Learning vector quantization

In computer science, learning vector quantization (LVQ) is a prototype-based supervised classification algorithm. LVQ is the supervised counterpart of vector quantization systems. LVQ can be understood as a special case of an artificial neural network, more precisely, it applies a winner-take-all Hebbian learning-based approach. It is a precursor to self-organizing maps (SOM) and related to neural gas and the k-nearest neighbor algorithm (k-NN). LVQ was invented by Teuvo Kohonen.

In computer science, learning vector quantization (LVQ) is a prototype-based supervised classification algorithm. LVQ is the supervised counterpart of vector quantization systems. LVQ can be understood as a special case of an artificial neural network, more precisely, it applies a winner-take-all Hebbian learning-based approach. It is a precursor to self-organizing maps (SOM) and related to neural gas and the k-nearest neighbor algorithm (k-NN). LVQ was invented by Teuvo Kohonen. An LVQ system is represented by prototypes W=(w(i),...,w(n)) which are defined in the feature space of observed data. In winner-take-all training algorithms one determines, for each data point, the prototype which is closest to the input according to a given distance measure. The position of this so-called winner prototype is then adapted, i.e. the winner is moved closer if it correctly classifies the data point or moved away if it classifies the data point incorrectly. An advantage of LVQ is that it creates prototypes that are easy to interpret for experts in the respective application domain. LVQ systems can be applied to multi-class classification problems in a natural way. A key issue in LVQ is the choice of an appropriate measure of distance or similarity for training and classification. Recently, techniques have been developed which adapt a parameterized distance measure in the course of training the system, see e.g. (Schneider, Biehl, and Hammer, 2009) and references therein. LVQ can be a valuable aid in classifying text documents. ==Algorithm== The algorithms are presented as in. Set up: Let the data be denoted by x_i \in \R^D , and their corresponding labels by y_i \in \{1, 2, \dots, C\} . The complete dataset is \{(x_i, y_i)\}_{i=1}^N . The set of code vectors is w_j \in \R^D . The learning rate at iteration step t is denoted by \alpha_t . The hyperparameters w and \epsilon are used by LVQ2 and LVQ3. The original paper suggests \epsilon \in [0.1, 0.5] and w \in [0.2, 0.3] . Let d_j := \|x_i - w_j\|, d_k := \|x_i - w_k\| . If \min \left(\frac{d_j}{d_k}, \frac{d_k}{d_j}\right)>s , where s=\frac{1-w}{1+w} , then If w_j and x_i have the same class, and w_k and x_i have different classes, then w_j \leftarrow w_j + \alpha_t(x_i - w_j) and w_k \leftarrow w_k - \alpha_t(x_i - w_k) . If w_k and x_i have the same class, and w_j and x_i have different classes, then w_j \leftarrow w_j - \alpha_t(x_i - w_j) and w_k \leftarrow w_k + \alpha_t(x_i - w_k) . If w_j and w_k and x_i have the same class, then w_j \leftarrow w_j - \epsilon\alpha_t(x_i - w_j) and w_k \leftarrow w_k + \epsilon\alpha_t(x_i - w_k) . If w_k and x_i have different classes, and w_j and x_i have different classes, then the original paper simply does not explain what happens in this case, but presumably nothing happens in this case. Otherwise, skip. Note that condition \min \left(\frac{d_j}{d_k}, \frac{d_k}{d_j}\right)>s , where s=\frac{1-w}{1+w} , precisely means that the point x_i falls between two Apollonian spheres.

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