1. Unsupervised learning
Unsupervised learning is the process of finding
structure, patterns or correlation in the given data.
we distinguish:
Unsupervised Hebbian learning
Principal component analysis
Unsupervised competitive learning
Clustering
Data compression
PCA not treated: Oja’s rule performs normalized Hebbian learning
w_i (n+1) = w_i(n) + \eta y(n)(x_i(n) – y(n)w_i(n))
UCL is treated
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2. Unsupervised Competitive Learning
In unsupervised competitive learning the neurons
take part in some competition for each input. The
winner of the competition and sometimes some
other neurons are allowed to change their weights
In simple competitive learning only the winner is allowed to learn (change its weight).
In self-organizing maps other neurons in the
neighborhood of the winner may also learn.
Requires lateral inhibition: output neurons that fire inhibit others from firing
If a single neuron fires this is a form of classification. Each class
Determines a cluster of input vectors. The weight associated with
each output neuron can be interpreted as a representing the whole
Cluster.
Question: How many cluster does the data contain?
SOM also retain topological properties of the data.
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3. Applications Speech Recognition OCR, e.g. handwritten characters
Image compression
(using code-book vectors)
Texture maps
Classification of cloud pattern (cumulus etc.)
Contextual maps
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4. Network topology For simple competitive learning the network con-
sists of a single layer of
linear neurons each con-
nected to all inputs.
Lateral inhibition not indicated
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5. Definition of the Winner
There are various criteria to define which neuron i
becomes the winner of the competition for input x:
When the weights are normalized these criteria are identical as can be seen from the equation
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6. Training Set A training set for unsupervised learning consists
only of input vectors (no targets!)
Given a network with weight matrix W the training
set can be partitioned into clusters Xi according to the classification made by the network
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7. Simple Competitive Learning (incremental version)
Reinforcement learning (compare perceptron rule)
0 < alpha < 1
This technique is sometimes called
‘the winner takes it all’
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8. Convergence (incremental version)
Unless the learning parameter tends to 0,
the incremental version of simple compe-
titive learning does not convergence
In absence of convergence the weight
vectors oscillate around the centers of their
clusters
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9. Simple Competitive Learning (batch version)
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10. Cluster Means Let ni be the number of element in cluster Xi.
Then we define the mean mi of cluster i by
Hence in the batch version the weight is
given by
So the weights of the winning neuron are moved in the direction of the mean of its cluster.
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11. Data Compression The final value of the weight vectors are some-
times called code-book vectors. This nomencla-
ture stems from data compression applications.
Compress (encode)
Map vector x to code-word i = win (W, x)
Decompress (decode)
Map code-word i to code-book vector wi which is presumably close to the original vector x
Note that this is a form of lossy data compression
Compression to speed-up data-transport. Not encryption.
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12. Convergence (batch version)
In the batch version of simple competitive learning the weight vector wi can be shown to converge to the mean of the input vectors that have i as winning neuron
In fact the batch version is a gradient des-cent method that converges to a local minimum of a suitably chosen error func-tion
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13. Error Function
For a network with weight matrix W and training set we define the error function E(W) by
Let , then
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14. Gradients of the Error functions
Because
It follows that the gradient of the error in the
i-th cluster is given by
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15. Minima are Cluster Means
After termination of the learning algorithm all
gradients are zero, i.e. for all i, 1 · i · k ,
So after learning the weight vectors of the non-empty clusters have converged to the mean vectors of those clusters.
Note that learning stops in a local minimum, so better clusters may exist.
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16. Dead neurons & minima input 2 7 w1 w2 E winner 1 3 >11 3.5 local2 7 w1 w2 E
winner 1 3
>11
3.5
local
1/3
global -
<-3
Geval dan w1 = (0+2+7)/3 |w2 – 7| > |w1 -7|
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17. K-means clustering as SCL
K-means clustering is a popular statistical method to organize multi-dimensional data into K groups.
K-means clustering can be seen as an instance of simple competitive learning, where each neuron has its own learning rate.
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18. SCL (batch version)
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19. Move learning factor outside repetition
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20. Set individual learning rate
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21. Split such that
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22. Eliminate
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23. Introduce separate cluster variables
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24. Reuse mj :K-means clustering I
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25. K-means Clustering II
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26. Convergence of K-means Clustering
The convergence proof of the K-means
clustering algorithms involves showing two
facts
Reassigning a vector to a different cluster does not increase the error function
Updating the mean of a cluster does not increase the error function
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27. Reassigning a vector Assume vector x(p) moves from cluster j to clus-
ter i. Then it follows that
Hence
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28. Updating the Mean of a Cluster
Consider cluster Xi with old mean and new mean
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29. Non-optimal Stable Clusters3 6 8 10
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