Neural Networks

5/26/2018 Neural Networks

1/73 Negnevitsky, Pearson Education, 2005 1

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



Introduction, or how the brain works

Machine learning involves adaptive mechanismsthat 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.



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 nervecells, or basic information-processing units, called

neurons.

The human brain incorporates nearly 10 billionneurons and 60 trillion connections,synapses,

between them. By using multiple neurons

simultaneously, the brain can perform its functionsmuch faster than the fastest computers in existence

today.



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.



Biological neural network

Soma Soma

Synapse

Synapse

Dendrites

Axon

Synapse

Dendrites

Axon



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 otherwords, in neural networks, both data and its

processing are global rather than local.

Learning is a fundamental and essentialcharacteristic of biological neural networks. The

ease with which they can learn led to attempts to

emulate a biological neural network in a computer.



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 theneurons outgoing connection. The outgoing

connection splits into a number of branches that

transmit the same signal. The outgoing branchesterminate at the incoming connections of other

neurons in the network.



Architecture of a typical artificial neural network

Input Layer Output Layer

Middle Layer

InputS

ignals

Ou

tputS

ignals



Analogy between biological and

artificial neural networks

Biological Neural Network Artificial Neural Network

Soma

DendriteAxonSynapse

Neuron

InputOutputWeight



The neuron as a simple computing element

Diagram of aneuron

Neuron Y

Input Signals

x1

x2

xn

Output Signals

Y

Y

Y

w2

w1

wn

Weights



The neuron computes the weighted sum of the input

signals and compares the result with a threshold

value, q. If the net input is less than the threshold,

the neuron output is1. 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:

This type of activation function is called a sign

function.

n

iiiwxX

1

q1

q1

X

X

Y if,

if,



Activation functions of a neuron

Step function Sign function

+1

-1

0

+1

-1

0X

Y

X

Y

1 1

-1

0 X

Y

Sigmoidfunction

-1

0 X

Y

Linear function

0if,00if,1

XXYstep

0if,10if,1

XXYsign

Xsigmoid

eY

1

1 XYlinear



Can a single neuron learn a task?

In 1958, Frank Rosenblatt introduced a trainingalgorithm that provided the first procedure for

training a simple ANN: a perceptron.

The perceptron is the simplest form of a neuralnetwork. It consists of a single neuron with

adjustable synaptic weights and a hard limiter.



Single-layer two-input perceptron

Threshold

Inputs

x1

x2

Output

Y

Hard

Limiter

w2

w1

Linear

Combiner

q



ThePerceptron

The operation of Rosenblatts perceptron is basedon 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.



The aim of the perceptron is to classify inputs,

x1,x2, . . .,xn, into one of two classes, sayA1and A2.

In the case of an elementary perceptron, the n-

dimensional space is divided by a hyperplane intotwo decision regions. The hyperplane is defined by

the l inearly separable function:

01

q

n

iiiwx


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Negnevitsky, Pearson Education, 2005 17

Linear separability in the perceptrons

x1

x2

Class A 2

Class A 1

1

2

x1w1 +x2w2q = 0

(a) Two-input perceptron. (b) Three-input perceptron.

x2

x1

x3 x1w1+ x2w2+ x3w3q = 0

1 2


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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.


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If at iterationp, the actual output is Y(p) and the

desired output is Yd(p), then the error is given by:

wherep = 1, 2, 3, . . .

Iterationp here refers to thepth training examplepresented to the perceptron.

If the error, e(p), is positive, we need to increase

perceptron output Y(p), but if it is negative, weneed to decrease Y(p).

)()()( pYpYpe d


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)()()()1( pepxpwpw iii

a. .

wherep = 1, 2, 3, . . .

a 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 classificationtasks.

The perceptron learning rule


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Perceptrons training algorithm

Step 1: InitialisationSet initial weights w1, w2,, wnand threshold q

to random numbers in the range [0.5, 0.5].

If the error, e(p), is positive, we need to increaseperceptronoutput Y(p), but if it is negative, we

need to decrease Y(p).


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Perceptrons training algorithm (continued)

Step 2: Activation

Activate the perceptron by applying inputsx1(p),

x2(p),,xn(p) and desired output Yd(p).

Calculate the actual output at iterationp = 1

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

q

n

iii pwpxsteppY

1

)()()(


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Perceptrons training algorithm (continued)

Step 3: Weight training

Update the weights of the perceptron

where Dwi(p) is the weight correction at iterationp.

The weight correction is computed by the deltarule:

Step 4: Iteration

Increase iterationpby one, go back to Step 2 and

repeat the process until convergence.

)()()1( pwpwpw iii D

)()()( pepxpw ii aD .


