neural networks

Lecture 7
Artificial neural networks:
Supervised learning
 Introduction, or how the brain works
 The neuron as a simple computing element
 The perceptron
 Multilayer neural networks
 Accelerated learning in multilayer neural networks
 The Hopfield network
 Bidirectional associative memories (BAM)
 Summary
© Negnevitsky, Pearson Education, 2005

1

Introduction, or how the brain works
Machine learning involves adaptive mechanisms
that enable computers to learn from experience,
learn by example and learn by analogy. Learning
capabilities can improve the performance of an
intelligent system over time. The most popular
approaches to machine learning are artificial
neural networks and genetic algorithms. This
lecture is dedicated to neural networks.


2

 A neural network can be defined as a model of
reasoning based on the human brain. The brain
consists of a densely interconnected set of nerve
cells, or basic information-processing units, called
neurons.
 The human brain incorporates nearly 10 billion
neurons and 60 trillion connections, synapses,
between them. By using multiple neurons
simultaneously, the brain can perform its functions
much faster than the fastest computers in existence
today.

3

 Each neuron has a very simple structure, but an
army of such elements constitutes a tremendous
processing power.
 A neuron consists of a cell body, soma, a number
of fibers called dendrites, and a single long fiber
called the axon.


4

Biological neural network
Synapse
Axon

Soma
Dendrites

Synapse

Dendrites
Axon

Soma
Synapse


5

 Our brain can be considered as a highly complex,
non-linear and parallel information-processing
system.
 Information is stored and processed in a neural
network simultaneously throughout the whole
network, rather than at specific locations. In other
words, in neural networks, both data and its
processing are global rather than local.
 Learning is a fundamental and essential
characteristic of biological neural networks. The
ease with which they can learn led to attempts to
emulate a biological neural network in a computer.

6

 An artificial neural network consists of a number of
very simple processors, also called neurons, which
are analogous to the biological neurons in the brain.
 The neurons are connected by weighted links
passing signals from one neuron to another.
 The output signal is transmitted through the
neuron’s outgoing connection. The outgoing
connection splits into a number of branches that
transmit the same signal. The outgoing branches
terminate at the incoming connections of other
neurons in the network.

7

Input Signals

Output Signals

Architecture of a typical artificial neural network

Middle Layer
Input Layer


Output Layer

8

Analogy between biological and
artificial neural networks
Biological Neural Network
Soma
Dendrite
Axon
Synapse


Artificial Neural Network
Neuron
Input
Output
Weight

9

The neuron as a simple computing element
Diagram of a neuron
Input Signals
x1
x2

xn

Weights

Output Signals
Y

w1
w2

wn


Neuron

Y

Y
Y

10

 The neuron computes the weighted sum of the input
signals and compares the result with a threshold
value, θ. If the net input is less than the threshold,
the neuron output is –1. But if the net input is
greater than or equal to the threshold, the neuron
becomes activated and its output attains a value +1.
 The neuron uses the following transfer or activation
function:
n

X = ∑ xi wi
i =1

+1, if X ≥ θ
Y =
−1, if X < θ

 This type of activation function is called a sign
function.

11

Activation functions of a neuron
Step function

Sign function Sigmoid function Linear function

Y

Y

Y

Y

+1

+1

1

1

0

X

0

X

-1

-1

0
-1

1
,
step=  , if X ≥ 0 Y sign = +1 if X ≥ 0 Y sigmoid=
Y


0, if X < 0

−1, if X < 0


X

0

X

-1

1
1 + e− X

Y linear= X

12

Can a single neuron learn a task?
 In 1958, Frank Rosenblatt introduced a training
algorithm that provided the first procedure for
training a simple ANN: a perceptron.
 The perceptron is the simplest form of a neural
network. It consists of a single neuron with
adjustable synaptic weights and a hard limiter.


13

Single-layer two-input perceptron
Inputs
x1
w1

Linear
Combiner

Hard
Limiter

Output
Y

w2

x2


θ
Threshold
14

The Perceptron
 The operation of Rosenblatt’s perceptron is based
on the McCulloch and Pitts neuron model. The
model consists of a linear combiner followed by a
hard limiter.
 The weighted sum of the inputs is applied to the
hard limiter, which produces an output equal to +1
if its input is positive and −1 if it is negative.


