Download as PDF, PPTX
![5
Introduction…Competitive nets
Fausett
The weight update for output (or cluster) unit j is given as:
where x is the input vector, w.j is the weight vector for unit j,
and α the learning rate, decreases as learning proceeds.
Two methods of determining the closest weight vector to a
pattern vector are commonly used for self-organizing nets.
Both are based on the assumption that the weight vector
for each cluster (output) unit serves as an exemplar for the
input vectors that have been assigned to that unit during
learning.
– The first method of determining the winner uses the
squared Euclidean distance between the I/P vector
[ ]
old)()1(
old)()old()new(
.
...
j
jjj
wx
wxww
αα
α
−+
−+=](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-5-2048.jpg)
![10
MAXNET
[ ]k
M
k
yWy Γ=+1
−−
−−
−−
=
1..
.
.
..1
..1
εε
εε
εε
MW
( )
≥
<
=
0,
0,0
netnet
net
netf
p
10 << ε](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-10-2048.jpg)
![14
Example
Note that in step 2, the i/p to the function f is the total
i/p to node Aj from all nodes, including itself.
Some precautions should be incorporated to handle
the situation in which two or more units have the
same, maximal, input.
Example:
ε = .2 and the initial activations (input signals) are:
a1(0) = .2, a2(0) = .4 a3(0) = .4 a4(0) = .8
As the net iterates, the activations are:
a1(1) = f{a1(0) -.2 [a2(0) + a3(0) + a4(0)]} = f(-.12)=0
a1(1) = .0, a2(1) = .08 a3(1) = .32 a4(1) = .56
a1(2) = .0, a2(2) = .0 a3(2) = .192 a4(2) = .48
a1(3) = .0, a2(3) = .0 a3(3) = .096 a4(3) = .442
a1(4) = .0, a2(4) = .0 a3(4) = .008 a4(4) = .422
a1(5) = .0, a2(5) = .0 a3(5) = .0 a4(5) = .421
The only node
to remain on](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-14-2048.jpg)
![32
Hamming Network Fixed-weight competitive net
Example (see architecture n=4)
[ ]
[ ] patterns)(exemplar2casein thism1-1,-1,-1,)2(
1-1,-1,-1,)1(
:ectorsexemplar vGiven the
=−=
=
e
e
The Hamming net can be used to find the exemplar that is closest to each of the
bipolar i/p patterns, (1, 1, -1, -1), (1, -1, -1, -1), (-1, -1, -1, 1), and (-1, -1, 1, 1)
Step 1: Store the m exemplar vectors in the weights:
case)(in this4nodesinputofnumber
2/2
:biasestheInitialize
2
)(
rewhe
5.5.
5.5.
5.5.
5.5.
21
==
===
=
−−
−−
−−
−
=
n
nbb
je
wW i
ij](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-32-2048.jpg)
![41
Fausett
Kohonen Self-Organizing Maps (SOM)
Architecture:
* * {* (* [#] *) *} * *
{ } R=2 ( ) R=1 [ ] R=0
Linear array of cluster units
This figure represents the neighborhood of the unit designated by # in a 1D topology
(with 10 cluster units).
*******
*******
*******
***#***
*******
*******
*******
R=2
R=1
R=0
Neighbourhoods for rectangular grid (Each unit has 8 neighbours)](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-41-2048.jpg)
()( oldwxoldwneww ijiijij −+= α](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-44-2048.jpg)
()(
2
222
+
−+=
This set of weights has not
been modified.
=
12.9.
28.5.
76.6.
92.2.
W](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-48-2048.jpg)
()(
+
−+=
=
024.999.
055.630.
360.047.
980.016.
W
=
0.00.1
0.05.
5.00.
0.10.
