Unsupervised learning in soft computing

1. Unsupervised Learning

2. Contents
• Introduction
• Competitive Learning networks
• Kohenen self-organizing networks
• Learning vector quantization
• Hebbian learning

3. The main property of a neural network is an
ability to learn from its environment, and to
improve its performance through learning. So
far we have considered supervised or active
learning - learning with an external “teacher”
or a supervisor who presents a training set to the
network. But another type of learning also
exists: unsupervised learning.
Introduction

4.  In contrast to supervised learning, unsupervised or
self-organised learning does not require an
external teacher. During the training session, the
neural network receives a number of different
input patterns, discovers significant features in
these patterns and learns how to classify input data
into appropriate categories. Unsupervised
learning tends to follow the neuro-biological
organisation of the brain.
 Unsupervised learning algorithms aim to learn
rapidly and can be used in real-time.

5. In 1949, Donald Hebb proposed one of the key
ideas in biological learning, commonly known as
Hebb’s Law. Hebb’s Law states that if neuron i is
near enough to excite neuron j and repeatedly
participates in its activation, the synaptic connection
between these two neurons is strengthened and
neuron j becomes more sensitive to stimuli from
neuron i.
Hebbian learning

6. Hebb’s Law can be represented in the form of two
rules:
1. If two neurons on either side of a connection
are activated synchronously, then the weight of
that connection is increased.
2. If two neurons on either side of a connection
are activated asynchronously, then the weight
of that connection is decreased.
Hebb’s Law provides the basis for learning
without a teacher. Learning here is a local
phenomenon occurring without feedback from
the environment.

7.  In competitive learning, neurons compete among
themselves to be activated.
 While in Hebbian learning, several output neurons
can be activated simultaneously, in competitive
learning, only a single output neuron is active at
any time.
 The output neuron that wins the “competition” is
called the winner-takes-all neuron.
Competitive learning

8. Competitive learning neural networks

13.  The basic idea of competitive learning was
introduced in the early 1970s.
 In the late 1980s, Teuvo Kohonen introduced a
special class of artificial neural networks called
self-organizing feature maps. These maps are
based on competitive learning.

14. Our brain is dominated by the cerebral cortex, a
very complex structure of billions of neurons and
hundreds of billions of synapses. The cortex
includes areas that are responsible for different
human activities (motor, visual, auditory,
somatosensory, etc.), and associated with different
sensory inputs. We can say that each sensory
input is mapped into a corresponding area of the
cerebral cortex. The cortex is a self-organizing
computational map in the human brain.
What is a self-organizing feature map?

15. Kohenen Self-Organizing
Feature Maps
• Feature mapping converts a wide pattern
space into a typical feature space
• Apart from reducing higher dimensionality
it has to preserve the neighborhood
relations of input patterns

16. Feature-mapping Kohonen model
Input layer
Kohonenlayer
(a)
Input layer
Kohonenlayer
1 1
(b)
0
0

17. The Kohonen network
 The Kohonen model provides a topological
mapping. It places a fixed number of input
patterns from the input layer into a higher-
dimensional output or Kohonen layer.
 Training in the Kohonen network begins with the
winner’s neighborhood of a fairly large size. Then,
as training proceeds, the neighborhood size
gradually decreases.

18. Model: Horizontal & Vertical lines
Rumelhart & Zipser, 1985
• Problem – identify vertical or horizontal
signals
• Inputs are 6 x 6 arrays
• Intermediate layer with 8 units
• Output layer with 2 units
• Cannot work with one layer

19. Rumelhart & Zipser, Cntd
H V

20. Geometrical Interpretation
• So far the ordering of the output units
themselves was not necessarily informative
• The location of the winning unit can give us
information regarding similarities in the data
• We are looking for an input output mapping that
conserves the topologic properties of the
inputs  feature mapping
• Given any two spaces, it is not guaranteed that
such a mapping exits!

21.  In the Kohonen network, a neuron learns by
shifting its weights from inactive connections to
active ones. Only the winning neuron and its
neighborhood are allowed to learn. If a neuron
does not respond to a given input pattern, then
learning cannot occur in that particular neuron.
 The competitive learning rule defines the change
Dwij applied to synaptic weight wij as
where xi is the input signal and a is the learning
rate parameter.