15

 The aim of the perceptron is to classify inputs,
x1, x2, . . ., xn, into one of two classes, say
A1 and A2.
 In the case of an elementary perceptron, the ndimensional space is divided by a hyperplane into
two decision regions. The hyperplane is defined by
the linearly separable function:
n

∑ xi wi − θ = 0
i =1


16

Linear separability in the perceptrons
x2

x2
Class A 1
1
1

Class A 2

2

x1

x1

2
x 1w 1 + x 2w 2 −θ = 0
(a) Two-input perceptron.


x3

x1w 1 + x2w 2 + x3w 3 −θ = 0
(b) Three-input perceptron.

17

How does the perceptron learn its classification
tasks?
This is done by making small adjustments in the
weights to reduce the difference between the actual
and desired outputs of the perceptron. The initial
weights are randomly assigned, usually in the range
[−0.5, 0.5], and then updated to obtain the output
consistent with the training examples.


18

 If at iteration p, the actual output is Y(p) and the
desired output is Yd (p), then the error is given by:

e( p) = Yd ( p) − Y ( p)

where p = 1, 2,
3, . . .

Iteration p here refers to the pth training example
presented to the perceptron.
 If the error, e(p), is positive, we need to increase
perceptron output Y(p), but if it is negative, we
need to decrease Y(p).

19

The perceptron learning rule
wi ( p + 1) = wi ( p ) +  . xi ( p ) . e( p )
×
where p = 1, 2, 3, . . .
α is the learning rate, a positive constant less than
unity.
The perceptron learning rule was first proposed by
Rosenblatt in 1960. Using this rule we can derive
the perceptron training algorithm for classification
tasks.


20

Perceptron’s training algorithm
Step 1: Initialisation
Set initial weights w1, w2,…, wn and threshold
θ
to random numbers in the range [−0.5, 0.5].
If the error, e(p), is positive, we need to increase
perceptron output Y(p), but if it is negative, we
need to decrease Y(p).


21

Perceptron’s training algorithm (continued)
Step 2: Activation
Activate the perceptron by applying inputs x1(p),
x2(p),…, xn(p) and desired output Yd (p). Calculate
the actual output at iteration p = 1
 n

Y ( p ) = step  ∑ x i ( p ) w i ( p ) − θ 
 i =1




where n is the number of the perceptron inputs,
and step is a step activation function.


22

Perceptron’s training algorithm (continued)
Step 3: Weight training
Update the weights of the perceptron
wi ( p + 1) = wi ( p) + ∆wi ( p)
where ∆wi(p) is the weight correction at iteration p.
The weight correction is computed by the delta
rule:
.
∆wi ( p) = α ×xi ( p ) ×e( p)

Step 4: Iteration
Increase iteration p by one, go back to Step 2 and
repeat the process until convergence.

23

Example of perceptron learning: the logical operation AND
Epoch

Inputs

Desired
output
Yd

Initial
weights
w1
w2

Actual
output
Y

Error

Final
weights
w1
w2

x1

x2

1

0
0
1
1

0
1
0
1

0
0
0
1

0.3
0.3
0.3
0.2

− 0.1
− 0.1
− 0.1
− 0.1

0
0
1
0

0
0
−1
1

0.3
0.3
0.2
0.3

− 0.1
− 0.1
− 0.1
0.0

2

0
0
1
1

0
1
0
1

0
0
0
1

0.3
0.3
0.3
0.2

0.0
0.0
0.0
0.0

0
0
1
1

0
0
−1
0

0.3
0.3
0.2
0.2

0.0
0.0
0.0
0.0

3

0
0
1
1

0
1
0
1

0
0
0
1

0.2
0.2
0.2
0.1

0.0
0.0
0.0
0.0

0
0
1
0

0
0
−1
1

0.2
0.2
0.1
0.2

0.0
0.0
0.0
0.1

4

0
0
1
1

0
1
0
1

0
0
0
1

0.2
0.2
0.2
0.1

0.1
0.1
0.1
0.1

0
0
1
1

0
0
−1
0

0.2
0.2
0.1
0.1

0.1
0.1
0.1
0.1

5

0
0
1
1

0
1
0
1

0
0
0
1

0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1

0
0
0
1

0
0
0
0

0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1

Threshold: θ = 0.2; learning rate:

e

= 0.1
24

Two-dimensional plots of basic logical operations
x2

x2

x2

1

1

1
x1

x1
0

1

(a) AND (x1 ∩ x2)

0

1

(b) OR (x 1 ∪ x 2 )

x1
0

1

(c) Ex cl us iv e- OR
(x 1 ⊕ x 2 )

A perceptron can learn the operations AND and OR,
but not Exclusive-OR.

25

Multilayer neural networks
 A multilayer perceptron is a feedforward neural
network with one or more hidden layers.
 The network consists of an input layer of source
neurons, at least one middle or hidden layer of
computational neurons, and an output layer of
computational neurons.
 The input signals are propagated in a forward
direction on a layer-by-layer basis.


26

Input Signals

Output Signals

Multilayer perceptron with two hidden layers

Input
layer

First
hidden
layer


Second
hidden
layer

Output
layer

27

What does the middle layer hide?
 A hidden layer “hides” its desired output.
Neurons in the hidden layer cannot be observed
through the input/output behaviour of the network.
There is no obvious way to know what the desired
output of the hidden layer should be.
 Commercial ANNs incorporate three and
sometimes four layers, including one or two
hidden layers. Each layer can contain from 10 to
1000 neurons. Experimental neural networks may
have five or even six layers, including three or four
hidden layers, and utilise millions of neurons.

28

Back-propagation neural network
 Learning in a multilayer network proceeds the
same way as for a perceptron.
 A training set of input patterns is presented to the
network.
 The network computes its output pattern, and if
there is an error − or in other words a difference
between actual and desired output patterns − the
weights are adjusted to reduce this error.


29

 In a back-propagation neural network, the learning
algorithm has two phases.
 First, a training input pattern is presented to the
network input layer. The network propagates the
input pattern from layer to layer until the output
pattern is generated by the output layer.
 If this pattern is different from the desired output,
an error is calculated and then propagated
backwards through the network from the output
layer to the input layer. The weights are modified
as the error is propagated.

30

Three-layer back-propagation neural network
Input signals
x1
x2

xi

1
1

2

y2

k

yk

l

yl

1

2

2
i

y1

wij

j

wjk

m

xn

n

Input
layer

Hidden
layer

Output
layer

Error signals

31

The back-propagation training algorithm
Step 1: Initialisation
Set all the weights and threshold levels of the
network to random numbers uniformly
distributed inside a small range:
 2.4
2.4 
−
 F , + F ÷
÷
i
i 


where Fi is the total number of inputs of neuron i
in the network. The weight initialisation is done
on a neuron-by-neuron basis.

32

Step 2: Activation
Activate the back-propagation neural
network by applying inputs x1(p), x2(p),…, xn(p)
and desired outputs yd,1(p), yd,2(p),…, yd,n(p).
(a) Calculate the actual outputs of the neurons in
the hidden layer:
 n

y j ( p ) = sigmoid  ∑ xi ( p ) ×wij ( p ) − θ j 
 i =1




where n is the number of inputs of neuron j in the
hidden layer, and sigmoid is the sigmoid activation
function.

33

Step 2 : Activation (continued)
(b) Calculate the actual outputs of the neurons in
the output layer:
m

y k ( p ) = sigmoid  ∑ x jk ( p ) ×w jk ( p ) − θ k 
 j =1




where m is the number of inputs of neuron k in the
output layer.