W
The 1st
column is the average
of the 2 vects in cluster 1, and
the 2nd
col is the average of
the 2 vects in cluster 2
This set of weights has not
been modified.](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-52-2048.jpg)
![59
CreatingaSOM
newsom -- Create a self-organizing map
Syntax
net = newsom
net = newsom(PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND)
Description
net = newsom creates a new network with a dialog box.
net = newsom (PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND)
takes,
PR - R x 2 matrix of min and max values for R input elements.
Di - Size of ith layer dimension, defaults = [5 8].
TFCN - Topology function, default ='hextop'.
DFCN - Distance function, default ='linkdist'.
OLR - Ordering phase learning rate, default = 0.9.
OSTEPS - Ordering phase steps, default = 1000.
TLR - Tuning phase learning rate, default = 0.02;
TND - Tuning phase neighborhood distance, default = 1.
and returns a new self-organizing map.
The topology function TFCN can be hextop, gridtop, or randtop. The
distance function can be linkdist, dist, or mandist.](https://image.slidesharecdn.com/lect7-uwa-160515045203/75/Artificial-Neural-Networks-Lect7-Neural-networks-based-on-competition-59-2048.jpg)
Uploaded byMohammed Bennamoun
Artificial Neural Networks Lect7: Neural networks based on competition
This document summarizes key concepts about neural networks based on competition. It discusses fixed weight competitive networks including Maxnet, Mexican Hat, and Hamming Net. Maxnet uses winner-take-all competition where only the neuron with the largest activation remains on. The Mexican Hat network enhances contrast through excitatory connections to nearby neurons and inhibitory connections to farther neurons. Iterating the activations over time steps increases the activation of neurons with initially larger signals and decreases others. Kohonen self-organizing maps and their training in Matlab are also mentioned.
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Artificial Neural Networks Lect7: Neural networks based on competition
- 1. 1 CS407 Neural Computation Lecture7: Neural Networks Based on Competition. Lecturer: A/Prof. M. Bennamoun
- 2. 2 Neural Networks Basedon Competition. Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading
- 3. 3 Introduction…Competitive nets Fausett The mostextreme form of competition among a group of neurons is called “Winner Take All”. As the name suggests, only one neuron in the competing group will have a nonzero output signal when the competition is completed. A specific competitive net that performs Winner Take All (WTA) competition is the Maxnet. A more general form of competition, the “Mexican Hat” will also be described in this lecture (instead of a non-zero o/p for the winner and zeros for all other competing nodes, we have a bubble around the winner ). All of the other nets we discuss in this lecture use WTA competition as part of their operation. With the exception of the fixed-weight competitive nets (namely Maxnet, Mexican Hat, and Hamming net) all of the other nets combine competition with some form of learning to adjusts the weights of the net (i.e. the weights that are not part of any interconnections in the competitive layer).
- 4. 4 Introduction…Competitive nets Fausett The formof learning depends on the purpose for which the net is being trained: – LVQ and counterpropagation net are trained to perform mappings. The learning in this case is supervised. – SOM (used for clustering of input data): a common use of unsupervised learning. – ART are also clustering nets: also unsupervised. Several of the nets discussed use the same learning algorithm known as “Kohonen learning”: where the units that update their weights do so by forming a new weight vector that is a linear combination of the old weight vector and the current input vector. Typically, the unit whose weight vector was closest to the input vector is allowed to learn.
- 5. 5 Introduction…Competitive nets Fausett The weightupdate for output (or cluster) unit j is given as: where x is the input vector, w.j is the weight vector for unit j, and α the learning rate, decreases as learning proceeds. Two methods of determining the closest weight vector to a pattern vector are commonly used for self-organizing nets. Both are based on the assumption that the weight vector for each cluster (output) unit serves as an exemplar for the input vectors that have been assigned to that unit during learning. – The first method of determining the winner uses the squared Euclidean distance between the I/P vector [ ] old)()1( old)()old()new( . ... j jjj wx wxww αα α −+ −+=
- 6. 6 Introduction…Competitive nets Fausett and theweight vector and chooses the unit whose weight vector has the smallest Euclidean distance from the I/P vector. – The second method uses the dot product of the I/P vector and the weight vector. The dot product can be interpreted as giving the correlation between the I/P and weight vector. In general and for consistency, we will use Euclidean distance squared. Many NNs use the idea of competition among neurons to enhance the contrast in activations of the neurons In the most extreme situation, the case of the Winner-Take- All, only the neuron with the largest activation is allowed to remain “on”.
