A Comparison of Fuzzy ARTMAP

1. UNIVERSITÉ LA AL V A Comparison of Fuzzy ARTMAP and Multilayer Perceptron for Handwritten Digit Recognition Martin Busque Marc Parizeau October 31, 1997 Computer Vision and Systems Laboratory Universit Laval e Sainte-Foy (Qubec), Canada e G1K 7P4 Tel: (418) 656-2131, ext. 7912 Fax: (418) 656-3159 Email: parizeau@gel.ulaval.ca

2. A Comparison of Fuzzy ARTMAP and Multilayer Perceptron for Handwritten Digit Recognition Abstract This paper explores the features of the Fuzzy ARTMAP neural network classi

3. er in the context of on-line handwritten recognition. A Fuzzy ARTMAP classi

4. er is compared with a multilayer Perceptron trained using the standard backpropagation algorithm. Results show that although the recognition performance of both networks is similar, the convergence of the Fuzzy ARTMAP is both much faster and incrementaly stable than that of the Perceptron. In fact, it is shown experimentally that a Fuzzy ARTMAP can be trained in a single pass with almost no performance penalty when using the fast learning mode. The recognition experiments were conducted on handwritten digits extracted from the Unipen training dataset. Keywords: Neural Networks; Fuzzy ARTMAP; Multilayer Perceptron; Hand- written Digit Recognition; Unipen Dataset. 1 Introduction There are many ways to model a system for handwriting recognition. One of them is to use a neural network. Still, there exist many dierent kinds of neural networks. Among them, the multilayer Perceptron and the Fuzzy ARTMAP. The Perceptron is now a well-known architecture [1]. It is able to model a system, based on a set of inputs with the correponding desired outputs. It 1

5. appeared in the 1960's and has much evolved, especially since the development of the back-propagation algorithm in the 1980's. This supervised learning neural network has proven in practice to be quite ecient and has been used on a number of diverse problems. What particularly interests us in this discussion is the Fuzzy ARTMAP. This network, introduced much more recently (1991), is a supervised network like the Perceptron. But it has some very interesting additional properties, such as a very fast convergence and a capacity for incremental learning. Our objective is to compare the Fuzzy ARTMAP with the Perceptron on handwritten digit recognition. First, in Section 2, the Fuzzy ART will be presented. The Fuzzy ART is an unsupervised network which is the essential component of the Fuzzy ARTMAP. Then the Fuzzy ARTMAP itself will be explained in Section 3. We will discuss in Section 4 the experimental results obtained on handwritten digit recognition with both the Perceptron and the Fuzzy ARTMAP. 2 Fuzzy ART The Fuzzy ART is a neural network introduced by Carpenter, Grossberg and Rosen in 1991 [2]. It is a modi

6. ed version of the binary ART1 [3], which is notably able to accept analog fuzzy input patterns, i.e. vectors whose components are real numbers between 0 and 1. The Fuzzy ART is an unsupervised neural network capable of incremental learning, that is it can learn continouously without 2

7. F2 F1 Complement 0.4 0.8 0.6 0.2 coding 0.4 0.8 Input vector Figure 1: Sample Fuzzy ART network. forgetting what it has previously learned. A Fuzzy ART network is formed of two layers of neurons, the input layer F1 and the output layer F2 , as illustrated in Figure 1. Both layers have an activity pattern, schematized on the

8. gure with vertical bars of varying height. The layers are fully interconnected, each neuron being connected to every neuron on the other layer. Every connection is weighted by a number lying between 0 and 1. A neuron of F2 represents one category formed by the network and is caracterized by its weight vector wj (j is the index of the neuron) . The weight vector's size is equal to the dimension M of layer F1 . Initially all the weight vectors' components are

9. xed to 1. Until the weights of a neuron are modi

10. ed, we say that it is uncommited. Inversely, once a neuron's weights have been modi

11. ed, this neuron is said to be commited. The network uses a form of normalization called complement coding. The operation consists on taking the input vector and concatenating it with its com- 3

