Self Organising Neural Networks
Kohonen Networks.
A Problem with Neural Networks.
ART.
Beal, R. and Jackson, T. (1990). Ne...
Kohonen Self Organising Networks
Kohonen, T. (1982). Self–organized formation of topologically cor-
rect feature maps., Bi...
Basic Kohonen Network
Geometric arrangement of units.
Units respond to “part” of the environment.
Neighbouring units shoul...
Neighbourhoods in the Kohonen Network
Example in 2D.
Neighbourhood of winning unit called Æ .
4
Learning in the Kohonen Network
All units in Æ are updated.
dÛ
dØ
«´Øµ Ü ´Øµ   Û ´Øµ for ¾ Æ
¼ otherwise
where
dÛ
dØ
= cha...
Biological origins of the Neighbourhoods
Lateral interaction of the units.
Mexican Hat form:
-100 -80 -60 -40 -20 0 20 40 ...
Biological origins of the Neighbourhoods: Mals-
burg
Inhibitory connections:
Excitatory units
Inhibitory units
Excitatory ...
1-d example
1
4
3
2
5
5
4
2
1
3
5
2
1
3
4
4 15 3 2
4
1
3 2
5
8
2-d example: uniform density
8x8 units in 2D lattice
2 input lines.
Inputs between ·½ and  ½.
Input space:
+1
+1
-1
-1
9
2-d example: uniform density
10
2-d example: non-uniform density
Same 8x8 units in 2D lattice.
Same input space.
Different input distribution
+1
+1
-1
-1
...
2-d example: non-uniform density
12
2-d µ1-d example: dimension reduction
2-d input uniform density; 1-d output arrangement.
“Space-filling” (Peano) curves; ca...
Example Application of Kohonen’s Network
The Phonetic Typewriter
MP Filter A/D
FFT
Rules
Kohonen
Network
Problem: Classific...
A Problem with Neural Networks
Consider 3 network examples:
Kohonen Network.
Associative Network.
Feed Forward Back-propag...
Adaptive Resonance Theory
Grossberg, S. (1976a). Adaptive pattern classification and univer-
sal recoding I: Feedback, expe...
ART1
Orientingsubsystem
+
+
-
+
+ ( )
G
+ (Ø )
r-
+
Attentional subsystem
Input (Ü )
F2 units ( )
F1 units (Ü )
F1  F2 ful...
Summary of ART 1
(Lippmann, 1987). N = number of F1 units.
Step 1: Initialization
Ø ½ ½
½·Æ
Set vigilance parameter ¼ ½
St...
ART1: Example
INPUT
UNIT 1 UNIT 2
resonance
resonance
1st choice
reset
resonance
1st choice
reset
2nd choice
reset
resonan...
Summary
Simple?
Interesting biological parallels.
Diverse applications.
Extensions.
20
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redes neuronales tipo Som