- 7.
- 8. 8 Introduction…Competitive nets 1. Euclideandistance between two vectors 2. Cosine of the angle between two vectors ( ) ( )i t ii xxxxxx −−=− i i t xx xx =ψcos 21 ψψψ << T
- 9. 9 Neural Networks Basedon Competition. Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading
- 10. 10 MAXNET [ ]k M k yWy Γ=+1 −− −− −− = 1.. . . ..1 ..1 εε εε εε MW () ≥ < = 0, 0,0 netnet net netf p 10 << ε
- 11. 11 Hamming Network andMAXNET MAXNET – A recurrent network involving both excitatory and inhibitory connections • Positive self-feedbacks and negative cross-feedbacks – After a number of recurrences, the only non- zero node will be the one with the largest initializing entry from i/p vector
- 12. 12 • Maxnet Lateralinhibition between competitors Maxnet Fixed-weight competitive net Fausett > = otherwise0 0if )( :Maxnetfor thefunctionactivation xx xf Competition: • iterative process until the net stabilizes (at most one node with positive activation) – Step 0: • Initialize activations and weight (set ) where m is the # of nodes (competitors) ,/10 m<< ε
- 13. 13 Algorithm Step 1: Whilestopping condition is false do steps 2-4 – Step 2: Update the activation of each node: – Step 3: Save activation for use in next iteration: – Step 4: Test stopping condition: If more than one node has a nonzero activation, continue; otherwise, stop. − = = = otherwise if1 :weights nodeinput to)0( ε ji w Aa ij jj mj(old)a(old) –af(new)a jk kjj ,...,1for = = ∑≠ ε mj(new)a(old)a jj ,...,1for == the total i/p to node Aj from all nodes, including itself
- 14. 14 Example Note that instep 2, the i/p to the function f is the total i/p to node Aj from all nodes, including itself. Some precautions should be incorporated to handle the situation in which two or more units have the same, maximal, input. Example: ε = .2 and the initial activations (input signals) are: a1(0) = .2, a2(0) = .4 a3(0) = .4 a4(0) = .8 As the net iterates, the activations are: a1(1) = f{a1(0) -.2 [a2(0) + a3(0) + a4(0)]} = f(-.12)=0 a1(1) = .0, a2(1) = .08 a3(1) = .32 a4(1) = .56 a1(2) = .0, a2(2) = .0 a3(2) = .192 a4(2) = .48 a1(3) = .0, a2(3) = .0 a3(3) = .096 a4(3) = .442 a1(4) = .0, a2(4) = .0 a3(4) = .008 a4(4) = .422 a1(5) = .0, a2(5) = .0 a3(5) = .0 a4(5) = .421 The only node to remain on
- 15. 15 Neural Networks Basedon Competition. Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading
- 16. 16 Mexican Hat Fixed-weightcompetitive net Architecture Each neuron is connected with excitatory links (positively weighted) to a number of “cooperative neighbors” neurons that are in close proximity. Each neuron is also connected with inhibitory links (with negative weights) to a number of “competitive neighbors” neurons that are somewhat further away. There may also be a number of neurons, further away still, to which the neurons is not connected. All of these connections are within a particular layer of a neural net. The neurons receive an external signal in addition to these interconnections signals (just like Maxnet). This pattern of interconnections is repeated for each neuron in the layer. The interconnection pattern for unit Xi is as follows:
