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Introduction4.1
第1頁
Back-propagation Learning Rule (BP)4.5
Functional-Link Net4.9
設計Multilayer Perceptron with 1 Hidden Layer 解XOR的分類問題
4.2
On Hidden Nodes for Neural Nets4.7
Gradient and Gradient Descent Method in Optimization
4.3
Multilayer Perceptron (MLP) and Forward Computation
4.4
Experiment of XOR Classification & Discussions
4.6
Application - NETtalk：A Parallel Network That Learns to Read Aloud
4.8
4.1
4.2
第2頁
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4.1 Introduction
對於Figure 4.1-1 Exclusive-OR的問題，我們不能利
用一層的Perceptron或一層的分辨函數去求得分辨
的區域。但我們可利用多層的Perceptron去解問題
而得到答案。
Fig. 4.1-1. Exclusive OR problem.
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4.1 IntroductionConsider the Exclusive-OR (XOR) problem shown in
Figure 4.1-1. We cannot just use one layer of perceptron
or discriminant function to achieve classification.
However, we can solve this problem by using the
multilayer perceptron.
Fig. 4.1-1. Exclusive OR problem.
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• Weighting coefficient adjustment is based on
the gradient decent method.
Fig. 4.1-2 Two-layer Perceptron of neural network.
第5頁
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4.2 Multilayer Perceptron with 1 Hidden
Layer to solve XOR classification problem
XOR的分兩類、兩度空間的mapping、及2-2-1的
neural network 如下，我們要求得網路上的
weighting coefficients，解XOR分兩類的問題。
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XOR is a two-class
mapping in 2-D problem.
Its correspond 2-2-1 neural
network is shown in Fig.
4.1-3.
We need to determine the
weighting coefficients so
that the neural network can
solve XOR problem. Fig. 4.1-3 2-2-1的neural network
第8頁
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• In 2-D space, we can draw 2
parallel lines L1, L2 to
separate the 2 classes.
• As shown in 2-2-1 network,
there is only one hidden layer
(two hidden nodes) in order
to solve this XOR problem.
• The first layer can determine
two lines (each perceptron
has one )
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Line equations for L1, L2
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• Both L1, L2 can separate
the space into two halves
(+1 or -1)
• The perceptron in the
second layer can separate
the combination of +1 and
-1 (as inputs) into 2 classes.
• There are totally 4
combinations of (P1, P2):
(0,0), (0, 1), (1, 1), (1, 0)
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Node 1與node 2的activation function為hardlimiter，
output value為+1 or –1。設 , 各為 node 1與node
2的output value。
𝑃2𝑃1
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4 cases/combinations of (P1, P2)
Case 1：如果 L1 , L2 的分割空間如下圖，
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則由input training patterns到hidden nodes到desired
class outputs的結果如下表：
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我們將hidden node output及其desired output劃在另
一空間上，可以得到如下圖，為一『Not AND, Not
OR』的空間分割。
在此我們以 軸表示 perceptron之output， 軸
表示 perceptron之output。𝑃2
𝑃2𝑃1𝑃1
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全部的2-2-1網路及其weighting coefficients如下圖：
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Case 2：如果 , 的分割空間如下圖，𝐿2𝐿1
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則由input training patterns到hidden nodes到desired
class outputs的結果如下表：
第19頁
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將hidden node output及其desired output劃在另一空
間上，可以得到如下圖，為一『Not AND』的空間
分割。
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The 2-2-1 network and all the weighting coefficients
are shown below:：
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Case 3: 如果 , 的分割空間如下圖𝐿2𝐿1
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則由input training patterns到hidden nodes到desired
class outputs的結果如下表：
第23頁
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我們將hidden node output及其desired output劃在另
一空間上，可以得到如下圖，為一『OR』的空間
分割。
第24頁
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The 2-2-1 network and all the weighting coefficients
are shown below:
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Case 4：如果 , 的分割空間如下圖，𝐿2𝐿1
+
--
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則由input training patterns到hidden nodes到desired
class outputs的結果如下表：
第27頁
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我們將hidden node output及其desired output劃在另
一空間上，可以得到如下圖，為一『OR』的空間
分割。
第28頁
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The 2-2-1 network and all the weighting coefficients
are shown below:
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4.3 Gradient and Gradient Descent
Method in Optimization
The gradient descent method is used in the back-
propagation learning rule of the Perceptron and
Multilayer Perceptron.
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(A) Gradient
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Example 4-1
Fig.4.3-1 Function and gradient at different location.
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example 4-2
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Fig. 4.3-2 (a) Function, (b) Gradient at different locations.
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(B) Gradient descent method
Fig. 4.3-3 Optimal solution by gradient descent.
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(B) Gradient descent method
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Fig. 4.3-4 Function of 2
variables.
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Gradient descent method: to find the minimum of a
given function. Like down hill to the valley.
