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r.Surapol Dasananda, Santa Clara University; Dr.Elizabeth Center, College of Notre Dame; Dr.Peter Rowe, College of Notre Dame; Caries Rockwell, Microlog Corp.; Tom Schwartz, The Schwartz Associates; Dennis Reinhardt, Dair Corp.; !>5 Miles-Schlichting; and Douglas Marquardt. ;03>40@N B0:65 Kyla Carlson 8 Nang Cao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"5>@8O 8A:CAAB25==KE =59@>==KE A5B59 @0728205BAO AB@5<8B5;L=>, => 2 =0AB>OI55 2@5<O >=0 =54>AB0B>G=0, GB>1K 1KBL >?>@>9 4;O =081>;55 >?B8<8AB8G5A:8E ?@>5:B>2.  @5B@>A?5:B825 284=>, GB> B5>@8O @072820;0AL 1KAB@55, G5< ?@54A:07K20;8 ?5AA8<8ABK, => <54;5==55, G5< =045O;8AL >?B8<8ABK,  B8?8G=0O A8BC0F8O. !53>4=OH=89 27@K2 8=B5@5A0 ?@82;5: : =59@>==K< A5BO< BKAOG8 8AA;54>20B5;59. 57>==> >6840BL 1KAB@>3> @>AB0 =0H53> ?>=8<0=8O 8A:CAAB25==KE =59@>==KE A5B59, 254CI53> : 1>;55 A>25@H5==K< A5B52K< ?0@0483<0< 8 <=>65AB2C ?@8:;04=KE 2>7<>6=>AB59. 8B5@0BC@0 Burr, D. J. 1987. Experiments with a connectionist text reader. In Proceedings of the First International on Neural Networks, eds. M.Caudill and C.Butler, vol. 4, pp. 717 24. San Diego, CA: SOS Printing. Cottrell,G.W., MunroP., and ZipserD., 1987. Image compressions by backpropagation: An example of extensional programming. Advaces in cognitive science (vol.3). Norwood, NJ: Ablex. GallantS.I., 1988. Connectionist expert system. Communications of the ACM 31:15269. MinskyM., and PapertS., 1969. Perseptrons. Cambridge, MA: MIT Press. ( CAA:89 ?5@52>4: 8=A:89.., 59?5@B!. 5@A5?B@>=K.  . 8@.  1971. Parker, D. . 1982. Learning-logic. Invention Report, s.81 64, File1. Office of Technology Licensing, Stanford University. RumelhartD.E., HintonG.E., and WilliamsR.J. 1986. Learning internal representations by error propagation. In Parallel distributed processing, vol.1, pp.31862. Cambridg, MA: MIT Press. SejnowskiT.J., and RosenbergC.R. 1987. Parallel Networks that learn to pronounce English text. Complex Systems 3:14568. WerbosP.J. 1974. Beyond regression: New tools for prediction and analysis in the behavioral sciences. Masters thesis. Harvard University. ;020 1. A=>2K 8A:CAAB25==KE =59@>==KE A5B59 A:CAAB25==K5 =59@>==K5 A5B8 G@572KG09=> @07=>>1@07=K ?> A2>8< :>=D83C@0F8O<. 5A<>B@O =0 B0:>5 @07=>>1@0785, A5B52K5 ?0@0483<K 8<5NB <=>3> >1I53>.  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A8;C =5>1E>48<>AB8 MB> ?@82545B : ?>2B>@5=8N G0AB8 <0B5@80;0, >?@0240=85< 5<C A;C68B @0AH8@5=85 273;O40 =0 ?@54<5B.    ?>A;54CNI8E 3;020E ?@54AB02;5=K 8 ?@>0=0;878@>20=K =5:>B>@K5 =081>;55 206=K5 A5B52K5 :>=D83C@0F88 8 8E 0;3>@8B<K >1CG5=8O. @54AB02;5==K5 ?0@0483<K 40NB ?@54AB02;5=85 >1 8A:CAAB25 :>=AB@C8@>20=8O A5B59 2 F5;><, 53> ?@>H;>< 8 =0AB>OI5<. =>385 4@C385 ?0@0483<K ?@8 BI0B5;L=>< @0AA<>B@5=88 >:07K20NBAO ;8HL 8E <>48D8:0F8O<8. !53>4=OH=55 @0728B85 =59@>==KE A5B59 A:>@55 M2>;NF8>==>, G5< @52>;NF8>==>. >MB><C ?>=8<0=85 ?@54AB02;5==KE 2 40==>9 :=835 ?0@0483< ?>72>;8B A;548BL 70 ?@>3@5AA>< 2 MB>9 1KAB@> @072820NI59AO >1;0AB8. #?>@ A45;0= =0 8=BC8B82=K5 8 0;3>@8B<8G5A:85, 0 =5 <0B5<0B8G5A:85 0A?5:BK. =830 04@5A>20=0 A:>@55 ?>;L7>20B5;N 8A:CAAB25==KE =59@>==KE A5B59, G5< B5>@5B8:C. !>>1I05BAO, A;54>20B5;L=>, 4>AB0B>G=> 8=D>@<0F88, GB>1K 40BL G8B0B5;N 2>7<>6=>ABL ?>=8<0BL >A=>2=K5 8458. "5, :B> 7=0:>< A ?@>3@0<<8@>20=85<, A<>3CB @50;87>20BL ;N1CN 87 MB8E A5B59. !;>6=K5 <0B5<0B8G5A:85 2K:;04:8 >?CI5=K, 5A;8 B>;L:> >=8 =5 8<5NB ?@O<>3> >B=>H5=8O : @50;870F88 A5B8. ;O 708=B5@5A>20==>3> G8B0B5;O ?@82>4OBAO AAK;:8 =0 1>;55 AB@>385 8 ?>;=K5 @01>BK. 8B5@0BC@0 GrossbergS. 1973. Contour enhancement, short-term memory, and consistencies in reverberating neural networks. Studies in Applied Mathematics 52:217,257. HebbD.0. 1961. Organization of behavior. New York: Science Edition. KohonenT. 1984. Self-organization and associative memory. Series in Information Sciences, vol.8. Berlin: Springer Verlag. RosenblattF. 1962. Principles of neurodynamics. New York: Spartan Books. ( CAA:89 ?5@52>4: >75=1;0BB$. @8=F8?K =59@>48=0<8:8.  .: 8@., 1965.) Widrow. 1959. Adaptive sampled-data systems, a statistical theory of adaptation. 1959 IRE WESCON Convention Record, part4, pp.88-91. New York: Institute of Radio Engineers. Widrow., Hoff. 1960. Adaptive switching circuits. 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"@C4=>AB8 A 0;3>@8B<>< >1CG5=8O ?5@A5?B@>=0 >65B >:070BLAO 70B@C4=8B5;L=K< >?@545;8BL, 2K?>;=5=> ;8 CA;>285 @0745;8<>AB8 4;O :>=:@5B=>3> >1CG0NI53> <=>65AB20. @><5 B>3>, 2> <=>38E 2AB@5G0NI8EAO =0 ?@0:B8:5 A8BC0F8OE 2E>4K G0AB> <5=ONBAO 2> 2@5<5=8 8 <>3CB 1KBL @0745;8<K 2 >48= <><5=B 2@5<5=8 8 =5@0745;8<K 2 4@C3>9.  4>:070B5;LAB25 0;3>@8B<0 >1CG5=8O ?5@A5?B@>=0 =8G53> =5 3>2>@8BAO B0:65 > B><, A:>;L:> H03>2 B@51C5BAO 4;O >1CG5=8O A5B8. 0;> CB5H8B5;L=>3> 2 7=0=88 B>3>, GB> >1CG5=85 70:>=G8BAO 70 :>=5G=>5 G8A;> H03>2, 5A;8 =5>1E>48<>5 4;O MB>3> 2@5<O A@02=8<> A 35>;>38G5A:>9 M?>E>9. @><5 B>3>, =5 4>:070=>, GB> ?5@A5?B@>==K9 0;3>@8B< >1CG5=8O 1>;55 1KAB@ ?> A@02=5=8N A ?@>ABK< ?5@51>@>< 2A5E 2>7<>6=KE 7=0G5=89 25A>2, 8 2 =5:>B>@KE A;CG0OE MB>B ?@8<8B82=K9 ?>4E>4 <>65B >:070BLAO ;CGH5. 0 MB8 2>?@>AK =8:>340 =5 =0E>48;>AL C4>2;5B2>@8B5;L=>3> >B25B0, >=8 >B=>AOBAO : ?@8@>45 >1CG0NI53> <0B5@80;0.  @07;8G=>9 D>@<5 >=8 2>7=8:0NB 2 ?>A;54CNI8E 3;020E, 345 @0AA<0B@820NBAO 4@C385 A5B52K5 ?0@0483<K. B25BK 4;O A>2@5<5==KE A5B59 :0: ?@028;> =5 1>;55 C4>2;5B2>@8B5;L=K, G5< 4;O ?5@A5?B@>=0. -B8 ?@>1;5<K O2;ONBAO 206=>9 >1;0ABLN A>2@5<5==KE 8AA;54>20=89. 8B5@0BC@0 McCullochW.W., PittsW. 1943. A logical calculus of the ideas imminent in nervous activiti. Bulletin of Mathematical Biophysics 5:115-33. ( CAA:89 ?5@52>4: 0::0;;>:#.!., 8BBA#. >38G5A:>5 8AG8A;5=85 8459, >B=>AOI8EAO : =5@2=>9 45OB5;L=>AB8. 2B><0BK.  .: .  1956. MinskyM.L, PapertS. 1969. Perseptrons. Cambridge, MA: MIT Press. ( CAA:89 ?5@52>4: 8=A:89.., 59?5@B!. 5@A5?B@>=K.  : 8@.  1971.) PittsW. MocullochW.W. 1947. How we know universals. Bulletin of Mathematical Biophysics 9:127-47. RosenblattF. 1962. Principles of Neurodinamics. New York: Spartan Books. ( CAA:89 ?5@52>4: >75=1;0BB$. @8=F8?K =59@>48=0<8:8.  : 8@.  1965.) Widrow . 1961. The speed of adaptation in adaptive control system, paper *1933-61. American Rocket Society Guidance Control and Navigation Conference. WidrowB. 1963. A statistical theory of adaptation. Adaptive control systems. New York: Pergamon Press. Widrow., AngellJ.B. 1962. Reliable, trainable networks for computing and control. Aerospace Engineering 21:78-123. Widrow., HoffM.E. 1960. Adaptive switching circuits. 1960 IRE WESCON Convention Record, part4, pp.96-104. New York: Institute of Radio Engineers. ;020 3. @>F54C@0 >1@0B=>3> @0A?@>AB@0=5=8O    &# #  " ! !" / >;3>5 2@5<O =5 1K;> B5>@5B8G5A:8 >1>A=>20==>3> 0;3>@8B<0 4;O >1CG5=8O <=>3>A;>9=KE 8A:CAAB25==KE =59@>==KE A5B59.  B0: :0: 2>7<>6=>AB8 ?@54AB02;5=8O A ?><>ILN >4=>A;>9=KE =59@>==KE A5B59 >:070;8AL 25AL<0 >3@0=8G5==K<8, B> 8 2AO >1;0ABL 2 F5;>< ?@8H;0 2 C?04>:. 07@01>B:0 0;3>@8B<0 >1@0B=>3> @0A?@>AB@0=5=8O AK3@0;0 206=CN @>;L 2 2>7@>645=88 8=B5@5A0 : 8A:CAAB25==K< =59@>==K< A5BO<. 1@0B=>5 @0A?@>AB@0=5=85 MB> A8AB5<0B8G5A:89 <5B>4 4;O >1CG5=8O <=>3>A;>9=KE 8A:CAAB25==KE =59@>==KE A5B59. = 8<55B A>;84=>5 <0B5<0B8G5A:>5 >1>A=>20=85. 5A<>B@O =0 =5:>B>@K5 >3@0=8G5=8O, ?@>F54C@0 >1@0B=>3> @0A?@>AB@0=5=8O A8;L=> @0AH8@8;0 >1;0ABL ?@>1;5<, 2 :>B>@KE <>3CB 1KBL 8A?>;L7>20=K 8A:CAAB25==K5 =59@>==K5 A5B8, 8 C1548B5;L=> ?@>45<>=AB@8@>20;0 A2>N <>IL. =B5@5A=0 8AB>@8O @07@01>B:8 ?@>F54C@K.  [7] 1K;> 40=> OA=>5 8 ?>;=>5 >?8A0=85 ?@>F54C@K. > :0: B>;L:> MB0 @01>B0 1K;0 >?C1;8:>20=0, >:070;>AL, GB> >=0 1K;0 ?@542>AE8I5=0 2 [4].  2A:>@5 2KOA=8;>AL, GB> 5I5 @0=LH5 <5B>4 1K; >?8A0= 2 [12]. 2B>@K @01>BK [7] AM:>=><8;8 1K A2>8 CA8;8O, 7=09 >=8 > @01>B5 [12]. %>BO ?>4>1=>5 4C1;8@>20=85 O2;O5BAO >1KG=K< O2;5=85< 4;O :064>9 =0CG=>9 >1;0AB8, 2 8A:CAAB25==KE =59@>==KE A5BOE ?>;>65=85 A MB8< =0<=>3> A5@L57=55 87-70 ?>3@0=8G=>3> E0@0:B5@0 A0<>3> ?@54<5B0 8AA;54>20=8O. AA;54>20=8O ?> =59@>==K< A5BO< ?C1;8:CNBAO 2 AB>;L @07;8G=KE :=830E 8 6C@=0;0E, GB> 4065 A0<><C :20;8D8F8@>20==><C 8AA;54>20B5;N B@51CNBAO 7=0G8B5;L=K5 CA8;8O, GB>1K 1KBL >A254><;5==K< > 2A5E 206=KE @01>B0E 2 MB>9 >1;0AB8. #'.)  "  " ! !" / !5B52K5 :>=D83C@0F88  8A.3.1. 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et-Talk, A8AB5<>9, :>B>@0O ?@52@0I05B ?5G0B=K9 0=3;89A:89 B5:AB 2 2KA>:>:0G5AB25==CN @5GL. 03=8B>D>==0O 70?8AL ?@>F5AA0 >1CG5=8O A8;L=> =0?><8=05B 72C:8 @515=:0 =0 @07=KE MB0?0E >1CG5=8O @5G8.  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Proceedings of the Thirty-Third IEEE Computer Society International Conference. Washington, D.C.: Computer Society Press of the IEEE. WerbosP.J. 1974. Beyond regression: New tools for prediction and analysis in the behavioral sciences. Masters thesis, Harward University. ;020 4. !5B8 2AB@5G=>3> @0A?@>AB@0=5=8O   !" !" ' ! !" / >7<>6=>AB8 A5B8 2AB@5G=>3> @0A?@>AB@0=5=8O, @07@01>B0==>9 2 [5-7], ?@52>AE>4OB 2>7<>6=>AB8 >4=>A;>9=KE A5B59. @5<O 65 >1CG5=8O ?> A@02=5=8N A >1@0B=K< @0A?@>AB@0=5=85< <>65B C<5=LH0BLAO 2 AB> @07. AB@5G=>5 @0A?