- 17. 17 Mexican Hat Fixed-weightcompetitive net The size of the region of cooperation (positive connections) and the region of competition (negative connections) may vary, as may vary the relative magnitudes of the +ve and –ve weights and the topology of the regions (linear, rectangular, hexagonal, etc..) si = external activation w1 = excitatory connectionw2 = inhibitory connection
- 18. 18 Mexican Hat Fixed-weightcompetitive net The contrast enhancement of the signal si received by unit Xi is accomplished by iteration for several time steps. The activation of unit Xi at time t is given by: Where the terms in the summation are the weighted signal from other units cooperative and competitive neighbors) at the previous time step. 1 −+= ∑ + k kikii )(txw(t)sf(t)x inputs + - - - -
- 19. 19 Mexican Hat Fixed-weightcompetitive net Nomenclature R2=Radius of region of interconnections; Xi is connected to units Xi+k and Xi-k for k=1, …R2 R1=Radius of region with +ve reinforcement; R1< R2 wk = weights on interconnections between Xi and units Xi+k and Xi-k; wk is positive for 0 ≤ k ≤ R1 wk is negative for R1 < k ≤ R2 x = vector of activations x_old = vector of activations at previous time step t_max = total number of iterations of contrast enhancement s = external signal
- 20. 20 Mexican Hat Fixed-weightcompetitive net Algorithm As presented, the algorithm corresponds to the external signal being given only for the first iteration (step 1). Step 0 Initialize parameters t_max, R1, R2 as desired Initialize weights: Step 1 Present external signal s: Set iteration counter: t = 1 0x_old toInitialize ...,1for0 ...,0for0 212 11 +=< => = RRkC RkC wij ii xoldx = = = _ :n)...,1,i(forarrayinactivationSave x_old sx
- 21. 21 Mexican Hat Fixed-weightcompetitive net Algorithm Step 2 While t is less than t_max, do steps 3-7 Step 3 Compute net input (i = 1, …n) Step 4 Apply activation function f (ramp from 0 to x_max, slope 1): ∑∑ ∑ +−= + −= −− −= ++ ++= 2 1 1 1 1 2 1 2 1 21 ___ R Rk ki R Rk R Rk kikii oldxColdxColdxCx > ≤≤ < = maxmax_ max0 00 )( xifx xifx xif xf nixxx ii ,...,1)),0max(max,_min( ==
- 22. 22 Mexican Hat Fixed-weightcompetitive net Step 5 Save current activations in x_old: Step 6 Increment iteration counter: Step 7 Test stopping condition: If t < t_max, continue; otherwise, stop. Note: The +ve reinforcement from nearby units and –ve reinforcement from units that are further away have the effect of – increasing the activation of units with larger initial activations, – reducing the activations of those that had a smaller external signal. nixoldx ii ,...,1_ == 1+= tt
- 23. 23 Mexican Hat Example Let’sconsider a simple net with 7 units. The activation function for this net is: Step 0 Initialize parameters R1=1, R2 =2, C1 =.6, C2 =-.4 Step 1 t=0 The external signal s = (.0, .5, .8, 1, .8, .5, .0) Step 2 t=1, the update formulas used in step 3, are listed as follows for reference: )0.0,5.0,8.0,0.1,8.0,5.0,0.0()1(old_ :_oldinSave )0.0,5.0,8.0,0.1,8.0,5.0,0.0()0( = = x x x > ≤≤ < = 22 20 00 )( xif xifx xif xf The node to be enhanced
- 24.
