IE 585

Competitive Network – I

Hamming Net & Self-Organizing Map

2

Competitive Nets

Unsupervised• MAXNET• Hamming Net• Mexican Hat Net• Self-Organizing Map (SOM)• Adaptive Resonance Theory (ART)

Supervised• Learning Vector Quantization (LVQ)• Counterpropagation

3

Clustering Net

• Number of input neurons equal to the dimension of input vectors

• Each output neuron represents a cluster

the number of output neurons limits the number of clusters that can be performed

• The weight vector for an output neuron serves as a representative for the input patterns which the net has placed on that cluster

• The weight vector for the winning neuron is adjusted

4

Winner-Take-All

• The squared Euclidean distance is used to determine the closest weight vector to a pattern vector

• Only the neuron with the smallest Euclidean distance from the input vector is allowed to update

5

MAXNET

• Developed by Lippmann, 1987• Can be used as a subset to pick the node

whose input is the largest• Completely interconnected (including self-

connection)• Symmetric weights• No training• Weights are fixed

6

Architecture of MAXNET

1

1

1

1

/1

/1

/1

/1

/1

/1

ji

jiwij if /1

if 1

otherwise 0

0 if )(

FunctionTransfer

xxxf

7

Procedure of MAXNET

Initialize activations and weights

Update activation of each node

If more than one node has a nonzero activation, continue; otherwise, stop

newj

oldj

j

oldjij

newj

xx

xwfx

)(

8

Hamming Net

• Developed by Lippmann, 1987• A maximum likelihood classifier• used to determine which of several exemplar

vectors is most similar to an input vector• Exemplar vectors determine the weights of the

net• Measure of similarity between the input vector

and the stored exemplar vectors is (n – HD between the vectors)

9

Weights and Transfer Function of Hamming Net

in vector s0' of # 0 if

1 if

1

1

(0,1)binary 2

2

1

(-1,1)bipolar

b

i

iij

b

iij

wx

xw

nw

xw

otherwise 0

0 if )(

Function)(Identity

FunctionTransfer

xxxf

10

Architecture of Hamming Net

x1 x2 x3 x4

B B

y1 y2

MAXNET

11

Procedure of the Hamming Net

Initialize weights to store the m exemplar vectors

For each input vector x

compute

initialize activation for MAXNET

MAXNET iterates for find the best match exemplar

i

iijjY xwbnetj

jYj nety )0(

12

Hamming Net Example

13

Mexican Hat Net

• Developed by Kohonen, 1989• Positive weight with “cooperative neighboring”

neurons• Negative weight with “competitive neighboring”

neurons• Not connect with far away neurons

14

Teuvo Kohonen

• http://www.cis.hut.fi/teuvo/ (his own home page)

• published his work starting in 1984

• LVQ - learning vectorquantization

• SOM - self organizingmap

Professor atHelsinki Univ.Finland

15

SOM• Also called Topology-Preserving Maps or Self-Organizing

Feature Maps (SOFM)• “Winner Take All” learning (also called competitive learning)• winner has the minimum Euclidean distance• learning only takes place for winner• final weights are at the centroids of each cluster• Continuous inputs, continuous or 0/1 (winner take all)

outputs• No bias, fully connected• used for data mining and exploration• supervised version exists

16

Architecture of SOM Net

I

N

P

U

T

S

(a’s)

O

U

T

P

U

T

S

(y’s)

Input

Layer

W

n

17

Kohonen Learning Rule Derivation

2222 min wwaawa winner

w)a-(w

wa

w22 )(

E2

training.during decreasesusually and

7.01.0

)(

waw

18

Kohonen Learning

oldj

oldj

oldj

oldj

newj

wa

waw

www

.

..

..

)1(

][

19

Procedure of SOM

Initialize weights uniformly and normalize to unit lengthNormalize inputs to unit length Present an input vector x

calculate Euclidean distance between x and all Kohonen neuronsselect winning output neuron j (with the smallest distance)update the winning neuron

re-normalize weights to j (sometimes skipped)present next training vector

][ oldoldnew waww

20

Method

Normalize input vectors, a, by:

Normalize weight vectors, w, by:

Calculate distance from a to each w by:

iki

kiki

a

aa

2

iji

jiji

w

ww

2

i

jikij wad 2

21

Min d wins (this is the winning neuron)Update w of the min d neuron by:

Return to 2 and repeat for all input vectors a

Reduce if applicableRepeat until weights converge (stop

changing)

jioldkioldnew wawwjiji

22

SOM Example - 4 patterns

p1 1 1p2 1 0p3 0 1p4 0 0neuron1 neuron2 neuron3 neuron4 dist1 dist2 dist3 dist4w1 w2 w1 w2 w1 w2 w1 w2

0.2 0.3 0.5 0.4 0.1 0.7 0.5 0.6 1.13 0.61 0.9 0.410.2 0.3 0.5 0.4 0.1 0.7 0.625 0.7 0.73 0.41 1.3 0.6306250.2 0.3 0.625 0.3 0.1 0.7 0.625 0.7 0.53 0.880625 0.1 0.4806250.2 0.3 0.625 0.3 0.075 0.775 0.625 0.7 0.13 0.480625 0.60625 0.880625

0.15 0.225 0.625 0.3 0.075 0.775 0.625 0.7 1.323125 0.630625 0.90625 0.2306250.15 0.225 0.625 0.3 0.075 0.775 0.71875 0.775 0.773125 0.230625 1.45625 0.6797270.15 0.225 0.71875 0.225 0.075 0.775 0.71875 0.775 0.623125 1.117227 0.05625 0.5672270.15 0.225 0.71875 0.225 0.05625 0.83125 0.71875 0.775 0.073125 0.567227 0.694141 1.117227

0.1125 0.16875 0.71875 0.225 0.05625 0.83125 0.71875 0.775 1.478633 0.679727 0.919141 0.1297270.1125 0.16875 0.71875 0.225 0.05625 0.83125 0.789063 0.83125 0.816133 0.129727 1.581641 0.7354710.1125 0.16875 0.789063 0.16875 0.05625 0.83125 0.789063 0.83125 0.703633 1.313596 0.031641 0.6510960.1125 0.16875 0.789063 0.16875 0.042188 0.873438 0.789063 0.83125 0.041133 0.651096 0.764673 1.313596

0.084375 0.126563 0.789063 0.16875 0.042188 0.873438 0.789063 0.83125

=0.25

23

Movement of 4 weight clusters

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Neuron 1 Neuron 2

Neuron 3 Neuron 4

24

Adding a “conscience”

• prevents neurons from winning too many training vectors using a bias (b) factor

• winner had min (d-b) where

bj=10(1/n-fj) (n=# output neurons)

fjnew=fjold+0.0001(yj-fjold)

finitial=1/n

• for neurons that win, b becomes negative and for neurons that don’t win, b becomes positive

25

Supervised Version

• Same, except if the winning neuron is “correct” use same weight update:

wnew = wold + (a - wold) and

• if winning neuron is “incorrect” use:

wnew = wold - (a - wold)