34

Step 3: Weight training
Update the weights in the back-propagation
network propagating backward the errors associated
with output neurons.
(a) Calculate the error gradient for the neurons in
the output layer:
k ( p)

= yk ( p) ×1 − y k ( p ) ×ek ( p )

where ek ( p ) = yd ,k ( p ) − yk ( p )
Calculate the weight corrections:
∆w jk ( p) = ×y j ( p) × k ( p)

Update the weights at the output neurons:
w jk ( p + 1) = w jk ( p ) + ∆w jk ( p )

35

Step 3: Weight training (continued)
(b) Calculate the error gradient for the neurons in
the hidden layer:
l

j ( p)

= y j ( p ) × 1 − y j ( p )] ×∑ k ( p ) w jk ( p )
[
k =1

Calculate the weight corrections:
∆wij ( p ) =

×xi ( p ) × j ( p )

Update the weights at the hidden neurons:
wij ( p + 1) = wij ( p ) + ∆wij ( p )


36

Step 4: Iteration
Increase iteration p by one, go back to Step 2 and
repeat the process until the selected error criterion
is satisfied.
As an example, we may consider the three-layer
back-propagation network. Suppose that the
network is required to perform logical operation
Exclusive-OR. Recall that a single-layer perceptron
could not do this operation. Now we will apply the
three-layer net.


37

Three-layer network for solving the
Exclusive-OR operation
−1
θ3
x1

1

w13
w23

3

−1
w35

θ5
5

x2

2

w24
w24

Input
layer


y5

w45

4
θ4

−1
Hidden layer

Output
layer

38

 The effect of the threshold applied to a neuron in the
hidden or output layer is represented by its weight, θ,
connected to a fixed input equal to −1.
 The initial weights and threshold levels are set
randomly as follows:
w13 = 0.5, w14 = 0.9, w23 = 0.4, w24 = 1.0, w35 = −1.2,
w45 = 1.1, θ3 = 0.8, θ4 = −0.1 and θ5 = 0.3.


39

 We consider a training set where inputs x1 and x2 are
equal to 1 and desired output yd,5 is 0. The actual
outputs of neurons 3 and 4 in the hidden layer are
calculated as
0
0
0
y3 = sigmoid ( x1w13 + x2 w23 − θ3 ) = 1 / 1 + e −(1× .5+1× .4 −1× .8) = 0.5250
0
1
0
y4 = sigmoid ( x1w14 + x2 w24 − θ4 ) = 1 / 1 + e − (1× .9 +1× .0+1× .1) = 0.8808

 Now the actual output of neuron 5 in the output layer
is determined as:
1
1
0
y5 = sigmoid ( y3w35 + y4w45 − θ5 ) = 1/ 1+ e−(−0.5250×.2+0.8808×.1−1× .3) = 0.5097

 Thus, the following error is obtained:
e = yd ,5 − y5 = 0 − 0.5097 = −0.5097

40

 The next step is weight training. To update the
weights and threshold levels in our network, we
propagate the error, e, from the output layer
backward to the input layer.
 First, we calculate the error gradient for neuron 5 in
the output layer:
5

= y5 (1 − y5 ) e = 0.5097 × − 0.5097) ×( −0.5097) = −0.1274
(1

 Then we determine the weight corrections assuming
that the learning rate parameter, α, is equal to 0.1:
∆w35 = ×y3 × 5 = 0.1×0.5250 ×(−0.1274) = −0.0067
∆w45 = ×y 4 × 5 = 0.1 ×0.8808 ×(−0.1274 ) = −0.0112
∆θ5 = ×( −1) × 5 = 0.1 ×(−1) ×(−0.1274) = −0.0127

41

 Next we calculate the error gradients for neurons 3
and 4 in the hidden layer:
(
(
3 = y3 (1 − y3 ) × 5 ×w35 = 0.5250 ×(1 − 0.5250) × − 0.1274) × − 1.2) = 0.0381
4

= y4 (1 − y4 ) × 5 ×w45 = 0.8808 ×(1 − 0.8808) ×( − 0.127 4) × .1 = −0.0147
1

 We then determine the weight corrections:
∆w13 =
∆w23 =
∆ θ3 =
∆w14 =
∆w24 =
∆θ 4 =