12. plement. The resulting vector is presented to layer F1 . Therefore, the dimension M of layer F1 is the double of the input vector's dimension. Complement coding can be deactivated, in this case layer F1 will have the same dimension as the input vector. Unless speci

13. ed otherwise, we will always suppose that complement coding is active. The Fuzzy ART learns by placing hyperboxes in the M {dimensions hyperspace, 2 M being the size of layer F1 . As said earlier, each neuron of layer F2 represents a category formed by the network, and this category is de

14. ned by a box1 [2, section 5]. The position of the box in the space is encoded in the weight vector of the neuron. Because of complement coding and of the learning process we will explain later, the

15. rst half of the weight vector memorizes one corner of the box (the closest to the origin) and the other half memorizes the opposite one. Learning process When an input vector is presented to the network, it is

16. rst preprocessed into complement coding form. The resulting vector, I, represents a point in the space. The activity pattern on layer F1 is set equal to I. The choice function is then evaluated for each neuron of F2 . This function is de

17. ned by Tj = I+ jwj jj j ^ w (1) j where j j is the L1 norm of the vector, i.e. the sum of its components (jpj PM pi), ^ is the fuzzy AND operator ((p ^ q)i min(pi; qi)) and is the choice i =1 1 To simplify the discussion, we will sometimes drop the hyper pre

18. x and use the terms box and space to refer respectively to hyperbox and hyperspace. 4

19. parameter. The choice parameter must be greater than zero. It is usually chosen close to zero for a good performance [4, p.701]. This parameter ensures that when a point is enclosed in more than one hyperbox, the smallest one is selected. This is in fact the objective of the choice function: to choose the smallest box in which the point is included. A point is said to be in a box when the weight vector wj corresponding to this box is a fuzzy subset of the input vector I. If no such box exist, either the one which needs to be the less expanded to enclose the point is selected or a new one is created, i.e. the

20. rst uncommited neuron is chosen. Note that we de

21. ne the size of a box as the sum of its lengths in each dimension. Once a neuron is selected (the winning neuron ), a vigilance criteron is evaluated. The vigilance criterion causes the network to choose another box (neuron) if the one chosen is already too large. Thus the vigilance parameter controls the maximum size of hyperboxes [2, section 5]. Mathematically, the vigilance criterion is jI ^ wJ j (2) jIj where J is the index of the winning neuron on F2 and is the vigilance parameter, which lies in the interval [0; 1]. If the criterion is respected, the Fuzzy ART learns the input vector, else it selects the next neuron with the highest choice function and reevaluates the vigilance criterion. These two steps are repeated until the vigilance criterion is respected. At that moment, it is said that the network is in resonance. All components of the activity pattern on F2 , which is also the output vector of the Fuzzy ART, are set to 0 except for the index of the winning neuron which is set to 1. 5

22. When the Fuzzy ART enters resonance, it proceeds with the learning of the input vector I by modifying the weight vector wJ of the neuron as follows wJ =

23. (I ^ wJ ) + (1

24. )wJ (3) Here, J is the index of the winning neuron and

25. 2 [0; 1] is the learning rate. When

26. = 1, we are in fast learning. We may note that the weights are nonin- creasing through time, which means that a hyperbox can never shrink but only expand itself. With fast learning, equation 3 means that the box associated with the neuron is enlarged just enough to include the point represented by the input vector. If

27. is smaller than 1, the box will expand towards the point, its enlarge- ment being directly proportional to the size of

28. . When we set

29. 1, we use what is called the fast-commit slow-recode option. This consists in setting

30. = 1 when the neuron is not commited and using the real value of

31. ( 1) after it is commited. 3 Fuzzy ARTMAP The Fuzzy ARTMAP, introduced by Carpenter et al. in 1992 [4] is a supervised network which is composed of two Fuzzy ARTs. The Fuzzy ARTs are identi

32. ed as ARTa and ARTb . The parameters of these networks are designated respectively by the subscripts a and b. The two Fuzzy ARTs are interconnected by a series of connections between the F2 layers of ARTa and ARTb . The connections are weighted, i.e. a weight wij between 0 and 1 is associated with each one of them. These connections form what is called the map

