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Autocorrelators1

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ranjit banshpal
ranjit banshpal

This document discusses autocorrelators and heterocorrelators, specifically Kosko's discrete bidirectional associative memory (BAM) model. Autocorrelators store patterns as a connection matrix that is the sum of the outer products of the patterns. Their recall equation performs vector-matrix multiplication between the input and connection matrix. Kosko's BAM extends unidirectional autocorrelators to be bidirectional. It stores paired patterns (A,B) and recalls using recall equations that update A and B vectors iteratively until equilibrium is reached. The correlation matrix is calculated from the training pairs and an energy function aims to minimize energy for stored pattern retrieval.

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1 Autocorrelators Dept. of Computer Science & Engineering 2013-2014 Presented By: Ranjit R. Banshpal Mtech 1st sem (CSE) Roll NO.18 1 A seminar on G.H. Raisoni College of Engineering Nagpur 1
Autocorrelators Autocorrelators easily recognized by the title of Hopfield Associative Memory (HAM). First order autocorrelators obtain their connection matrix by multiplying a patterns element with every other patterns elements. A first order autocorrelator stores M bipolar pattern A1,A2,,Am by summing together m outer product as
Here, is a (p p) connection matrix and p The autocorrelators recall equation is vector-matrix multiplication, The recall equation is given by, =f ( ) Where Ai=(a1,a2,..,ap) and two parameter bipolar threshold function is, 1, if 留 > 0 f(留 , 硫 )= 硫, if 留 = 0 -1, if 留 < 0
Working of an autocorrelator Consider the following pattern, A1=(-1,1,-1,1) A2=(1,1,1,-1) A3=(-1,-1,-1,1) The connection matrix, 3 1 3 -3 41 14 1 3 1 -1 3 1 3 -3 -3 -1 -3 3
Recognition of stored patterns The autocorrelator is presented stored pattern A2=(1,1,1,-1) With the help of recall equation = f ( 3 + 1 + 3 + 3, 1 ) = 1 = f ( 6, 1 ) = 1 = f ( 10, 1 ) = 1 = f ( -10, 1 ) = -1
Recognition of noisy patterns Consider a vector A=(1,1,1,1) which is a distorted presentation of one among the store pattern With the help of Hamming Distance measure we can find noisy vector pattern The Hamming distance (HD) of vector X from Y, given X=(x1,x2,.,xn) and Y=(y1,y2,.,yn) is given by, HD( x, y ) =
Heterocorrelators : Koskos discrete BAM Bidirectional associative memory (BAM) is two level nonlinear neural network Kosko extended the unidirectional to bidirectional processes. Noise does not affect performance There are N training pairs {(A1,B1),(A2,B2),.,(Ai,Bi),,(An, Bn)} where Ai=(ai1,ai2,.,ain) Bi =(bi1,bi2,,bip)
Here, aij or bij is either ON or OFF state In binary mode, ON = 1 and OFF = 0 and In bipolar mode, ON = 1 and OFF = -1 Formula for correlation matrix is, Recall equations, Starting with (留, 硫) as the initial condition, we determine the finite sequence (留, 硫 ),(留, 硫),.., until equilibrium point (留F, 硫 F ) is reached. Here , 硫 = (留M) 留 = (硫 MT)
陸(F) = G = g1, g2, ., gn F = ( f1,f2,.,f n) 1 if fi > 0 0 (binary) gi = , fi < 0 -1 (bipolar) previous gi, fi = 0
Addition and Deletion of Pattern Pairs If given set of pattern pairs (Xi, Yi) for i=1,2,.,n Then we can be added (X,Y) or can be deleted (Xj,Yj) from the memory model. In the case of addition, In the case of deletion,
Energy function for BAM The value of the energy function for particular pattern has to occupy a minimum point in energy landscape, Adding new patterns do not destroy previously stored patterns.
Hopfield propose an energy function as, E(A) = -AMAT Kosko propose an energy function as, E(A,B)= - AMBT If energy function for any point (留, 硫) is given by E = - 留M硫T If energy E evaluate using the coordinates of the pair (Ai,Bi),
Working of Koskos BAM Step 1: converting to bipolar forms Step 2: The matrix M is calculated as, Step 3: Retrieve the associative pair 硫 = (留M) 留 = (硫 MT)
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Autocorrelators1

