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with = rj/a. Equation (4.5) implies that inputs for which zi,p <u k i (t - I)/ have then- corresponding weights Uki decreased by a value proportional to the output value ok,p. When zi,p > uki(t l)/ , weight uki is increased proportional to o k,p . Sejnowski proposed another way to formulate Hebb's postulate, using the covariance correlation of the neuron activation values [Sejnowski 1977]: = ri[(zif - Zi)(okf - Ok)} with
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Another variant of the Hebbian learning rule uses the correlation in the changes in activation values over consecutive time steps. For this learning rule, referred to as differential Hebbian learning, &uki(t) = ri&Zi(t)&ok(t - 1) where Azj(t) = Zif(t) - ziip(t - 1)
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(4.9) (4.10) (4.11)
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Aot(t - 1) = oktp(t - 1) - oktp(t - 2)
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Principal Component Learning Rule
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Principal component analysis (PCA) is a statistical technique used to transform a data space into a smaller space of the most relevant features. The aim is to project the original I-dimensional space onto an I-dimensional linear subspace, where / < /, such that the variance in the data is maximally explained within the smaller I-dimensional space. Features (or inputs) that have little variance are thereby removed. The principal components of a data set are found by calculating the covariance (or correlation) matrix of the data patterns, and by getting the minimal set of orthogonal vectors (the eigenvectors) that span the space of the Covariance matrix. Given the set of orthogonal vectors, any vector in the space can be constructed with a linear combination of the eigenvectors. Oja developed the first principal components learning rule, with the aim of extracting the principal components from the input data [Oja 1982]. Oja's principal
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components learning is an extension of the Hebbian learning rule, referred to as normalized Hebbian learning, to include a feedback term to constrain weights. In doing so, principal components could be extracted from the data. The weight change is given as
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The first term corresponds to standard Hebbian learning (refer to equation (4.2)), while the second term is a forgetting factor to prevent weight values from becoming unbounded. The value of the learning rate, 77, above is important to ensure convergence to a stable state. If n is too large, the algorithm will not converge due to numerical unstability. If n is too small, convergence is extremely slow. Usually, the learning rate is time dependent, starting with a large value which decays gradually as training progresses. To ensure numerical stability of the algorithm, the learning rate rjk(t] for output unit ok must satisfy the inequality:
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where \k is the largest eigenvalue of the covariance matrix, Cz, of the inputs to the unit [Oja and Karhuner 1985]. A good initial value is given as nk(0) = 1 / [ 2 Z T Z ] , where Z is the input matrix. Cichocki and Unbehauen provided an adaptive learning rate which utilizes a forgetting factor, 7, as follows [Cichocki and Unbehauen 1993]:
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nk(*) =
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Usually, 0.9 < 7 < 1. The above can be adapted to allow the same learning rate for all the weights in the following way:
Sanger developed another principal components learning algorithm, similar to that of Oja, referred to as generalized Hebbian learning [Sanger 1989]. The only difference is the inclusion of more feedback information and a decaying learning rate n(t): ,p -ok,p
, Uji(t - l)a,,p]
For more information on principal component learning, the reader is referred to the summary in [Haykin 1994].
Learning Vector Quantizer-I
One of the most frequently used unsupervised clustering algorithms is the learning vector quantizer (LVQ) developed by Kohonen [Kohonen 1995]. While several versions of LVQ exist, this section considers the unsupervised version, LVQ-I. Ripley defined clustering algorithms as those algorithms where the purpose is to divide a set on n observations into m groups such that members of the same group are more alike than members of different groups [Ripley 1996]. The aim of a clustering algorithm is therefore to construct clusters of similar input vectors (patterns), where similarity is usually measured in terms of Euclidean distance. LVQ-I performs such clustering. The training process of LVQ-I to construct clusters is based on competition. Referring to Figure 4.1, each output unit ok represents a single cluster. The competition is among the cluster output units. During training, the cluster unit whose weight vector is the "closest" to the current input pattern is declared as the winner. The corresponding weight vector and that of neighbor units are then adjusted to better resemble the input pattern. The "closeness" of an input pattern to a weight vector is usually measured using the Euclidean distance. The weight update is given as 0 otherwise (4.13)
where n ( t ) is a decaying learning rate, and K k , p (t) is the set of neighbors of the winning cluster unit ok for pattern p. It is, of course, not strictly necessary that LVQ-I makes use of a neighborhood function, thereby updating only the weights of the winning output unit. An illustration of clustering, as done by LVQ-I, is given in Figure 4.2. The input space, defined by two input units z\ and 221 is represented in Figure 4.2(a), while Figure 4.2(b) illustrates the LVQ-I network architecture required to form the clusters. Note that although only three classes exist, four output units are necessary -