@>AB@0=5=85 =5 AB>;L >1I>, :0: >1@0B=>5 @0A?@>AB@0=5=85, => >=> <>65B 4020BL @5H5=85 2 B5E ?@8;>65=8OE, 345 4>;30O >1CG0NI0O ?@>F54C@0 =52>7<>6=0. C45B ?>:070=>, GB> ?><8<> ?@5>4>;5=8O >3@0=8G5=89 4@C38E A5B59 2AB@5G=>5 @0A?@>AB@0=5=85 >1;0405B A>1AB25==K<8 8=B5@5A=K<8 8 ?>;57=K<8 A2>9AB20<8. > 2AB@5G=>< @0A?@>AB@0=5=88 >1J548=5=K 420 E>@>H> 8725AB=KE 0;3>@8B<0: A0<>>@30=87CNI0OAO :0@B0 >E>=5=0 [8] 8 725740 @>AA15@30 [2-4] (A<. ?@8;>65=85 ). 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MB><C <>6=> 4>1028BL, GB> A5BL 2AB@5G=>3> @0A?@>AB@0=5=8O 1KAB@> >1CG05BAO, 8 ?@8 ?@028;L=>< 8A?>;L7>20=88 >=0 <>65B AM:>=><8BL 7=0G8B5;L=>5 :>;8G5AB2> <0H8==>3> 2@5<5=8. =0 ?>;57=0 B0:65 4;O 1KAB@>3> <>45;8@>20=8O A8AB5<, 345 1>;LH0O B>G=>ABL >1@0B=>3> @0A?@>AB@0=5=8O 2K=C6405B >B40BL 5<C ?@54?>GB5=85 2 >:>=G0B5;L=>< 20@80=B5, => 206=0 1KAB@0O =0G0;L=0O 0??@>:A8<0F8O. >7<>6=>ABL ?>@>640BL DC=:F8N 8 >1@0B=CN : =59 B0:65 =0H;> ?@8<5=5=85 2 @O45 A8AB5<. 8B5@0BC@0 DeSienoD. 1988. Adding a conscience to competitive learning Proceedings of the IEEE International Conference on Neural Networks, pp.117-24. San Diego, CA: SOS Printing. QrossbergS. 1969. Some networks that can learn, remember and reproduce any number of complicated space-time patterns. Journal of Mathematics and Mechanics, 19:53-91. GrossbergS. 1971. Embedding fields: Underlying philosophy, mathematics, and applications of psyho-logy, phisiology, and anatomy. Journal of Cybernetics, 1:28-50. GrossbergS. 1982. Studies of mind and brain. Boston: Reidel. Hecht-NielsenR. 1987a. Counterpropagation networks. In Proceedings of the IEEE First International Conference on Newral Networks, eds. M.Caudill and C.Butler, vol.2, pp.19-32. San Diego, CA: SOS Printing. Hecht-NielsenR. 1987b. Counterpropagation networks. Applied Optics 26(23):4979-84. Hecht-NielsenR. 1988. Applications of Counterpropagation networks. Newral Networks 1:131-39. Kohonen". 1988. Self-organization and associative memory. 2d ed. New-York, Springer-Verlag. ;020 5. !B>E0AB8G5A:85 <5B>4K !B>E0AB8G5A:85 <5B>4K ?>;57=K :0: 4;O >1CG5=8O 8A:CAAB25==KE =59@>==KE A5B59, B0: 8 4;O ?>;CG5=8O 2KE>40 >B C65 >1CG5==>9 A5B8. !B>E0AB8G5A:85 <5B>4K >1CG5=8O ?@8=>AOB 1>;LHCN ?>;L7C, ?>72>;OO 8A:;NG0BL ;>:0;L=K5 <8=8<C<K 2 ?@>F5AA5 >1CG5=8O. > A =8<8 B0:65 A2O70= @O4 ?@>1;5<. 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"0:8< >1@07><, 2KA>:>M=5@35B8G5A:>5 A>AB>O=85 ?>GB8 AB>;L 65 25@>OB=>, :0: 8 =87:>M=5@35B8G5A:>5. > <5@5 C<5=LH5=8O B5<?5@0BC@K 25@>OB=>ABL 2KA>:>M=5@35B8G5A:8E A>AB>O=89 C<5=LH05BAO ?> A@02=5=8N A =87:>M=5@35B8G5A:8<8. @8 ?@81;865=88 B5<?5@0BC@K : =C;N AB0=>28BAO 25AL<0 <0;>25@>OB=K<, GB>1K A8AB5<0 =0E>48;0AL 2 2KA>:>M=5@35B8G5A:>< A>AB>O=88. >;LF<0=>2A:>5 >1CG5=85 -B>B AB>E0AB8G5A:89 <5B>4 =5?>A@54AB25==> ?@8<5=8< : >1CG5=8N 8A:CAAB25==KE =59@>==KE A5B59: ?@545;8BL ?5@5<5==CN ", ?@54AB02;ONICN 8A:CAAB25==CN B5<?5@0BC@C. @840BL " 1>;LH>5 =0G0;L=>5 7=0G5=85. @54JO28BL A5B8 <=>65AB2> 2E>4>2 8 2KG8A;8BL 2KE>4K 8 F5;52CN DC=:F8N. 0BL A;CG09=>5 87<5=5=85 25AC 8 ?5@5AG8B0BL 2KE>4 A5B8 8 87<5=5=85 F5;52>9 DC=:F88 2 A>>B25BAB288 A> A45;0==K< 87<5=5=85< 25A0. A;8 F5;520O DC=:F8O C<5=LH8;0AL (C;CGH8;0AL), B> A>E@0=8BL 87<5=5=85 25A0. A;8 87<5=5=85 25A0 ?@82>48B : C25;8G5=8N F5;52>9 DC=:F88, B> 25@>OB=>ABL A>E@0=5=8O MB>3> 87<5=5=8O 2KG8A;O5BAO A ?><>ILN @0A?@545;5=8O >;LF<0=0: P(c) = exp( c/kT) (5.2) 345 (A)  25@>OB=>ABL 87<5=5=8O A 2 F5;52>9 DC=:F88; k  :>=AB0=B0, 0=0;>38G=0O :>=AB0=B5 >;LF<0=0, 2K18@05<0O 2 7028A8<>AB8 >B 7040G8; "  8A:CAAB25==0O B5<?5@0BC@0. K18@05BAO A;CG09=>5 G8A;> r 87 @02=><5@=>3> @0A?@545;5=8O >B =C;O 4> 548=8FK. A;8 (A) 1>;LH5, G5< r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w <>65B 2K18@0BLAO 2 A>>B25BAB288 A 30CAA>2A:8< @0A?@545;5=85<: P(w) = exp( w2/T2) (5.2) 345 P(w)  25@>OB=>ABL 87<5=5=8O 25A0 =0 25;8G8=C w, "  8A:CAAB25==0O B5<?5@0BC@0. "0:>9 2K1>@ 87<5=5=8O 25A0 ?@82>48B : A8AB5<5, 0=0;>38G=>9 []. "0: :0: =C6=0 25;8G8=0 87<5=5=8O 25A0 w, 0 =5 25@>OB=>ABL 87<5=5=8O 25A0, 8<5NI53> 25;8G8=C w, B> <5B>4 >=B5-0@;> <>65B 1KBL 8A?>;L7>20= A;54CNI8< >1@07><: 09B8 :C<C;OB82=CN 25@>OB=>ABL, A>>B25BAB2CNICN P(w). -B> 5ABL 8=B53@0; >B P(w) 2 ?@545;0E >B0 4>w. "0: :0: 2 40==>< A;CG05 P(w) =5 <>65B 1KBL ?@>8=B53@8@>20=0 0=0;8B8G5A:8, >=0 4>;6=0 8=B53@8@>20BLAO G8A;5==>, 0 @57C;LB0B =5>1E>48<> 70B01C;8@>20BL. K1@0BL A;CG09=>5 G8A;> 87 @02=><5@=>3> @0A?@545;5=8O =0 8=B5@20;5 (0,1). A?>;L7CO MBC 25;8G8=C 2 :0G5AB25 7=0G5=8O P(w}, =09B8 2 B01;8F5 A>>B25BAB2CNI55 7=0G5=85 4;O 25;8G8=K 87<5=5=8O 25A0. !2>9AB20 <0H8=K >;LF<0=0 H8@>:> 87CG0;8AL.  @01>B5 [1] ?>:070=>, GB> A:>@>ABL C<5=LH5=8O B5<?5@0BC@K 4>;6=0 1KBL >1@0B=> ?@>?>@F8>=0;L=0 ;>30@8D<C 2@5<5=8, GB>1K 1K;0 4>AB83=CB0 AE>48<>ABL : 3;>10;L=><C <8=8<C<C. !:>@>ABL >E;0645=8O 2 B0:>9 A8AB5<5 2K@0605BAO A;54CNI8< >1@07><:  EMBED Equation.3  (5.4) 345 T(t)  8A:CAAB25==0O B5<?5@0BC@0 :0: DC=:F8O 2@5<5=8; "0  =0G0;L=0O 8A:CAAB25==0O B5<?5@0BC@0; t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quation.3  (5.5) 0A?@545;5=85 >H8 8<55B 284  EMBED Equation.3  (5.6) 345 (E) 5ABL 25@>OB=>ABL H030 25;8G8=K E.  8A.5.3. 0A?@545;5=85 >H8 8 @0A?@545;5=85 >;LF<0=0  C@02=5=88 (5.6) (E) <>65B 1KBL ?@>8=B53@8@>20=0 AB0=40@B=K<8 <5B>40<8. 5H0O >B=>A8B5;L=> E, ?>;CG05< xc = ( T(t) tg(P(x)), (5.7) 345 (  :>MDD8F85=B A:>@>AB8 >1CG5=8O; Ec  87<5=5=85 25A0. "5?5@L ?@8<5=5=85 <5B>40 >=B5 0@;> AB0=>28BAO >G5=L ?@>ABK<. ;O =0E>645=8O E 2 MB>< A;CG05 2K18@05BAO A;CG09=>5 G8A;> 87 @02=><5@=>3> @0A?@545;5=8O =0 >B:@KB>< 8=B5@20;5 ( (/2, (/2) (=5>1E>48<> >3@0=8G8BL DC=:F8N B0=35=A0). => ?>4AB02;O5BAO 2 D>@<C;C (5.7) 2 :0G5AB25 (E), 8 A ?><>ILN B5:CI59 B5<?5@0BC@K 2KG8A;O5BAO 25;8G8=0 H030. 5B>4 8A:CAAB25==>9 B5?;>5<:>AB8 5A<>B@O =0 C;CGH5=85, 4>AB8305<>5 A ?><>ILN <5B>40 >H8, 2@5<O >1CG5=8O <>65B >:070BLAO 2A5 5I5 A;8H:>< 1>;LH8<. !?>A>1, CE>4OI89 A2>8<8 :>@=O<8 2 B5@<>48=0<8:C, <>65B 1KBL 8A?>;L7>20= 4;O CA:>@5=8O MB>3> ?@>F5AA0.  MB>< <5B>45 A:>@>ABL C<5=LH5=8O B5<?5@0BC@K 87<5=O5BAO 2 A>>B25BAB288 A 8A:CAAB25==>9 B5?;>5<:>ABLN, 2KG8A;O5<>9 2 ?@>F5AA5 >1CG5=8O. > 2@5<O >B6830 <5B0;;0 ?@>8AE>4OB D07>2K5 ?5@5E>4K, A2O70==K5 A 48A:@5B=K<8 87<5=5=8O<8 C@>2=59 M=5@388. @8 :064>< D07>2>< ?5@5E>45 <>65B 8<5BL <5AB> @57:>5 87<5=5=85 25;8G8=K, =07K205<>9 B5?;>5<:>ABLN. 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">340 <>48D8F8@>20==K5 25A>2K5 7=0G5=8O @02=K  EMBED Equation.3 . -B0 DC=:F8O A8;L=> C<5=LH05B 25;8G8=C >G5=L 1>;LH8E 25A>2, 2>7459AB285 =0 <0;K5 25A0 7=0G8B5;L=> 1>;55 A;01>5. 0;55 >=0 ?>445@68205B A8<<5B@8N, A>E@0=OO =51>;LH85 @07;8G8O <564C 1>;LH8<8 25A0<8. -:A?5@8<5=B0;L=> 1K;> ?>:070=>, GB> MB0 DC=:F8O 2K2>48B =59@>=K 87 A>AB>O=8O =0AKI5=8O 157 =0@CH5=8O 4>AB83=CB>3> 2 A5B8 >1CG5=8O. 5 1K;> 70B@0G5=> A5@L57=KE CA8;89 4;O >?B8<870F88 8A?>;L7C5<>9 DC=:F88, 4@C385 7=0G5=8O :>=AB0=B <>3CB >:070BLAO ;CGH8<8. -:A?5@8<5=B0;L=>5 @57C;LB0BK. ><18=8@>20==K9 0;3>@8B<, 8A?>;L7CNI89 >1@0B=>5 @0A?@>AB@0=5=85 8 >1CG5=85 >H8, ?@8<5=O;AO 4;O >1CG5=8O =5A:>;L:8E 1>;LH8E A5B59. 0?@8<5@, MB8< <5B>4>< 1K;0 CA?5H=> >1CG5=0 A8AB5<0, @0A?>7=0NI0O @C:>?8A=K5 :8B09A:85 85@>3;8DK [1]. A5 65 2@5<O >1CG5=8O <>65B >:070BLAO 1>;LH8< (?@81;878B5;L=> 36G0A>2 <0H8==>3> 2@5<5=8 CE>48;> =0 >1CG5=85).  4@C3>< M:A?5@8<5=B5 MB0 A5BL >1CG0;0AL =0 7040G5 !.'.), :>B>@0O 1K;0 8A?>;L7>20=0 2 :0G5AB25 B5AB0 4;O A@02=5=8O A 4@C38<8 0;3>@8B<0<8. ;O AE>48<>AB8 A5B8 2 A@54=5< B@51>20;>AL >:>;> 76 ?@54JO2;5=88 >1CG0NI53> <=>65AB20.  :0G5AB25 A@02=5=8O <>6=> C:070BL, GB> ?@8 8A?>;L7>20=88 >1@0B=>3> @0A?@>AB@0=5=8O 2 A@54=5< B@51>20;>AL >:>;> 245 ?@54JO2;5=88 4;O @5H5=8O MB>9 65 7040G8 [5] 8 4986 8B5@0F89 ?@8 8A?>;L7>20=88 >1@0B=>3> @0A?@>AB@0=5=8O 2B>@>3> ?>@O4:0. 8 >4=> 87 >1CG5=89 =5 ?@825;> : ;>:0;L=><C <8=8<C<C, > :>B>@KE A>>1I0;>AL 2 [5]. >;55 B>3>, =8 >4=> 87 160 >1CG5=89 =5 >1=0@C68;> =5>6840==KE ?0B>;>389, A5BL 2A5340 ?@028;L=> >1CG0;0AL. -:A?5@8<5=BK 65 A G8AB>9 <0H8=>9 >H8 ?@825;8 : 7=0G8B5;L=> 1>;LH8< 2@5<5=0< >1CG5=8O. 0?@8<5@, ?@8 ( = 0,002 4;O >1CG5=8O A5B8 2 A@54=5< B@51>20;>AL >:>;> 2284 ?@54JO2;5=88 >1CG0NI53> <=>65AB20. 1AC645=85 ><18=8@>20==0O A5BL, 8A?>;L7CNI0O >1@0B=>5 @0A?@>AB@0=5=85 8 >1CG5=85 >H8, >1CG05BAO 7=0G8B5;L=> 1KAB@55, G5< :064K9 87 0;3>@8B<>2 2 >B45;L=>AB8, 8 >B=>A8B5;L=> =5GC2AB28B5;L=0 : 25;8G8=0< :>MDD8F85=B>2. !E>48<>ABL : 3;>10;L=><C <8=8<C<C 30@0=B8@C5BAO 0;3>@8B<>< >H8, 2 A>B=OE M:A?5@8<5=B>2 ?> >1CG5=8N A5BL =8 @07C =5 ?>?040;0 2 ;>2CH:8 ;>:0;L=KE <8=8<C<>2. @>1;5<0 A5B52>3> ?0@0;8G0 1K;0 @5H5=0 A ?><>ILN 0;3>@8B<0 A5;5:B82=>3> A60B8O 25A>2, :>B>@K9 >15A?5G8; AE>48<>ABL 2> 2A5E ?@54JO2;5==KE B5AB>2KE 7040G0E 157 ACI5AB25==>3> C25;8G5=8O >1CG0NI53> 2@5<5=8. 5A<>B@O =0 B0:85 >1=0456820NI85 @57C;LB0BK, <5B>4 5I5 =5 8AA;54>20= 4> :>=F0, >A>15==> =0 1>;LH8E 7040G0E. =0G8B5;L=> 1>;LH0O @01>B0 ?>B@51C5BAO 4;O >?@545;5=8O 53> 4>AB>8=AB2 8 =54>AB0B:>2. 8B5@0BC@0 GemanS., GemanD. 1984. 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WassermannP.D. 1988. Combined backpropagation/Cauchi machine. Neural Networks. Abstracts of the First INNS Meeting, Boston 1988, vol.1, p.556. Elmsford, NY. Pergamon Press. ;020 6. !5B8 %>?D8;40 !5B8, @0AA<>B@5==K5 2 ?