Stochastic Training Rule
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SOM training is based on a competitive learning strategy. Assume I-dimensional input vectors zp, where the subscript p denotes a single training pattern. The first step of the training process is to define a map structure, usually a two-dimensional grid (refer to Figure 4.3). The map is usually square, but can be of any rectangular shape. The number of elements (neurons) in the map is less than the number of training patterns. Ideally, the number of neurons should be less than or equal to the number of independent training patterns. With each neuron on the map is associated an I-dimensional weight vector which forms the centroid of one cluster. Larger cluster groupings are formed by grouping
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CHAPTER 4. UNSUPERVISED LEARNING NEURAL NETWORKS
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Figure 4.3: Self-organizing map together "similar" neighboring neurons. Initialization of the codebook vectors can occur in various ways: Assign random values to each weight w^j = (w k j 1 ,w k j 2 , , w K J I ] , with K the number of rows and J the number of columns of the map. The initial values are bounded by the range of the corresponding input parameter. While random initialization of weight vectors is simple to implement, this form of initialization introduces large variance components into the map which increases training time. Assign to the codebook vectors randomly selected input patterns. That is,
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wkj = Zp
with p~ 7(1, P). This approach may lead to premature convergence, unless weights are perturbed with small random values. Find the principal components of the input space, and initialize the codebook vectors to reflect these principal components.
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4.5. SELF-ORGANIZING FEATURE MAPS
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A different technique of weight initialization is due to Su et a/., where the objective is to define a large enough hyper cube to cover all the training patterns [Su et al. 1999]. The algorithm starts by finding the four extreme points of the map by determining the four extreme training patterns. Firstly, two patterns are found with the largest inter-pattern Euclidean distance. A third pattern is located at the furthest point from these two patterns, and the fourth pattern with largest Euclidean distance from these three patterns. These four patterns form the corners of the map. Weight values of the remaining neurons are found through interpolation of the four selected patterns, in the following way: Weights of boundary neurons are initialized as
j-n + di,
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K WKJ ^ij n 1N , -. = K -i (k-l) + wij
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for all j = 2, , J - 1 and k = 2, , K - 1. The remaining codebook vectors are initialized as
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The standard training algorithm for SOMs is stochastic, where codebook vectors are updated after each pattern is presented to the network. For each neuron, the associated codebook vector is updated as Wkj (t + l)= wkj (t) + hmn,kj (t) [zp - wkj (t)] (4.19)
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where mn is the row and column index of the winning neuron. The winning neuron is found by computing the Euclidean distance from each codebook vector to the input vector, and selecting the neuron closest to the input vector. That is,
The function h m n , k j ( t ) in equation (4.19) is referred to as the neighborhood function. Thus, only those neurons within the neighborhood of the winning neuron mn have their codebook vectors updated. For convergence, it is necessary that hmn,kj (t) > 0 when t -> oo.
CHAPTER 4. UNSUPERVISED LEARNING NEURAL NETWORKS
The neighborhood function is usually a function of the distance between the coordinates of the neurons as represented on the map, i.e.
hmn,kj(t) = h(\\Cmn - Ckj\\2,t)
with the coordinates Cmn,Ckj G K2. With increasing value of ||c^ CjH2, (that is, neuron kj is further away from the winning neuron mn), hmn,kj > 0. The neighborhood can be defined as a square or hexagon. However, the smooth Gaussian kernel is mostly used: hmn,kj(t) = n ( t ) e 2-2 > (4.20)
where rj(t) is the learning rate factor and cr(t) is the width of the kernel. Both n(t) and a(t) are monotonically decreasing functions. The learning process is iterative, continuing until a "good" enough map has been found. The quantization error is usually used as an indication of map accuracy, defined as the sum of Euclidean distances of all patterns to the codebook vector of the winning neuron, i.e.