×x1 × 3 = 0.1 × ×0.0381 = 0.0038
1
×x2 × 3 = 0.1 × ×0.0381 = 0.0038
1
×( −1) × 3 = 0.1 ×( −1) ×0.0381 = −0.0038
×x1 × 4 = 0.1 × ×(− 0.0147 ) = −0.0015
1
×x2 × 4 = 0.1 × ×(−0.0147 ) = −0.0015
1
×( −1) × 4 = 0.1 ×( −1) ×( −0 .0147 ) = 0.0015


42

 At last, we update all weights and threshold:
w13 = w13 + ∆ w13 = 0 . 5 + 0 . 0038 = 0 .5038
w14 = w14 + ∆ w14 = 0 . 9 − 0 . 0015 = 0 .8985
w 23 = w 23 + ∆ w 23 = 0 . 4 + 0 . 0038 = 0 .4038
w 24 = w 24 + ∆ w 24 = 1 . 0 − 0 . 0015 = 0 .9985
w 35 = w35 + ∆ w35 = − 1 . 2 − 0 . 0067 = − 1 . 2067
w 45 = w 45 + ∆ w 45 = 1 . 1 − 0 . 0112 = 1 .0888
θ 3 = θ 3 + ∆ θ 3 = 0 . 8 − 0 .0038 = 0 . 7962
θ 4 = θ 4 + ∆ θ 4 = − 0 . 1 + 0 . 0015 = − 0 .0985
θ 5 = θ 5 + ∆ θ 5 = 0 . 3 + 0 . 0127 = 0 . 3127

 The training process is repeated until the sum of
squared errors is less than 0.001.

43

Learning curve for operation Exclusive-OR
10

Sum-Squared Network Error for 224 Epochs

1

Sum-Squared Error

10 0

10 -1

10 -2

10 -3

10 -4

0

50


100
Epoch

150

200
44

Final results of three-layer network learning
Inputs

x1

x2

1
0
1
0

1
1
0
0

Desired
output

Actual
output

yd

y5
0.0155
0.9849
0.9849
0.0175

0
1
1
0


e

Sum of
squared
errors

0.0010

45

Network represented by McCulloch-Pitts model
for solving the Exclusive-OR operation
−1
+1.5

x1

1

+1.0

3

−1
−2.0

+1.0

+0.5

5
x2

2

+1.0
+1.0

y5

+1.0

4
+0.5
−1


46

Decision boundaries
x2

x2

x2

x1 + x 2 – 1.5 = 0

x 1 + x2 – 0.5 = 0

1

1

1
x1

x1
0

1
(a)

0

1
(b)

x1
0

1
(c)

(a) Decision boundary constructed by hidden neuron 3;
(b) Decision boundary constructed by hidden neuron 4;
(c) Decision boundaries constructed by the complete
three-layer network

47

Accelerated learning in multilayer
neural networks
 A multilayer network learns much faster when the
sigmoidal activation function is represented by a
hyperbolic tangent:
2a
tan h
Y
=
−a
1 + e −bX
where a and b are constants.
Suitable values for a and b are:
a = 1.716 and b = 0.667

48

 We also can accelerate training by including a
momentum term in the delta rule:

∆w jk ( p) =

×∆w jk ( p − 1) + ×y j ( p ) × k ( p )

where β is a positive number (0 ≤ β < 1) called the
momentum constant. Typically, the momentum
constant is set to 0.95.
This equation is called the generalised delta rule.

49

Learning with momentum for operation Exclusive-OR
10

Training for 126 Epochs

2

10 1
10 0
10 -1
10 -2
10 -3
10 -4

0

20

40

60
Epoch

80

100

120

1.5
Learning Rate

1
0.5
0
-0.5
-1

0

20

40

60

80

100

120

140

Epoch

50

Learning with adaptive learning rate
To accelerate the convergence and yet avoid the
danger of instability, we can apply two heuristics:

Heuristic 1
If the change of the sum of squared errors has the
same algebraic sign for several consequent epochs,
then the learning rate parameter, α, should be
increased.

