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Supervised Learning Networks
Supervised Learning Networks
• Linear perceptron networks
• Multi-layer perceptrons
• Mixture of experts
• Decision-based neural networks
• Hierarchical neural networks
Two-Level:
(b) Linear perceptron networks
(c) decision-based neural network.
(d) mixture of experts network.
Hierarchical Neural Network Structures
Three-Level:
(e) experts-in-class network.
(f) classes-in-expert network.
One-Level:
(a) multi-layer perceptrons.
Hierarchical Structure of NN
•1-level hierarchy: BP
•2-level hierarchy: MOE,DBNN
•3-level hierarchy: PDBNN
“Synergistic Modeling and Applications of Hierarchical Fuzzy Neural Networks”,
by S.Y. Kung, et al., Proceedings of the IEEE, Special Issue on Computational Intelligence, Sept. 1999
All Classes in One Net
multi-layer perceptron
Divide-and-conquer principle: divide the task into modules and then integrate the individual results into a collective decision.
Modular Structures (two-level)
Two typical modular networks:
(1) mixture-of-experts (MOE) which utilizes the expert-level modules,
(2) decision-based neural networks (DBNN) based on the class-level modules.
Each expert serves the function of
(1) extracting local features and
(2) making local recommendations.
The rules in the gating network are used to decide how to combine recommendations from several local experts, with corresponding degree of confidences.
Expert-level (Rule-level) Modules:
mixture of experts network
Class-level modules are natural basic partitioning units, where each module specializes in distinguishing its own class from the others.
Class-level modules:
In contrast to expert-level partitioning, this OCON structure facilitates a global (or mutual) supervised training scheme. In global inter-class supervised learning, any dispute over a pattern region by (two or more) competing classes may be effectively resolved by resorting to the teacher's guidance.
Decision Based Neural Network
Depending on the order used, two kinds of hierarchical networks:
•one has an experts-in-class construct and
•another a classes-in-expert Construct.
Three-level hierarchical structures:
Apply the divide-and-conquer principle twice:
one time on the expert-level and another on the class-level.
Classes-in-Expert Network
Experts-in-Class Network
Multilayer Back-Propagation Networks
A BP Multi-Layer Perceptron(MLP) possesses adaptive learning abilities to estimate sampled functions, represent these samples, encode structural knowledge, and inference inputs to outputs via association.
Its main strength lies in its (sufficiently large number of ) hidden units, thus a large number of interconnections.
The MLP neural networks enhance the ability to learn and generalize from training data. Namely, MLP can approximate almost any function.
BP Multi-Layer Perceptron(MLP)
A 3-Layer Network
Neuron Units: Activation Function
Linear Basis Function (LBF)
RBF NN is More Suitable for Probabilistic Pattern Classification
MLP RBFHyperplane Kernel function
The probability density function (also called conditional density function or likelihood) of the k-th class is defined as
kCxp |
The centers and widths of the RBF Gaussian kernels are deterministic functions of the training data;
RBF BP Neural Network
•According to Bays’ theorem, the posterior prob. is
xp
CPCxpxCP kk
k
||
where P(Ck) is the prior prob. and
RBF Output as Probability Function
'
'
'|k
kk
CPCxpxp
kM
jk CjPjxpCxp |||
1
)1|(xp
)(xp
)|( Mxp)2|(xp
kk
M
jk CPCjPjxpxp
1
||
M
j
kk
k
M
j
jPjxp
CPCjPjxp
1
1
|
||
kk
M
jk CPCjPjxpxp
1
||
M
j
kk
k
M
j
jPjxp
CPCjPjxp
1
1
|
||
jPjP
jPjxp
CPCjPjxp
xCP M
j
M
jkk
k
1
''
1
|
||
|
M
jjkj
M
jk
M
j
M
j
kk
xw
xjPjCP
jPjxp
jPjxp
jP
CPCjP
1
1
1
''1
||
|
||
RBF output jx posterior prob. of the j-th set of
features in the input .
weight wkj posterior prob. of class membership, giventhe presence of the j-th set of features .
jPjP
jPjxp
CPCjPjxp
xCP M
j
M
jkk
k
1
''
1
|
||
|
MLPs are highly non-linear in the parameter space gradient descent local minima
RBF networks solve this problem by dividing the learning into two independent processes.
1. Use the K-mean algorithm to find ci and determine weights w using the least square method
2. RBF learning by gradient descent
RBF networks MLP
Learning speed Very Fast Very Slow
Convergence Almost guarantee Not guarantee
Response time Slow Fast
Memoryrequirement
Very large Small
Hardwareimplementation
IBM ZISC036Nestor Ni1000
Intel 80170NX
Generalization Usually better Usually poorer
Comparison of RBF and MLP
xp
K-means
K-NearestNeighbor
BasisFunctions
LinearRegression
ci
ci
i
A w
RBF learning process
RBF networks implement the function
s x w w x ci i ii
M
( ) ( )
0
1
wi i and ci can be determined separately
Fast learning algorithm Basis function types
22
2
2
1)(
)2
exp()(
rr
rr
Finding the RBF Parameters
(1 ) Use the K-mean algorithm to find ci
1
2
2
2
1
1
Centers and widths found by K-means and K-NN
Use K nearest neighbor rule to find the function width
2
1
1
K
kiki cc
K
k-th nearest neighbor of ci
The objective is to cover the training points so that a smooth fit of the training samples can be achieved
For Gaussian basis functions
s x w w x c
w wx c
p i i p ii
M
ipj ij
ijj
n
i
M
( )
exp( )
01
0
2
211 2
Assume the variance across each dimension are equal
M
i
n
jijpj
iip cxwwxs
1 1
220 )(
2
1exp)(
ipipi cxa
To write in matrix form, let
a x c
s x w a a
pi i p i
p i pii
M
p
where ( )
00 1
s x
s x
s x
a a a
a a a
a a a
w
w
wN
M
M
N N NM M
( )
( )
( )
`
1
2
11 12 1
21 22 2
1 2
0
1
1
1
1
sAw
Aws
Determining weights w using the least square method
E d w x cp j jj
M
p jp
N
0
2
1
where dp is the desired output for pattern p
E
E
T
T T
( ) ( )
( )
d Aw d Aw
wA A A dSet w
0 1
(2) RBF learning by gradient descent
Let and i p
pj ij
ijj
n
p p pxx c
e x d x s x( ) exp ( ) ( ) ( )
1
2
2
21
E e x pp
N
1
2 1
2
( ) .
we have
E
w
E E
ci ij ij
, , and
Apply
we have the following update equations
w t w t e x x i M
w t w t e x i
t t e x w x x c t
c t c t e x w x x c t
i i w p i pp
N
i i w pp
N
ij ij p i i p pj ij ijp
N
ij ij c p i i p pj ij ijp
N
( ) ( ) ( ) ( ) , , ,
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
1 1 2
1 0
1
1
1
1
2 3
1
2
1
when
when
Elliptical Basis Function networks
)}()(2
1exp{)( 1
jpjT
jppj xxx
j
j
: function centers
: covariance matrix
1
x1
2 M
x2 xn
J
jpjkjpk xwxy
0
)()(
y W D W = +
y x1( )
y xK ( )
EBF Vs. RBF networks
RBFN with 4 centers EBFN with 4 centers
MatLab Assignment #3: RBF BP Network to separate 2 classes
RBF BP with 4 hidden units EBF BP with 4 hidden units
ratio=2:1