Barbara Hammer

Generalized relevance learning vector quantization

We propose a new scheme for enlarging generalized learning vector quantization (GLVQ) with weighting factors for the input dimensions. The factors allow an appropriate scaling of the input dimensions according to their relevance. They are adapted automatically during training according to the specific classification task whereby training can be interpreted as stochastic gradient descent on an appropriate error function. This method leads to a more powerful classifier and to an adaptive metric with little extra cost compared to standard GLVQ. Moreover, the size of the weighting factors indicates the relevance of the input dimensions. This proposes a scheme for automatically pruning irrelevant input dimensions. The algorithm is verified on artificial data sets and the iris data from the UCI repository. Afterwards, the method is compared to several well known algorithms which determine the intrinsic data dimension on real world satellite image data.

Adaptive relevance matrices in learning vector quantization

We propose a new matrix learning scheme to extend relevance learning vector quantization (RLVQ), an efficient prototype-based classification algorithm, toward a general adaptive metric. By introducing a full matrix of relevance factors in the distance measure, correlations between different features and their importance for the classification scheme can be taken into account and automated, and general metric adaptation takes place during training. In comparison to the weighted Euclidean metric used in RLVQ and its variations, a full matrix is more powerful to represent the internal structure of the data appropriately. Large margin generalization bounds can be transferred to this case, leading to bounds that are independent of the input dimensionality. This also holds for local metrics attached to each prototype, which corresponds to piecewise quadratic decision boundaries. The algorithm is tested in comparison to alternative learning vector quantization schemes using an artificial data set, a benchmark multiclass problem from the UCI repository, and a problem from bioinformatics, the recognition of splice sites for C.elegans.

Neural maps in remote sensing image analysis

We study the application of self-organizing maps (SOMs) for the analyses of remote sensing spectral images. Advanced airborne and satellite-based imaging spectrometers produce very high-dimensional spectral signatures that provide key information to many scientific investigations about the surface and atmosphere of Earth and other planets. These new, sophisticated data demand new and advanced approaches to cluster detection, visualization, and supervised classification. In this article we concentrate on the issue of faithful topological mapping in order to avoid false interpretations of cluster maps created by an SOM. We describe several new extensions of the standard SOM, developed in the past few years: the growing SOM, magnification control, and generalized relevance learning vector quantization, and demonstrate their effect on both low-dimensional traditional multi-spectral imagery and approximately 200-dimensional hyperspectral imagery.

Supervised Neural Gas with General Similarity Measure

Prototype based classification offers intuitive and sparse models with excellent generalization ability. However, these models usually crucially depend on the underlying Euclidian metric; moreover, online variants likely suffer from the problem of local optima. We here propose a generalization of learning vector quantization with three additional features: (I) it directly integrates neighborhood cooperation, hence is less affected by local optima; (II) the method can be combined with any differentiable similarity measure whereby metric parameters such as relevance factors of the input dimensions can automatically be adapted according to the given data; (III) it obeys a gradient dynamics hence shows very robust behavior, and the chosen objective is related to margin optimization.

Batch and median neural gas

Neural Gas (NG) constitutes a very robust clustering algorithm given Euclidean data which does not suffer from the problem of local minima like simple vector quantization, or topological restrictions like the self-organizing map. Based on the cost function of NG, we introduce a batch variant of NG which shows much faster convergence and which can be interpreted as an optimization of the cost function by the Newton method. This formulation has the additional benefit that, based on the notion of the generalized median in analogy to Median SOM, a variant for non-vectorial proximity data can be introduced. We prove convergence of batch and median versions of NG, SOM, and k-means in a unified formulation, and we investigate the behavior of the algorithms in several experiments.

Recursive self-organizing network models

Self-organizing models constitute valuable tools for data visualization, clustering, and data mining. Here, we focus on extensions of basic vector-based models by recursive computation in such a way that sequential and tree-structured data can be processed directly. The aim of this article is to give a unified review of important models recently proposed in literature, to investigate fundamental mathematical properties of these models, and to compare the approaches by experiments. We first review several models proposed in literature from a unifying perspective, thereby making use of an underlying general framework which also includes supervised recurrent and recursive models as special cases. We shortly discuss how the models can be related to different neuron lattices. Then, we investigate theoretical properties of the models in detail: we explicitly formalize how structures are internally stored in different context models and which similarity measures are induced by the recursive mapping onto the structures. We assess the representational capabilities of the models, and we shortly discuss the issues of topology preservation and noise tolerance. The models are compared in an experiment with time series data. Finally, we add an experiment for one context model for tree-structured data to demonstrate the capability to process complex structures.

A general framework for unsupervised processing of structured data

We propose a general framework for unsupervised recurrent and recursive networks. This proposal covers various popular approaches like standard self organizing maps (SOM), temporal Kohonen maps, resursive SOM, and SOM for structured data. We define Hebbian learning within this general framework. We show how approaches based on an energy function, like neural gas, can be transferred to this abstract framework so that proposals for new learning algorithms emerge.

Merge SOM for temporal data

The recent merging self-organizing map (MSOM) for unsupervised sequence processing constitutes a fast, intuitive, and powerful unsupervised learning model. In this paper, we investigate its theoretical and practical properties. Particular focus is put on the context established by the self-organizing MSOM, and theoretic results on the representation capabilities and the MSOM training dynamic are presented. For practical studies, the context model is combined with the neural gas vector quantizer to obtain merging neural gas (MNG) for temporal data. The suitability of MNG is demonstrated by experiments with artificial and real-world sequences with one- and multi-dimensional inputs from discrete and continuous domains.

Topographic mapping of large dissimilarity data sets

Topographic maps such as the self-organizing map (SOM) or neural gas (NG) constitute powerful data mining techniques that allow simultaneously clustering data and inferring their topological structure, such that additional features, for example, browsing, become available. Both methods have been introduced for vectorial data sets; they require a classical feature encoding of information. Often data are available in the form of pairwise distances only, such as arise from a kernel matrix, a graph, or some general dissimilarity measure. In such cases, NG and SOM cannot be applied directly. In this article, we introduce relational topographic maps as an extension of relational clustering algorithms, which offer prototype-based representations of dissimilarity data, to incorporate neighborhood structure. These methods are equivalent to the standard (vectorial) techniques if a Euclidean embedding exists, while preventing the need to explicitly compute such an embedding. Extending these techniques for the general case of non-Euclidean dissimilarities makes possible an interpretation of relational clustering as clustering in pseudo-Euclidean space. We compare the methods to well-known clustering methods for proximity data based on deterministic annealing and discuss how far convergence can be guaranteed in the general case. Relational clustering is quadratic in the number of data points, which makes the algorithms infeasible for huge data sets. We propose an approximate patch version of relational clustering that runs in linear time. The effectiveness of the methods is demonstrated in a number of examples.

Distance learning in discriminative vector quantization

Discriminative vector quantization schemes such as learning vector quantization (LVQ) and extensions thereof offer efficient and intuitive classifiers 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 letter, 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, 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.