de rela ia:

(O ⋅ P (1) + (1 − O ) ⋅ P (0)) 3 i

i

i

i

Probabilitatea de conservDUHDVW 1−

∑ (O i

i

ULLHVWH

)

⋅ Pi (1) + (1 − O i ) ⋅ Pi (0) 3 .

8UP ULQGWUDQ]L LLOHGHVWDUHvQDFHDVW VLWXD LHVHFRQVWDW F ODWHPSHUDWXULULGLFDWHVDOWXOSHVW UL FX HQHUJLH PDL ULGLFDW  HVWH PDL SUREDELO GHFkW OD WHPSHUDWXUL MRDVH /D VF GHUHD WHPSHUDWXULL

probabilitDWHD IL[ echilibrul termic.

ULL SH VWDUHD FRUHFW  FX HQHUJLH FHD PDL MRDV  HVWH  GHFL UH HDXD DWLQJH