1. x0 xn w0 wn o
Threshold units
SOM

2. Self-Organizing Maps

3. Inputs
Neurons
Teuvo Kohonen

4. Self-Organizing Maps : Origins
Ideas first introduced by C. von der Malsburg (1973), developed and refined by T. Kohonen (1982)
Neural network algorithm using unsupervised competitive learning
Primarily used for organization and visualization of complex data
Biological basis: ‘brain maps’

5. Self-Organizing Maps SOM - Architecture 2d array of neurons
Lattice of neurons (‘nodes’) accepts and responds to set of input signals
Responses compared; ‘winning’ neuron selected from lattice
Selected neuron activated together with ‘neighbourhood’ neurons
Adaptive process changes weights to more closely resemble inputs
2d array of neurons
Set of input signals
(connected to all neurons in lattice)
Weighted synapses x1 x2 x3 xn
...
wj1
wj2
wj3
wjn j

6. Self-Organizing Maps Randomly initialise all weights
SOM – Algorithm Overview
Randomly initialise all weights
Select input vector x = [x1, x2, x3, … , xn]
Compare x with weights wj for each neuron j to determine winner
Update winner so that it becomes more like x, together with the winner’s neighbours
Adjust parameters: learning rate & ‘neighbourhood function’
Repeat from (2) until the map has converged (i.e. no noticeable changes in the weights) or pre-defined no. of training cycles have passed

7. Initialisation
Randomly initialise the weights

8. Finding a Winner Euclidean distance
Find the best-matching neuron w(x), usually the neuron whose weight vector has smallest Euclidean distance from the input vector x
The winning node is that which is in some sense ‘closest’ to the input vector
‘Euclidean distance’ is the straight line distance between the data points, if they were plotted on a (multi-dimensional) graph
Euclidean distance between two vectors a and b,
a = (a1,a2,…,an), b = (b1,b2,…bn), is calculated as:
Euclidean distance

9. wj(t +1) = wj(t) + (t) (x)(j,t) [x - wj(t)]
Weight Update
SOM Weight Update Equation
wj(t +1) = wj(t) + (t) (x)(j,t) [x - wj(t)]
“The weights of every node are updated at each cycle by adding
Current learning rate × Degree of neighbourhood with respect to winner × Difference between current weights and input vector
to the current weights”
Example of (t) Example of (x)(j,t)
L. rate
No. of cycles
x-axis shows distance from winning node
y-axis shows ‘degree of neighbourhood’ (max. 1)

10. Kohonen’s Algorithm
jth input
Winner ith

11. Neighborhoods Square and hexagonal grid with
neighborhoods based on box distance
Grid-lines are not shown

12. i
One-dimensional i
Two-dimensional
Neighborhood of neuron i

14. A neighborhood function (i, k) indicates how closely neurons i and k in the output layer are connected to each other.
Usually, a Gaussian function on the distance between the two neurons in the layer is used: 
position of i
position of k

16. Clustering of the Self Organising Map
A simple toy example
Clustering of the Self Organising Map

17. However, instead of updating only the winning neuron i
However, instead of updating only the winning neuron i*, all neurons within a certain neighborhood Ni* (d), of the winning neuron are updated using the Kohonen rule. Specifically, we adjust all such neurons i Ni* (d), as follow
Here the neighborhood Ni* (d), contains the indices for all of the neurons that lie within a radius d of the winning neuron i*.

18. Topologically Correct Maps
The aim of unsupervised self-organizing
learning is to construct a topologically correct
map of the input space.

19. Self Organizing Map
Determine the winner (the neuron of which the weight vector has the smallest distance to the input vector)
Move the weight vector w of the winning neuron towards the input i
Before learning i w
After learning i w

20. Network Features Input nodes are connected to every neuron
The “winner” neuron is the one whose weights are most “similar” to the input
Neurons participate in a “winner-take-all” behavior
The winner output is set to 1 and all others to 0
Only weights to the winner and its neighbors are adapted

21. P 1 2 3 4 5 6 7 8 9 wi

22. P1 P2 1 2 3 4 5 6 7 8 9
wi2
wi1

23. output
input (n-dimensional)
winner

25. Example I: Learning a one-dimensional representation of a two-dimensional (triangular) input space:20
100
10000
25000
1000

26. Some nice illustrations

27. Some nice illustrations

28. Some nice illustrations

29. Self Organizing Map
Impose a topological order onto the competitive neurons (e.g., rectangular map)
Let neighbors of the winner share the “prize” (The “postcode lottery” principle)
After learning, neurons with similar weights tend to cluster on the map

30. Conclusion Advantages
SOM is Algorithm that projects high-dimensional data onto a two-dimensional map.
The projection preserves the topology of the data so that similar data items will be mapped to nearby locations on the map.
SOM still have many practical applications in pattern recognition, speech analysis, industrial and medical diagnostics, data mining
Disadvantages
Large quantity of good quality representative training data required
No generally accepted measure of ‘quality’ of a SOM
e.g. Average quantization error (how well the data is classified)

31. Topologies (gridtop, hextop, randtop)
pos = gridtop(3,2)
pos =
plotsom (pos)
pos = gridtop(2,3)
pos =
plotsom (pos)

32. pos = gridtop(8,10);
plotsom(pos)

33. pos = hextop(2,3)
pos =

34. pos = hextop(3,2)
pos =
plotsom(pos)

35. pos = hextop(8,10);
plotsom(pos)

36. pos = randtop(2,3)
pos =

37. pos = randtop(3,2)
pos =

38. pos = randtop(8,10);
plotsom(pos)

39. Distance Funct. (dist, linkdist, mandist, boxdist)
pos2 = [ 0 1 2; 0 1 2]
pos2 =
D2 = dist(pos2)
D2 =

41. pos = gridtop(2,3)
pos =
plotsom(pos)
d = boxdist(pos)
d =

42. pos = gridtop(2,3)
pos =
plotsom(pos)
d=linkdist(pos)
d =

43. The Manhattan distance between two vectors x and y is calculated as
D = sum(abs(x-y))
Thus if we have
W1 = [ 1 2; 3 4; 5 6]
W1 =
and
P1= [1;1]
P1 = 1
then we get for the distances
Z1 = mandist(W1,P1)
Z1 = 5 9

44. A One-dimensional Self-organizing Map
angles = 0:2*pi/99:2*pi;
P = [sin(angles); cos(angles)];
plot(P(1,:),P(2,:),'+r')

45. net = newsom([-1 1;-1 1],[30]); net.trainParam.epochs = 100;
net = train(net,P);
plotsom(net.iw{1,1},net.layers{1}.distances)
The map can now be used to classify inputs, like [1; 0]:
Either neuron 1 or 10 should have an output of 1, as the above input vector was at one end of the presented input space. The first pair of numbers indicate the neuron, and the single number indicates its output.
p = [1;0];
a = sim (net, p)
a =
(1,1)

46. x = -4:0.01:4
P = [x;x.^2];
plot(P(1,:),P(2,:),'+r')
net = newsom([-10 10;0 20],[10 10]);
net.trainParam.epochs = 100;
net = train(net,P);
plotsom(net.iw{1,1},net.layers{1}.distances)