  • 1. 1 Autocorrelators Dept. of Computer Science & Engineering 2013-2014 Presented By: Ranjit R. Banshpal Mtech 1st sem (CSE) Roll NO.18 1 A seminar on G.H. Raisoni College of Engineering Nagpur 1
  • 2. Autocorrelators Autocorrelators easily recognized by the title of Hopfield Associative Memory (HAM). First order autocorrelators obtain their connection matrix by multiplying a patterns element with every other patterns elements. A first order autocorrelator stores M bipolar pattern A1,A2,,Am by summing together m outer product as
  • 3. Here, is a (p p) connection matrix and p The autocorrelators recall equation is vector-matrix multiplication, The recall equation is given by, =f ( ) Where Ai=(a1,a2,..,ap) and two parameter bipolar threshold function is, 1, if 留 > 0 f(留 , 硫 )= 硫, if 留 = 0 -1, if 留 < 0
  • 4. Working of an autocorrelator Consider the following pattern, A1=(-1,1,-1,1) A2=(1,1,1,-1) A3=(-1,-1,-1,1) The connection matrix, 3 1 3 -3 41 14 1 3 1 -1 3 1 3 -3 -3 -1 -3 3
  • 5. Recognition of stored patterns The autocorrelator is presented stored pattern A2=(1,1,1,-1) With the help of recall equation = f ( 3 + 1 + 3 + 3, 1 ) = 1 = f ( 6, 1 ) = 1 = f ( 10, 1 ) = 1 = f ( -10, 1 ) = -1
  • 6. Recognition of noisy patterns Consider a vector A=(1,1,1,1) which is a distorted presentation of one among the store pattern With the help of Hamming Distance measure we can find noisy vector pattern The Hamming distance (HD) of vector X from Y, given X=(x1,x2,.,xn) and Y=(y1,y2,.,yn) is given by, HD( x, y ) =
  • 7. Heterocorrelators : Koskos discrete BAM Bidirectional associative memory (BAM) is two level nonlinear neural network Kosko extended the unidirectional to bidirectional processes. Noise does not affect performance There are N training pairs {(A1,B1),(A2,B2),.,(Ai,Bi),,(An, Bn)} where Ai=(ai1,ai2,.,ain) Bi =(bi1,bi2,,bip)
  • 8. Here, aij or bij is either ON or OFF state In binary mode, ON = 1 and OFF = 0 and In bipolar mode, ON = 1 and OFF = -1 Formula for correlation matrix is, Recall equations, Starting with (留, 硫) as the initial condition, we determine the finite sequence (留, 硫 ),(留, 硫),.., until equilibrium point (留F, 硫 F ) is reached. Here , 硫 = (留M) 留 = (硫 MT)
  • 9. 陸(F) = G = g1, g2, ., gn F = ( f1,f2,.,f n) 1 if fi > 0 0 (binary) gi = , fi < 0 -1 (bipolar) previous gi, fi = 0
  • 10. Addition and Deletion of Pattern Pairs If given set of pattern pairs (Xi, Yi) for i=1,2,.,n Then we can be added (X,Y) or can be deleted (Xj,Yj) from the memory model. In the case of addition, In the case of deletion,
  • 11. Energy function for BAM The value of the energy function for particular pattern has to occupy a minimum point in energy landscape, Adding new patterns do not destroy previously stored patterns.
  • 12. Hopfield propose an energy function as, E(A) = -AMAT Kosko propose an energy function as, E(A,B)= - AMBT If energy function for any point (留, 硫) is given by E = - 留M硫T If energy E evaluate using the coordinates of the pair (Ai,Bi),
  • 13. Working of Koskos BAM Step 1: converting to bipolar forms Step 2: The matrix M is calculated as, Step 3: Retrieve the associative pair 硫 = (留M) 留 = (硫 MT)