• Unsupervised neural networks, equivalent to clustering.
• Two layers – input and output
– The input layer represents the input variables.
– The output layer: neurons arranged in a single line
(one-dimensional) or a two-dimensional grid.
• Main feature – weights

• Learning means adopting the weights.
• Each output receives inputs through the weights.
 weight vector has the same dimensionality as the input vector
• The output of each neuron is its activation – weighted sum of inputs (i.e. linear activation function).
u = x
1 w
11
+ x
2 w
21 w
11 w
21
2

• The objective of learning: project highdimensional data onto 1D or 2D output neurons.
• Each neuron incrementally learns to represent a cluster of data.
• The weights are adjusted incrementally – the weights of neurons are called codebook vectors (codebooks).

• The so-called competitive learning (winnertakes-all) .
• Competitive learning will be demonstrated on simple 1D network with two inputs.
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

• First, number of output neurons (i.e. clusters) must be selected.
– Not always known, do reasonable estimate, it is better to use more, not used neurons can be eliminated later.
• Then initialize weights.
– e.g., small random values
– Or randomly choose some input vectors and use their values for the weights.
• Then competitive learning can begin.

• The activation for each output neuron is calculated as weighted sum of inputs.
• e.g., the activation of the output neuron 1 is n u
1
= w
11 x
1
+ w
21 x
2
. Generally u j
  i

1 w x ij i
• Activation is the dot product between input vector x and weight vector w j
.
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

• Dot product is not only , but also j
 n  i

1 w x ij i u x w j
 j cos
 x w
θ
• If |x| = |w j
| = 1, then u j
= cosθ.
• The closer these two vectors are (i.e. the smaller θ is), the bigger the u j is (cos 0 = 1).

• Say it again, and loudly:
The closer the weight and input vectors are, the bigger the neuron activation is. Dan na
Hrad.
• A simple measure of the closeness – Euclidean distance between x and w j
.
d j x w j
 i n 

1
 x i
 w ij

2

• Scale the input vector so that its length is equal to one. |x|=1
• An input is presented to the network.
• Scale weight vectors of individual output neurons to the unit length. |w|=1
• Calculate, how close the input vector x is to each weight vector w neurons).
j
(j is 1 … # output
• The neuron which codebook is closest to the input vector becomes a winner ( BMU , Best
Matching Unit).
• Its weights will be updated.

• The weight vector w is updated so that it moves closer to the input x. d Δw j
 
 x w j

  d j x
β – learning rate
Δw w

• Conceptually similar to online/batch learning
• Recursive learning:
– update weights of the winning neuron after each presentation of input vector
• Batch learning:
– note the weight update for each input vector
– the average weight adjustment for each output neuron is done after the whole epoch
• When to terminate learning?
– mean distance between neurons and inputs they represent is at a minimum, distance stops changing

Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

epoch
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

Topology is not preserved.
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

Teuvo Kohonen

• SOM, also Self-Organizing Feature Map
(SOFM), Kohonen neural network.
• Inspired by the function of brain:
– Different brain regions correspond to specific aspects of human activities.
– These regions are organized such that tasks of similar nature (e.g. speech and vision) are controlled by regions that are in spatial proximity each to other.
– This is called topology preservation .

• In SOM learning, not only the winner, but also the neighboring neurons adapt their weights.
• Neurons closer to the winner adjust weights more than further neurons.
• Thus we need
1. to define the neighborhood
2. to define a way how much neighboring neurons adapt their weights

2 1 1 2
2
1
1
2
3 neighborhood radius r
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

• Follows similar manner of standard winner-takesall competitive training.
• However, new rule is used for weight changes.
• Suppose, that the BMU is at position {i win
, j win
} on the 2D map.
• Then all codebook vectors of BMU and neighbors are adjusted to w ’ j according to w
' j
 w j
 
NS

 j where NS is the neighbor strength varying with the distance to the BMU. β is learning rate.

• When using neighbor features, all neighbor codebooks are shifted towards the input vector.
• However, BMU updates most, and the further away the neighbor neuron is, the less its weights update.
• The NS function tells us how the weight adjustment decays with distance from the winner.