33. eld F ab. The map

34. eld has two 6

35. xab F ab ab wJ ARTa ARTb ya yb xa xb A 0.8 0.4 0.2 0.6 0.6 0.4 B a 0.8 0.4 Figure 2: Sample Fuzzy ARTMAP network. parameters (

36. ab and ab ) and an output vector xab . The input vector a of ARTa is put in complement coding form, resulting in vector A. Complement coding is not necessary in ARTb so we present the input vector B directly to this network. Figure 2 presents the structure of the Fuzzy ARTMAP. The weights of the map

37. eld's connections are illustrated by a vertical bar with a height proportional to the size of the weight. The weights of the map

38. eld are all initialized to 1. Training In order to train the Fuzzy ARTMAP, we present to ARTa a vector representing a data pattern and to ARTb a vector which is the desired output corresponding to this pattern. The network uses a form of hypothesis testing. When it receives the

39. rst vector, it deduces to which category it should belong. With the second 7

40. vector, the Fuzzy ARTMAP can either con

41. rm or reject the hypothesis, in which case the process is repeated with a new one. During the training process, learning is deactivated in ARTa (

42. a = 0) and is controlled by the Fuzzy ARTMAP. The vigilance parameter of ARTa , a , varies during learning. We identify the initial value of a as the baseline vigilance a . The vigilance of ARTb, b, is set to 1 to perfectly distinguish the desired output vectors. When vectors A and B are presented to ARTa and ARTb , both networks soon enter resonance. A vigilance criterion is then evaluated to verify if the winning neuron of ARTa corresponds to the desired output vector presented to ARTb. This criterion is jyb ^ wJ j ab (4) jybj ab where yb is the output vector of ARTb (the activity pattern on F2b), J is the index of the winning neuron on F2a, wJ corresponds to the weights of the connections ab to the J th neuron of F2a and ab 2]0; 1] is the vigilance parameter of the map

43. eld. Note that if the map

44. eld uses fast learning, the value of this fraction will either be 0 or 1. If the criterion is not respected, the vigilance of ARTa is increased just enough to select another winning neuron (a jA ^ wJ j=jAj, see equation 2) and the vector A is repropagated in ARTa . When the vigilance criterion is respected, the map

45. eld learns the association between vectors A and B by modifying its weigths as follows wJ =

46. abxab + (1

47. ab)wJ ab ab (5) 8

48. The weights in ARTa are also modi

49. ed as in equation 3. The vigilance of ARTa is set back to its baseline value a . In terms of hypexboxes, the map

50. eld of the Fuzzy ARTMAP associates a category number to each box created by ARTa . This number is the index of the associated neuron of F2b. Consequently, each category learned by the Fuzzy ARTMAP is represented by many rectangles. In other words it learna to distinguish the data by mapping boxes into the space, enclosing each category with a certain number of these boxes. Classifying Once the Fuzzy ARTMAP is trained, it can be used as a classi

51. er. In this case, the ARTb network is not used. We present a single input vector to ARTa which is propagated until resonance, with a (temporary) vigilance a = 0. Thus, the

52. rst category selected by the choice function is accepted. Learning in this network is also temporarily deactivated (

53. a = 0). The output vector of the map

54. eld is then set to xab = wJ ab (6) where J is the index of the winning node on F2a. If the map

55. eld used fast learning (

56. ab = 1), the output vector contains only 0's except a 1. The index of this component is the number of the category in which the input vector A has been classi

57. ed. The use of the map

58. eld is thus to associate a category number to each neuron of ARTa 's F2 layer (F2a), i.e. to each box in the hyperspace. 9

59. If fast learning was not used, one neuron of F2a can correspond to many categories with dierent degrees. One way to determine the category number could be to select the index of the maximal component. If needed, the desired output vector can be restored from the weights of the neuron of F2b whose index is the category number. 4 Experimental results We used the multilayer Perceptron and the Fuzzy ARTMAP to recognize handwritten digits taken from the

60. fth release of the Unipen training set, section 1a2. All the samples in the database were identi