- 25. 25 Mexican Hat Example Step4 Step 5-7 Bookkeeping for next iteration Step 3: t=2 Step 4: Step 5-7 Bookkeeping for next iteration 196.0)0.0(6.0)38.0(6.0)06.1(4.0 39.0)0.0(6.0)38.0(6.0)06.1(6.0)16.1(4.0 14.1)0.0(4.0)38.0(6.0)06.1(6.0)16.1(6.0)06.1(4.0 66.1)38.0(4.0)06.1(6.0)16.1(6.0)06.1(6.0)38.0(4.0 14.1)06.1(4.0)16.1(6.0)06.1(6.0)38.0(6.0)0.0(4.0 39.0)16.1(4.0)06.1(6.0)38.0(6.0)0.0(6.0 196.0)06.1(4.0)38.0(6.0)0.0(6.0 7 6 5 4 3 2 1 −=++−= =+++−= =−+++−= =−+++−= =−+++−= =−++= −=−+= x x x x x x x )0.0,38.0,06.1,16.1,06.1,38.0,0.0(=x )0.0,39.0,14.1,66.1,14.1,39.0,0.0(=x The node Which has been enhanced
- 26. 26 Neural Networks Basedon Competition. Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading
- 27. 27 Hamming Network Fixed-weightcompetitive net Hamming distance of two vectors is the number of components in which the vector differ. For 2 bipolar vectors, of dimension n, The Hamming net uses Maxnet as a subnet to find the unit with the largest input (see figure on next slide). yx and distanceHammingshorterlargerlarger )(5.0 2 ),HD(distanceHamming andindifferentbitsofnumberis andinagreementinbitsofnumberis:where ⇒⇒⋅ +⋅= −=⋅ =−= −=⋅ a na na and d a da yx yx yx yx yx yx yx 0 1 0 and 0 1 1
- 28. 28 Hamming Network Fixed-weightcompetitive net Architecture The sample architecture shown assumes input vectors are 4- tuples, to be categorized as belonging to one of 2 classes.
- 29. 29 Hamming Network Fixed-weightcompetitive net Algorithm Given a set of m bipolar exemplar vectors e(1), e(2), …, e(m), the Hamming net can be used to find the exemplar that is closest to the bipolar i/p vector x. The net i/p y_inj to unit Yj gives the number of components in which the i/p vector and the exemplar vector for unit Yje(j) agree (n minus the Hamming distance between these vectors). Nomenclature: n number of input nodes, number of components of any I/p vector m number of o/p nodes, number of exemplar vectors e(j) the jth exemplar vector: ( ))(,),(),()( 1 jejejej ni LL=e
- 30. 30 Hamming Network Fixed-weightcompetitive net Algorithm Step 0 To store the m exemplar vectors, initialise the weights and biases: Step 1 For each vector x, do steps 2-4 Step 2 Compute the net input to each unit Yj: Step 3 Initialise activations for Maxnet Step 4 Maxnet iterates to find the best match exemplar. nbjew jiij 5.0),(5.0 == mjwxbiny i ijijj ,,1for_ L=+= ∑ mjinyy jj ,,1for_)0( L==
- 31. 31 Hamming Network Fixed-weightcompetitive net Upper layer: Maxnet – it takes the y_in as its initial value, then iterates toward stable state (the best match exemplar) – one output node with highest y_in will be the winner because its weight vector is closest to the input vector jY
- 32. 32 Hamming Network Fixed-weightcompetitive net Example (see architecture n=4) [ ] [ ] patterns)(exemplar2casein thism1-1,-1,-1,)2( 1-1,-1,-1,)1( :ectorsexemplar vGiven the =−= = e e The Hamming net can be used to find the exemplar that is closest to each of the bipolar i/p patterns, (1, 1, -1, -1), (1, -1, -1, -1), (-1, -1, -1, 1), and (-1, -1, 1, 1) Step 1: Store the m exemplar vectors in the weights: case)(in this4nodesinputofnumber 2/2 :biasestheInitialize 2 )( rewhe 5.5. 5.5. 5.5. 5.5. 21 == === = −− −− −− − = n nbb je wW i ij