@54K4CI8E 3;020E, =5 8<5;8 >1@0B=KE A2O759, B.5. A2O759, 84CI8E >B 2KE>4>2 A5B59 8 8E 2E>40<. BACBAB285 >1@0B=>9 A2O78 30@0=B8@C5B 157CA;>2=CN CAB>9G82>ABL A5B59. =8 =5 <>3CB 2>9B8 2 @568<, :>340 2KE>4 15A?@5@K2=> 1;C6405B >B A>AB>O=8O : A>AB>O=8N 8 =5 ?@83>45= : 8A?>;L7>20=8N. > MB> 25AL<0 65;0B5;L=>5 A2>9AB2> 4>AB8305BAO =5 15A?;0B=>, A5B8 157 >1@0B=KE A2O759 >1;040NB 1>;55 >3@0=8G5==K<8 2>7<>6=>ABO<8 ?> A@02=5=8N A A5BO<8 A >1@0B=K<8 A2O7O<8. 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KE>4 B0:>3> =59@>=0 @025= 548=8F5, 5A;8 2725H5==0O AC<<0 2KE>4>2 A 4@C38E =59@>=>2 1>;LH5 ?>@>30 Tj, 2 ?@>B82=>< A;CG05 >=0 @02=0 =C;N. = 2KG8A;O5BAO A;54CNI8< >1@07><:  EMBED Equation.3 , (6.1) OUT, = 1, 5A;8 NETj>"j, OUT. = 0, 5A;8 NETj<"j, OUT =5 87<5=O5BAO, 5A;8 NETj = "j,  8A.6.1. 4=>A;>9=0O A5BL A >1@0B=K<8 A2O7O<8. C=:B8@=K5 ;8=88 >1>7=0G0NB =C;52K5 25A0 !>AB>O=85 A5B8  MB> ?@>AB> <=>65AB2> B5:CI8E 7=0G5=89 A83=0;>2 OUT >B 2A5E =59@>=>2.  ?5@2>=0G0;L=>9 A5B8 %>?D8;40 A>AB>O=85 :064>3> =59@>=0 <5=O;>AL 2 48A:@5B=K5 A;CG09=K5 <><5=BK 2@5<5=8, 2 ?>A;54CNI59 @01>B5 A>AB>O=8O =59@>=>2 <>3;8 <5=OBLAO >4=>2@5<5==>. "0: :0: 2KE>4>< 18=0@=>3> =59@>=0 <>65B 1KBL B>;L:> =>;L 8;8 548=8F0 (?@><56CB>G=KE C@>2=59 =5B), B> B5:CI55 A>AB>O=85 A5B8 O2;O5BAO 42>8G=K< G8A;><, :064K9 18B :>B>@>3> O2;O5BAO A83=0;>< OUT =5:>B>@>3> =59@>=0. $C=:F8>=8@>20=85 A5B8 ;53:> 287C0;878@C5BAO 35><5B@8G5A:8. 0 @8A.6.20 ?>:070= A;CG09 42CE =59@>=>2 2 2KE>4=>< A;>5, ?@8G5< :064>9 25@H8=5 :204@0B0 A>>B25BAB2C5B >4=> 87 G5BK@5E A>AB>O=89 A8AB5<K (00, 01, 10, 11). 0 @8A.6.21 ?>:070=0 B@5E=59@>==0O A8AB5<0, ?@54AB02;5==0O :C1>< (2 B@5E<5@=>< ?@>AB@0=AB25), 8<5NI8< 2>A5<L 25@H8=, :0640O 87 :>B>@KE ?><5G5=0 B@5E18B>2K< 18=0@=K< G8A;><.  >1I5< A;CG05 A8AB5<0 A n =59@>=0<8 8<55B 2n @07;8G=KE A>AB>O=89 8 ?@54AB02;O5BAO n-<5@=K< 38?5@:C1><.  8A.6.20. 20 =59@>=0 ?>@>640NB A8AB5<C A G5BK@L<O A>AB>O=8<8  8A.6.21. "@8 =59@>=0 ?>@>640NB A8AB5<C A 2>A5<LN A>AB>O=8O<8 >340 ?>405BAO =>2K9 2E>4=>9 25:B>@, A5BL ?5@5E>48B 87 25@H8=K 2 25@H8=C, ?>:0 =5 AB018;878@C5BAO. #AB>9G820O 25@H8=0 >?@545;O5BAO A5B52K<8 25A0<8, B5:CI8<8 2E>40<8 8 25;8G8=>9 ?>@>30. A;8 2E>4=>9 25:B>@ G0AB8G=> =5?@028;5= 8;8 =5?>;>=, B> A5BL AB018;878@C5BAO 2 25@H8=5, 1;8609H59 : 65;05<>9. #AB>9G82>ABL 0: 8 2 4@C38E A5BOE, 25A0 <564C A;>O<8 2 MB>9 A5B8 <>3CB @0AA<0B@820BLAO 2 2845 <0B@8FK W.  @01>B5 [2] ?>:070=>, GB> A5BL A >1@0B=K<8 A2O7O<8 O2;O5BAO CAB>9G82>9, 5A;8 55 <0B@8F0 A8<<5B@8G=0 8 8<55B =C;8 =0 3;02=>9 4803>=0;8, B.5. 5A;8 wij = wji 8 wii = 0 4;O 2A5E i. #AB>9G82>ABL B0:>9 A5B8 <>65B 1KBL 4>:070=0 A ?><>ILN M;530=B=>3> <0B5<0B8G5A:>3> <5B>40. >?CAB8<, GB> =0945=0 DC=:F8O, :>B>@0O 2A5340 C1K205B ?@8 87<5=5=88 A>AB>O=8O A5B8.  :>=F5 :>=F>2 MB0 DC=:F8O 4>;6=0 4>AB8GL <8=8<C<0 8 ?@5:@0B8BL 87<5=5=85, 30@0=B8@CO B5< A0<K< CAB>9G82>ABL A5B8. "0:0O DC=:F8O, =07K205<0O DC=:F859 O?C=>20, 4;O @0AA<0B@8205<KE A5B59 A >1@0B=K<8 A2O7O<8 <>65B 1KBL 22545=0 A;54CNI8< >1@07><:  EMBED Equation.3  (6.2) 345   8A:CAAB25==0O M=5@38O A5B8; wij  25A >B 2KE>40 =59@>=0 i : 2E>4C =59@>=0 j; OUTj  2KE>4 =59@>=0 j; Ij  2=5H=89 2E>4 =59@>=0 j; "j  ?>@>3 =59@>=0 j. 7<5=5=85 M=5@388 , 2K720==>5 87<5=5=85< A>AB>O=8O j-=59@>=0, 5ABL  EMBED Equation.3  (6.3) 345 OUTj  87<5=5=85 2KE>40 j-3> =59@>=0. >?CAB8<, GB> 25;8G8=0 NET =59@>=0 j 1>;LH5 ?>@>30. ">340 2K@065=85 2 A:>1:0E 1C45B ?>;>68B5;L=K<, 0 87 #@02=5=8O (6.1) A;54C5B, GB> 2KE>4 =59@>=0j 4>;65= 87<5=8BLAO 2 ?>;>68B5;L=CN AB>@>=C (8;8 >AB0BLAO 157 87<5=5=8O). -B> 7=0G8B, GB> OUT. <>65B 1KBL B>;L:> ?>;>68B5;L=K< 8;8 =C;5< 8  4>;6=> 1KBL >B@8F0B5;L=K<. !;54>20B5;L=>, M=5@38O A5B8 4>;6=0 ;81> C<5=LH8BLAO, ;81> >AB0BLAO 157 87<5=5=8O. 0;55, 4>?CAB8<, GB> 25;8G8=0 NET <5=LH5 ?>@>30. ">340 25;8G8=0 OUTj <>65B 1KBL B>;L:> >B@8F0B5;L=>9 8;8 =C;5<. !;54>20B5;L=>, >?OBL M=5@38O 4>;6=0 C<5=LH8BLAO 8;8 >AB0BLAO 157 87<5=5=8O.  >:>=G0B5;L=>, 5A;8 25;8G8=0 NET @02=0 ?>@>3C, j @02=0 =C;N 8 M=5@38O >AB05BAO 157 87<5=5=8O. -B> ?>:07K205B, GB> ;N1>5 87<5=5=85 A>AB>O=8O =59@>=0 ;81> C<5=LH8B M=5@38N, ;81> >AB028B 55 157 87<5=5=8O. ;03>40@O B0:><C =5?@5@K2=><C AB@5<;5=8N : C<5=LH5=8N M=5@38O 2 :>=F5 :>=F>2 4>;6=0 4>AB83=CBL <8=8<C<0 8 ?@5:@0B8BL 87<5=5=85. > >?@545;5=8N B0:0O A5BL O2;O5BAO CAB>9G82>9. !8<<5B@8O A5B8 O2;O5BAO 4>AB0B>G=K<, => =5 =5>1E>48<K< CA;>285< 4;O CAB>9G82>AB8 A8AB5<K. <55BAO <=>3> CAB>9G82KE A8AB5< (=0?@8<5@, 2A5 A5B8 ?@O<>3> 459AB28O!), :>B>@K5 5<C =5 C4>2;5B2>@ONB. >6=> ?@>45<>=AB@8@>20BL ?@8<5@K, 2 :>B>@KE =57=0G8B5;L=>5 >B:;>=5=85 >B A8<<5B@88 <>65B ?@82>48BL : =5?@5@K2=K< >AF8;;OF8O<. 4=0:> ?@81;865==>9 A8<<5B@88 >1KG=> 4>AB0B>G=> 4;O CAB>9G82>AB8 A8AB5<. AA>F80B82=0O ?0<OBL '5;>25G5A:0O ?0<OBL 0AA>F80B82=0, B.5. =5:>B>@>5 2>A?><8=0=85 <>65B ?>@>640BL 1>;LHCN A2O70==CN A =8< >1;0ABL. 0?@8<5@, =5A:>;L:> <C7K:0;L=KE B0:B>2 <>3CB 2K720BL F5;CN 30<<C GC2AB25==KE 2>A?><8=0=89, 2:;NG0O ?597068, 72C:8 8 70?0E8. 0?@>B82, >1KG=0O :><?LNB5@=0O ?0<OBL O2;O5BAO ;>:0;L=> 04@5AC5<>9, ?@54JO2;O5BAO 04@5A 8 872;5:05BAO 8=D>@<0F8O ?> MB><C 04@5AC. !5BL A >1@0B=>9 A2O7LN D>@<8@C5B 0AA>F80B82=CN ?0<OBL. >4>1=> G5;>25G5A:>9 ?0<OB8 ?> 7040==>9 G0AB8 =C6=>9 8=D>@<0F88 2AO 8=D>@<0F8O 872;5:05BAO 87 ?0<OB8. 'B>1K >@30=87>20BL 0AA>F80B82=CN ?0<OBL A ?><>ILN A5B8 A >1@0B=K<8 A2O7O<8, 25A0 4>;6=K 2K18@0BLAO B0:, GB>1K >1@07>2K20BL M=5@35B8G5A:85 <8=8<C<K 2 =C6=KE 25@H8=0E 548=8G=>3> 38?5@:C10. %>?D8;4 @07@01>B0; 0AA>F80B82=CN ?0<OBL A =5?@5@K2=K<8 2KE>40<8, 87<5=ONI8<8AO 2 ?@545;0E >B +1 4>  1, A>>B25BAB2CNI8E 42>8G=K< 7=0G5=8O<0 8 1, 0?><8=05<0O 8=D>@<0F8O :>48@C5BAO 42>8G=K<8 25:B>@0<8 8 E@0=8BAO 2 25A0E A>3;0A=> A;54CNI59 D>@<C;5:  EMBED Equation.3  (6.4) 345 B  G8A;> 70?><8=05<KE 2KE>4=KE 25:B>@>2; d  =><5@ 70?><8=05<>3> 2KE>4=>3> 25:B>@0; OUTi,j  i-:><?>=5=B0 70?><8=05<>3> 2KE>4=>3> 25:B>@0. -B> 2K@065=85 <>65B AB0BL 1>;55 OA=K<, 5A;8 70<5B8BL, GB> 25A>2>9 <0AA82 W <>65B 1KBL =0945= 2KG8A;5=85< 2=5H=53> ?@>872545=8O :064>3> 70?><8=05<>3> 25:B>@0 A A0<8< A>1>9 (5A;8 B@51C5<K9 25:B>@ 8<55B n :><?>=5=B, B> MB0 >?5@0F8O >1@07C5B <0B@8FC @07<5@>< ? E ?) 8 AC<<8@>20=85< <0B@8F, ?>;CG5==KE B0:8< >1@07><. -B> <>65B 1KBL 70?8A0=> 2 2845  EMBED Equation.3 , (6.5) 345 Di  i-9 70?><8=05<K9 25:B>@-AB@>:0. 0: B>;L:> 25A0 7040=K, A5BL <>65B 1KBL 8A?>;L7>20=0 4;O ?>;CG5=8O 70?><=5==>3> 2KE>4=>3> 25:B>@0 ?> 40==><C 2E>4=><C 25:B>@C, :>B>@K9 <>65B 1KBL G0AB8G=> =5?@028;L=K< 8;8 =5?>;=K<. ;O MB>3> 2KE>40< A5B8 A=0G0;0 ?@840NB 7=0G5=8O MB>3> 2E>4=>3> 25:B>@0. 0B5< 2E>4=>9 25:B>@ C18@05BAO 8 A5B8 ?@54>AB02;O5BAO 2>7<>6=>ABL @0AA;018BLAO, >?CAB82H8AL 2 1;8609H89 3;C1>:89 <8=8<C<. !5BL 84CI0O ?> ;>:0;L=><C =0:;>=C DC=:F88 M=5@388, <>65B 1KBL 70E20G5=0 ;>:0;L=K< <8=8<C<><, =5 4>AB83=C2 =08;CGH53> 2 3;>10;L=>< A<KA;5 @5H5=8O. 5?@5@K2=K5 A8AB5<K  @01>B5 [7] @0AA<>B@5=K <>45;8 A =5?@5@K2=>9 0:B820F8>==>9 DC=:F859 F, B>G=55 <>45;8@CNI59 18>;>38G5A:89 =59@>=.  >1I5< A;CG05 MB> S->1@07=0O 8;8 ;>38AB8G5A:0O DC=:F8O  EMBED Equation.3 , (6.6) 345 (  :>MDD8F85=B, >?@545;ONI89 :@CB87=C A83<>840;L=>9 DC=:F88. A;8 ( 25;8:>, F ?@81;8605BAO : >?8A0==>9 @0=55 ?>@>3>2>9 DC=:F88. 51>;LH85 7=0G5=8O ( 40NB 1>;55 ?>;>389 =0:;>=. 0: 8 4;O 18=0@=KE A8AB5<, CAB>9G82>ABL 30@0=B8@C5BAO, 5A;8 25A0 A8<<5B@8G=K, B.5. wij = wji 8 wii = 0 ?@8 2A5E i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exp( E/kT), 345   M=5@38O A8AB5<K; k  ?>AB>O==0O >;LF<0=0; "  B5<?5@0BC@0. BAN40 <>6=> 2845BL, GB> 8<55BAO :>=5G=0O 25@>OB=>ABL B>3>, GB> A8AB5<0 >1;0405B 2KA>:>9 M=5@3859 4065 ?@8 =87:8E B5<?5@0BC@0E. !E>4=K< >1@07>< 8<55BAO =51>;LH0O, => 2KG8A;O5<0O 25@>OB=>ABL, GB> G09=8: A 2>4>9 =0 >3=5 70<5@7=5B, ?@5645 G5< 70:8?5BL. !B0B8AB8G5A:>5 @0A?@545;5=85 M=5@389 ?>72>;O5B A8AB5<5 2KE>48BL 87 ;>:0;L=KE <8=8<C<>2 M=5@388.  B> 65 2@5<O 25@>OB=>ABL 2KA>:>M=5@35B8G5A:8E A>AB>O=89 1KAB@> C<5=LH05BAO A> A=865=85< B5<?5@0BC@K. !;54>20B5;L=>, ?@8 =87:8E B5<?5@0BC@0E 8<55BAO A8;L=0O B5=45=F8O 70=OBL =87:>M=5@35B8G5A:>5 A>AB>O=85. !B0B8AB8G5:85 A5B8 %>?D8;40 A;8 ?@028;0 87<5=5=8O A>AB>O=89 4;O 18=0@=>9 A5B8 %>?D8;40 7040=K AB0B8AB8G5A:8, 0 =5 45B5@<8=8@>20==>, :0: 2 C@02=5=88 (6.1), B> 2>7=8:05B A8AB5<0, 8<8B8@CNI0O >B683. ;O 55 @50;870F88 22>48BAO 25@>OB=>ABL 87<5=5=8O 25A0 :0: DC=:F8O >B 25;8G8=K, =0 :>B>@CN 2KE>4 =59@>=0 OUT ?@52KH05B 53> ?>@>3. CABL Ek = NETk  (k, 345 NETk  2KE>4 NET =59@>=0 k; (  ?>@>3 =59@>=0 k, 8  EMBED Equation.3 , (>B<5BLB5 25@>OB=>AB=CN DC=:F8N >;LF<0=0 2 7=0<5=0B5;5), 345 "  8A:CAAB25==0O B5<?5@0BC@0.  AB0488 DC=:F8>=8@>20=8O 8A:CAAB25==>9 B5<?5@0BC@5 " ?@8?8AK205BAO 1>;LH>5 7=0G5=85, =59@>=K CAB0=02;820NBAO 2 =0G0;L=>< A>AB>O=88, >?@545;O5<>< 2E>4=K< 25:B>@><, 8 A5B8 ?@54>AB02;O5BAO 2>7<>6=>ABL 8A:0BL <8=8<C< M=5@388 2 A>>B25BAB288 A =865A;54CNI59 ?@>F54C@>9: @8?8A0BL A>AB>O=8N :064>3> =59@>=0 A 25@>OB=>ABLN @k 7=0G5=85 548=8F0, 0 A 25@>OB=>ABLN 1 @k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quation.3 , B.5. ?> 2A5<C <=>65AB2C >1CG0NI8E 25:B>@>2 2KG8A;8BL 25@>OB=>ABL B>3>, GB> 7=0G5=8O >1>8E =59@>=>2 @02=K 548=8F5. 2. 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'B>1K C4>2;5B2>@8BL CA;>28N CAB>9G82>AB8, 2KE>4 =59@>=0 =5 A>548=O;AO A>?