61. ed with one of the following labels depending on their quality : Good, Ok and Bad. Good samples are perfectly recognizable, Ok samples are of a lower quality but should still be recognizable and Bad samples are either mis-labelled3, mis-segmented4, mis-scaled5 or would be very dicult to recognize by a human. The data set for our experiments is composed of a training set and a test set. We

62. rst selected randomly 300 Good and Ok samples of each digit (0-9) to form the 3000 samples training set. The test set consists of the rest of the database (3519 samples), approximately 11% of them being Bad samples. To transform the data into a vectorial space, we used a fuzzy representation based on a segmentation of the digits into circular arcs and straight lines [5]. 2 For more information on the Unipen database, see http://hwr.nici.kun.nl/unipen/. 3 For example a 1 labelled as a 0. 4 A single digit is segmented into two dierent digits. 5 Two dierent scales are applied on the X and Y axis. 10

63. Each vector representing a data sample has 49 components. Brie y, each digit is divided in 6 regions having 7 fuzzy characteristics related to the orientation and curvature of the segments in those regions. There are 7 more fuzzy variables related to the whole digit. Multilayer Perceptron We will now describe our results with the multilayered Perceptron using the de- scribed vectorial representation and data sets. We used a Perceptron with one hidden layer of 100 neurons. The ouput layer has 10 neurons, one neuron for each digit class. We trained this Perceptron using the standard back-propagation algorithm with a learning rate = 0:5 and a momentum = 0:5 [1]. We

64. rst presented to the network the data set in a random order. After 200 epochs, an RMS error of 0.0613 was obtained and the training set was correctly classi

65. ed at 99.97% (single error). The training took a CPU time of 1494 seconds (24.9 minutes) on a Sun Ultra-sparc 30. We then obtained a classi

66. cation rate on the test set of 93.0%. We measured the CPU time to compare this result with the performance of the Fuzzy ARTMAP. We also trained the Perceptron with the data in order, from digits 0 to 9. After 100 training epochs, an RMS error of 0.0212 was obtained. When we used the network to classify the test set, the rate of the data that was correctly classi

67. ed was extremely low, 9.21%. This is not surprising because it is well known that the multilayer Perceptron is not capable of incremental learning. In fact it only remembers the few last data it was trained with, in this case the samples of nines. 11

68. Table 1: Experimental results with the Fuzzy ARTMAP trained until perfect classi

69. cation of the training set when

70. a = 1 and the training set is presented in order. ARTa Classi

71. cation rate (%) Cumulated a Epochs categories Training set Test set CPU time (s) 0.60 1 88 98.2 86.5 7.3 2 92 99.8 86.7 14 3 93 100 86.8 21 0.65 1 137 99.0 88.2 12 2 140 99.97 88.0 22 3 141 100 88.0 32 0.70 1 219 99.6 90.6 26 2 222 100 90.6 42 0.75 1 332 99.8 92.0 45 2 333 100 92.0 69 0.80 1 502 99.9 91.8 79 2 503 100 91.8 115 0.85 1 774 99.97 92.5 129 2 775 100 92.5 185 0.90 1 1150 100 91.9 232 Indeed, all elements of the test set but two were eectively classi

72. ed as nines. Fuzzy ARTMAP We now look at the work performed by a Fuzzy ARTMAP on the same data sets we used with the multilayer Perceptron. We

73. xed a number of parameters of the Fuzzy ARTMAP. For example, we always used fast learning (

74. ab = 1) and a vigilance parameter ab = 1 in the map

75. eld. Other

76. xed parameters are a = b = 0:01,

77. b = 1, b = 1. Thus the two varying parameters in our experiments are

78. a and a , the learning rate and baseline vigilance of the ARTa 12

79. module. We trained the Fuzzy ARTMAP by presenting the training set in order, from the zeros to the nines. The learning rate of ARTa was set to

80. a = 1. The network was trained until it perfectly classi

81. ed the training set. Table 1 presents the results with dierent baseline vigilances a . These results show that values of a between 0.75 and 0.90 are the ones yielding the best correct classi