- 33. 33 Hamming Network Fixed-weightcompetitive net Example For the vector x=(1,1, -1, -1), let’s do the following steps: These values represent the Hamming similarity because (1, 1, -1, -1) agrees with e(1)= (1,-1, -1, -1) in the 1st , 3rd, and 4th components and because (1,1, -1, -1) agrees with e(2)=(-1,-1,-1,1) in only the 3rd component Since y1(0) > y2(0), Maxnet will find that unit Y1 has the best match exemplar for input vector x = (1, 1, -1, -1). 112_ 312_ 222 111 =−=+= =+=+= ∑ ∑ i ii i ii wxbiny wxbiny 1)0( 3)0( 2 1 = = y y
- 34. 34 Hamming Network Fixed-weightcompetitive net Example For the vector x=(1,-1, -1, -1), let’s do the following steps: Note that the input agrees with e(1) in all 4 components and agrees with e(2) in the 2nd and 3rd components Since y1(0) > y2(0), Maxnet will find that unit Y1 has the best match exemplar for input vector x = (1, -1, -1, -1). 202_ 422_ 222 111 =+=+= =+=+= ∑ ∑ i ii i ii wxbiny wxbiny 2)0( 4)0( 2 1 = = y y
- 35. 35 Hamming Network Fixed-weightcompetitive net Example For the vector x=(-1,-1, -1, 1), let’s do the following steps: Note that the input agrees with e(1) in the 2nd and 3rd components and agrees with e(2) in all four components Since y2(0) > y1(0), Maxnet will find that unit Y2 has the best match exemplar for input vector x = (-1, -1, -1, 1). 422_ 202_ 222 111 =+=+= =+=+= ∑ ∑ i ii i ii wxbiny wxbiny 4)0( 2)0( 2 1 = = y y
- 36. 36 Hamming Network Fixed-weightcompetitive net Example For the vector x=(-1,-1, 1, 1), let’s do the following steps: Note that the input agrees with e(1) in the 2nd component and agrees with e(2) in the 1st, 2nd, and 4th components Since y2(0) > y1(0), Maxnet will find that unit Y2 has the best match exemplar for input vector x = (-1, -1, 1, 1). 312_ 112_ 222 111 =+=+= =−=+= ∑ ∑ i ii i ii wxbiny wxbiny 3)0( 1)0( 2 1 = = y y
- 37. 37 Neural Networks Basedon Competition. Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading
- 38. 38 Kohonen Self-Organizing Maps(SOM) Fausett Teuvo Kohonen, born: 11-07- 34, Finland Author of the book: “Self- organizing maps”, Springer (1997) SOM = “Topology preserving maps” SOM assume a topological structure among the cluster units This property is observed in the brain, but not found in other ANNs
- 39. 39 Kohonen Self-Organizing Maps(SOM) Fausett
- 40. 40 Fausett Kohonen Sefl-Organizing Maps(SOM) Architecture:
- 41. 41 Fausett Kohonen Self-Organizing Maps(SOM) Architecture: * * {* (* [#] *) *} * * { } R=2 ( ) R=1 [ ] R=0 Linear array of cluster units This figure represents the neighborhood of the unit designated by # in a 1D topology (with 10 cluster units). ******* ******* ******* ***#*** ******* ******* ******* R=2 R=1 R=0 Neighbourhoods for rectangular grid (Each unit has 8 neighbours)
- 42. 42 Fausett Kohonen Self-Organizing Maps(SOM) Architecture: ****** ****** ****** **#*** ****** ****** ****** R=2 R=1 R=0 Neighbourhoods for hexagonal grid (Each unit has 6 neighbours). In each of these illustrations, the “winning unit” is indicated by the symbol # and the other units are denoted by * NOTE: What if the winning unit is on the edge of the grid? • Winning units that are close to the edge of the grid will have some neighbourhoods that have fewer units than those shown in the respective figures • Neighbourhoods do not “wrap around” from one side of the grid to the other; “missing” units are simply ignored.