Draback: settled at local minimum
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(C) Chain Rule
Given E(z(y(x))),𝜕𝑦
𝜕𝑥
𝜕𝑧
𝜕𝑦
𝜕𝐸
𝜕𝑧
𝜕𝐸
𝜕𝑥=
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4.4 Multilayer Perceptron (MLP)
and Forward Computation
Fig. 4.4-1 Two-layer perceptron of neural network.
第42頁
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• The system is shown in Fig. 4.4-1. The index
of the node in input layer is i.
• There are I+1 inputs, the (J+1) node index in
the hidden layer is j; the K node index in the
output layer is k.
• Input vector has I features: x1, x2, …, xI . We
must have a 1 as the input at the input layer,
that is a linear combination of the variable.
And the same at each hidden layer.
• Fig. 4.4-2 shows the mathematical computation
第43頁
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Fig. 4.4-2 Two-layer perceptron of neural network.
I
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• w1 is a matrix, whose jth row denotes the
weighting coefficients connecting the input y
to the jth hidden node in the first layer
• w2 is the matrix, whose jth row denotes the
weighting coefficients connecting the output o1
in the first layer to the jth hidden node in the
second layer
第45頁
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At input layer, = , = 1
At hidden layer, j-th node,
𝑓 𝑠𝑗jo
𝑠𝑗 =
𝑖=1
𝐼+1
𝑤𝑗𝑖𝑜𝑖
𝑜𝐼+1𝑥𝑖𝑜𝑖
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4.5 Back-propagation Learning Rule (BP)
• In the learning, the adjustment of the
weighting coefficients propagates from
output layer to the inside hidden layers, so
it is called back-propagation (BP) learning
algorithm.
• The multi-layer perceptron is also called
back-propagation learning neural network
(BPNN)
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Fig. 4.5-1 Back-propagation learning in multilayer perceptron.
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4.5.1 Analysis(A) One hidden layer case (I-J-K network)
Given an input training pattern vector x and its desired
output vector d, we can get the real output vector o.
Consider the error1
2| 𝐝 − 𝐨 𝑡 |𝟐 , the sum of the
squared-error E,
𝑤𝑘𝑗
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4.5.1 Analysis(A) One hidden layer case (I-J-K network)
Fig. 4.5-2 Adjust weighting
coefficient 𝑤𝑘𝑗 between output and
hidden layer.
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1
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Fig. 4.5-3 Adjust weighting coefficient 𝑤𝑗𝑖 between hidden and input layers.
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(B)Two hidden layers case (B-I-J-K network)
Fig. 4.5-4 Adjust weighting between hidden and input layers.𝑤𝑖𝑏
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(B) Two hidden layers case (B-I-J-K network)
3. Adjust weighting 𝑤𝑖𝑏 between hidden (index i) and
input (index b) layers.
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(c) Three hidden layers case (A-B-I-J-K) and n
hidden layers case
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4.5.2 Back-propagation learning algorithm of one-
hidden layer perceptron (I)
Algorithm：具一層隱藏層的認知器之由後傳遞學
習法則(I) (輸入一個樣本就算誤差)
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(A) Major steps of algorithm
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(B)Flowchart of programming
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4.5.3 Back-propagation learning algorithm of
one-hidden layer perceptron (II)
Algorithm：具一層隱藏層的認知器之由後傳遞學
習法則 (II) (Calculate the error after one iteration
is complete，輸入一個iteration才算誤差)
Input: N augmented vectors y1, …, yN, and the
corresponding output vector d1, …, dN.
Output: The weighting coefficients wji, wkj for the first
and second layers, respectively.
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Steps:
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(A) Major steps of algorithm
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(B)Flowchart of programming
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4.6 Experiment of XOR Classification &
Discussions• The 2-3-2 (2 inputs, 1 hidden layer with 3
hidden nodes, and 2 output nodes, truly 3-4-2)
multilayer perceptron is applied to the
classification of XOR problem.
• The training samples are the 4 points in XOR.
• After learning, we input the test points of the
square area and get the two classification
regions in the following.
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Fig. 4.6-1 Classification result of Exclusive-OR using 2-3-2 perceptron.
(a) Error vs. number of iterations (Error threshold is set at 0.01).
(b) Two class regions.
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Fig. 4.6-1 Classification result of Exclusive-OR using 2-3-2 perceptron.
(a) Error vs. number of iterations (Error threshold is set at 0.01).
(b) Two class regions. (續)
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討論：
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7. Universal Approximation Theorem
The theorem can be applied to multilayer perceptrons,
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Multilayer perceptron and decision regions in
2-D space
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4.7 On Hidden Nodes for Neural Nets
Mirchandani與Cao於1989年發表 “On hidden nodes
for neural nets”之論文，最主要的貢獻為發表了定
理，
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Theorem: For the input of d dimensions, number of
hidden nodes is H, the maximum number M of linearly
separable regions can determined as
= = + + +
where = 0，H < k
( = )!
( )! !
H
H k k( , )C H k
( , )C H k
( , )C H d
( ,1)C H( ,0)C H 0
,d
k
C H k
,M H d
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