@>B82;5=85< A 53> A>1AB25==K< 2E>4><, 0 25A0 1@0;8AL A8<<5B@8G=K<8, B.5. A>?@>B82;5=85 >B 2KE>40 =59@>=0 i : 2E>4C =59@>=0 j 8<5;> BC 65 25;8G8=C, GB> 8 A>?@>B82;5=85 >B 2KE>40 =59@>=0 j : 2E>4C =59@>=0 i. 0<5B8<, GB> CA8;8B5;8 8<5NB ?@O<>9 8 8=25@B8@>20==K9 2KE>4K. -B> ?>72>;O5B A ?><>ILN >1KG=KE ?>;>68B5;L=KE A>?@>B82;5=89 @50;87>2K20BL 8 B5 A;CG08, :>340 25A0 4>;6=K 1KBL >B@8F0B5;L=K<8. 0 @8A.6.4 ?>:070=K 2A5 2>7<>6=K5 A>?@>B82;5=8O, ?@8 MB>< =8:>340 =5 2>7=8:05B =5>1E>48<>AB8 ?@8A>548=OBL :0: ?@O<>9, B0: 8 8=25@B8@>20==K9 2KE>4K =59@>=0 : 2E>4C 4@C3>3> =59@>=0.  @50;L=>9 A8AB5<5 :064K9 CA8;8B5;L >1;0405B :>=5G=K< 2E>4=K< A>?@>B82;5=85< 8 2E>4=>9 5<:>ABLN, GB> 4>;6=> CG8BK20BLAO ?@8 @0AG5B5 48=0<8G5A:>9 E0@0:B5@8AB8:8. ;O CAB>9G82>AB8 A5B8 =5 B@51C5BAO @025=AB20 MB8E ?0@0<5B@>2 4;O 2A5E CA8;8B5;59 8 8E A8<<5B@8G=>AB8. "0: :0: MB8 ?0@0<5B@K 2;8ONB ;8HL =0 2@5<O ?>;CG5=8O @5H5=8O, 0 =5 =0 A0<> @5H5=85, 4;O C?@>I5=8O 0=0;870 >=8 8A:;NG5=K. @54?>;0305BAO, GB> 8A?>;L7C5BAO ?>@>3>20O DC=:F8O (?@545; A83<>840;L=>9 DC=:F88 ?@8 (, AB@5<OI5<AO : 15A:>=5G=>AB8). 0;55, 2A5 2KE>4K 87<5=ONBAO 2 =0G0;5 48A:@5B=KE 8=B5@20;>2 2@5<5=8, =07K205<KE M?>E0<8.  =0G0;5 :064>9 M?>E8 8AA;54C5BAO AC<<0 2E>4>2 :064>3> =59@>=0. A;8 >=0 1>;LH5 ?>@>30, 2KE>4 ?@8=8<05B 548=8G=>5 7=0G5=85, 5A;8 <5=LH5  =C;52>5. 0 ?@>BO65=88 M?>E8 2KE>4K =59@>=>2 =5 87<5=ONBAO.  8A.6.4. '5BK@5E18B>2K9 0=0;>3>-F8D@>2>9 ?@5>1@07>20B5;L, 8A?>;L7CNI89 A5BL %>?D8;40 &5;LN O2;O5BAO B0:>9 2K1>@ A>?@>B82;5=89 (25A>2), GB> =5?@5@K2=> @0ABCI55 =0?@O65=85 X, ?@8;>65==>5 : >4=>2E>4>2><C B5@<8=0;C, ?>@>6405B <=>65AAB2> 87 G5BK@5E 2KE>4>2, ?@54AB02;ONI8E 42>8G=CN 70?8AL G8A;0, 25;8G8=0 :>B>@>3> ?@81;865==> @02=0 2E>4=><C =0?@O65=8N (@8A.6.5). ?@545;8< A=0G0;0 DC=:F8N M=5@388 A;54CNI8< >1@07><:  EMBED Equation.3 , (6.7) 345 X  2E>4=>5 =0?@O65=85. >340  <8=8<878@>20=>, B> ?>;CG0NBAO =C6=K5 2KE>4K. 5@2>5 2K@065=85 2 A:>1:0E <8=8<878@C5BAO, :>340 42>8G=>5 G8A;>, >1@07>20==>5 2KE>40<8, =081>;55 1;87:> (2 A@54=5:204@0B8G=>< A<KA;5) : 0=0;>3>2>9 25;8G8=5 2E>40 X. B>@>5 2K@065=85 2 A:>1:0E >1@0I05BAO 2 =C;L, :>340 2A5 2KE>4K @02=K1 8;80, B5< A0<K< =0:;04K20O >3@0=8G5=85, GB> 2KE>4K ?@8=8<0NB B>;L:> 42>8G=K5 7=0G5=8O. 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"0: :0: A5BL 87 n 42>8G=KE =59@>=>2 <>65B 8<5BL 2n A>AB>O=89, B> 8AA;54>20B5;8 1K;8 C482;5=K, >1=0@C682, GB> <0:A8<0;L=0O 5<:>ABL ?0<OB8 >:070;0AL 7=0G8B5;L=> <5=LH59. 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E 48=0<8G5A:>5 ?>2545=85 >B:@K205B =>2K5 8=B5@5A=K5 2>7<>6=>AB8 8 AB028B A?5F8D8G5A:85 ?@>1;5<K. 0: >B<5G05BAO 2 3;.9, MB8 2>7<>6=>AB8 8 ?@>1;5<K A>E@0=ONBAO ?@8 @50;870F88 =59@>==KE A5B59 2 2845 >?B8G5A:8E A8AB5<. 8B5@0BC@0 Abu-MostafaY.S., St. Jacques, J. 1985. Information capacity of the Hopfield model. IEEE Transactions on Information Theory 31(4):461-64. CohenM.A., GrossbergS.G. 1983. Absolute stability of global pattern formation and parallel memory storage by compatitive neural networks. IEEE Transactions on Systems, Man and Cybernetics 13:815-26. QareyM.R., JohnsonD.S. 1979. Computers and intrac-tality. New York: W.H. Freeman. GrossbergS. 1987. The adapptive brain, vol.1 and2. Amsterdam: North-Holland. HintonG.E., SejnowskiT.J. 1986. Learning and relearning in Boltzmann machines. In Parallel distributed processing, vol.1, pp.282-317. Cambridge, MA: MIT Press. HorfieldJ.J. 1982. Neural networks and physical systems with emergent collective computational abilities. 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AA>F80B82=0O ?0<OBL, @0AA<>B@5==0O 2 3;.6, O2;O5BAO, AB@>3> 3>2>@O, 02B>0AA>F80B82=>9, MB> >7=0G05B, GB> >1@07 <>65B 1KBL 7025@H5= 8;8 8A?@02;5=, => =5 <>65B 1KBL 0AA>F88@>20= A 4@C38< >1@07><. 0==K9 D0:B O2;O5BAO @57C;LB0B>< >4=>C@>2=52>9 AB@C:BC@K 0AA>F80B82=>9 ?0<OB8, 2 :>B>@>9 25:B>@ ?>O2;O5BAO =0 2KE>45 B5E 65 =59@>=>2, =0 :>B>@K5 ?>ABC?05B 2E>4=>9 25:B>@. 2C=0?@02;5==0O 0AA>F80B82=0O ?0<OBL () O2;O5BAO 35B5@>0AA>F80B82=>9; 2E>4=>9 25:B>@ ?>ABC?05B =0 >48= =01>@ =59@>=>2, 0 A>>B25BAB2CNI89 2KE>4=>9 25:B>@ 2K@010BK205BAO =0 4@C3>< =01>@5 =59@>=>2. 0: 8 A5BL %>?D8;40,  A?>A>1=0 : >1>1I5=8N, 2K@010BK20O ?@028;L=K5 @50:F88, =5A<>B@O =0 8A:065==K5 2E>4K. @><5 B>3>, <>3CB 1KBL @50;87>20=K 040?B82=K5 25@A88 , 2K45;ONI85 MB0;>==K9 >1@07 87 70HC<;5==KE M:75<?;O@>2. -B8 2>7<>6=>AB8 A8;L=> =0?><8=0NB ?@>F5AA <KH;5=8O G5;>25:0 8 ?>72>;ONB 8A:CAAB25==K< =59@>==K< A5BO< A45;0BL H03 2 =0?@02;5=88 <>45;8@>20=8O <>730.  ?>A;54=8E ?C1;8:0F8OE [9,12] ?@54AB02;5=> =5A:>;L:> D>@< @50;870F88 42C=0?@02;5==>9 0AA>F80B82=>9 ?0<OB8. 0: 1>;LH8=AB2> 206=KE 8459, 87;>65==K5 2 MB8E @01>B0E 8458 8<5NB 3;C1>:85 :>@=8; =0?@8<5@, 2 @01>B5 @>AA15@30 [6] ?@54AB02;5=K =5:>B>@K5 206=K5 4;O  :>=F5?F88.  40==>9 @01>B5 AAK;:8 ?@82>4OBAO =5 A F5;LN @07@5H5=8O 2>?@>A0 > ?@8>@8B5B5 8AA;54>20B5;LA:8E @01>B, 0 8A:;NG8B5;L=> 4;O >A25I5=8O 8E 2:;040 2 8AA;54>20B5;LA:CN B5<0B8:C. !" #"#    8A.7.1. >=D83C@0F8O 42C=0?@02;5==>9 0AA>F80B82=>9 ?0<OB8 0 @8A.7.1 ?@82545=0 107>20O :>=D83C@0F8O . -B0 :>=D83C@0F8O ACI5AB25==> >B;8G05BAO >B 8A?>;L7C5<>9 2 @01>B5 [9]. =0 2K1@0=0 B0:8< >1@07><, GB>1K ?>4G5@:=CBL AE>4AB2> A A5BO<8 %>?D8;40 8 ?@54CA<>B@5BL C25;8G5=8O :>;8G5AB20 A;>52. 0 @8A.7.1 2E>4=>9 25:B>@  >1@010BK205BAO <0B@8F59 25A>2 W A5B8, 2 @57C;LB0B5 G53> 2K@010BK205BAO 25:B>@ 2KE>4=KE A83=0;>2 =59@>=>2 . 5:B>@  70B5< >1@010BK205BAO B@0=A?>=8@>20==>9 <0B@8F59 Wt 25A>2 A5B8, :>B>@0O 2K@010BK205B =>2K5 2KE>4=K5 A83=0;K, ?@54AB02;ONI85 A>1>9 =>2K9 2E>4=>9 25:B>@ . -B>B ?@>F5AA ?>2B>@O5BAO 4> B5E ?>@, ?>:0 A5BL =5 4>AB83=5B AB018;L=>3> A>AB>O=8O, 2 :>B>@>< =8 25:B>@ , =8 25:B>@  =5 87<5=ONBAO. 0<5B8<, GB> =59@>=K 2 A;>OE1 8 2 DC=:F8>=8@CNB, :0: 8 2 4@C38E ?0@0483<0E, 2KG8A;OO AC<<C 2725H5==KE 2E>4>2 8 2KG8A;OO ?> =59 7=0G5=85 DC=:F88 0:B820F88 F. -B>B ?@>F5AA <>65B 1KBL 2K@065= A;54CNI8< >1@07><:  EMBED Equation.3  (7.1) 8;8 2 25:B>@=>9 D>@<5:  = F(AW), (7.2) 345   25:B>@ 2KE>4=KE A83=0;>2 =59@>=>2 A;>O2,   25:B>@ 2KE>4=KE A83=0;>2 =59@>=>2 A;>O1, W  <0B@8F0 25A>2 A2O759 <564C A;>O<81 8 2, F  DC=:F8O 0:B820F88. =0;>38G=> A = F(BWt) (7.3) 345 Wt O2;O5BAO B@0=A?>78F859 <0B@8FK W. 0: >B<5G5=> 2 3;.1, @>AA15@3 ?>:070; ?@58<CI5AB20 8A?>;L7>20=8O A83<>840;L=>9 (;>38AB8G5A:>9) DC=:F88 0:B820F88  EMBED Equation.3  (7.3) 345 OUTi  2KE>4 =59@>=0 i, NETi  2725H5==0O AC<<0 2E>4=KE A83=0;>2 =59@>=0 i, (  :>=AB0=B0, >?@545;ONI0O AB5?5=L :@8287=K.  ?@>AB59H8E 25@A8OE  7=0G5=85 :>=AB0=BK ( 2K18@05BAO 1>;LH8<, 2 @57C;LB0B5 G53> DC=:F8O 0:B820F88 ?@81;8605BAO : ?@>AB>9 ?>@>3>2>9 DC=:F88.  40;L=59H8E @0AAC645=8OE 1C45< ?@54?>;030BL, GB> 8A?>;L7C5BAO ?>@>3>20O DC=:F8O 0:B820F88. @8<5< B0:65, GB> ACI5AB2C5B ?0<OBL 2=CB@8 :064>3> =59@>=0 2 A;>OE1 8 2 8 GB> 2KE>4=K5 A83=0;K =59@>=>2 87<5=ONBAO >4=>2@5<5==> A :064K< B0:B>< A8=E@>=870F88, >AB020OAL ?>AB>O==K<8 <564C MB8<8 B0:B0<8. "0:8< >1@07><, ?>2545=85 =59@>=>2 <>65B 1KBL >?8A0=> A;54CNI8<8 ?@028;0<8: OUTi(n+1) = 1, 5A;8 NETi(n)>0, OUTi(n+l) = 0, 5A;8 NETi(n)<0, OUTi(n+l) = OUT(n), 5A;8 NETi(n) = 0, 345 OUTi(n) ?@54AB02;O5B A>1>9 25;8G8=C 2KE>4=>3> A83=0;0 =59@>=0 i 2 <><5=B 2@5<5=8 ?. 0<5B8<, GB> :0: 8 2 >?8A0==KE @0=55 A5BOE A;>90 =5 ?@>872>48B 2KG8A;5=89 8 =5 8<55B ?0<OB8; >= O2;O5BAO B>;L:> A@54AB2>< @0A?@545;5=8O 2KE>4=KE A83=0;>2 A;>O2 : M;5<5=B0< <0B@8FK Wt. !!" +% !!&& >;3>2@5<5==0O ?0<OBL (8;8 0AA>F80F88) @50;87C5BAO 2 25A>2KE <0AA820E W 8 Wt. 064K9 >1@07 A>AB>8B 87 42CE 25:B>@>2: 25:B>@0 A, O2;ONI53>AO 2KE>4>< A;>O1, 8 25:B>@0 B, 0AA>F88@>20==>3> >1@070, O2;ONI53>AO 2KE>4>< A;>O2. ;O 2>AAB0=>2;5=8O 0AA>F88@>20==>3> >1@070 25:B>@ A 8;8 53> G0ABL :@0B:>2@5<5==> CAB0=02;820NBAO =0 2KE>40E A;>O 1. 0B5< 25:B>@ A C40;O5BAO 8 A5BL ?@82>48BAO 2 AB018;L=>5 A>AB>O=85, 2K@010BK20O 0AA>F88@>20==K9 25:B>@ B =0 2KE>45 A;>O2. 0B5< 25:B>@ B 2>7459AB2C5B G5@57 B@0=A?>=8@>20==CN <0B@8FC Wt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quation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i 8<5NB @07<5@=>ABL B0:CN 65, :0: 8 25:B>@K i. 04> >B<5B8BL, GB> MB> =5 O2;O5BAO =5>1E>48<K< CA;>285< 4;O @01>BK 0;3>@8B<0; 0AA>F80F88 <>3CB 1KBL AD>@<8@>20=K 8 <564C 25:B>@0<8 @07;8G=>9 @07<5@=>AB8. AE>4=K9 25:B>@AA>F88@>20==K9 25:B>@8=0@=0O 25@A8OA1 = (1,0,0)B1 = (0,0,1)A 1 = (1, 1, 1)B 1 = ( 1, 1,1)A2 = (0,1,0)B2 = (0,1,0)A 1 = ( 1,1, 1)B 1 = ( 1,1, 1)A3 = (0,0,1)B3 = (1,0,0)A 1 = ( 1, 1,1)B 1 = (1, 1, 1) KG8A;O5< 25A>2CN <0B@8FC W = A 1t B 1 + A 2t B 2 + A 3t B 3  1 11+1 11+ 111= 1 1311 1 11 1 1 11 13 111 11 111 1 13 1 1 0;55 ?@8:;04K20O 2E>4=>9 25:B>@  = (1,0,0), 2KG8A;O5< 2KE>4=>9 25:B>@  O = A1t W = (1,0,0) x1 13=( 1, 1,3) 13 13 1 1 A?>;L7CO ?>@>3>2>5 ?@028;> bi = 1, 5A;8 oi > 0, bi = 0, 5A;8 oi < 0, bi = 0, =5 87<5=O5BAO, 5A;8 oi = 0 2KG8A;O5< B 1 = (0,0,1), GB> O2;O5BAO B@51C5<>9 0AA>F80F859. 0B5<, ?>4020O 25:B>@  1 G5@57 >1@0B=CN A2O7L =0 2E>4 ?5@2>3> A;>O : Wt ?>;CG05< O = B 1 Wt = (0,0,1) x1 13=(3, 1, 1) 13 13 1 1 GB> 405B 7=0G5=85 (1,0,0) ?>A;5 ?@8<5=5=8O ?>@>3>2>9 DC=:F88, >1@07CO 25;8G8=C 25:B>@0 A1. -B>B ?@8<5@ ?>:07K205B, :0: 2E>4=>9 25:B>@ A A 8A?>;L7>20=85< <0B@8FK W ?@>872>48B 2KE>4=>9 25:B>@ B.  A2>N >G5@54L 25:B>@ B A 8A?>;L7>20=85< <0B@8FK Wt ?