Heuristic 2
If the algebraic sign of the change of the sum of
squared errors alternates for several consequent
epochs, then the learning rate parameter, α, should be
decreased.

51

 Adapting the learning rate requires some changes
in the back-propagation algorithm.
 If the sum of squared errors at the current epoch
exceeds the previous value by more than a
predefined ratio (typically 1.04), the learning rate
parameter is decreased (typically by multiplying
by 0.7) and new weights and thresholds are
calculated.
 If the error is less than the previous one, the
learning rate is increased (typically by multiplying
by 1.05).

52

Learning with adaptive learning rate
Sum-Squared Erro

10


2

10 1
10 0
10 -1
10 -2
10 -3
10 -4

0

10

20

30

40

50
60
Epoch

70

80

90

100

1
Learning Rate

0. 8
0. 6
0. 4
0. 2
0

0

20


40

60
Epoch

80

100

120

53

Learning with momentum and adaptive learning rate
Sum-Squared Erro

10


2

10 1
10 0
10 -1
10 -2
10 -3
10 -4

0

10

0

10

20

30

40
Epoch

50

60

70

80

Learning Rate

2.5
2
1.5
1
0.5
0

20

30

40

50

60

70

80

90

Epoch

54

The Hopfield Network
 Neural networks were designed on analogy with
the brain. The brain’s memory, however, works
by association. For example, we can recognise a
familiar face even in an unfamiliar environment
within 100-200 ms. We can also recall a
complete sensory experience, including sounds
and scenes, when we hear only a few bars of
music. The brain routinely associates one thing
with another.


55

 Multilayer neural networks trained with the backpropagation algorithm are used for pattern
recognition problems. However, to emulate the
human memory’s associative characteristics we
need a different type of network: a recurrent
neural network.
 A recurrent neural network has feedback loops
from its outputs to its inputs. The presence of
such loops has a profound impact on the learning
capability of the network.


56

 The stability of recurrent networks intrigued
several researchers in the 1960s and 1970s.
However, none was able to predict which network
would be stable, and some researchers were
pessimistic about finding a solution at all. The
problem was solved only in 1982, when John
Hopfield formulated the physical principle of
storing information in a dynamically stable
network.


57

x1

1

y1

x2

2

y2

xi

i

yi

xn

n

yn


Output Signals

Input Signals

Single-layer n-neuron Hopfield network

58

 The Hopfield network uses McCulloch and Pitts
neurons with the sign activation function as its
computing element:
+1, if X > 0
sign 
Y
= −1, if X < 0
 Y , if X = 0



59

 The current state of the Hopfield network is
determined by the current outputs of all neurons, y1,
y2, . . ., yn.
Thus, for a single-layer n-neuron network, the state
can be defined by the state vector as:
 y1 
 y 
 2 
Y =




 yn 




60

 In the Hopfield network, synaptic weights between
neurons are usually represented in matrix form as
follows:
W=

M

T
YmYm − M I
∑

m=1

where M is the number of states to be memorised
by the network, Ym is the n-dimensional binary
vector, I is n × n identity matrix, and superscript T
denotes matrix transposition.


61

Possible states for the three-neuron
Hopfield network
y2
(−1,1, −1)

(1, 1, −1)

(1, 1, 1)

(−1, 1, 1)

y1
0
(1,−1,−1)

(−1,−1,−1)

y3

(−1,−1, 1)


(1,−1, 1)
62

 The stable state-vertex is determined by the weight
matrix W, the current input vector X, and the
threshold matrix θ . If the input vector is partially
incorrect or incomplete, the initial state will converge
into the stable state-vertex after a few iterations.
 Suppose, for instance, that our network is required to
memorise two opposite states, (1, 1, 1) and (−1, −1, −1).
Thus,
1
Y1 = 1

1


− 1
Y2 = − 1
 
− 1
 

T
or Y1 = 1 1 1

T
Y2 = − 1 − 1 − 1

where Y1 and Y2 are the three-dimensional vectors.

