Learning vector quantization (LVQ) schemes constitute intuitive, powerful classification heuristics with numerous successful applications but, so far, limited theoretical background. We study LVQ rigorously within a simplifying model situation: two competing prototypes are trained from a sequence of examples drawn from a mixture of Gaussians. Concepts from statistical physics and the theory of on-line learning allow for an exact description of the training dynamics in highdimensional feature space. The analysis yields typical learning curves, convergence properties, and achievable generalization abilities. This is also possible for heuristic training schemes which do not relate to a cost function. We compare the performance of several algorithms, including Kohonen’s LVQ1 and LVQ+/-, a limiting case of LVQ2.1. The former shows close to optimal performance, while LVQ+/- displays divergent behavior. We investigate how early stopping can overcome this difficulty. Furthermore, we study a crisp version of robust soft LVQ, which was recently derived from a statistical formulation. Surprisingly, it exhibits relatively poor generalization. Performance improves if a window for the selection of data is introduced; the resulting algorithm corresponds to cost function based LVQ2. The dependence of these results on the model parameters, for example, prior class probabilities, is investigated systematically, simulations confirm our analytical findings. Keywords: prototype based classification, learning vector quantization, Winner-Takes-All algorithms, on-line learning, competitive learning 1.

We have earlier introduced a principle for learning metrics, which shows how metric-based methods can be made to focus on discriminative properties of data. The main applications are in supervising unsupervised learning to model interesting variation in data, instead of modeling all variation as plain unsupervised learning does. The metrics are derived by approximations to an information-geometric formulation. In this paper, we review the theory, introduce better approximations to the distances, and show how to apply them in two different kinds of unsupervised methods: prototype-based and pairwise distance-based. The two examples are self-organizing maps and multidimensional scaling (Sammon’s mapping).

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Discriminative vector quantization schemes such as learning vector quantization (LVQ) and extensions thereof offer efficient and intuitive classifiers which are based on the representation of classes by prototypes. The original methods, however, rely on the Euclidean distance corresponding to the assumption that the data can be represented by isotropic clusters. For this reason, extensions of the methods to more general metric structures have been proposed such as relevance adaptation in generalized LVQ (GLVQ) and matrix learning in GLVQ. In these approaches, metric parameters are learned based on the given classification task such that a data driven distance measure is found. In this article, we consider full matrix adaptation in advanced LVQ schemes; in particular, we introduce matrix learning to a recent statistical formalization of LVQ, robust soft LVQ, and we compare the results on several artificial and real life data sets to matrix learning in GLVQ, which is a derivation of LVQ-like learning based on a (heuristic) cost function. In all cases, matrix adaptation allows a significant improvement of the classification accuracy. Interestingly, however, the principled behavior of the models with respect to prototype locations and extracted matrix dimensions shows several characteristic differences depending on the data sets.

In this tutorial paper about mathematical aspects of neural networks, we will focus on two directions: on the one hand, we will motivate standard mathematical questions and well studied theory of classical neural models used in machine learning. On the other hand, we collect some recent theoretical results (as of beginning of 2003) in the respective areas. Thereby, we follow the dichotomy offered by the overall network structure and restrict ourselves to feedforward networks, recurrent networks, and self-organizing neural systems, respectively.

Self-organization constitutes an important paradigm in machine learning with successful applications e.g. for data- and web-mining. However, so far most approaches have been proposed for data contained in a fixed and finite dimensional vector space. We will focus on extensions for more general data structures like sequences and tree structures in this article. Various extensions of the standard self-organizing map (SOM) to sequences or tree structures have been proposed in the literature: the temporal Kohonen map, the recursive SOM, and SOM for structured data (SOMSD), for example. These methods enhance the standard SOM by recursive connections. We define in this article a general recursive dynamic which enables the recursive processing of complex data structures based on recursively computed internal representations of the respective context. The above mechanisms of SOMs for structures are special cases of the proposed general dynamic, furthermore, the dynamic covers the supervised case of recurrent and recursive networks, too. The general framework offers a uniform notation for training mechanisms such as Hebbian learning and the transfer of alternatives such as vector quantization or the neural gas algorithm to structure processing networks. The formal definition of the recursive dynamic for structure processing unsupervised networks allows the transfer of theoretical issues from the SOM literature to the structure processing case. One can formulate general cost functions corresponding to vector quantization, neural gas, and a modification of SOM for the case of structures. The cost functions can be compared to Hebbian learning which can be interpreted as an approximation of a stochastic gradient descent. We derive as an alternative the exact gradien...

We present a new variant of Generalized Learning Vector Quantization (GRLVQ) in a computer vision scenario. A version with incrementally added prototypes is used for the non-trivial case of high-dimensional object recognition. Training is based upon a generic set of standard visual features, the learned input weights are used for iterative feature pruning. Thus, prototypes and input space are altered simultaneously, leading to very sparse and task-specific representations. The effectiveness of the approach and the combination of the incremental variant together with pruning was tested on the Coil100 database. It exhibits excellent performance with regard to codebook size, feature selection and recognition accuracy. Key words: object recognition, relevance learning, feature selection, incremental learning vector quantization, adaptive metric 1

Relevance Learning Vector Quantization (RLVQ) (introduced in [1]) is a variation of Learning Vector Quantization (LVQ) which allows a heuristic determination of relevance factors for the input dimensions. The method is based on Hebbian learning and defines weighting factors of the input dimensions which are automatically adapted to the specific problem. These relevance factors increase the overall performance of the LVQ algorithm. At the same time, relevances can be used for feature ranking and input dimensionality reduction. We introduce

We extend soft nearest neighbor classification to fuzzy classification with adaptive class labels. The adaptation follows a gradient descent on a cost function. Further, it is applicable for general distance measures, in particular task specific choices and relevance learning for metric adaptation can be done. The performance of the algorithm is shown on synthetical as well as on real life data taken from proteomic research.

Abstract. The issue of Automatic Relevance Determination (ARD) has attracted attention over the last decade for the sake of efficiency and accuracy of classifiers, and also to extract knowledge from discriminant functions adapted to a given data set. Based on Learning Vector Quantization (LVQ), we recently proposed an approach to ARD utilizing genetic algorithms. Another approach is the Generalized Relevance LVQ which has been shown to outperform other algorithms of the LVQ family. In the following we present a unique description of a number of LVQ algorithms and compare them concerning their classification accuracy and their efficacy. For this purpose a real world data set consisting of spontaneous EEG and EOG during overnight-driving is employed to detect so-called microsleep events. Results show that relevance learning can improve classification accuracies, but do not reach the performance of Support Vector Machines. The computational costs for the best performing classifiers are exceptionally high and exceed basic LVQ1 by a factor of 10 4.