@>872>48B 25:B>@ A, B0:8< >1@07>< 2 A8AB5<5 D>@<8@C5BAO CAB>9G82>5 A>AB>O=85 8 @57>=0=A.  >1;0405B A?>A>1=>ABLN : >1>1I5=8N. 0?@8<5@, 5A;8 =57025@H5==K9 8;8 G0AB8G=> 8A:065==K9 25:B>@ ?>405BAO 2 :0G5AB25 A, A5BL 8<55B B5=45=F8N : 2K@01>B:5 70?><=5==>3> 25:B>@0 B, :>B>@K9 2 A2>N >G5@54L AB@5<8BAO 8A?@028BL >H81:8 2 A. >7<>6=>, 4;O MB>3> ?>B@51C5BAO =5A:>;L:> ?@>E>4>2, => A5BL AE>48BAO : 2>A?@>872545=8N 1;8609H53> 70?><=5==>3> >1@070. !8AB5<K A >1@0B=>9 A2O7LN <>3CB 8<5BL B5=45=F8N : :>;510=8O<; MB> >7=0G05B, GB> >=8 <>3CB ?5@5E>48BL >B A>AB>O=8O : A>AB>O=8N, =8:>340 =5 4>AB830O AB018;L=>AB8.  [9] 4>:070=>, GB> 2A5  157CA;>2=> AB018;L=K ?@8 ;N1KE 7=0G5=8OE 25A>2 A5B8. -B> 206=>5 A2>9AB2> 2>7=8:05B 87 >B=>H5=8O B@0=A?>=8@>20=8O <564C 42C<O 25A>2K<8 <0B@8F0<8 8 >7=0G05B, GB> ;N1>9 =01>@ 0AA>F80F89 <>65B 1KBL 87CG5= 157 @8A:0 2>7=8:=>25=8O =5AB018;L=>AB8. !CI5AB2C5B 2708<>A2O7L <564C  8 @0AA<>B@5==K<8 2 3;.6 A5BO<8 %>?D8;40. A;8 25A>20O <0B@8F0 W O2;O5BAO :204@0B=>9 8 A8<<5B@8G=>9, B> W=Wt.  MB>< A;CG05, 5A;8 A;>81 8 2 O2;ONBAO >4=8< 8 B5< 65 =01>@>< =59@>=>2,  ?@52@0I05BAO 2 02B>0AA>F80B82=CN A5BL %>?D8;40. !", /" 0: 8 A5B8 %>?D8;40,  8<55B >3@0=8G5=8O =0 <0:A8<0;L=>5 :>;8G5AB2> 0AA>F80F89, :>B>@K5 >=0 <>65B B>G=> 2>A?@>8725AB8. A;8 MB>B ;8<8B ?@52KH5=, A5BL <>65B 2K@01>B0BL =525@=K9 2KE>4=>9 A83=0;, 2>A?@>872>4O 0AA>F80F88, :>B>@K< =5 >1CG5=0.  @01>B5 [9] ?@82545=K >F5=:8, 2 A>>B25BAB288 A :>B>@K<8 :>;8G5AB2> 70?><=5==KE 0AA>F80F89 =5 <>65B ?@52KH0BL :>;8G5AB20 =59@>=>2 2 <5=LH5< A;>5. @8 MB>< ?@54?>;0305BAO, GB> 5<:>ABL ?0<OB8 <0:A8<878@>20=0 ?>A@54AB2>< A?5F80;L=>3> :>48@>20=8O, ?@8 :>B>@>< :>;8G5AB2> :><?>=5=B A> 7=0G5=8O<8 +1 @02=> :>;8G5AB2C :><?>=5=B A> 7=0G5=8O<8 1 2 :064>< 18?>;O@=>< 25:B>@5. -B0 >F5=:0 >:070;0AL A;8H:>< >?B8<8AB8G=>9. 01>B0 [13] ?> >F5=:5 5<:>AB8 A5B59 %>?D8;40 <>65B 1KBL ;53:> @0AH8@5=0 4;O . >6=> ?>:070BL, GB> 5A;8 L 25:B>@>2 2K1@0=K A;CG09=> 8 ?@54AB02;5=K 2 C:070==>9 2KH5 D>@<5, 8 5A;8 L <5=LH5 G5< n/(21og2 ?), 345 ?  :>;8G5AB2> =59@>=>2 2 =08<5=LH5< A;>5, B>340 2A5 70?><=5==K5 >1@07K, 70 8A:;NG5=85< <0;>9 G0AB8, <>3CB 1KBL 2>AAB0=>2;5=K. 0?@8<5@, 5A;8 ?=1024, B>340 L 4>;6=> 1KBL <5=LH5 51. A;8 2A5 >1@07K 4>;6=K 2>AAB0=02;820BLAO, L 4>;6=> 1KBL <5=LH5 re/(41og2 ?), B> 5ABL <5=LH5 25. -B8, A:>@55 >7040G820NI85, @57C;LB0BK ?>:07K20NB, GB> 1>;LH85 A8AB5<K <>3CB 70?><8=0BL B>;L:> C<5@5==>5 :>;8G5AB2> 0AA>F80F89.  @01>B5 [7] ?>:070=>, GB>  <>65B 8<5BL 4> 2n AB018;L=KE A>AB>O=89, 5A;8 ?>@>3>2>5 7=0G5=85 " 2K18@05BAO 4;O :064>3> =59@>=0. "0:0O :>=D83C@0F8O, :>B>@CN 02B>@K =0720;8 =53><>35==>9 , O2;O5BAO @0AH8@5=85< 8AE>4=>9 3><>35==>9 , 2 :>B>@>9 2A5 ?>@>38 1K;8 =C;52K<8. >48D8F8@>20==0O ?5@540B>G=0O DC=:F8O =59@>=0 ?@8=8<05B 2 MB>< A;CG05 A;54CNI89 284: OUTi(n+l) = l, 5A;8 NETi(n) > Ti, OUTi(n+l) = l, 5A;8 NETi(n) < Ti, OUTi(n+l) = OUTi(n), 5A;8 NETi(n) = Ti, 345 OUTi(t)  2KE>4 =59@>=0 i 2 <><5=B 2@5<5=8 t. >A@54AB2>< 2K1>@0 A>>B25BAB2CNI53> ?>@>30 4;O :064>3> =59@>=0 :>;8G5AB2> AB018;L=KE A>AB>O=89 <>65B 1KBL A45;0=> ;N1K< 2 480?07>=5 >B 1 4> 2, 345 ? 5ABL :>;8G5AB2> =59@>=>2 2 <5=LH5< A;>5.  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"5>@5<0 >E5=-@>AA15@30 [1] ?>:07K205B, GB> B0:0O A5BL O2;O5BAO 157CA;>2=> AB018;L=>9, 5A;8 25A>2K5 <0B@8FK A8<<5B@8G=K. 0 ?@0:B8:5 A5B8 >1KG=> AB018;L=K 4065 2 A;CG05 >BACBAB28O A8<<5B@88 25A>2KE <0B@8F. 4=0:> =58725AB=>, :0:85 >A>15==>AB8 25A>2KE <0B@8F <>3CB ?@825AB8 : =5CAB>9G82>AB8 DC=:F8>=8@>20=8O A5B8. .' 3@0=8G5==0O 5<:>ABL ?0<OB8 , ;>6=K5 >B25BK 8 =5:>B>@0O =5?@54A:07C5<>ABL ?>2545=8O ?@825;8 : @0AA<>B@5=8N 55 :0: CAB0@52H59 <>45;8 8A:CAAB25==KE =59@>==KE A5B59. -B>B 2K2>4 >?@545;5==> O2;O5BAO ?@56452@5<5==K<.  8<55B <=>3> ?@58<CI5AB2: >=0 A>2<5AB8<0 A 0=0;>3>2K<8 AE5<0<8 8 >?B8G5A:8<8 A8AB5<0<8; 4;O =55 1KAB@> AE>4OBAO :0: ?@>F5AA >1CG5=8O B0:, 8 ?@>F5AA 2>AAB0=>2;5=8O 8=D>@<0F88; >=0 8<55B ?@>ABCN 8 8=BC8B82=> ?@82;5:0B5;L=CN D>@<C DC=:F8>=8@>20=8O.  A2O78 A 1KAB@K< @0728B85< B5>@88 <>3CB 1KBL =0945=K <5B>4K, >1JOA=ONI85 ?>2545=85  8 @07@5H0NI85 55 ?@>1;5<K. 8B5@0BC@0 CohenM., GrossbergS. 1983. 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CAB0=02;8205BAO 2 10000, :@8B5@89 AE>4AB20 C4>2;5B2>@O5BAO 8 0;3>@8B< >1CG5=8O CAB0=02;8205B 2B>@CN 8 B@5BLN 548=8FK 25:B>@>2 "2 8 2 2 =C;L, @07@CH0O 70?><=5==K9 >1@07. 0AHB018@>20=85 25A>2 bij ?@54>B2@0I05B MB> =565;0B5;L=>5 ?>2545=85. @54?>;>68<, GB> 2 2K@065=88 (8.2) 8A?>;L7C5BAO 7=0G5=85 L=2, B5< A0<K< >?@545;OO A;54CNICN D>@<C;C:  EMBED Equation.3  =0G5=8O 25:B>@>2 1C4CB B>340 AB@5<8BLAO : 25;8G8=0< >4020O =0 2E>4 A5B8 25:B>@ %1 , ?>;CG8< 2>71C640NI55 2>7459AB285 1,0 4;O =59@>=01 2 A;>5 @0A?>7=020=8O 8 4;O =59@>=02; B0:8< >1@07><, =59@>=1 (?@028;L=K9) 2K83@05B A>@52=>20=85. =0;>38G=> ?@54JO2;5=85 25:B>@0 %2 2K7>25B C@>25=L 2>71C645=8O 1,0 4;O =59@>=01 8 3/2 4;O =59@>=02, B5< A0<K< A=>20 ?@028;L=> 2K18@0O ?>1548B5;O. =8F80;870F8O 25A>2 bij =8F80;870F8O 25A>2 bij <0;K<8 7=0G5=8O<8 O2;O5BAO ACI5AB25==>9 4;O :>@@5:B=>3> DC=:F8>=8@>20=8O A8AB5< APT. A;8 >=8 A;8H:>< 1>;LH85, 2E>4=>9 25:B>@0 :>B>@K9 @0=55 1K; 70?><=5=, 1C45B A:>@55 0:B82878@>20BL =5A2O70==K9 =59@>=, G5< @0=55 >1CG5==K9. K@065=85 (8.1), >?@545;ONI55 =0G0;L=K5 7=0G5=8O 25A>2, ?>2B>@O5BAO 745AL 4;O A?@02:8  EMBED Equation.3  4;O 2A5E i, j, (8.1) #AB0=>2:0 MB8E 25A>2 2 <0;K5 25;8G8=K 30@0=B8@C5B, GB> =5A2O70==K5 =59@>=K =5 1C4CB ?>;CG0BL 2>71C645=8O 1>;LH53>, G5< >1CG5==K5 =59@>=K 2 A;>5 @0A?>7=020=8O. A?>;L7CO ?@54K4CI89 ?@8<5@ A L=2, B=5 8 bij<1/3, ?@>872>;L=> CAB0=>28< bij=1/6. ! B0:8<8 25A0<8 ?@54JO2;5=85 25:B>@0, :>B>@><C A5BL 1K;0 @0=55 >1CG5=0, ?@82545B : 1>;55 2KA>:><C C@>2=N 0:B820F88 4;O ?@028;L=> >1CG5==>3> =59@>=0 2 A;>5 @0A?>7=020=8O, G5< 4;O =5A2O70==>3> =59@>=0. 0?@8<5@, 4;O =5A2O70==>3> =59@>=0 %1 1C45B ?@>872>48BL 2>71C645=85 1/6, 2 B> 2@5<O :0: %2 1C45B ?@>872>48BL 2>71C645=85 ; 8 B> 8 4@C3>5 =865 2>71C645=8O 4;O >1CG5==KE =59@>=>2. >8A:. >65B ?>:070BLAO, GB> 2 >?8A0==KE 0;3>@8B<0E >BACBAB2C5B =5>1E>48<>ABL =0;8G8O D07K ?>8A:0 70 8A:;NG5=85< A;CG0O, :>340 4;O 2E>4=>3> 25:B>@0 4>;65= 1KBL @0A?@545;5= =>2K9 =5A2O70==K9 =59@>=. -B> =5 A>2A5< B0:; ?@54JO2;5=85 2E>4=>3> 25:B>@0, AE>4=>3>, => =5 01A>;NB=> 845=B8G=>3> >4=><C 87 70?><=5==KE >1@07>2, <>65B ?@8 ?5@2>< 8A?KB0=88 =5 >15A?5G8BL 2K1>@ =59@>=0 A;>O @0A?>7=020=8O A C@>2=5< AE>4AB20 1>;LH8< @, E>BO B0:>9 =59@>= 1C45B ACI5AB2>20BL. 0: 8 2 ?@54K4CI5< ?@8<5@5, ?@54?>;>68<, GB> A5BL >1CG05BAO A;54CNI8< 42C< 25:B>@0<: X1 = 1 0 0 0 0 X2 = 1 1 1 0 0 A 25:B>@0<8 25A>2 i, >1CG5==K<8 A;54CNI8< >1@07>< B1 = 1 0 0 0 0 B2 = 0 0 "5?5@L ?@8;>68< 2E>4=>9 25:B>@ X3 = 11000.  MB>< A;CG05 2>71C645=85 =59@>=01 2 A;>5 @0A?>7=020=8O 1C45B 1,0, 0 =59@>=0 2 B>;L:> 2/3. 59@>=1 2K945B ?>1548B5;5< (E>BO >= =5 ;CGH8< >1@07>< A>>B25BAB2C5B 2E>4=><C 25:B>@C), 25:B>@ ! ?>;CG8B 7=0G5=85 11000, S 1C45B @02=> . A;8 C@>25=L AE>4AB20 CAB0=>2;5= 2 3/4, =59@>=1 1C45B 70B>@<>65= 8 =59@>=2 2K83@05B A>ABO70=85. ! AB0=5B @02=K< 11000, S AB0=5B @02=K<1, :@8B5@89 AE>4AB20 1C45B C4>2;5B2>@5= 8 ?>8A: 70:>=G8BAO. "5>@5<K APT  @01>B5 [2] 4>:070=K =5:>B>@K5 B5>@5<K, ?>:07K20NI85 E0@0:B5@8AB8:8 A5B59 APT. '5BK@5 @57C;LB0B0, ?@82545==K5 =865, O2;ONBAO >4=8<8 87 =081>;55 206=KE: >A;5 AB018;870F88 ?@>F5AA0 >1CG5=8O ?@54JO2;5=85 >4=>3> 87 >1CG0NI8E 25:B>@>2 (8;8 25:B>@0 A ACI5AB25==K<8 E0@0:B5@8AB8:0<8 :0B53>@88) 1C45B 0:B82878@>20BL B@51C5<K9 =59@>= A;>O @0A?>7=020=8O 157 ?>8A:0. -B0 E0@0:B5@8AB8:0 ?@O<>3> 4>ABC?0 >?@545;O5B 1KAB@K9 4>ABC? : ?@5420@8B5;L=> 87CG5==K< >1@070<. @>F5AA ?>8A:0 O2;O5BAO CAB>9G82K<. >A;5 >?@545;5=8O 2K83@02H53> =59@>=0 2 A5B8 =5 1C45B 2>71C645=89 4@C38E =59@>=>2 2 @57C;LB0B5 87<5=5=8O 25:B>@>2 2KE>40 A;>O A@02=5=8O !; B>;L:> A83=0; A1@>A0 <>65B 2K720BL B0:85 87<5=5=8O. @>F5AA >1CG5=8O O2;O5BAO CAB>9G82K<. 1CG5=85 =5 1C45B 2K7K20BL ?5@5:;NG5=8O A >4=>3> 2>71C645==>3> =59@>=0 A;>O @0A?>7=020=8O =0 4@C3>9. @>F5AA >1CG5=8O :>=5G5=. N10O ?>A;54>20B5;L=>ABL ?@>872>;L=KE 2E>4=KE 25:B>@>2 1C45B ?@>872>48BL AB018;L=K9 =01>@ 25A>2 ?>A;5 :>=5G=>3> :>;8G5AB20 >1CG0NI8E A5@89; ?>2B>@ONI85AO ?>A;54>20B5;L=>AB8 >1CG0NI8E 25:B>@>2 =5 1C4CB ?@82>48BL : F8:;8G5A:><C 87<5=5=8N 25A>2. .' !5B8 APT O2;ONBAO 8=B5@5A=K< 8 206=K< 284>< A8AB5<. =8 A?>A>1=K @5H8BL 48;5<<C AB018;L=>AB8-?;0AB8G=>AB8 8 E>@>H> @01>B0NB A 4@C38E B>G5: 7@5=8O. @E8B5:BC@0 APT A:>=AB@C8@>20=0 ?> ?@8=F8?C 18>;>38G5A:>3> ?>4>18O; MB> >7=0G05B, GB> 55 <5E0=87<K 2> <=>3>< A>>B25BAB2CNB <5E0=87<0< <>730 (:0: <K 8E ?>=8<05<). 4=0:> >=8 <>3CB >:070BLAO =5 2 A>AB>O=88 <>45;8@>20BL @0A?@545;5==CN ?0<OBL, :>B>@CN <=>385 @0AA<0B@820NB :0: 206=CN E0@0:B5@8AB8:C DC=:F89 <>730. -:75<?;O@K APT ?@54AB02;ONB A>1>9 101CH:8=K C75;:8; ?>B5@O >4=>3> C7;0 @07@CH05B 2AN ?0<OBL. 0<OBL <>730, =0?@>B82, @0A?@545;5=0 ?> 25I5AB2C <>730, 70?><=5==K5 >1@07K <>3CB G0AB> ?5@568BL 7=0G8B5;L=K5 D878G5A:85 ?>2@5645=8O <>730 157 ?>;=>9 8E ?>B5@8. 065BAO ;>38G=K< 87CG5=85 0@E8B5:BC@, A>>B25BAB2CNI8E =0H5<C ?>=8<0=8N >@30=870F88 8 DC=:F89 <>730. '5;>25G5A:89 <>73 ?@54AB02;O5B ACI5AB2CNI55 4>:070B5;LAB2> B>3> D0:B0, GB> @5H5=85 ?@>1;5<K @0A?>7=020=8O >1@07>2 2>7<>6=>. 065BAO @07C<=K< M<C;8@>20BL @01>BC <>730, 5A;8 <K E>B8< ?>2B>@8BL 53> @01>BC. 4=0:> :>=B@0@3C<5=B>< O2;O5BAO 8AB>@8O ?