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Example of perceptron learning: the logical operationANDInputs

x1 x2

0

01

1

0

10

1

0

00

EpochDesiredoutputYd

1

Initialweights

w1

w2

1

0.3

0.30.3

0.2

0.1

0.1

0.1

0.1

0

01

0

Actualoutput

Y

Error

e

0

01

1

Finalweightsw

1 w

2

0.3

0.30.2

0.3

0.1

0.1

0.1

0.0

00

11

0

1

01

00

0

2

1

0.30.3

0.30.2

0

0

11

00

10

0.30.3

0.20.2

0.00.0

0.00.0

0

0

1

1

0

1

0

1

0

0

0

3

1

0.2

0.2

0.2

0.1

0.00.0

0.00.0

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

0

0

11

0

1

01

0

0

0

4

1

0.2

0.2

0.20.1

0.1

0.1

0.10.1

0

0

11

0

0

10

0.2

0.2

0.10.1

0.1

0.1

0.10.1

00

11

0

1

01

00

0

5

1

0.10.1

0.10.1

0.10.1

0.10.1

0

0

01

00

0

0.10.1

0.10.1

0.10.1

0.10.1

0

Threshold: q= 0.2; learning rate: = 0.1


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Two-dimensional plots of basic logical operations

A perceptron can learn the operationsAND and OR,but notExclusive-OR.

x1

x2

1

1

x1

x2

1

1

(b) OR (x1 x2)

x1

x2

1

1

(c) Exclusive-OR

(x1 x2)

00 0

(a) AND (x1x2)


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Multilayerneuralnetworks

A multilayer perceptron is a feedforward neuralnetwork with one or more hidden layers.

The network consists of an input layer of source

neurons, at least one middle or hidden layer ofcomputational neurons, and an output layer of

computational neurons.

The input signals are propagated in a forwarddirection on a layer-by-layer basis.


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Multilayer perceptron with two hidden layers

Input

layer

First

hidden

layer

Second

hidden

layerOutput

layer

InputSi

gnals

Ou

tputS

ignals


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What does the middle layer hide? A hidden layer hides its desired output.

Neurons in the hidden layer cannot be observedthrough 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 mayhave five or even six layers, including three or

four hidden layers, and utilise millions of neurons.


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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.


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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 outputlayer to the input layer. The weights are modified

as the error is propagated.


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Three-layer back-propagation neural network

Input

layer

xi

x1

x2

xn

1

2

i

n

Output

layer

1

2

k

l

yk

y1

y2

yl

Input signals

Error signals

wjk

Hidden

layer

wij

1

2

j

m


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The back-propagation training algorithm

Step 1: Initialisation

Set all the weights and threshold levels of thenetwork to random numbers uniformly

distributed inside a small range:

whereFiis the total number of inputs of neuron iin the network. The weight initialisation is done

on a neuron-by-neuron basis.

ii FF

4.2,

4.2


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Step 2: Activation

Activate the back-propagation neural network by

applying inputsx1(p),x

2(p),,x

n(p) and desired

outputsyd,1(p),yd,2(p),,yd,n(p).

(a) Calculate the actual outputs of the neurons in

the hidden layer:

where n is the number of inputs of neuronj in thehidden layer, andsigmoid is thesigmoid activation

function.

q

j

n

iijij pwpxsigmoidpy

1

)()()(


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(b) Calculate the actual outputs of the neurons in

the output layer:

Step 2: Activation (continued)

where m is the number of inputs of neuron k in the

output layer.

q

k

m

jjkjkk pwpxsigmoidpy

1

)()()(


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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:

where

Calculate the weight corrections:

Update the weights at the output neurons:

)()()1( pwpwpw jkjkjk D

)()(1)()( pepypyp kkkk

)()()( , pypype kkdk

)()()( ppypw kjjk D


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(b) Calculate the error gradient for the neurons in

the hidden layer:

Step 3: Weight training (continued)

Calculate the weight corrections:

Update the weights at the hidden neurons:

)()()(1)()(1

][ pwppypyp jkl

kkjjj

)()()( ppxpw jiij D

)()()1( pwpwpw ijijij D


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Step 4: Iteration

Increase iterationpby 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.


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Three-layer network for solving the

Exclusive-OR operation

y55

x1 31

x2

Input

layer

Output

layer

Hiddenlayer

42

q3

w13

w24

w23

w24

w35

w45

q4

q5

1

1

1


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The effect of the threshold applied to a neuron in the

hidden or output layer is represented by its weight, q,

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, q3= 0.8, q4= 0.1 and q5= 0.3.