 -

D
n
competitio
the
loses
neuron
if
,
0
n
competitio
the
wins
neuron
if
),
(
j
j
w
x
w
ij
i
ij
a

22.  The overall effect of the competitive learning rule
resides in moving the synaptic weight vector Wj of
the winning neuron j towards the input pattern X.
The matching criterion is equivalent to the
minimum Euclidean distance between vectors.
 The Euclidean distance between a pair of n-by-1
vectors X and Wj is defined by
where xi and wij are the ith elements of the vectors
X and Wj, respectively.
2
/
1
1
2
)
(








-

-
 

n
i
ij
i
j w
x
d W
X

23.  To identify the winning neuron, jX, that best
matches the input vector X, we may apply the
following condition:
where m is the number of neurons in the Kohonen
layer.
,
j
j
min
j W
X
X -
 j = 1, 2, . . .,m

24.  Suppose, for instance, that the 2-dimensional input
vector X is presented to the three-neuron Kohonen
network,
 The initial weight vectors, Wj, are given by







12
.
0
52
.
0
X







81
.
0
27
.
0
1
W 






70
.
0
42
.
0
2
W 






21
.
0
43
.
0
3
W

25.  We find the winning (best-matching) neuron jX
using the minimum-distance Euclidean criterion:
 Neuron 3 is the winner and its weight vector W3 is
updated according to the competitive learning rule.
2
21
2
2
11
1
1 )
(
)
( w
x
w
x
d -

-
 73
.
0
)
81
.
0
12
.
0
(
)
27
.
0
52
.
0
( 2
2

-

-

2
22
2
2
12
1
2 )
(
)
( w
x
w
x
d -

-
 59
.
0
)
70
.
0
12
.
0
(
)
42
.
0
52
.
0
( 2
2

-

-

2
23
2
2
13
1
3 )
(
)
( w
x
w
x
d -

-
 13
.
0
)
21
.
0
12
.
0
(
)
43
.
0
52
.
0
( 2
2

-

-

0.01
)
43
.
0
52
.
0
(
1
.
0
)
( 13
1
13 
-

-

D w
x
w
0.01
)
21
.
0
12
.
0
(
1
.
0
)
( 23
2
23 -

-

-

D w
x
w

26.  The updated weight vector W3 at iteration (p + 1)
is determined as:
 The weight vector W3 of the wining neuron 3
becomes closer to the input vector X with each
iteration.













-








D



20
.
0
44
.
0
01
.
0
0.01
21
.
0
43
.
0
)
(
)
(
)
1
( 3
3
3 p
p
p W
W
W

27. Measures of similarity
Distance Normalized scalar product

28. Kohenen Self-Organizing
networks

29. Neighbourhood Shapes
* * * * * * *
* * * * * * *
* * * * * * *
* * * * * * *
* * * * * * *
* * * * * * *
* * * # * * *
r=2
r=1
r=0
* * *
* * *
* *
* *
#
*
* *
r=2
r=1
r=0
Square neighbourhood Hexagonal neighbourhood

30. • Also known as Kohenen Feature maps or
topology-preserving maps
• Learning procedure of Kohenen feature maps is
similar to that of competitive learning networks.
• Similarity (dissimilarity) measure is selected and
the winning unit is considered to be the one with
the largest (smallest) activation
• The weights of the winning neuron as well as the
neighborhood around the winning units are
adjusted.
• Neighborhood size decreases slowly with every
iteration.

31. Training of kohenon self organizing
network
1. Select the winning output unit as the one with
the largest similarity measure between all wi
and xi . The winning unit c satisfies the
equation
||x-wc||=min||x-wi|| where the index c refers to
the winning unit (Euclidean distance)
2. Let NBc denote a set of index corresponding to
a neighborhood around winner c. The weights
of the winner and it neighboring units are
updated by
Δwi=ɳ(x-wi) iεNBc

32. Start
Initialize
Weights,
Learning rate
Initialize
Topological
Neighborhood
params
For
Each i/p
X
For i=1 to n
For j=1 to m
D(j)=Ʃ(xi-wij)2
Winning unit index J is computed
D(J)=minimum
Update Weights of winning
unit
continue
continue
Reduce learning
rate
Reduce radius
of network
Test
(t+1)
Is reduced
Stop

33. Problem
• Construct a kohenen self-organizing map
to cluster the four given vectors [0 0 1 1],
[1 0 0 0], [0 1 1 0],[0 0 01]. The number of
cluster to be formed is two. Initial learning
rate is 0.5
• Initial weights =












3
.
0
8
.
0
5
.
0
6
.
0
7
.
0
4
.
0
9
.
0
2
.
0

34. Solution to the problem
Input vector Winner weights
[0 0 1 1] D(1) [0.1 0.2 0.8 0.9]
[1 0 0 0] D(2) [0.95 0.35 0.25 0.15]
[0 1 1 0] D(1) [0.05 0.6 0.9 0.95]
[0 0 0 1] D(1) [0.025 0.3 0.45 0.975]

35. Some Observations
• Ordering phase (initial period of adaptation) :
learning rate should be close to unity
• Learning rate should be decreased linearly,
exponentially or inversely with iteration over the first
1000 epochs while maintaining its value above 0.1
• Convergence phase: learning rate should be
maintained at around 0.01 for a large number of
epochs
– may typically run into many tens of thousands of
epochs
• During the ordering phase Nk
IJ shrinks linearly with k
to finally include only a few neurons
• During the convergence phase Nk
IJ may comprise
only one or no neighbours

36. Simulation Example
The data employed in the
experiment comprised
500 points distributed
uniformly over the bipolar
square [−1, 1] × [−1, 1]
The points thus describe
a geometrically square
topology