>;5B>2; G5;>25: =5 A<>3 >B>@20BLAO >B 75<;8 4> B5E ?>@, ?>:0 =5 ?5@5AB0; 8<8B8@>20BL 42865=8O :@K;L52 8 ?>;5B ?B8F. 8B5@0BC@0 CarpenterG., GrossbergS. 1986. Neural dynamics of category learning and recognition: Attention; memory consolidation and amnesia. In Brain Structure, Learning and Memory (AAAS Symposium Series), eds. J.Davis., R.Newburgh and E.Wegman. CarpenterG., GrossbergS. 1987. A massively parallel architecture for a self-organizing neural pattern recognition machine. Computing Vision. Graphics, and Image Processing 37:54-115. CarpenterG., GrossbergS. 1987 ART-2: Self-organization of stable category recognition codes for analog input patterns. Applied Optics 26(23):4919-30. CrossbergS. 1987. Competitive learning: From interactive activation to adaptive resonanse. Cognitive Science 11:23-63. LippmanR.P. 1987. An introduction to computing with neurals nets. IEEE Transactions on Acosufics, Speech and Signal Processing, April, pp.4-22. ;020 9. ?B8G5A:85 =59@>==K5 A5B8 A?>;L7>20=85 8 >1CG5=85 =59@>==KE A5B59 B@51C5B 2 >A=>2=>< 42CE B8?>2 >?5@0F89 =04 40==K<8: 2KG8A;5=89 8 ?5@540G8 40==KE. KG8A;8B5;L=K5 DC=:F88 ;53:> 8 ?@>AB> 2K?>;=ONBAO M;5:B@>==K<8 A8AB5<0<8. -;5<5=BK 8=B53@0;L=KE F5?59 @01>B0NB 2 =0=>A5:C=4=KE 8=B5@20;0E. @><5 B>3>, >=8 8<5NB @07<5@K, 87<5@O5<K5 2 <8:@>=0E, 8 <>3CB 8<5BL AB>8<>ABL <5=55 A>B>9 F5=B0 70 25=B8;L. 040G8 ?5@540G8 40==KE @5H0NBAO =5 ?@>AB>. -;5:B@>==K5 A83=0;K 2 8=B53@0;L=KE A5BOE B@51CNB =0;8G8O :>=45=A0B>@>2 4;O ?5@540G8 A83=0;>2 >B 25=B8;O : 25=B8;N. %>BO :>=45=A0B>@K 8<5NB <8:@>==K5 @07<5@K, 70=8<05<>5 ?@>AB@0=AB2> (A CG5B>< ?@>AB@0=AB20, =5>1E>48<>3> 4;O 87>;OF88 >4=>3> :>=45=A0B>@0 >B 4@C3>3>) <>65B AB0BL =0AB>;L:> 1>;LH8<, GB> =0 ?;0AB8=5 :@5<=8O =5 >AB0=5BAO <5AB0 4;O @07<5I5=8O 2KG8A;8B5;L=KE F5?59. 5A<>B@O =0 B>, GB> ACI5AB2C5B B5E=>;>38O @50;870F88 >1K:=>25==KE F8D@>2KE :><?LNB5@>2 2 2845 1>;LH8E DC=:F8>=0;L=KE 1;>:>2 A >B=>A8B5;L=> =51>;LH8< :>;8G5AB2>< :>=45=A0B>@>2, MB0 B5E=>;>38O =5 3>48BAO 2 A;CG05 <0AA>2>3> ?0@0;;5;87<0. =0;>38G=>5 @5H5=85 4;O 8A:CAAB25==KE =59@>==KE A5B59 2 =0AB>OI55 2@5<O =58725AB=>. >I=>ABL =59@>==KE A5B59 >?@545;O5BAO 1>;LH8< :>;8G5AB2>< A2O759; 27OBK5 >B45;L=> M;5<5=BK 8<5NB >B=>A8B5;L=> <0;K5 2KG8A;8B5;L=K5 2>7<>6=>AB8. !5@L57=CN ?@>1;5<C ?@54AB02;O5B 4>AB865=85 B@51C5<>9 A2O70==>AB8 2 M;5:B@>==KE F5?OE.  [10] ?@54?>;0305BAO, GB> ?;>B=>ABL :>=45=A0B>@>2 2 42C<5@=>9 A8AB5<5 4>;6=0 C<5=LH0BLAO >1@0B=> ?@>?>@F8>=0;L=> :204@0BC @0AAB>O=8O >B =59@>=0-8AB>G=8:0; 2 ?@>B82=>< A;CG05 >BACBAB2C5B 2>7<>6=>ABL @50;870F88 A8AB5<K 2 2845 8=B53@0;L=KE F5?59. -B> >3@0=8G5=85 8<55B >A>1>5 7=0G5=85 ?@8 @50;870F88 A5B59 A ?>;=K<8 2708<=K<8 A2O7O<8. 50;870F8O =59@>==KE A5B59 2 2845 >?B8G5A:8E A8AB5< ?>72>;O5B @5H8BL MBC ?@>1;5<C. 708<=>5 A>548=5=85 =59@>=>2 A ?><>ILN A25B>2KE ;CG59 =5 B@51C5B 87>;OF88 <564C A83=0;L=K<8 ?CBO<8, A25B>2K5 ?>B>:8 <>3CB ?@>E>48BL >48= G5@57 4@C3>9 157 2708<=>3> 2;8O=8O. >;55 B>3>, A83=0;L=K5 ?CB8 <>3CB 1KBL @0A?>;>65=K 2 B@5E 87<5@5=8OE. (=B53@0;L=K5 F5?8 O2;ONBAO ACI5AB25==> ?;0=0@=K<8 A =5:>B>@>9 @5;L5D=>ABLN, >1CA;>2;5==>9 <=>65AB2>< A;>52.) ;>B=>ABL ?CB59 ?5@540G8 >3@0=8G5=0 B>;L:> @07<5@0<8 8AB>G=8:>2 A25B0, 8E 4825@35=F859 8 @07<5@0<8 45B5:B>@0. >B5=F80;L=> MB8 @07<5@K <>3CB 8<5BL 25;8G8=C 2 =5A:>;L:> <8:@>=. 0:>=5F, 2A5 A83=0;L=K5 ?CB8 <>3CB @01>B0BL >4=>2@5<5==>, B5< A0<K< >15A?5G820O >3@><=K9 B5<? ?5@540G8 40==KE.  @57C;LB0B5 A8AB5<0 A?>A>1=0 >15A?5G8BL ?>;=K9 =01>@ A2O759, @01>B0NI8E A> A:>@>ABLN A25B0. ?B8G5A:85 =59@>==K5 A5B8 <>3CB B0:65 >15A?5G8BL 206=K5 ?@58<CI5AB20 ?@8 ?@>2545=88 2KG8A;5=89. 5;8G8=0 A8=0?B8G5A:8E A2O759 <>65B 70?><8=0BLAO 2 3>;>3@0<<0E A 2KA>:>9 AB5?5=LN ?;>B=>AB8; =5:>B>@K5 >F5=:8 40NB B5>@5B8G5A:89 ?@545; 2 1012 18B =0 :C18G5A:89 A0=B8<5B@. %>BO B0:85 7=0G5=8O =0 ?@0:B8:5 =5 4>AB83=CBK, ACI5AB2CNI89 C@>25=L ?;>B=>AB8 ?0<OB8 >G5=L 2KA>:. @><5 B>3>, 25A0 <>3CB <>48D8F8@>20BLAO 2 ?@>F5AA5 @01>BK A5B8, >1@07CO ?>;=>ABLN 040?B82=CN A8AB5<C. #G8BK20O MB8 ?@58<CI5AB20, <>6=> 7040BL 2>?@>A, ?>G5<C =0@O4C A >?B8G5A:8<8 =59@>==K<8 A5BO<8 2>>1I5 @0AA<0B@820NBAO 4@C385 A?>A>1K @50;870F88.  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KE>4>< A8AB5<K O2;O5BAO AC?5@?>78F8O 25:B>@=>-<0B@8G=KE ?@>872545=89 2E>4=>3> 25:B>@0 8 70?><=5==KE >1@07F>2KE 25:B>@>2. -B>B >?B8G5A:89 >1@07 2K@010BK205BAO >?B8G5A:>9 <0B@8F59 2=CB@5==8E A2O759 8 ?@8:;04K205BAO : <0AA82C >?B8G5A:8E =59@>=>2, 70<K:0O :>=BC@ >1@0B=>9 A2O78.  8A.9.8. 0AA82 >?B8G5A:8E =59@>=>2 1AC645=85 >?B8G5A:>9 <0B@8FK 2=CB@5==8E A2O759 745AL A8;L=> C?@>I5=>; 45B0;8 @50;870F88 2:;NG0NB A;>6=CN >?B8G5A:CN B5E=8:C DC=:F8>=8@>20=8O, 2KE>4OICN 70 @0<:8 40==>9 @01>BK. =B5@5ACNI85AO G8B0B5;8 >BAK;0NBAO : @01>B0< [12] 8 [5].  8A.9.9. !83<>840;L=0O DC=:F8O 0:B820F88 .' ?B8G5A:85 =59@>==K5 A5B8 ?@54;030NB >3@><=K5 2K3>4K A B>G:8 7@5=8O A:>@>AB8 8 ?;>B=>AB8 2=CB@5==8E A2O759. =8 <>3CB 1KBL 8A?>;L7>20=K (2 B>9 8;8 8=>9 D>@<5) 4;O @50;870F88 A5B59 D0:B8G5A:8 A ;N1>9 0@E8B5:BC@>9.  =0AB>OI55 2@5<O >3@0=8G5=8O M;5:B@>==>->?B8G5A:8E CAB@>9AB2 A>740NB <=>65AB2> A5@L57=KE ?@>1;5<, :>B>@K5 4>;6=K 1KBL @5H5=K ?@5645, G5< >?B8G5A:85 =59@>==K5 A5B8 ?>;CG0B H8@>:>5 ?@8<5=5=85. 4=0:> CG8BK20O, GB> 1>;LH>5 :>;8G5AB2> ?@52>AE>4=KE 8AA;54>20B5;59 @01>B05B =04 MB>9 ?@>1;5<>9, 0 B0:65 1>;LHCN ?>445@6:C A> AB>@>=K 2>5==KE, <>6=> =045OBLAO =0 1KAB@K9 ?@>3@5AA 2 MB>9 >1;0AB8. 8B5@0BC@0 Abu-MostafaY.S., PsaltisD. 1987. 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"5E=8:0 <8:@>?@>1 ?>72>;O5B ?@>25AB8 8AA;54>20=8O 2=CB@5==59 M;5:B@>E8<88 C7;>2, >4=0:> B@C4=> :>=B@>;8@>20BL >4=>2@5<5==> 1>;LH>5 :>;8G5AB2> C7;>2 8 =01;N40BL 8E 2708<>459AB285. 0:>=5F, MB8G5A:85 A>>1@065=8O 70?@5I0NB <=>385 206=K5 M:A?5@8<5=BK, :>B>@K5 <>3CB 1KBL 2K?>;=5=K B>;L:> =0 ;N4OE. >;LH>5 7=0G5=85 8<5;8 M:A?5@8<5=BK =04 682>B=K<8, >4=0:> 682>B=K5 =5 >1;040NB A?>A>1=>ABO<8 G5;>25:0 >?8AK20BL A2>8 2?5G0B;5=8O. 5A<>B@O =0 MB8 >3@0=8G5=8O, <=>3>5 1K;> 87CG5=> 1;03>40@O 1;5ABOI5 704C<0==K< M:A?5@8<5=B0<. 0?@8<5@, 2 [1] >?8A0= M:A?5@8<5=B, 2 :>B>@>< :>BOB0 2K@0I820;8AL 2 287C0;L=>< >:@C65=88, A>AB>OI5< B>;L:> 87 3>@87>=B0;L=KE G5@=KE 8 15;KE ?>;>A. 725AB=>, GB> >?@545;5==K5 >1;0AB8 :>@K GC2AB28B5;L=K : C3;C >@85=B0F88, ?>MB><C C MB8E :>B>2 =5 @0728;8AL =59@>=K, GC2AB28B5;L=K5 : 25@B8:0;L=K< ?>;>A0<. -B>B @57C;LB0B =02>48B =0 <KA;L, GB> <>73 <;5:>?8B0NI8E =5 O2;O5BAO ?>;=>ABLN ?@54CAB0=>2;5==K< 4065 =0 ?@8<8B82=>< C@>2=5 @0A?>7=020=8O >@85=B0F88 ;8=89. 0?@>B82, >= ?>AB>O==> A0<>>@30=87C5BAO, >A=>2K20OAL =0 >?KB5. 0 <8:@>A:>?8G5A:>< C@>2=5 >1=0@C65=>, GB> =59@>=K >1;040NB :0: 2>>71C640NI8<8, B0: 8 B>@<>7OI8<8 A8=0?A0<8. 5@2K5 AB@5<OBAO : 2>71C645=8N =59@>=0; ?>A;54=85 ?>402;ONB 53> 2>71C645=85 (A<. ?@8;>65=85 ). -B> =02>48B =0 <KA;L, GB> <>73 040?B8@C5BAO ;81> 87<5=5=85< 2>7459AB28O MB8E A8=0?A>2, ;81> A>740=85< 8;8 @07@CH5=85< A8=0?A>2 2 @57C;LB0B5 2>7459AB28O >:@C60NI59 A@54K. 0==>5 ?@54?>;>65=85 >AB05BAO ?>:0 38?>B57>9 A >3@0=8G5==K< D878>;>38G5A:8< ?>4B25@645=85<. 4=0:> 8AA;54>20=8O, ?@>2545==K5 2 @0<:0E MB>9 38?>B57K, ?@825;8 : A>740=8N F8D@>2KE <>45;59, =5:>B>@K5 87 :>B>@KE ?>:07K20NB 70<5G0B5;L=K5 A?>A>1=>AB8 : 040?B82=><C @0A?>7=020=8N >1@07>2. "  A=>2K20OAL =0 B5:CI8E 7=0=8OE 0=0B><88 8 D878>;>388 <>730, 2 @01>B5 [2] @07@01>B0= :>3=8B@>=, 38?>B5B8G5A:0O <>45;L A8AB5<K 2>A?@8OB8O G5;>25:0. ><?LNB5@=K5 <>45;8, 8AA;54>20==K5 2 [2], ?@>45<>=AB@8@>20;8 2?5G0B;ONI85 A?>A>1=>AB8 040?B82=>3> @0A?>7=020=8O >1@07>2, ?>1C640O D878>;>3>2 8AA;54>20BL A>>B25BAB2CNI85 <5E0=87<K <>730. -B> 2708<=> CA8;820NI55 2708<>459AB285 <564C 8A:CAAB25==K<8 =59@>==K<8 A5BO<8, D878>;>3859 8 ?A8E>;>3859 <>65B >:070BLAO A@54AB2><, ?>A@54AB2>< :>B>@>3> 1C45B A> 2@5<5=5< 4>AB83=CB> ?>=8<0=85 <5E0=87<>2 <>730. !B@C:BC@0 >3=8B@>= :>=AB@C8@C5BAO 2 2845 A;>52 =59@>=>2, A>548=5==KE A8=0?A0<8. 0: ?>:070=> =0 @8A.10.1, ?@54A8=0?B8G5A:89 =59@>= 2 >4=>< A;>5 A2O70= A ?>ABA8=0?B8G5A:8< =59@>=>< 2 A;54CNI5< A;>5. <5NBAO 420 B8?0 =59@>=>2: 2>71C640NI85 C7;K, :>B>@K5 AB@5<OBAO 2K720BL 2>71C645=85 ?>ABA8=0?B8G5A:>3> C7;0, 8 B>@<>7OI85 C7;K, :>B>@K5 B>@<>7OB MB> 2>71C645=85. >71C645=85 =59@>=0 >?@545;O5BAO 2725H5==>9 AC<<>9 53> 2>71C640NI8E 8 B>@<>7OI8E 2E>4>2, >4=0:> 2 459AB28B5;L=>AB8 <5E0=87< O2;O5BAO 1>;55 A;>6=K<, G5< ?@>AB>5 AC<<8@>20=85.  8A.10.1. @5A8=0?B8G5A:85 8 ?>ABA8=0?B8G5A:85 =59@>=K 0 @8A.10.2 ?>:070=>, GB> :064K9 =59@>= A2O70= B>;L:> A =59@>=0<8 2 A>A54=59 >1;0AB8, =07K205<>9 >1;0ABLN A2O78. -B> >3@0=8G5=85 >1;0AB8 A2O78 A>3;0AC5BAO A 0=0B><859 7@8B5;L=>9 :>@K, 2 :>B>@>9 @54:> A>548=ONBAO <564C A>1>9 =59@>=K, @0A?>;030NI85AO 4@C3 >B 4@C30 =0 @0AAB>O=88 1>;55 >4=>3> <8;;8<5B@0.  @0AA<0B@8205<>9 <>45;8 =59@>=K C?>@O4>G5=K 2 2845 A;>52 A> A2O7O<8 >B >4=>3> A;>O : A;54CNI5<C. -B> B0:65 0=0;>38G=> ?>A;>9=>9 AB@C:BC@5 7@8B5;L=>9 :>@K 8 4@C38E G0AB59 3>;>2=>3> <>730.  8A.10.2. 1;0ABL A2O759 =59@>=0 1CG5=85 "0: :0: :>3=8B@>= @50;87>20= 2 2845 <=>3>A;>9=>9 A5B8, 2>7=8:0NB A;>6=K5 ?@>1;5<K >1CG5=8O, A2O70==K5 A 2K1@0==>9 AB@C:BC@>9. 2B>@ >B25@3 C?@02;O5<>5 >1CG5=85, :0: 18>;>38G5A:8 =5?@024>?>4>1=>5, 8A?>;L7CO 270<5= MB>3> >1CG5=85 157 CG8B5;O. >;CG0O >1CG0NI89 =01>@ 2E>4=KE >1@07>2, A5BL A0<>>@30=87C5BAO ?