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We consider a training set where inputsx1 andx2are

equal to 1 and desired outputyd,5is 0. The actual

outputs of neurons 3 and 4 in the hidden layer are

calculated as

Now the actual output of neuron 5 in the output layer

is determined as:

Thus, the following error is obtained:

5250.01/1)( )8.014.015.01(32321313 q ewxwxsigmoidy

8808.01/1)(

)1.010.119.01(

42421414 q

ewxwxsigmoidy

5097.01/1)( )3.011.18808.02.15250.0(

54543535 q

ewywysigmoidy

5097.05097.0055, yye d


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The next step is weight training. To update the

weights and threshold levels in our network, we

propagate the error, e, from the output layerbackward to the input layer.

First, we calculate the error gradient for neuron 5 in

the output layer:

Then we determine the weight corrections assuming

that the learning rate parameter, a, is equal to 0.1:

1274.05097).0(0.5097)(10.5097)1( 555 eyy

0112.0)1274.0(8808.01.05445 D yw

0067.0)1274.0(5250.01.05335 D yw

0127.0)1274.0()1(1.0)1( 55 qD


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Next we calculate the error gradients for neurons 3

and 4 in the hidden layer:

We then determine the weight corrections:

0381.0)2.1(0.1274)(0.5250)(10.5250)1( 355333 wyy

0.0147.114)0.127(0.8808)(10.8808)1( 455444 wyy

0038.00381.011.03113 D xw0038.00381.011.03223 D xw

0038.00381.0)1(1.0)1( 33 qD

0015.0)0147.0(11.04114 D

xw 0015.0)0147.0(11.04224 D xw0015.0)0147.0()1(1.0)1( 44 qD


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At last, we update all weights and threshold:

The training process is repeated until the sum of

squared errors is less than 0.001.

5038.00038.05.0131313 D www

8985.00015.09.0141414 D www

4038.00038.04.0232323 D www

9985.00015.00.1242424 D www

2067.10067.02.1353535

D www

0888.10112.01.1454545 D www

7962.00038.08.0333 qDqq

0985.00015.01.0444 qDqq

3127.00127.03.0555 qDqq


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Learning curve for operation Exclusive-OR

0 50 100 150 200

10 1

Epoch

Sum-Squared Network Error for 224 Epochs

100

10 -1

10 -2

10 -3

10-4

Sum-SquaredError


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Final results of three-layer network learning

Inputs

x1 x2

1

01

0

1

10

0

0

11

Desiredoutput

yd

0

0.0155

Actualoutput

y5 e

Sum ofsquared

errors

0.98490.9849

0.0175

0.0010


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Network represented by McCulloch-Pitts model

for solving the Exclusive-OR operation

y55

x1 31

x2 42

+1.0

1

1

1+1.0

+1.0

+1.0

+1.5

+1.0

+0.5

+0.52.0

i i i


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Decision boundaries

(a) Decisionboundary constructed by hidden neuron 3;(b) Decision boundary constructed by hidden neuron 4;

(c) Decision boundaries constructed by the complete

three-layer network

x1

x2

1

(a)

1

x2

1

1

(b)

00

x1

+ x21.5 = 0 x

1+ x

2 0.5 = 0

x1 x1

x2

1

1

(c)

0


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Accelerated learning in multilayer

neural networks

A multilayer network learns much faster when thesigmoidal activation function is represented by a

hyperbolic tangent:

where a and bare constants.

Suitable values for a and b are:

a = 1.716 and b = 0.667

ae

aYbX

htan

1

2


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We also can accelerate training by including a

momentum term in the delta rule:

where b is a positive number (0 b 1) called themomentum constant. Typically, the momentum

constant is set to 0.95.

This equation is called the generalised delta rule.

)()()1()( ppypwpw kjjkjk DD

L i ith t f ti E l i OR


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Learning with momentum for operation Exclusive-OR

0 20 40 60 80 100 120

10-4

10-2

100

102

Epoch

Training for 126 Epochs

0 100 140

-1

-0.5

0

0. 5

1

1. 5

Epoch

10-3

101

10-1

20 40 60 80 120

Learning

Rate

L i ith d ti l i t


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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 thelearning rate parameter, a, should be increased.

Heuristic 2

If the algebraic sign of the change of the sum ofsquared errors alternates for several consequent

epochs, then the learning rate parameter, a, should be

decreased.


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Adapting the learning rate requires some changesin the back-propagation algorithm.

If the sum of squared errors at the current epochexceeds the previous value by more than a

predefined ratio (typically 1.04), the learning rate

parameter is decreased (typically by multiplyingby 0.7) and new weights and thresholds arecalculated.

If the error is less than the previous one, thelearning rate is increased (typically by multiplyingby 1.05).