- 43. 43 Training SOM Networks Whichnodes are trained? How are they trained? – Training functions – Are all nodes trained the same amount? Two basic types of training sessions? – Initial formation – Final convergence How do you know when to stop? – Small number of changes in input mappings – Max # epochs reached
- 44. 44 Fausett Kohonen Self-Organizing Maps(SOM) Algorithm: Step 0. Initialise weights wij (randomly) Set topological neighbourhood parameters Set learning rate parameters Step 1 While stopping condition is false, do steps 2-8. Step 2 For each i/p vector x, do steps 3-5 Step 3 For each j, compute: Step 4 Find index J such that D(J) is a minimum Step 5 For all units j within a specified neighbourhood of J, and for all i: Step 6 Update learning rate. Step 7 Reduce radius of topological neighbourhood at specified time Step 8 Test stopping condition ∑ −= i iij xwJD 2 )()( )]([)()( oldwxoldwneww ijiijij −+= α
- 45. 45 Fausett Kohonen Self-Organizing Maps(SOM) Alternative structure for reducing R and α: The learning rate α is a slowly decreasing function of time (or training epochs). Kohonen indicates that a linearly decreasing function is satisfactory for practical computations; a geometric decrease would produce similar results. The radius of the neighbourhood around a cluster unit also decreases as the clustering process progresses. The formation of a map occurs in 2 phases: The initial formation of the correct order and the final convergence The second phase takes much longer than the first and requires a small value for the learning rate. Many iterations through the training set may be necessary, at least in some applications.
- 46. 46 Fausett Kohonen Self-Organizing Maps(SOM) Applications and Examples: The SOM have had may applications: Computer generated music (Kohonen 1989) Traveling salesman problem Examples: A Kohonen SOM to cluster 4 vectors: – Let the 4 vectors to be clustered be: s(1) = (1, 1, 0, 0) s(2) = (0, 0, 0, 1) s(3) = (1, 0, 0, 0) s(4) = (0, 0, 1, 1) - The maximum number of clusters to be formed is m=2 (4 input nodes, 2 output nodes) Suppose the learning rate is α(0)=0.6 and α(t+1) = 0.5 α(t) With only 2 clusters available, the neighborhood of node J (step 4) is set so that only one cluster update its weights at each step (i.e. R=0)
- 47. 47 Fausett Kohonen Self-Organizing Maps(SOM) Examples: Step 0 Initial weight matrix Initial radius: R=0 Initial learning rate: α(0)=0.6 Step 1 Begin training Step 2 For s(1) = (1, 1, 0, 0) , do steps 3-5 Step 3 = 3.9. 7.5. 4.6. 8.2. W 98.0)03(.)07(.)14(.)18(.)2( 86.1)09(.)05(.)16(.)12(.)1( 2222 2222 =−+−+−+−= =−+−+−+−= D D
- 48. 48 Fausett Kohonen Self-Organizing Maps(SOM) Examples: Step 4 The i/p vector is closest to o/p node 2, so J=2 Step 5 The weights on the winning unit are updated This gives the weight matrix: ii iiii xoldw oldwxoldwneww 6.0)(4.0 )]([6.0)()( 2 222 + −+= This set of weights has not been modified. = 12.9. 28.5. 76.6. 92.2. W
- 49. 49 Fausett Kohonen Self-Organizing Maps(SOM) Examples: Step 2 For s(2) = (0, 0, 0, 1) , do steps 3-5 Step 3 2768.2)112(.)028(.)076(.)092(.)2( 66.0)19(.)05(.)06(.)02(.)1( 2222 2222 =−+−+−+−= =−+−+−+−= D D Step 4 The i/p vector is closest to o/p node 1, so J=1 Step 5 The weights on the winning unit are updated This gives the weight matrix: = 12.96. 28.20. 76.24. 92.08. W This set of weights has not been modified.
- 50. 50 Fausett Kohonen Self-Organizing Maps(SOM) Examples: Step 2 For s(3) = (1, 0, 0, 0) , do steps 3-5 Step 3 6768.0)012(.)028(.)076(.)192(.)2( 8656.1)096(.)02(.)024(.)108(.)1( 2222 2222 =−+−+−+−= =−+−+−+−= D D Step 4 The i/p vector is closest to o/p node 2, so J=2 Step 5 The weights on the winning unit are updated This gives the weight matrix: = 048.96. 112.20. 304.24. 968.08. W This set of weights has not been modified.