>A@54AB2>< 87<5=5=8O A8;K A8=0?B8G5A:8E A2O759. @8 MB>< >BACBAB2CNB ?@5420@8B5;L=> >?@545;5==K5 2KE>4=K5 >1@07K, ?@54AB02;ONI85 B@51C5<CN @50:F8N A5B8, >4=0:> A5BL A0<>=0AB@08205BAO A F5;LN @0A?>7=020=8O 2E>4=KE >1@07>2 A 70<5G0B5;L=>9 B>G=>ABLN. ;3>@8B< >1CG5=8O :>3=8B@>=0 O2;O5BAO :>=F5?BC0;L=> ?@82;5:0B5;L=K<.  7040==>9 >1;0AB8 A;>O >1CG05BAO B>;L:> =081>;55 A8;L=> 2>71C645==K9 =59@>=. 2B>@ A@02=8205B MB> A M;8B=K< >1CG5=85<, ?@8 :>B>@>< >1CG0NBAO B>;L:> C<=K5 M;5<5=BK. "5 =59@>=K, :>B>@K5 C65 E>@>H> >1CG5=K, GB> 2K@0605BAO A8;>9 8E 2>71C645=8O, ?>;CG0B ?@8@0I5=85 A8;K A2>8E A8=0?A>2 A F5;LN 40;L=59H53> CA8;5=8O A2>53> 2>71C645=8O. 0 @8A.10.3 ?>:070=>, GB> >1;0AB8 A2O78 A>A54=8E C7;>2 7=0G8B5;L=> ?5@5:@K20NBAO. -B> @0AB>G8B5;L=>5 4C1;8@>20=85 DC=:F89 >?@024K205BAO 2708<=>9 :>=:C@5=F859 <564C 1;8609H8<8 C7;0<8. 065 5A;8 C7;K 2 =0G0;L=K9 <><5=B 8<5NB 01A>;NB=> 845=B8G=K9 2KE>4, =51>;LH85 >B:;>=5=8O 2A5340 8<5NB <5AB>; >48= 87 C7;>2 2A5340 1C45B 8<5BL 1>;55 A8;L=CN @50:F8N =0 2E>4=>9 >1@07, G5< A>A54=85. 3> A8;L=>5 2>71C645=85 1C45B >:07K20BL A45@6820NI55 2>7459AB285 =0 2>71C645=85 A>A54=8E C7;>2, 8 B>;L:> 53> A8=0?AK 1C4CB CA8;820BLAO; A8=0?AK A>A54=8E C7;>2 >AB0=CBAO =587<5==K<8. >71C640NI89 =59@>=. @C1> 3>2>@O, 2KE>4 2>71C640NI53> =59@>=0 2 :>3=8B@>=5 >?@545;O5BAO >B=>H5=85< 53> 2>71C640NI8E 2E>4>2 : B>@<>7OI8< 2E>40<. -B0 =5>1KG=0O DC=:F8O 8<55B 206=K5 ?@58<CI5AB20, :0: ?@0:B8G5A:85, B0: 8 B5>@5B8G5A:85.  8A.10.3. 1;0ABL A2O78 A >1;0ABLN :>=:C@5=F88 !C<<0@=K9 2>71C640NI89 2E>4 2 =59@>= 2725H5==>9 AC<<>9 2E>4>2 >B 2>71C640NI8E ?@54H5AB2CNI5< A;>5. =0;>38G=> AC<<0@=K9 2E>4 / O2;O5BAO 2725H5==>9 AC<<>9 2E>4>2 >B 2A5E B>@<>7OI8E =59@>=>2.  A8<2>;L=>< 2845  EMBED Equation.3 ,  EMBED Equation.3 , 345 ai  25A i-3> 2>71C640NI53> A8=0?A0, ui  2KE>4 i-3> 2>71C640NI53> =59@>=0, bj  25A j-3> B>@>7OI53> A8=0?A0, vj  2KE>4 j-3> B>@>7OI53> =59@>=0. 0<5B8<, GB> 25A0 8<5NB B>;L:> ?>;>68B5;L=K5 7=0G5=8O. KE>4 =59@>=0 70B5< 2KG8A;O5BAO A;54CNI8< >1@07><:  EMBED Equation.3  OUT = NET, ?@8 NETe"0, OUT = 0, ?@8 NET<0. @54?>;030O, GB> NET 8<55B ?>;>68B5;L=>5 7=0G5=85, MB> <>6=> 70?8A0BL A;54CNI8< >1@07><:  EMBED Equation.3  >340 B>@<>7OI89 2E>4 <0; (I << 1), OUT <>65B 1KBL 0??@>:A8<8@>20=> :0: OUT =   I, GB> A>>B25BAB2C5B 2K@065=8N 4;O >1KG=>3> ;8=59=>3> ?>@>3>2>3> M;5<5=B0 (A =C;52K< ?>@>3><). ;3>@8B< >1CG5=8O :>3=8B@>=0 ?>72>;O5B 25A0< A8=0?A>2 2>7@0AB0BL 157 >3@0=8G5=89. ;03>40@O >BACBAB28N <5E0=87<0 C<5=LH5=8O 25A>2 >=8 ?@>AB> 2>7@0AB0NB 2 ?@>F5AA5 >1CG5=8O.  >1KG=KE ;8=59=KE ?>@>3>2KE M;5<5=B0E MB> ?@825;> 1K : ?@>872>;L=> 1>;LH><C 2KE>4C M;5<5=B0.  :>3=8B@>=5 1>;LH85 2>71C640NI85 8 B>@<>7OI85 2E>4K @57C;LB8@CNBAO 2 >3@0=8G820NI59 D>@<C;5 2840:  EMBED Equation.3 , 5A;8 E >> 1 8 I >> 1.  40==>< A;CG05 OUT >?@545;O5BAO >B=>H5=85< 2>71C640NI8E 2E>4>2 : B>@<>7OI8< 2E>40<, 0 =5 8E @07=>ABLN. "0:8< >1@07><, 25;8G8=0 OUT >3@0=8G8205BAO, 5A;8 >10 2E>40 2>7@0AB0NB 2 >4=>< 8 B>< 65 480?07>=5 X. @54?>;>682, GB> MB> B0:,  8 I <>6=> 2K@078BL A;54CNI8< >1@07><:  = @%, I = qX, p,q  :>=AB0=BK, 8 ?>A;5 =5:>B>@KE ?@5>1@07>20=89  EMBED Equation.3 . -B0 DC=:F8O 2>7@0AB05B ?> 70:>=C 515@0-$5E=5@0, :>B>@K9 G0AB> 8A?>;L7C5BAO 2 =59@>D878>;>388 4;O 0??@>:A8<0F88 =5;8=59=KE A>>B=>H5=89 2E>40/2KE>40 A5=A>@=KE =59@>=>2. @8 8A?>;L7>20=88 MB>3> A>>B=>H5=8O =59@>= :>3=8B@>=0 2 B>G=>AB8 M<C;8@C5B @50:F8N 18>;>38G5A:8E =59@>=>2. -B> 45;05B 53> :0: <>I=K< 2KG8A;8B5;L=K< M;5<5=B><, B0: 8 B>G=>9 <>45;LN 4;O D878>;>38G5A:>3> <>45;8@>20=8O. ">@<>7OI85 =59@>=K.  :>3=8B@>=5 A;>9 A>AB>8B 87 2>71C640NI8E 8 B>@<>7OI8E C7;>2. 0: ?>:070=> =0 @8A.10.4, =59@>= A;>O2 8<55B >1;0ABL A2O78, 4;O :>B>@>9 >= 8<55B A8=0?B8G5A:85 A>548=5=8O A =01>@>< 2KE>4>2 =59@>=>2 2 A;>51. =0;>38G=> 2 A;>51 ACI5AB2C5B B>@<>7OI89 =59@>=, 8<5NI89 BC 65 >1;0ABL A2O78. !8=0?B8G5A:85 25A0 B>@<>7OI8E C7;>2 =5 87<5=ONBAO 2 ?@>F5AA5 >1CG5=8O; 8E 25A0 70@0=55 CAB0=>2;5=K B0:8< >1@07><, GB> AC<<0 25A>2 2 ;N1>< 87 B>@<>7OI8E =59@>=>2 @02=0 548=8F5.  A>>B25BAB288 A MB8<8 >3@0=8G5=8O<8, 2KE>4 B>@<>7OI53> C7;0 INHIB O2;O5BAO 2725H5==>9 AC<<>9 53> 2E>4>2, :>B>@K5 2 40==>< A;CG05 ?@54AB02;ONB A>1>9 A@54=55 0@8D<5B8G5A:>5 2KE>4>2 2>71C640NI8E =59@>=>2, : :>B>@K< >= ?>4A>548=5=. "0:8< >1@07><,  8A.10.4. !;>8 :>3=8B@>=0  EMBED Equation.3 , 345  EMBED Equation.3 , ci  2>71C640NI89 25A i. @>F54C@0 >1CG5=8O. 0: >1JOA=O;>AL @0=55, 25A0 2>71C640NI8E =59@>=>2 87<5=ONBAO B>;L:> B>340, :>340 =59@>= 2>71C645= A8;L=55, G5< ;N1>9 87 C7;>2 2 >1;0AB8 :>=:C@5=F88. A;8 MB> B0:, 87<5=5=85 2 ?@>F5AA5 >1CG5=8O ;N1>3> 87 53> 25A>2 <>65B 1KBL >?@545;5=> A;54CNI8< >1@07><: ai = qcjuj, 345 Aj  B>@<>7OI89 25A A2O78 =59@>=0 j 2 A;>51 A B>@<>7OI8< =59@>=>< i, 8j  2KE>4 =59@>=0 j 2 A;>51, 0i  2>71C640NI89 25A i, q - =>@<8@CNI89 :>MDD8F85=B >1CG5=8O. 7<5=5=85 B>@<>7OI8E 25A>2 =59@>=0 i 2 A;>52 ?@>?>@F8>=0;L=> >B=>H5=8N 2725H5==>9 AC<<K 2>71C640NI8E 2E>4>2 : C42>5==><C B>@<>7OI5<C 2E>4C. KG8A;5=8O ?@>2>4OBAO ?> D>@<C;5  EMBED Equation.3 . >340 2>71C645==KE =59@>=>2 2 >1;0AB8 :>=:C@5=F88 =5B, 4;O 87<5=5=8O 25A>2 8A?>;L7CNBAO 4@C385 2K@065=8O. -B> =5>1E>48<>, ?>A:>;L:C ?@>F5AA >1CG5=8O =0G8=05BAO A =C;52K<8 7=0G5=8O<8 25A>2; ?>MB><C ?5@2>=0G0;L=> =5B 2>71C645==KE =59@>=>2 =8 2 >4=>9 >1;0AB8 :>=:C@5=F88, 8 >1CG5=85 ?@>872>48BLAO =5 <>65B. > 2A5E A;CG0OE, :>340 ?>1548B5;O 2 >1;0AB8 :>=:C@5=F88 =59@>=>2 =5B, 87<5=5=85 25A>2 =59@>=>2 2KG8A;O5BAO A;54CNI8< >1@07><: ai = q cjuj, bi = q INHIB, 345 q  ?>;>68B5;L=K9 >1CG0NI89 :>MDD8F85=B <5=LH89, G5< q. @82545==0O AB@0B538O =0AB@>9:8 30@0=B8@C5B, GB> C7;K A 1>;LH>9 @50:F859 70AB02;ONB 2>71C640NI85 A8=0?AK, :>B>@K<8 >=8 C?@02;ONB, C25;8G820BLAO A8;L=55, G5< B>@<>7OI85 A8=0?AK.  =0>1>@>B, C7;K, 8<5NI85 <0;CN @50:F8N, 2K7K20NB <0;>5 2>7@0AB0=85 2>71C640NI8E A8=0?A>2, => 1>;LH55 .2>7@0AB0=85 B>@<>7OI8E A8=0?A>2. "0:8< >1@07><, 5A;8 C75;1 2 A;>51 8<55B 1>;LH89 2KE>4, A8=0?A 01 2>7@0AB5B 1>;LH5, G5< A8=0?A b1.  =0>1>@>B, C7;K, 8<5NI85 <0;K9 2KE>4, >15A?5G0B <0;CN 25;8G8=C 4;O ?@8@0I5=8O 0i. 4=0:> 4@C385 C7;K 2 >1;0AB8 A2O78 1C4CB 2>71C640BLAO, B5< A0<K< C25;8G820O A83=0; INHIB 8 7=0G5=8O bi.  ?@>F5AA5 >1CG5=8O 25A0 :064>3> C7;0 2 A;>52 =0AB@0820NBAO B0:8< >1@07><, GB> 2<5AB5 >=8 A>AB02;ONB H01;>=, A>>B25BAB2CNI89 >1@070<, :>B>@K5 G0AB> ?@54JO2;ONBAO 2 ?@>F5AA5 >1CG5=8O. @8 ?@54JO2;5=88 AE>4=>3> >1@070 H01;>= A>>B25BAB2C5B 5<C 8 C75; 2K@010BK205B 1>;LH>9 2KE>4=>9 A83=0;. !8;L=> >B;8G0NI89AO >1@07 2K@010BK205B <0;K9 2KE>4 8 >1KG=> ?>402;O5BAO :>=:C@5=F859. 0B5@0;L=>5 B>@<>65=85. 0 @8A.10.4 ?>:070=>, GB> :064K9 =59@>= A;>O2 ?>;CG05B ;0B5@0;L=>5 B>@<>65=85 >B =59@>=>2, @0A?>;>65==KE 2 53> >1;0AB8 :>=:C@5=F88. ">@<>7OI89 =59@>= AC<<8@C5B 2E>4K >B 2A5E =59@>=>2 2 >1;0AB8 :>=:C@5=F88 8 2K@010BK205B A83=0;, AB@5<OI89AO : B>@<>65=8N F5;52>3> =59@>=0. -B>B <5B>4 O2;O5BAO MDD5:B=K<, => A 2KG8A;8B5;L=>9 B>G:8 7@5=8O <54;5==K<. = >E20BK205B 1>;LHCN A8AB5<C A >1@0B=>9 A2O7LN, 2:;NG0NICN :064K9 =59@>= 2 A;>5; 4;O 53> AB018;870F88 <>65B ?>B@51>20BLAO 1>;LH>5 :>;8G5AB2> 2KG8A;8B5;L=KE 8B5@0F89. ;O CA:>@5=8O 2KG8A;5=89 2 @01>B5 [2] 8A?>;L7C5BAO >AB@>C<=K9 <5B>4 CA:>@5==>3> ;0B5@0;L=>3> B>@<>65=8O (@8A.10.5). 45AL 4>?>;=8B5;L=K9 C75; ;0B5@0;L=>3> B>@<>65=8O >1@010BK205B 2KE>4 :064>3> 2>71C640NI53> C7;0 4;O <>45;8@>20=8O B@51C5<>3> ;0B5@0;L=>3> B>@<>65=8O. !=0G0;0 >= >?@545;O5B A83=0;, @02=K9 AC<<0@=><C B>@<>7OI5<C 2;8O=8N 2 >1;0AB8 :>=:C@5=F88:  8A.10.5. #A:>@5==>5 B>@<>65=85  EMBED Equation.3 , 345 OUTi  2KE>4 i-3> =59@>=0 2 >1;0AB8 :>=:C@5=F88, gi  25A A2O78 >B MB>3> =59@>=0 : ;0B5@0;L=>-B>@<>7OI5<C =59@>=C; gi 2K1@0=K B0:8< >1@07><, GB>  EMBED Equation.3 . 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AB>8<>ABL 2KG8A;5=89 C<5=LH0;0AL 2 420 @070 :064K5 420-B@8 3>40, B5=45=F8O, :>B>@0O, ?> 2A59 25@>OB=>AB8, A>E@0=8BAO 2 B5G5=85 :0: <8=8<C< 1;8609H8E 45AOB8 ;5B. 5A<>B@O =0 B>, GB> <=>385 ?>4E>4K, :0702H85AO =5@50;87C5<K<8 =5A:>;L:> ;5B =0704, O2;ONBAO >1I5?@8=OBK<8 A53>4=O 8 <>3CB >:070BLAO B@8280;L=K<8 G5@57 =5A:>;L:> ;5B, @50;870F8O <>45;59 =5>:>3=8B@>=0 =0 C=825@A0;L=KE :><?LNB5@0E O2;O5BAO 15A?5@A?5:B82=>9. 5>1E>48<> 4>AB83=CBL BKAOG5:@0B=KE C;CGH5=89 AB>8<>AB8 8 ?@>872>48B5;L=>AB8 :><?LNB5@>2 70 AG5B A?5F80;870F88 0@E8B5:BC@K 8 2=54@5=8O B5E=>;>388 !!, GB>1K A45;0BL =5>:>3=8B@>= ?@0:B8G5A:>9 A8AB5<>9 4;O @5H5=8O A;>6=KE ?@>1;5< @0A?>7=020=8O >1@07>2, >4=0:> =8 MB0, =8 :0:0O-;81> 4@C30O <>45;L 8A:CAAB25==KE =59@>==KE A5B59 =5 4>;6=K >B25@30BLAO B>;L:> =0 >A=>20=88 8E 2KA>:8E 2KG8A;8B5;L=KE B@51>20=89. 8B5@0BC@0 Blakemore!., CooperG.F. 1970. Development of the brain depends on the visual enviroment. Nature 228(5270):477 78. Fukushima. 1975. 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A=>2K20OAL =0 D878>;>38G5A:8E 8 ?A8E>;>38G5A:8E 8AA;54>20=8OE, %M11 2 [2] 8=BC8B82=> 2K428=C; 38?>B57C > B><, :0:8< >1@07>< <>65B >1CG0BLAO =01>@ 18>;>38G5A:8E =59@>=>2. 3> B5>@8O ?@54?>;0305B B>;L:> ;>:0;L=>5 2708<>459AB285 <564C =59@>=0<8 ?@8 >BACBAB288 3;>10;L=>3> CG8B5;O; A;54>20B5;L=>, >1CG5=85 O2;O5BAO =5C?@02;O5<K<, 5A<>B@O =0 B> GB> 53> @01>B0 =5 2:;NG05B <0B5<0B8G5A:>3> 0=0;870, 8458, 87;>65==K5 2 =59, =0AB>;L:> OA=K 8 =5?@8=C645==K, GB> ?>;CG8;8 AB0BCA C=825@A0;L=KE 4>?CI5=89. 