37. SOFM Simulation

38. SOFM Simulation

39. SOFM Simulation

40. Simulation Notes
• Initial value of the neighbourhood radius r = 6
– Neighbourhood is initially a square of width 12
centered around the winning neuron IJ
• Neighbourhood width contracts by 1 every 200
epochs
• After 1000 epochs, neighbourhood radius
maintained at 1
– Means that the winning neuron and its four adjacent
neurons are designated to update their weights on all
subsequent iterations
– Can also let this value go to zero which means that
eventually, during the learning phase only the winning
neuron updates its weights

41. Inference
Initial random weights

42. Network after 100 iterations
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1
W(2,j)
W(1,j)

43. Network after 1000 iterations
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1
W(2,j)
W(1,j)

44. Network after 10,000 iterations
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1
W(2,j)
W(1,j)

45. Cluster Visualisation with ANNs
• Another demo (from http://www.ai-junkie.com/ann/som/som5.html): self-
organisation of small coloured blocks on the basis of their RGB colour
values.
• It can be used for practical purposes in mapping world poverty, for example,
when measured by a complex series of variables (e.g. health, nutrition,
education, water supply etc.)
• All of these are forms of dimensionality reduction – take complex
multivariate data and reduce it to two (or N) dimensions.

46. Limitations of competitive learning
• Weights are initialized to random values which
might be far from any input vector and it never
gets updated
– Can be prevented by initializing the weights to
samples from the input data itself, thereby ensuring
that all weights get updated when all the input
patterns are presented
– Or else the weights of winning as well as losing
neurons can be updated by tuning the learning
constant by using a significantly smaller learning rate
for the losers. This is called as leaky learning
– Note:- Changing η is generally desired. An initial value of η explores the data
space widely. Later on progressively smaller value refines the weights.

47. Limitations of competitive learning
• Lacks the capability to add new clusters
when deemed necessary
• If ɳ is constant –no stability of clusters
• If ɳ is decreasing with time may become
too small to update cluster centers
• This is called as stability-plasticity
dilemma (Solved using adaptive
resonance theory (ART))

48. • If the output units are arranged in the form of a vector or
matrix then the weights of winners as well as
neighbouring losers can be updated. (Kohenen feature
maps)
• After learning the input space is divided into a number of
disjoint clusters. These cluster centers are known as
template or code book
• For any input pattern presented we can use an
appropriate code book vector (Vector Quantization)
• This vector quantization is used in data compression in
IP and communication systems.

49. Learning Vector Quantization
LVQ

50. LVQ
• Recall that a Kohonen SOM is a clustering technique, which can be
used to provide insight into the nature of data. We can transform this
unsupervised neural network into a supervised LVQ neural network.
• The network architecture is just like a SOM, but without a topological
structure.
• Each output neuron represents a known category (e.g. apple, pear,
orange).
• Input vector = x=(x1,x2…..xn)
• Weight vector for the jth output neuron wj=( w1j,w2j,….wnj)
• Cj= Category represented by the jth neuron. This is pre-assigned.
• T = Correct category for input
• Define Euclidean distance between the input vector and the weight
vector of the jth neuron as: Ʃ(xi-wij)2

51. • It is an adaptive data classification method
based on training data with desired class
information
• It is actually a supervised training method
but employs unsupervised data-clustering
techniques to preprocess the data set and
obtain cluster centers
• Resembles a competitive learning network
except that each output unit is associated
with a class.

52. Network representation of LVQ

53. Possible data distributions and
decision boundaries

54. LVQ learning algorithm
• Step 1: Initialize the cluster centers by a
clustering method
• Step 2: Label each cluster by the voting method
• Step 3: Randomly select a training input vector x
and find k such that ||x-wk|| is a minimum
• Step 4: If x and wk belong to the same class
update wk by
)
( wk
x
wk -

D 
else
)
( wk
x
wk -
-

D 

55. • The parameters used for the training
process of a LVQ include the following
– x=training vector (x1,x2,……xn)
– T=category or class for the training vector x
– wj= weight vector for j th output unit
(w1j,…wij….wnj)
– cj= cluster or class or category associated
with jth output unit
– The Euclidean distance of jth output unit is
D(j)=Ʃ(xi-wij)2

56. Start Initialize weight
Learning rate
For each i/p
x
A
B
Y Calculate winner
Winner = min D(j)
If T=Cj
Input T
wj(n)=wj(o) +
ɳ[x-wj(o)]
wj(n)=wj(o) -
ɳ[x-wj(o)]
Y N
Reduce ɳ
ɳ(t+1)=0.5 ɳ(t)
If ɳ reduces
negligible
Stop
Y
B
A

57. Problem
• Construct and test and LVQ net with five
vectors assigned to two classes. The
given vectors along with the classes are
as shown in the table below
Vector Class
[0 0 1 1] 1
[1 0 0 0] 2
[0 0 0 1] 2
[1 1 0 0] 1
[0 1 1 0] 1