82. cation rate on the test set. We can note that increasing the baseline vigilance a causes the Fuzzy ARTMAP to form a higher number of ARTa categories. Because the maximal size of the boxes formed by a Fuzzy ART is smaller when the vigilance is higher, ARTa has to create a greater number of boxes, or categories. A remarkable point in those results is that the number of training epochs and the corresponding CPU time are both much lower for the Fuzzy ARTMAP than for the multilayer Perceptron. Still, the best classi

83. cation rate obtained with both networks are very close to one another (93.0% vs. 92.5%). Moreover, it is not needed to perfectly classify the training set to obtain good rates on the test set. In fact, the test set correct classi

84. cation rate is almost always maximal in only one epoch. Changing the order of the training set has a small but non negligible in uence on the results. The test set classi

85. cation rates may vary by about 1%. We were able to obtain, by randomly ordering the training set, a correct classi

86. cation rate on the test set of 92.7% (with a = 0:85) in one epoch (and a CPU time of 184 s). Table 2 presents results obtained when varying the learning rate of ARTa ,

87. a. We used a = 0:85, which yielded the best result with

88. a = 1, and we presented the 13

89. Table 2: Experimental results with the Fuzzy ARTMAP trained until perfect classi

90. cation of the training set with

91. a 1 when a = 0:85 and the training set is presented in a random order. ARTa Classi

92. cation rate (%) Cumulated

93. a Epochs categories Training set Test set CPU time (s) 0.8 4 1137 100 93.0 621 0.6 4 1077 100 92.1 570 0.4 11 963 100 93.1 967 0.2 11 958 100 93.0 905 data in a random order at each training epoch. The test set correct classi

94. cation rate was increased up to 93.1%. However, this implies more training epochs and, since the number of ARTa categories is higher, a larger ARTa network. Finally, to verify that Fuzzy ARTMAP can do incremental learning, we trained it in two separate steps, using fast learning and again a = 0:85. First, we performed training with the samples of all digits except the nines. After two epochs, the training set was perfectly learned and the test set correct classi

95. cation rate on digits 0 to 8 was 93.1%. Secondly, we trained the same network exclusively on the nines. It then correctly classi

96. ed the whole test set at 92.4%. It is now clear that this network can learn new data without forgetting what it was previously trained with. In fact, only 7 more samples of digits between 0 and 8 were incor- rectly classi

97. ed (on a total of 3197 samples). Thus the Fuzzy ARTMAP is capable of incremental learning, which is not the case for the multilayer Perceptron. 14

98. 5 Conclusion We have seen the structure of the Fuzzy ARTMAP and its essential component, the Fuzzy ART. We performed handwritten digit recognition with the multilayer Perceptron and the Fuzzy ARTMAP and compared the performance of these two neural networks. We saw that they yield similar recognition rates, but Fuzzy ARTMAP learning process is a lot faster. Moreover, Fuzzy ARTMAP is capable of incrementaly stable learning. We presented the best results we obtained so far but we believe the performance of the Fuzzy ARTMAP may still be improved. Nevertheless, it is clear that Fuzzy ARTMAP is a neural network that has attractive features for use in pattern recognition. 15

99. References [1] Simon Haykin. Neural Networks: A comprehensive foundation. IEEE Com- puter Society Press, 1994. [2] Gail A. Carpenter, Stephen Grossberg, and David B. Rosen. Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system. Neural Networks, 4:759{771, 1991. [3] Gail A. Carpenter and Stephen Grossberg. A massively parallel architecture for a self-organizing neural pattern recognition machine. Computer Vision, Graphics and Image Processing, 37:54{115, 1987. [4] Gail A. Carpenter, Stephen Grossberg, Natalya Markuzon, John H. Reynolds, and David B. Rosen. Fuzzy ARTMAP: A neural network architecture for incremental supervised learning of analog multidimensional maps. IEEE Trans- actions on Neural Networks, 3:698{713, 1992. [5] Jean-Franois Hbert, Marc Parizeau, and Nadia Ghazzali. A new fuzzy ge- c e ometric representation for isolated character recognition. Submitted to the 14th International Conference on Pattern Recognition (ICPR '98). 16

A Comparison of Fuzzy ARTMAP

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A Comparison of Fuzzy ARTMAP