3> :=830 AB0;0 :;0AA8G5A:>9 8 H8@>:> 87CG05BAO A?5F80;8AB0<8, 8<5NI8<8 A5@L57=K9 8=B5@5A 2 MB>9 >1;0AB8. ;3>@8B< >1CG5=8O %M110 > ACI5AB2C %M11 ?@54?>;>68;, GB> A8=0?B8G5A:>5 A>548=5=85 42CE =59@>=>2 CA8;8205BAO, 5A;8 >10 MB8 =59@>=0 2>71C645=K. -B> <>6=> ?@54AB028BL :0: CA8;5=85 A8=0?A0 2 A>>B25BAB288 A :>@@5;OF859 C@>2=59 2>71C645==KE =59@>=>2, A>548=O5<KE 40==K< A8=0?A><. > MB>9 ?@8G8=5 0;3>@8B< >1CG5=8O %M110 8=>340 =07K205BAO :>@@5;OF8>==K< 0;3>@8B<><. 45O 0;3>@8B<0 2K@0605BAO A;54CNI8< @025=AB2><: wij(t+1) = wij(t) + NETi NETj, 345 wij(t)  A8;0 A8=0?A0 >B =59@>=0 i : =59@>=C j 2 <><5=B 2@5<5=8 t; NETi  C@>25=L 2>71C645=8O ?@54A8=0?B8G5A:>3> =59@>=0; NETj  C@>25=L 2>71C645=8O ?>ABA8=0?B8G5A:>3> =59@>=0. >=F5?F8O %M110 >B25G05B =0 A;>6=K9 2>?@>A, :0:8< >1@07>< >1CG5=85 <>65B ?@>2>48BLAO 157 CG8B5;O.  <5B>45 %M110 >1CG5=85 O2;O5BAO 8A:;NG8B5;L=> ;>:0;L=K< O2;5=85<, >E20BK20NI8< B>;L:> 420 =59@>=0 8 A>548=ONI89 8E A8=0?A; =5 B@51C5BAO 3;>10;L=>9 A8AB5<K >1@0B=>9 A2O78 4;O @0728B8O =59@>==KE >1@07>20=89. >A;54CNI55 8A?>;L7>20=85 <5B>40 %M110 4;O >1CG5=8O =59@>==KE A5B59 ?@825;> : 1>;LH8< CA?5E0<, => =0@O4C A MB8< ?>:070;> >3@0=8G5==>ABL <5B>40; =5:>B>@K5 >1@07K ?@>AB> =5 <>3CB 8A?>;L7>20BLAO 4;O >1CG5=8O MB8< <5B>4><.  @57C;LB0B5 ?>O28;>AL 1>;LH>5 :>;8G5AB2> @0AH8@5=89 8 =>2>22545=89, 1>;LH8=AB2> 87 :>B>@KE 2 7=0G8B5;L=>9 AB5?5=8 >A=>20=> =0 @01>B5 %M110. 5B>4 A83=0;L=>3> >1CG5=8O %M110 0: <K 2845;8, 2KE>4 NET ?@>AB>3> 8A:CAAB25==>3> =59@>=0 O2;O5BAO 2725H5==>9 AC<<>9 53> 2E>4>2. -B> <>65B 1KBL 2K@065=> A;54CNI8< >1@07><:  EMBED Equation.3  345 NETj  2KE>4 NET =59@>=0 j; OUTi  2KE>4 =59@>=0 i;wij  25A A2O78 =59@>=0 i A =59@>=>< j. >6=> ?>:070BL, GB> 2 MB>< A;CG05 ;8=59=0O <=>3>A;>9=0O A5BL =5 O2;O5BAO 1>;55 <>I=>9, G5< >4=>A;>9=0O A5BL; @0AA<0B@8205<K5 2>7<>6=>AB8 A5B8 <>3CB 1KBL C;CGH5=K B>;L:> 22545=85< =5;8=59=>AB8 2 ?5@540B>G=CN DC=:F8N =59@>=0. >2>@OB, GB> A5BL, 8A?>;L7CNI0O A83<>840;L=CN DC=:F8N 0:B820F88 8 <5B>4 >1CG5=8O %M110, >1CG05BAO ?> A83=0;L=><C <5B>4C %M110.  MB>< A;CG05 C@02=5=85 %M110 <>48D8F8@C5BAO A;54CNI8< >1@07><:  EMBED Equation.3  wij(t+1) = wij(t) + OUTi OUTj 345 wij(t)  A8;0 A8=0?A0 >B =59@>=0 i : =59@>=C j 2 <><5=B 2@5<5=8 t; OUTi  2KE>4=>9 C@>25=L ?@5A8=0?B8G5A:>3> =59@>=0 @02=K9 F(NETi); OUTj  2KE>4=>9 C@>25=L ?>ABA8=0?B8G5A:>3> =59@>=0 @02=K9 F(NET). 5B>4 48DD5@5=F80;L=>3> >1CG5=8O %M110 5B>4 A83=0;L=>3> >1CG5=8O %M110 ?@54?>;0305B 2KG8A;5=85 A25@B:8 ?@54K4CI8E 87<5=5=89 2KE>4>2 4;O >?@545;5=8O 87<5=5=8O 25A>2. 0AB>OI89 <5B>4, =07K205<K9 <5B>4>< 48DD5@5=F80;L=>3> >1CG5=8O %M110, 8A?>;L7C5B A;54CNI55 @025=AB2>: wij(t+1) = wij(t) + [OUTi(t)  OUTi(t 1)][ OUTj(t)  OUTj(t 1)], 345 wij(t)  A8;0 A8=0?A0 >B =59@>=0 i : =59@>=C j 2 <><5=B 2@5<5=8 t; OUTi(t)  2KE>4=>9 C@>25=L ?@5A8=0?B8G5A:>3> =59@>=0 2 <><5=B 2@5<5=8 t; OUTj(t)  2KE>4=>9 C@>25=L ?>ABA8=0?B8G5A:>3> =59@>=0 2 <><5=B 2@5<5=8 t.  8A..1. !5BL =AB0@ @>AA15@30 %+  +%+ + =>3> >1I8E 8459, 8A?>;L7C5<KE 2 8A:CAAB25==KE =59@>==KE A5BOE, ?@>A;56820NBAO 2 @01>B0E @>AA15@30; 2 :0G5AB25 ?@8<5@0 <>6=> C:070BL :>=D83C@0F88 2E>4=KE 8 2KE>4=KE 72574 [I], 8A?>;L7C5<K5 2> <=>38E A5B52KE ?0@0483<0E. E>4=0O 725740, :0: ?>:070=> =0 @8A..1, A>AB>8B 87 =59@>=0, =0 :>B>@K9 ?>405BAO 3@C??0 2E>4>2 G5@57 A8=0?A8G5A:85 25A0. KE>4=0O 725740, ?>:070==0O =0 @8A..2, O2;O5BAO =59@>=><, C?@02;ONI8< 3@C??>9 25A>2. E>4=K5 8 2KE>4=K5 72574K <>3CB 1KBL 2708<=> A>548=5=K 2 A5B8 ;N1>9 A;>6=>AB8; @>AA15@3 @0AA<0B@8205B 8E :0: <>45;L >?@545;5==KE 18>;>38G5A:8E DC=:F89. 84 72574K >?@545;O5B 55 =0720=85, >4=0:> 72574K >1KG=> 87>1@060NBAO 2 A5B8 8=0G5.  8A..2. !5BL CBAB0@ @>AA15@30 1CG5=85 2E>4=>9 72574K E>4=0O 725740 2K?>;=O5B @0A?>7=020=85 >1@07>2, B.5. >=0 >1CG05BAO @5038@>20BL =0 >?@545;5==K9 2E>4=>9 25:B>@ % 8 =8 =0 :0:>9 4@C3>9. -B> >1CG5=85 @50;87C5BAO ?CB5< =0AB@>9:8 25A>2 B0:8< >1@07><, GB>1K >=8 A>>B25BAB2>20;8 2E>4=><C 25:B>@C. KE>4 2E>4=>9 72574K >?@545;O5BAO :0: 2725H5==0O AC<<0 55 2E>4>2, :0: MB> >?8A0=> 2 ?@54K4CI8E @0745;0E. ! 4@C3>9 B>G:8 7@5=8O, 2KE>4 <>6=> @0AA<0B@820BL :0: A25@B:C 2E>4=>3> 25:B>@0 A 25A>2K< 25:B>@><, <5@C AE>4AB20 =>@<0;87>20==KE 25:B>@>2. !;54>20B5;L=>, =59@>= 4>;65= @5038@>20BL =081>;55 A8;L=> =0 2E>4=>9 >1@07, :>B>@><C 1K; >1CG5=. @>F5AA >1CG5=8O 2K@0605BAO A;54CNI8< >1@07><: wi(t+1) = wi(t) + ([xi  wi(t)], 345 wi  25A 2E>40 Ei; Ei  i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wi(t+1) = wi(t) + ([yi  wi(t)], 345 ( ?@54AB02;O5B A>1>9 =>@<8@CNI89 :>MDD8F85=B >1CG5=8O, :>B>@K9 2 =0G0;5 ?@81;878B5;L=> @025= 548=8F5 8 ?>AB5?5==> C<5=LH05BAO 4> =C;O 2 ?@>F5AA5 >1CG5=8O. 0: 8 2 A;CG05 2E>4=>9 72574K, 25A0 2KE>4=>9 72574K, ?>AB5?5==> =0AB@0820NBAO =04 <=>65AB2>< 25:B>@>2, ?@54AB02;ONI8E A>1>9 >1KG=K5 20@80F88 8450;L=>3> 25:B>@0.  MB>< A;CG05 2KE>4=>9 A83=0; =59@>=>2 ?@54AB02;O5B A>1>9 AB0B8AB8G5A:CN E0@0:B5@8AB8:C >1CG0NI53> =01>@0 8 <>65B 2 459AB28B5;L=>AB8 AE>48BLAO 2 ?@>F5AA5 >1CG5=8O : 8450;L=><C 25:B>@C ?@8 ?@54JO2;5=88 B>;L:> 8A:065==KE 25@A89 25:B>@0. #'  !"   1957 3. >75=1;0BB [4] @07@01>B0; <>45;L, :>B>@0O 2K720;0 1>;LH>9 8=B5@5A C 8AA;54>20B5;59. 5A<>B@O =0 =5:>B>@K5 >3@0=8G5=8O 55 8AE>4=>9 D>@<K, >=0 AB0;0 >A=>2>9 4;O <=>38E A>2@5<5==KE, =081>;55 A;>6=KE 0;3>@8B<>2 >1CG5=8O A CG8B5;5<. 5@A5?B@>= O2;O5BAO =0AB>;L:> 206=K<, GB> 2AO 3;.2 ?>A2OI5=0 53> >?8A0=8N; >4=0:> MB> >?8A0=85 O2;O5BAO :@0B:8< 8 ?@82>48BAO 2 D>@<0B5, =5A:>;L:> >B;8G=>< >B 8A?>;L7C5<>3> 2 [4]. 5@A5?B@>= O2;O5BAO 42CEC@>2=52>9, =5@5:C@@5=B=>9 A5BLN, 284 :>B>@>9 ?>:070= =0 @8A..3. =0 8A?>;L7C5B 0;3>@8B< >1CG5=8O A CG8B5;5<; 4@C38<8 A;>20<8, >1CG0NI0O 2K1>@:0 A>AB>8B 87 <=>65AB20 2E>4=KE 25:B>@>2, 4;O :064>3> 87 :>B>@KE C:070= A2>9 B@51C5<K9 25:B>@ F5;8. ><?>=5=BK 2E>4=>3> 25:B>@0 ?@54AB02;5=K =5?@5@K2=K< 480?07>=>< 7=0G5=89; :><?>=5=BK 25:B>@0 F5;8 O2;ONBAO 42>8G=K<8 25;8G8=0<8 (0 8;8 1). >A;5 >1CG5=8O A5BL ?>;CG05B =0 2E>45 =01>@ =5?@5@K2=KE 2E>4>2 8 2K@010BK205B B@51C5<K9 2KE>4 2 2845 25:B>@0 A 18=0@=K<8 :><?>=5=B0<8.  8A..3. 4=>A;>8=0O =59@>==0O A5BL 1CG5=85 >ACI5AB2;O5BAO A;54CNI8< >1@07><: 0=4><878@CNBAO 2A5 25A0 A5B8 2 <0;K5 25;8G8=K. 0 2E>4 A5B8 ?>405BAO 2E>4=>9 >1CG0NI89 25:B>@ % 8 2KG8A;O5BAO A83=0; NET >B :064>3> =59@>=0, 8A?>;L7CO AB0=40@B=>5 2K@065=85  EMBED Equation.3 . KG8A;O5BAO 7=0G5=85 ?>@>3>2>9 DC=:F88 0:B820F88 4;O A83=0;0 NET >B :064>3> =59@>=0 A;54CNI8< >1@07><: OUTj = 1, 5A;8 NETj 1>;LH5 G5< ?>@>3j, OUTj = 0  2 ?@>B82=>< A;CG05. 45AL j ?@54AB02;O5B A>1>9 ?>@>3, A>>B25BAB2CNI89 =59@>=C j (2 ?@>AB59H5< A;CG05, 2A5 =59@>=K 8<5NB >48= 8 B>B 65 ?>@>3). KG8A;O5BAO >H81:0 4;O :064>3> =59@>=0 ?>A@54AB2>< 2KG8B0=8O ?>;CG5==>3> 2KE>40 87 B@51C5<>3> 2KE>40: errorj = targetj  OUTj. 064K9 25A <>48D8F8@C5BAO A;54CNI8< >1@07><: Wij(t+1) = wij(t) +(xierrorj. >2B>@ONBAO H038 A> 2B>@>3> ?> ?OBK9 4> B5E ?>@, ?>:0 >H81:0 =5 AB0=5B 4>AB0B>G=> <0;>9. " #'/ # #-%$$ 0: <K 2845;8, ?5@A5?B@>= >3@0=8G8205BAO 18=0@=K<8 2KE>40<8. #84@>C 2<5AB5 A> ABC45=B>< C=825@A8B5B0 %>DD>< @0AH8@8;8 0;3>@8B< >1CG5=8O ?5@A5?B@>=0 =0 A;CG09 =5?@5@K2=KE 2KE>4>2, 8A?>;L7CO A83<>840;L=CN DC=:F8N [5,6]. @><5 B>3>, >=8 @07@01>B0;8 <0B5<0B8G5A:>5 4>:070B5;LAB2> B>3>, GB> A5BL ?@8 >?@545;5==KE CA;>28OE 1C45B AE>48BLAO : ;N1>9 DC=:F88, :>B>@CN >=0 <>65B ?@54AB028BL. 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"0: :0: 2A5 25:B>@K =>@<0;87CNBAO 2 25:B>@K A 548=8G=>9 4;8=>9, >=8 <>3CB @0AA<0B@820BLAO :0: B>G:8 =0 ?>25@E=>AB8 548=8G=>9 38?5@AD5@K.  ?@>F5AA5 >1CG5=8O 3@C??0 A>A54=8E 25A>2KE B>G5: ?5@5<5I05BAO 1;865 : B>G:5 2E>4=>3> 25:B>@0. @54?>;0305BAO, GB> 2E>4=K5 25:B>@K D0:B8G5A:8 3@C??8@CNBAO 2 :;0AAK 2 A>>B25BAB288 A 8E ?>;>65=85< 2 25:B>@=>< ?@>AB@0=AB25. ?@545;5==K9 :;0AA 1C45B 0AA>F88@>20BLAO A >?@545;5==K< =59@>=><, ?5@5<5I0O 53> 25A>2>9 25:B>@ 2 =0?@02;5=88 F5=B@0 :;0AA0 8 A?>A>1AB2CO 53> 2>71C645=8N ?@8 ?>O2;5=88 =0 2E>45 ;N1>3> 25:B>@0 40==>3> :;0AA0. >A;5 >1CG5=8O :;0AA8D8:0F8O 2K?>;=O5BAO ?>A@54AB2>< ?>40G8 =0 2E>4 A5B8 8A?KBC5<>3> 25:B>@0, 2KG8A;5=8O 2>71C645=8O 4;O :064>3> =59@>=0 A ?>A;54CNI8< 2K1>@>< =59@>=0 A =082KAH8< 2>71C645=85< :0: 8=48:0B>@0 ?@028;L=>9 :;0AA8D8:0F88. 8B5@0BC@0 GrossbergS. 1974. Classical and instrumental learning by neural networks. Progress in theoretical biology, vol.3, pp.51 141. New York: Academic Press. HebbD.O. 1949. Organization of behavior. New York: Science Editions. KohonenT. 1984. Self organization and associative memory. Series in Information Sciences, vol.8. Berlin: Springer verlag. RosenblattR. 1959. Principles of neurodynamics. New York: Spartan Books. WidrowB. 1959 Adaptive sampled data systems, a statistical theory of adaptation. 1959. IRE WESCON Convention Record, part 4. New York: Institute of Radio Engineers. Widrow., HoffM. 1960. Adaptive switching circuits. I960. IRE WESCON Convention Record. New York: Institute of Radio Engineers.      HYPERLINK "http://globalteka.ru/" globalteka.ru - @0<>B=>5 8 :0G5AB25==>5 2K?>;=5=85 2A5E 284>2 =0CG=KE @01>B. !:84:8, >@838=0;L=>ABL, :>=B@>;L ?;0380B0, ?@O<>5 >1I5=85 A 02B>@><. 0:070BL 2K?>;=5=85 CG51=>9 8;8 =0CG=>9 @01>BK <>6=> 745AL:  HYPERLINK "http://globalteka.ru/order.html" http://globalteka.ru/order.html PAGE 1 PAGE 1 8A. 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