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redes neuronales tipo Som

  1. 1. Self Organising Neural Networks Kohonen Networks. A Problem with Neural Networks. ART. Beal, R. and Jackson, T. (1990). Neural Computing: An Introduction. Chapters 5 & 7. Adam Hilger, NY. Hertz, J., Krogh, A. and Palmer, R. (1991). Introduction to the Theory of Neural Computation. Chapter 9. Addison–Wesley. NY. Grossberg, S. (1987). Competitive Learning: from interactive acti- vation to adaptive resonance. Cognitive Science, 11: 23–63. 1
  2. 2. Kohonen Self Organising Networks Kohonen, T. (1982). Self–organized formation of topologically cor- rect feature maps., Biological Cybernetics, 43: 59–69. An abstraction from earlier models (e.g. Malsburg, 1973). The formation of feature maps (introducing a geo- metric layout). Popular and useful. Can be traced to biologically inspired origins. Why have topographic mappings? – Minimal wiring – Help subsequent processing layers. Example: Xenopus retinotectal mapping (Price & Will- shaw 2000, p121). 2
  3. 3. Basic Kohonen Network Geometric arrangement of units. Units respond to “part” of the environment. Neighbouring units should respond to similar parts of the environment. Winning unit selected by: Ü   Û min Ü   Û where Û is the weight vector of winning unit, and Ü is the input pattern. and Neighbourhoods... 3
  4. 4. Neighbourhoods in the Kohonen Network Example in 2D. Neighbourhood of winning unit called Æ . 4
  5. 5. Learning in the Kohonen Network All units in Æ are updated. dÛ dØ «´Øµ Ü ´Øµ   Û ´Øµ for ¾ Æ ¼ otherwise where dÛ dØ = change in weight over time. «´Øµ = time dependent learning parameter. Ü ´Øµ = input component at time Ø. Û ´Øµ = weight from input to unit at time Ø. ¯ Geometrical effect: move weight vector closer to in- put vector. ¯ « is strongest for winner and can decrease with dis- tance. Also decreases over time for stability. 5
  6. 6. Biological origins of the Neighbourhoods Lateral interaction of the units. Mexican Hat form: -100 -80 -60 -40 -20 0 20 40 60 80 100 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 10 20 30 40 0 10 20 30 40 -1 0 1 2 3 6
  7. 7. Biological origins of the Neighbourhoods: Mals- burg Inhibitory connections: Excitatory units Inhibitory units Excitatory units Inhibitory units Excitatory connections: Implements winner-take-all processing. 7
  8. 8. 1-d example 1 4 3 2 5 5 4 2 1 3 5 2 1 3 4 4 15 3 2 4 1 3 2 5 8
  9. 9. 2-d example: uniform density 8x8 units in 2D lattice 2 input lines. Inputs between ·½ and  ½. Input space: +1 +1 -1 -1 9
  10. 10. 2-d example: uniform density 10
  11. 11. 2-d example: non-uniform density Same 8x8 units in 2D lattice. Same input space. Different input distribution +1 +1 -1 -1 11
  12. 12. 2-d example: non-uniform density 12
  13. 13. 2-d µ1-d example: dimension reduction 2-d input uniform density; 1-d output arrangement. “Space-filling” (Peano) curves; can solve Travelling Salesman Problem. init wts epoch 10 epoch 500 epoch 700 13
  14. 14. Example Application of Kohonen’s Network The Phonetic Typewriter MP Filter A/D FFT Rules Kohonen Network Problem: Classification of phonemes in real time. Pre and post processing. Network trained on time sliced speech wave forms. Rules needed to handle co-articulation effects. 14
  15. 15. A Problem with Neural Networks Consider 3 network examples: Kohonen Network. Associative Network. Feed Forward Back-propagation. Under the situation: Network learns environment (or I/O relations). Network is stable in the environment. Network is placed in a new environment. What happens: Kohonen Network won’t learn. Associative Network OK. Feed Forward Back-propagation Forgets. called The Stability/Plasticity Dilemma. 15
  16. 16. Adaptive Resonance Theory Grossberg, S. (1976a). Adaptive pattern classification and univer- sal recoding I: Feedback, expectation, olfaction, illusions. Biological Cybernetics, 23: 187–202. a “neural network that self–organize[s] stable pat- tern recognition codes in real time, in response to arbitrary sequences of input patterns”. ART1 (1976). Localist representation, binary patterns. ART2 (1987). Localist representation, analog patterns. ART3 (1990). Distributed representation, analog pat- terns. Desirable properties: plastic + stable biological mechanisms analytical math foundation 16
  17. 17. ART1 Orientingsubsystem + + - + + ( ) G + (Ø ) r- + Attentional subsystem Input (Ü ) F2 units ( ) F1 units (Ü ) F1  F2 fully connected, excitatory ( ). F2  F1 fully connected, excitatory (Ø ). Pattern of activation on F1 and F2 called Short Term Memory. Weight representations called Long Term Memory. Localist representations of binary input patterns. 17
  18. 18. Summary of ART 1 (Lippmann, 1987). N = number of F1 units. Step 1: Initialization Ø ½ ½ ½·Æ Set vigilance parameter ¼ ½ Step 2: apply new input (binary Ü ) Step 3: compute F2 activation Æ ½ Ü Step 4: find best matching node , where . Step 5: vigilance test Æ ½ Ü Ì ¡ Æ ½ Ø Ü Is Ì ¡ If no, go to step 6. If yes go to step 7. Step 6: mismatch/reset: set ¼ and go to step 4. Step 7: resonance — adapt best match Ø Ø Ü Ø · ÈÆ ½ Ø Ü Step 8: Re-enable all F2 units and go to step 2 18
  19. 19. ART1: Example INPUT UNIT 1 UNIT 2 resonance resonance 1st choice reset resonance 1st choice reset 2nd choice reset resonance 1st choice reset 2nd choice resetreset 3rd choice resonance UNIT 3 UNIT 4 1st choice resonance 1st choice reset 2nd choice resonance 1st choice reset 2nd choice reset 3rd choice resetreset 4th choice resonance UNIT 5 F2 UNITS REPRESENT: 19
  20. 20. Summary Simple? Interesting biological parallels. Diverse applications. Extensions. 20

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