Slide by Johan Everts

Linear
Exponential
NS

Exp
 kd ij
Gaussian
NS

Exp
 d ij
2
2

2

source: http://www.cis.hut.fi/projects/somtoolbox/download/sc_shots2.shtml

• Large neighborhood – proper placement of neurons in the initial stage to broadly represent spatial organization of input data.
• Further refinement – subsequent shrinking of the neighborhood.
σ
0
• A size of large starting neighborhood is reduced with iterations.
… initial neighborhood size
  t

0

1
 t T
   t

0
Exp
  t T
 linear decay exponential decay
σ t
… neighborhood size at iteration t
T … total number of iterations bringing neighborhood to zero (i.e., only winner remains)
NS

Exp
 d ij
2
Exp

 d ij
2
2

2 
2


0
Exp
  t T
 
2



• The step length (learning rate 𝛽 ) is also reduce with iterations.
• Two common forms: linear or exponential decay
 t
 
0

1
 t T
  t
 
0
Exp
  t T

𝑇 … constant bringing 𝛽 to zero (or small value)
• Strategy: start with relatively high β, decrease gradually, but remain above 0.0
1
Weight update incorporating learning rate and neighborhood decay w j
   w j
 t
         w j
 t

1


• SOM in a batch mode with no neigborhood is equivalent to k-means
• The use of neighborhood leads to topology preservation
– Regions closer in input space are represented by neurons closer in a map.

• Two phases
1. ordering
2. convergence
• Ordering
– neighborhood and learning rate are reduced to small values
– topological ordering
– starting 𝛽 high, gradually decrease, remain above 0.01
– neighborhood – cover whole output layer
• Convergence
– fine tuning with the shrunk neighborhood
– small non-zero (~0.01) learning rate, NS no more than
1 st neighborghood

Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

NS

Exp
 
0.1
d
  t

2
3
 t
 t
 
0

1
 t T
 neighborhood drops to 0 after 3 iterations

topology preservation takes effect very quickly
Complete training
Converged after 40 epochs.
Epochs
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

2
3
1
4
5
6
• All vectors have found cluster centers
• Except one
• Solution: add one more neuron
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

1
2
3
4
5
6
7
Sandhya Samarasinghe, Neural Networks for Applied
Sciences and Engineering, 2006

• Play with http://www.demogng.de/
• Choose 'The self-organizing map'. Following update formulas are used:
NS

Exp
 d ij
2
2

2
 t
 i
 f i
 t t max
 t
 i
 f i
 t t max neighborhood size learning rate

A self-organizing feature map from a square source space to a square (grid) target space.
Duda, Hart, Stork, Pattern Classification, 2000

Some initial (random) weights and the particular sequence of patterns (randomly chosen) lead to kinks in the map; even extensive further training does not eliminate the kink. In such cases, learning should be restarted with randomized weights and possibly a wider window function and slower decay in learning.
Duda, Hart, Stork, Pattern Classification, 2000

• Iris data set
– 150 patterns, 4 attributes, 3 classes (Setosa – 1,
Versicolor – 2, Virginica – 3)
– more than 2 dimensions, so all data can not be vizualized in a meaningful way
– SOM can be used not only to cluster input data, but also to explore the relationships between different attributes.
• SOM structure
– 8x8, hexagonal, exp decay of learning rate β (β init
0.5, T max
= 20x150 = 3000), NS: Gaussian
=

virginica versicolor setosa
What can be learned?
• petal length and width have structure similar to the class panel
 low length correlates with low width and these relate to the Versicolor class
• sepal width – very different pattern
• class panel – boundary between Virginica and Setosa – classes overlap
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006

• Since we have class labels, we can assess the classification accuracy of the map.
• So first we train the map using all 150 patterns.
• And then we present input patterns individually again and note the winning neuron.
– The class to which the input belongs is the class associated with this BMU codebook vector (see previous slide, Class panel).
– Only the winner decides classification.

Only winner decides the classification
Vers (2) – 100% accuracy
Set (1) – 86%
Virg (3) – 88%
Overall accuracy = 91.3%
Neighborhood of size 2 decides the classification
Vers (2) – 100% accuracy
Set (1) – 90%
Virg (3) – 94%
Overall accuracy = 94.7%
Sandhya Samarasinghe, Neural Networks for Applied Sciences and Engineering, 2006