Neural Network. Sensors Sorter

Transkript

1 CSC Neural Networks Simple Neural Nets for Pattern Recognition 1

2 Apple-Banana Sorter Neural Network Sensors Sorter Apples Bananas 2

3 Prototype Vectors Measurement vector p = [shape, texture, weight] T Shape: {1: round, -1: elliptical} Texture: {1: smooth, -1: rough} Weight: {1: > 1 lb, -1: < 1 lb} Prototype banana p 1 = [-1, 1, -1] T Prototype apple p 2 = [1, 1, -1] T 3

4 Perceptron p R*1 W S*R a S*1 + n S*1 R 1 b S*1 S a = hardlims (Wp + b) 4

5 Two-Input Case p 2 p 1 w 1,1 n a n > 0 2 W n < 0 p 2 w 1,2 1 b -2 2 p 1 Can classify input vectors into two categories. a = hardlims ([ 1 2 ]p + (-2)) where b = -2 Decision boundary Wp + b = 0 [ 1 2 ]p + (-2) = 0 or p 1 + 2p 2 2 = 0 Recognize only linear separable patterns. w 1, 1 = 1 w 1, 2 = 2 5

6 Apple/Banana Example How many neurons are required? There are only two categories and single Perceptron enough to distinguish apples and bananas. Vector inputs are three-dimensional (R=3). Perceptron equation a = hardlims w 1 1 p 1, w 1, 2 w 1, 3 p 2 p 3 + b Goal choose the bias b and elements of the weight matrix so that the perceptron will be able to distinguish between apples and bananas. 6

7 Apple/Banana Example p 3 p 1 p 2 p 2 (apple) p 1 (banana) The bias determines the position of the boundary. The decision boundary should separate the prototype vectors (symmetrically). (p 1 = 0) Weight vector should be orthogonal to decision boundary point in the direction of the vector which should produce an output of 1. 7

8 Apple/Banana Example p 1 p 2 p 3 Weight vector [-1 0 0] T Output 1 for bananas Bias b = 0 Equation of the decision boundary p 2 (apple) p 1 (banana) p p = 0 p 3 8

9 Testing the Network Banana: 1 a = hardlims = 1 1( b anana) Apple: 1 a = hardlims = 1 1( apple) Rough Banana: 1 a = hardlims = 1 1( b anana) 9

10 Summary Designed the network graphically. What about the problems with high dimensional input spaces? Learning algorithms to train networks by using a set of examples. 10

11 Hamming Network Designed to solve binary pattern recognition problems. Uses both feed-forward forward an recurrent (feed-back) layers. Objective To decide which prototype vector is closest to the input vector. This decision is indicated by the output of the recurrent layer. 11

12 Hamming Network 12

13 Hamming Network Number of neurons in the first layer = Number of neurons in the second layer There is one neuron in the recurrent layer for each prototype pattern. Only one neuron produces a nonzero output when the recurrent layer converges. This neuron indicates the prototype pattern that is closest to the input vector. 13

14 Feedforward Layer Performs a correlation, or inner product, between each of prototype patterns and the input pattern. Weight matrix set to the prototype patterns Each element of the bias vector is equal to R,, where R is the number of elements in the input vector. For Apple/Banana example S = 2 W 1 p 1 T = = b 1 = R = R p 2 T

15 Feedforward Layer Output of the feedforward layer p 1 T a 1 = W 1 p + b 1 = p + 3 = T p 3 2 p T 1 p + 3 T p 2p + 3 Inner product of two vectors Largest when vectors point in the same direction. Smallest when they point in the opposite directions. Adding R to the inner product guarantee that outputs of the feedforward layer can never be negative. The neuron with the largest output will correspond to the prototype pattern that is closest in Hamming distance to input pattern. 15

16 Hamming Distance The Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. E.g. The Hamming distance between: and is and is 3. "toned"" and "roses"" is 3. 16

17 Recurrent Layer Known as a competitive layer. The neurons are initialized with the outputs of the feedforward layer. Compete with each other (neuron) to determine a winner. After the competition only one neuron will have a non- zero output. The winning neuron indicates which category of input is presented to the net. 17

18 W 2 = 1 ε ε ε 1 Recurrent Layer The weight matrix of the recurrent layer < S 1 S no of neurons in the recurrent layer An iteration of the recurrent layer proceeds as follows: ( ) poslin 1 ε a 2 t + 1 a 2 = ( t) = ε 1 poslin a 1 2 t 2 ( ) εa2( t) 2 2 a 2( t) εa1( t) Each element is reduced by the same fraction of the other. The larger element will be reduced by less, and the smaller element will be reduced by more. Output of each neuron become zero except the one with the largest initial value. 18

19 Hamming Operation First Layer Input (Rough Banana) p = a 1 = = ( 1 + 3) = ( 1 + 3)

20 Hamming Operation Second Layer a 2 ( 1) = poslin( W 2 a 2 ( 0) ) = poslin poslin 3 = 0 0 a 2 ( 2) = poslin( W 2 a 2 ( 1) ) = poslin poslin 3 =

21 Exercise Suppose that we want to distinguish between bananas and pineapples: p 1 = [-1 1-1] (Banana) p 2 = [-1-1 1] (Pineapple) (i) Design a perceptron to recognize these patterns. (ii) Design a Hamming network to recognize these patterns. 21

22 Lab Work - 1 Design a Hamming Network to recognize the following Arabic Numerals. 22

23 Lab Work 1 Write a Matlab program to obtain the output of the designed Hamming Network. Test the designed network with the following patterns. 23