- 51. 51 Fausett Kohonen Self-Organizing Maps(SOM) Examples: Step 2 For s(4) = (0, 0, 1, 1) , do steps 3-5 Step 3 724.2)1048(.)1112(.)0304(.)0968(.)2( 7056.0)196(.)12(.)024(.)008(.)1( 2222 2222 =−+−+−+−= =−+−+−+−= D D Step 4 The i/p vector is closest to o/p node 1, so J=1 Step 5 The weights on the winning unit are updated This gives the weight matrix: Step 6 Reduce the learning rate α = 0.5(0.6) = 0.3 = 048.984. 112.680. 304.096. 968.032. W This set of weights has not been modified.
- 52. 52 Fausett Kohonen Self-Organizing Maps(SOM) Examples: The weights update equations are now The weight matrix after the 2nd epoch of training is: Modifying the learning rate and after 100 iterations (epochs), the weight matrix appears to be converging to: iij ijiijij xoldw oldwxoldwneww 3.0)(7.0 )]([3.0)()( + −+= = 024.999. 055.630. 360.047. 980.016. W = 0.00.1 0.05. 5.00. 0.10. W The 1st column is the average of the 2 vects in cluster 1, and the 2nd col is the average of the 2 vects in cluster 2 This set of weights has not been modified.
- 53. 53 Example of aSOM That Use a 2-D Neighborhood Character Recognition – Fausett, pp. 176-178
- 54. 54 Traveling Salesman Problem(TSP) by SOM The aim of the TSP is to find a tour of a given set of cities that is of minimum length. A tour consists of visiting each city exactly once and returning to the starting city. The net uses the city coordinates as input (n=2); there are as many cluster units as there are cities to be visited. The net has a linear topology (with the first and last unit also connected). Fig. shows the Initial position of cluster units and location of cities
- 55. 55 Traveling Salesman Problem(TSP) by SOM Fig. Shows the result after 100 epochs of training with R=1 (learning rate decreasing from 0.5 to 0.4). Fig. Shows the final tour after 100 epochs of training with R=0 Two candidate solutions: ADEFGHIJBC ADEFGHIJCB
- 56. 56 Neural Networks Basedon Competition. Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading
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- 59. 59 CreatingaSOM newsom -- Createa self-organizing map Syntax net = newsom net = newsom(PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND) Description net = newsom creates a new network with a dialog box. net = newsom (PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND) takes, PR - R x 2 matrix of min and max values for R input elements. Di - Size of ith layer dimension, defaults = [5 8]. TFCN - Topology function, default ='hextop'. DFCN - Distance function, default ='linkdist'. OLR - Ordering phase learning rate, default = 0.9. OSTEPS - Ordering phase steps, default = 1000. TLR - Tuning phase learning rate, default = 0.02; TND - Tuning phase neighborhood distance, default = 1. and returns a new self-organizing map. The topology function TFCN can be hextop, gridtop, or randtop. The distance function can be linkdist, dist, or mandist.
- 60. 60 Neural Networks Basedon Competition. Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading Introduction Fixed weight competitive nets – Maxnet – Mexican Hat – Hamming Net Kohonen Self-Organizing Maps (SOM) SOM in Matlab References and suggested reading
- 61. 61 Suggested Reading. L. Fausett, “Fundamentalsof Neural Networks”, Prentice-Hall, 1994, Chapter 4. J. Zurada, “Artificial Neural systems”, West Publishing, 1992, Chapter 7.
- 62. 62 References: These lecture noteswere based on the references of the previous slide, and the following references 1. Berlin Chen Lecture notes: Normal University, Taipei, Taiwan, ROC. http://140.122.185.120 2. Dan St. Clair, University of Missori-Rolla, http://web.umr.edu/~stclair/class/classfiles/cs404_fs02/Misc/CS 404_fall2001/Lectures/ Lecture 14.