- Autocorrelators were introduced as a theoretical concept by Donald Hebb in 1949 and were further analyzed by Amari in the 1970s. They store bipolar patterns by summing their outer products to create a connection matrix. - The autocorrelator recalls patterns using a vector-matrix multiplication recall equation. For example, it can correctly recall the stored pattern A2 from a distorted input pattern A' by reducing the noise through the computation. - Bidirectional associative memories (BAMs) are two-layer neural networks based on earlier autocorrelator models. They can store and recall pairs of patterns using a correlation matrix and recall equations to iteratively update input-output pairs until equilibrium is

2. Autocorrelator easily recognized
by the title of Hope Field
Association Memory(HAM).
Autocorrelator were introduced as
a Theoretical notation by DONALD
HEBB(1949) and Rigiorously
analysis by AMARI(1972,1977).

3. A first order auto correlated stores M
bipolar pattern A1,A2,A3,…..Am by
summing together M outer product as,
T= ∑ [Ai][Aj]
Here,
T=[tij] is a (p*p) connection matrix and
Ai È {-1,1}.
The Autocorrelator is recalls Equation is
Vector matrix Multipulication.

4. Consider the following pattern’s
A1=(-1,1,-1,1)
A2=(1,1,1,-1)
A3=(-1,-1,-1,1)
The Connection Matrix:-
3 1 3 -3
T = ∑ [Ai][Aj]= 1 3 1 -1
3 1 3 -3
-3 -1 -3 3

5. The autocorrelation is presented stored
pattern A2=(1,1,1,-1) with the help of recall
Equation,
a1new=f(3+1+3+3,1)=1
a2new=f(6,1)=1
a3new=f(10,1)=1
a4new=f(-10,1)=-1
This is indeed the vector, retrieval of A3=(-1,-
1,-1,1)
(a1new,a2new,a3new,a4new)=(-1,-1,-1,1) yield
the same vector.

6.  Consider a vector A’=(1,1,1,1) which is a
distorted presentation of one among the
stored patterns.
 The Hamming distance (HD) of a vector X
from Y, given X=(x1,x2,x3,….xn) and
Y=(y1,y2,y3,….yn).
HD (x,y) = ∑|xi-yi|
 Thus the HD of A’ from each of the pattern
in the stored set,
HD(A’ , A1) = 4
HD(A’ ,A2) =2

7.  It is evident that they vector A’ is closer to A2
and therefore resembles it, or in other word, is a
noisy version of A2,
 The computation are,
A2=(1, 1, 1, -1)
( a1, a2, a3, a4 ) = ( f(4,1), f(4,1), f(4,1), f(-4,1) )
= ( 1, 1, 1, -1 )
= A2.
Hence in the case of partial vectors, an
autocorrelation results in refinement of the
pattern or removal of noisy to retrieve the
closest matching stored pattern

8.  As KOSKO and others (1987a, 1987b),
Curz. Jr. and Stubberud (1987) have
noted, the bidirectional associative
memory (BAM) is two level non-linear
neural network based on earlier studies
and models of associative memory.
 Kosko extended the unidirectional
autoassociators to bidirectional
processes.

9.  There are N training pairs ,
{ (A1, B1), (A2, B2) ….(Ai, Bi)….(An,Bn)}
where,
Ai = ( ai1, ai2, …,ain )
Bi = ( bi1,bi2,…,bin )
Here aij or bij is either in the ON or OFF
state.
 In the binary mode, ON = 1 and OFF = 0
and in the bipolar mode, ON = 1 and OFF =
-1. We frame the correlation matrix

10.  To retrieve the nearest (Ai , Bi) pair given
any pair (α,β) the recall equation are as
follows:
 Starting with (α,β) as the initial condition,
We determine a finite sequence (α’,β’),
(α’’,β’’),… until an equilibrium point (αf,βf),
is reached.
Here, β’=Φ(αM)
α’=Φ(βM)
Φ(F)=G=g1,g2,…,gn
F=(f1,f2,…,fn)

11. 1 if fi>0
0(binary)
gi = -1(bipolor) , fi<0
Previous gi , fi=0
 One important performance attribute of
discrete BAM is its ability to recall stored
pairs particularly in the presence of noise.

12.  Given a set of pattern pairs (Xi,Yi), for i =
1, 2, …, n and their correlation matrix M, a
pair (X’,Y’) can be added or an existing
pair (Xj,Yj) can be erased or deleted from
the memory model.
 In case of addition, the new correlation
matrix M is,
M = X1Y1+X2Y2+…+XnYn+XY

13.  In the case of deletion, we subtract the
matrix corresponding to (Xj, Yj) from the
matrix M,
(New) M=M-(Xj,Yj)
 The addition and deletion of information
contributes to the functioning of the
system as a typical human memory
exhibiting learning and forgetfulness.

14.  The stability of a BAM can be proved by
identifying a Lyapunov or energy function
E with each state (A,B).
 In the autoassociative case, Hopfield
identified an appropriate E (actually,
Hopfield defined half this quantity) as,
E(A) = -AMA
 Kosko proposed an energy function,
E(A,B) = -AMB

15.  Kosko proved that each cycle of decoding
lowers the energy E if the energy function
for any point ( α,β ) is given by,
E = -αMβ ,
 However, if the energy E evaluated using
the coordinates of the pair (Ai, Bi),
(i.e),
E = -AiMBi
 Eventhought one starts with α = Ai. In this
aspect, Kosko’s encoding method does not
ensure that the stored pairs are at a local
mimima.

16. Thank you!!!