polynomial discrete time cellular neural networks eduardo gomez-ramirez † giovanni egidio...
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Polynomial Discrete Time Cellular Neural Networks
Eduardo Gomez-Ramirez † Giovanni Egidio Pazienza‡
† LIDETEA, POSGRADO E INVESTIGACION Universidad La Salle – México, D.F. ‡ Department d’Electronica, EALS
Universitat “Ramon Llull” – Barcelona, Spain
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Outline Cellular Neural Networks (CNN)
Introduction and Objective Genetic Algorithms (GA) Polynomial Discrete Time CNNs
(PDTCNNs) XOR Problem Game of Life
Learning vs Design Conclusions and future workIntro CNN & GA Polyn. CNN XOR GoL Conclusions
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CNN: Introduction CNN for complex task (linearly
nonseparable data)
Multilayer CNNs Include more degrees of freedom for the
output state of each layer Search in a finite set of templates
Single layer: Polynomial CNNs
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Improve the representation power of a single layer CNN including a simple nonlinear term to solve problems with linearly nonseparable data (XOR)
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
Objective
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CNN: mathematical model
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
The simplified mathematical model is:
where xc is the state of the cell, uc the input and yc the output
iubtyatxdt
dx dcd
dcd
cc
)()(
)1)(1)((2
1 txtxy ccc
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CNN: Activation Function
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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CNN: Block Diagram
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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CNN: Discrete Model
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
Computing x(∞), the model can be represented as
iubnyanx dcd
dcd
c )()(
0)(,1
0)(,1)(
nx
nxny
c
c
c
using the following activation function
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Steps: Crossover C(Fg) Mutation M(*) Adding random parent Ag()
GA: main steps proposed
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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dc
bc
da
ba
FC
dcMbaMM
MF
g
g 212
1 ,,
GA: Crossover
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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125.0125.
000
000
A
125.125.0
000
000
B I=0
125.00
000
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A
125.125.125.
000
000
B I=0
2121
2211
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2121
2211
1111
1
cccc
bbbb
aaaa
cccc
bbbb
aaaa
P
Individual 1
Individual 2
GA: Crossover
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
a1
b1
c1
a2
b2
c2
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GA: Mutation
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
mij
mijmij PrF
PrFPFM
)(
)(),(
where rU(0,1) is a random variable with uniform distribution defined on a probability space (,,P),
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GA: Mutation (resolution)
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
'2
'1
'2
'1
'2
'2
'1
'1
'2
'2
'2
'2
'2
'1
'2
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P
125.0125.
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A
125.125.0
000
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B I=0
125.00
000
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A
125.125.125.
000
000
B I=0
Individual 1
Individual 2
a1
b1
c1
a2
b2
c2
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Population=sons + sons mutated
g
g
g
C FA
M C F
ggn
pgg AOnAS pmin,
GA: Selecting Parents
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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)(1 wF
FF
g
gg
GA: Adding Random Parent
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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THEOREM 1: (Weierstrass’s Approximation Theorem)
Let g be a continuous real valued function defined on a closed interval [a,b]. Then, given any positive, there exists a polynomial y (which may depend on ) with real coefficients such that:
For every x [a,b].
Polynomial Discrete Time Cellular Neural Network
)()( xyxg
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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THEOREM 2 *: Any Boolean Function of n-variables can be realized using a Polynomial Threshold gates of order sn.The quadratic threshold gate can be defined:
And s is the number of inputs and T is the threshold constant.
Polynomial Discrete Time Cellular Neural Network
otherwise
Txxwxwify
n
i
n
ijiijii
0
11 1
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
* N. J. Nilsson. The Mathematical Foundations of Learning Machines. McGraw Hill, New York, 1990.
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PDTCNN: the model (I)
cdd
cNd
dcd
cNd
dcd
c iyuguBkyAkxrr
),()()()()(
0)1(1
0)1(1))1(()(
kxf
kxifkxfky
c
ccc
)0(
)0(
)0(
),(
)(
)(
)(
cd
cNd
cd
cd
cNd
cd
dd
cNd
cd
dd
yuP
uyP
yuP
yug
r
r
r
)(
)(
)(
),(
)(
)(
)(
kyuP
ukyP
kyuP
yug
cd
cNd
cd
cd
cNd
cd
dd
cNd
cd
dd
r
r
r
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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PDTCNN: the model (II)
)()(),()()(
kuQkyPyug d
cNd
cd
d
cNd
cd
dd
rr
cdd
cNd
dcd
cNd
dcd
c iyuguBkyAkxrr
),()()()()(
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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PDTCNN: Solving XOR problemSome papers: Z. Yang, Y. Nishio, A. Ushida,
Templates and algorithms for two-layer cellular neural networks. IJCNN’02, 2002.
F. Chen, G. He, G. Chen & X. Xu,
Implementation of Arbitrary Boolean Functions via CNN. CNNA’06, 2006.
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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PDTCNN:Solving XOR problem M. Balsi, Generalized CNN: Potentials of
a CNN with Non-Uniform Weights. CNNA-92, 2002 .