63

 The 3 × 3 identity matrix I is
1 0 0
I = 0 1 0


0 0 1



 Thus, we can now determine the weight matrix as
follows:
1
−1
1 0 0 0
W = 1 1 1 1 + −1 −1 −1 −1 − 2 0 1 0 = 2

 

 
1
−1
0 0 1 2

 

 

2
0
2

2
2

0


 Next, the network is tested by the sequence of input
vectors, X1 and X2, which are equal to the output (or
target) vectors Y1 and Y2, respectively.

64

 First, we activate the Hopfield network by applying
the input vector X. Then, we calculate the actual
output vector Y, and finally, we compare the result
with the initial input vector X.
0

Y1 = sign 2
2


2
0
2

2 1 0  1
 1 − 0  = 1
2       
0 1 0  1
     

0

Y2 = sign 2
2


2
0
2

2 −1 0 −1
 −1 − 0 = −1
2      
0 −1 0 −1
      


65

 The remaining six states are all unstable. However,
stable states (also called fundamental memories) are
capable of attracting states that are close to them.
 The fundamental memory (1, 1, 1) attracts unstable
states (−1, 1, 1), (1, −1, 1) and (1, 1, −1). Each of
these unstable states represents a single error,
compared to the fundamental memory (1, 1, 1).
 The fundamental memory (−1, −1, −1) attracts
unstable states (−1, −1, 1), (−1, 1, −1) and (1, −1, −1).
 Thus, the Hopfield network can act as an error
correction network.

66

Storage capacity of the Hopfield network
 Storage capacity is or the largest number of
fundamental memories that can be stored and
retrieved correctly.
 The maximum number of fundamental memories
Mmax that can be stored in the n-neuron recurrent
network is limited by
M max = 0.15n


67

Bidirectional associative memory (BAM)
 The Hopfield network represents an autoassociative
type of memory − it can retrieve a corrupted or
incomplete memory but cannot associate this memory
with another different memory.
 Human memory is essentially associative. One thing
may remind us of another, and that of another, and so
on. We use a chain of mental associations to recover
a lost memory. If we forget where we left an
umbrella, we try to recall where we last had it, what
we were doing, and who we were talking to. We
attempt to establish a chain of associations, and
thereby to restore a lost memory.

68

 To associate one memory with another, we need a
recurrent neural network capable of accepting an
input pattern on one set of neurons and producing
a related, but different, output pattern on another
set of neurons.
 Bidirectional associative memory (BAM), first
proposed by Bart Kosko, is a heteroassociative
network. It associates patterns from one set, set A,
to patterns from another set, set B, and vice versa.
Like a Hopfield network, the BAM can generalise
and also produce correct outputs despite corrupted
or incomplete inputs.

69

BAM operation
x1(p)

x1(p+1)

1
1

x2 (p)

2

xi (p)

i

y1(p)

2

y2(p)

j

yj(p)

m

xn(p)

1
2

xi(p+1)

i

xn(p+1)
Output
layer

(a) Forward direction.

y1(p)

2

y2(p)

j

yj(p)

m

x2(p+1)

ym(p)

n

Input
layer

1

ym(p)

n

Input
layer

Output
layer

(b) Backward direction.
70

The basic idea behind the BAM is to store
pattern pairs so that when n-dimensional vector
X from set A is presented as input, the BAM
recalls m-dimensional vector Y from set B, but
when Y is presented as input, the BAM recalls X.


71

 To develop the BAM, we need to create a
correlation matrix for each pattern pair we want to
store. The correlation matrix is the matrix product
of the input vector X, and the transpose of the
output vector YT. The BAM weight matrix is the
sum of all correlation matrices, that is,

W=

M

T
Xm Ym
∑

m=1

where M is the number of pattern pairs to be stored
in the BAM.

72

Stability and storage capacity of the BAM
 The BAM is unconditionally stable. This means that
any set of associations can be learned without risk of
instability.

 The maximum number of associations to be stored in
the BAM should not exceed the number of
neurons in the smaller layer.
 The more serious problem with the BAM is
incorrect convergence. The BAM may not
always produce the closest association. In fact, a
stable association may be only slightly related to
the initial input vector.

73

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