L i ith d ti l i t


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Learning with adaptive learning rate

0 10 20 30 40 50 60 70 80 90 100Epoch

Tr aining for 103Epochs

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Epoch

10-4

10 -2

100

102

10-3

10 1

10-1

Sum-SquaredErro

LearningRate

L i ith t d d ti l i t


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Learning with momentum and adaptive learning rate

0 10 20 30 40 50 60 70 80Epoch

Trainingfor 85 Epochs

0 10 20 30 40 50 60 70 80 900

0.5

1

2.5

Epoch

10 -4

10 -2

10 0

10 2

10-3

101

10 -1

1.5

2

Sum-SquaredE

rro

Learnin

gRate

Th H fi ld N t k


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The Hopfield Network

Neural networks were designed on analogy with

the brain. The brains 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 completesensory experience, including sounds and scenes,

when we hear only a few bars of music. The brain

routinely associates one thing with another.


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Multilayer neural networks trained with the back-

propagation algorithm are used for pattern

recognition problems. However, to emulate thehuman memorys 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.


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The stability of recurrent networks intrigued

several researchers in the 1960s and 1970s.

However, none was able to predict which networkwould 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.

Si l l n H fi ld t k


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Single-layer n-neuron Hopfield network

xi

x1

x2

xn

yi

y1

y2

yn

1

2

i

n


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The Hopfield network uses McCulloch and Pitts

neurons with thesign activation function as its

computing element:

XY

X

X

Ysign

if,

if,1

0if,1


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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:

ny

y

y

2

1

Y


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In the Hopfield network, synaptic weights between

neurons are usually represented in matrix form as

follows:

whereMis the number of states to be memorised

by the network, Ymis the n-dimensional binary

vector, I is nnidentity matrix, and superscript T

denotes matrix transposition.

IYYW MM

m

Tmm

1

Possible states for the three neuron


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Possible states for the three-neuron

Hopfield network

y1

y2

y3

(1,1,1)(1,1,1)

(1,1,1) (1,1,1)

(1, 1, 1)(1,1,1)

(1, 1,1)(1,1,1)

0

The stable state vertex is determined by the weight


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The stable state-vertex is determined by the weight

matrix W, the current input vector X, and the

threshold matrix . 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,

where Y1and Y2are the three-dimensional vectors.

or

11

1

1Y

1

1

1

2Y 111

1 TY 111

2

TY

The 3 3 identity matrix I is


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The 3 3 identity matrix I is

Thus, we can now determine the weight matrix as

follows:

Next, the network is tested by the sequence of input

vectors, X1and X2, which are equal to the output (or

target) vectors Y1and Y2, respectively.

100

010

001

I

100

010

001

2111

1

1

1

111

1

1

1

W

022

202

220

First we activate the Hopfield network by applying


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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.

11

1

00

0

11

1

022202

220

1 signY

1

1

1

0

0

0

1

1

1

022

202

220

2 signY

The remaining six states are all unstable However


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The remaining six states are all unstable. However,stable states (also calledfundamental 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 ofthese 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 errorcorrection network.

St it f th H fi ld t k


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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

Mmaxthat can be stored in the n-neuron recurrent

network is limited by

nMmax 0.15

Bidirectional associative memory (BAM)


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Bidirectional associative memory (BAM)

The Hopfield network represents an autoassociative

type of memory it can retrieve a corrupted orincomplete memory but cannot associate this memorywith another different memory.

Human memory is essentially associative. One thingmay remind us of another, and that of another, and soon. We use a chain of mental associations to recovera lost memory. If we forget where we left an

umbrella, we try to recall where we last had it, whatwe were doing, and who we were talking to. We

attempt to establish a chain of associations, andthereby to restore a lost memory.

To associate one memory with another we need a


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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, setA,

to patterns from another set, setB, and vice versa.

Like a Hopfield network, the BAM can generaliseand also produce correct outputs despite corrupted

or incomplete inputs.

BAM operation


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BAM operation

yj(p)

y1(p)

y2(p)

ym(p)

1

2

j

m

Output

layer

Input

layer

xi(p)

x1

x2

(p)

(p)

xn(p)

2

i

n

1

xi(p+1)

x1(p+1)

x2(p+1)

xn(p+1)

yj(p)

y1(p)

y2(p)

ym(p)

1

2

j

m

Output

layer

Input

layer

2

i

n

1

(a) Forward direction. (b) Backward direction.


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The basic idea behind the BAM is to storepattern pairs so that when n-dimensional vector

X from setA is presented as input, the BAM

recalls m-dimensional vector Y from setB, but

when Y is presented as input, the BAM recalls X.


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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 productof 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,

whereM is the number of pattern pairs to be stored

in the BAM.

Tm

M

m

mYXW

1

Stability and storage capacity of the BAM


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Stability and storage capacity of the BAM

The BAM is unconditionally stable. This means that

any set of associations can be learned without risk ofinstability.

The maximum number of associations to be stored in

the BAM should not exceed the number ofneurons in the smaller layer.

The more serious problem with the BAM is

incorrect convergence. The BAM may notalways produce the closest association. In fact, a

stable association may be only slightly related to

the initial input vector

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Neural Networks