E. Bilgili, I. C. Göknar and O. N. Ucan, Cellular neural network with trapezoidal activation function. Int. J. Circ. Theor. Appl., 2005
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Learning parameters Initialpop=20000 Number of fathers=7 Maximum number of random
parents to be add = 3 Kpro=0.8 Increment=1 Mutation Probability=0.15
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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PDTCNN:First Scheme U:uij=xijxij+1
1401
150140
iP
BA
0300
010030
iP
BA
0100
010010
iP
BA
)()()( 1 kyuukykx ccccc
)(
)(
)(
),(
)(
)(
)(
kyuP
ukyP
kyuP
yug
cd
cNd
cd
cd
cNd
cd
dd
cNd
cd
dd
r
r
r
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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PDTCNN:Second Scheme U:uij=xijyij
2281
000000
iP
BA
b)
1151
000001
iP
BAc)
0 1 0 0 0 0
0 1 0 0
A B
P i
)()()( kyukykx cccc
)0(
)0(
)0(
),(
)(
)(
)(
cd
cNd
cd
cd
cNd
cd
dd
cNd
cd
dd
yuP
uyP
yuP
yug
r
r
r
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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The Game of Life (I) The Game of Life (GoL) is a totalistic cellular
automaton consisting in a two-dimensional grid cells, that may be either alive (black) or dead (white).
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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The Game of Life (II) The state of each cell varies according to the
following rules: Birth: a cell that is dead at time t becomes
alive at time t + 1 only if exactly 3 of its neighbors were alive at time t;
Survival: a cell that was living at time t will remain alive at t + 1 if and only if it had exactly 2 or 3 alive neighbors at time t.
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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The Game of Life (III) Every sufficient well-stated mathematical problem can be reduced to a question about Life;
It is possible to make a life computer (logic gates, storage etc.);
Life is universal: it can be programmed to perform any desired calculation;
Given a large enough Life space and enough time, self-reproducing animals will emerge...
The whole universe is a CA! (E.Fredkin, MIT).Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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The Game of Life – NOT gate
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
A
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CNN & GoL Multilayer CNN (Chua, Roska) – 1990 Activation function (Chua, Roska) –
1990 CNN-UM (Roska,Chua) -1990 CNN Universal Cells (Dogaru, Chua) –
1999
Simplicity vs. Computational power
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Polynomial CNN (I)
),()()( ddede
ede
d yugiubnyanx
0)(,1
0)(,1)(
nx
nxny
d
d
d
What’s g(ud,yd)?
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Polynomial CNN (II) In the simplest case g(ud, yd) is a second
degree polynomial, whose general form is
2
0
2 ))()()()((),(i
iedei
iedei
dd yqupyug
200 )()()1()( ed
eed
e yqp
)()()()( 11ed
eed
e yqup
)1()()()( 22
2ed
eed
e qup
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Polynomial CNN (III)
Thanks to some considerations we find that
)()()()()()()( 12
0ded
eed
eed
eed
ed yupyqiubnyanx
000
00
000
caA
ppp
pcp
ppp
bbb
bbb
bbb
B
000
00
000
0 0cqQ
ppp
pcp
ppp
ppp
ppp
ppp
P
111
111
111
1
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Polynomial CNN (IV)
pcppccccd uypbuypbnx )()()(
iyqya cccc 2
uc and appear in the state equation
direct link with totalistic Cellular Automata
pu
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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GoL: Rules (I)
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GoL: Rules (II)
Rule 1: a cell will be alive at least 3 of the 9 cells in its 3 × 3 neigh. are alive
Black pixel= +1
White pixel= -1pixel centr. = 1 (black)
Σ neigh. = -2 (5 w, 2 b)
next state = -1 (white)
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GoL: Rules (III)
Rule 2: a cell will be alive if at most 3 of its 8 neighbors are alive
Black pixel= +1
White pixel= -1pixel centr. = 1 (black)
Σ neigh. = -2 (5 w, 2 b)
next state = 1 (black)
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Design algorithm (I) First iteration: we try to perform the first rule (a
cell will be alive at least 3 of the 9 cells in its 3 × 3 neigh. are alive)
If Y(0)=0
bc=1 bp=1 i=3
pcppccccd uypbuypbnx )()()(
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Design algorithm (II) Second iteration: we try to accomplish the
second rule (a cell will be alive if at most 3 of its 8 neighbors are alive)
pcpcccd uypuypnx )1()1()(
32 cccc yqya
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Design algorithm (III) Hyp: pc=0
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Templates found using learning
Coming soon...
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Conclusions (I) In general:
In some cases it is possible to reduce a multilayer DTCNN to a single layer PDTCNN
Thanks to the GoL we can explore the capacity of PDTCNNs for Universal Machine
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Conclusions (II) About learning:
The resolution used reduces the search space
The step “Add random parent” improves the behavior to avoid local minimas
About design We give a simple algorithm to design
templates for the Polynomial CNN
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Future Work Implementations of mathematical
morphology functions with PDTCNNs
Intro CNN & GA Polyn. CNN XOR GoL Conclusions
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Polynomial Discrete Time Cellular Neural Networks
Eduardo Gomez-Ramirez Giovanni Egidio Pazienza
[email protected]@salle.url.edu