Topology representing networks
Neural Networks
Dimension reduction by local principal component analysis
Neural Computation
Mixtures of probabilistic principal component analyzers
Neural Computation
Self-Organizing Maps
Physical Models of Neural Networks
Physical Models of Neural Networks
Clustering Algorithms
A unified framework for model-based clustering
The Journal of Machine Learning Research
Improved learning of Riemannian metrics for exploratory analysis
Neural Networks - 2004 Special issue: New developments in self-organizing systems
Self-organizing maps and clustering methods for matrix data
Neural Networks - 2004 Special issue: New developments in self-organizing systems
Learning an optimal distance metric in a linguistic vector space
Systems and Computers in Japan
Neural Networks - 2006 Special issue: Advances in self-organizing maps--WSOM'05
A Unified Continuous Optimization Framework for Center-Based Clustering Methods
The Journal of Machine Learning Research
Prototype based fuzzy classification in clinical proteomics
International Journal of Approximate Reasoning
KI '07 Proceedings of the 30th annual German conference on Advances in Artificial Intelligence
Growing hierarchical principal components analysis self-organizing map
ISNN'06 Proceedings of the Third international conference on Advances in Neural Networks - Volume Part I
Topology preservation in self-organizing feature maps: exact definition and measurement
IEEE Transactions on Neural Networks
IEEE Transactions on Neural Networks
Self-organizing mixture networks for probability density estimation
IEEE Transactions on Neural Networks
Self-organizing maps, vector quantization, and mixture modeling
IEEE Transactions on Neural Networks
Coupled principal component analysis
IEEE Transactions on Neural Networks
Survey of clustering algorithms
IEEE Transactions on Neural Networks
`Neural-gas' network for vector quantization and its application to time-series prediction
IEEE Transactions on Neural Networks
Relevance learning in unsupervised vector quantization based on divergences
WSOM'11 Proceedings of the 8th international conference on Advances in self-organizing maps
Accelerating kernel neural gas
ICANN'11 Proceedings of the 21th international conference on Artificial neural networks - Volume Part I
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The self-organizing map (SOM) and neural gas (NG) and generalizations thereof such as the generative topographic map constitute popular algorithms to represent data by means of prototypes arranged on a (hopefully) topology representing map. Most standard methods rely on the Euclidean metric, hence the resulting clusters tend to have isotropic form and they cannot account for local distortions or correlations of data. For this reason, several proposals exist in the literature which extend prototype-based clustering towards more general models which, for example, incorporate local principal directions into the winner computation. This allows to represent data faithfully using less prototypes. In this contribution, we establish a link of models which rely on local principal components (PCA), matrix learning, and a formal cost function of NG and SOM which allows to show convergence of the algorithm. For this purpose, we consider an extension of prototype-based clustering algorithms such as NG and SOM towards a more general metric which is given by a full adaptive matrix such that ellipsoidal clusters are accounted for. The approach is derived from a natural extension of the standard cost functions of NG and SOM (in the form of Heskes). We obtain batch optimization learning rules for prototype and matrix adaptation based on these generalized cost functions and we show convergence of the algorithm. The batch optimization schemes can be interpreted as local principal component analysis (PCA) and the local eigenvectors correspond to the main axes of the ellipsoidal clusters. Thus, this approach provides a cost function associated to proposals in the literature which combine SOM or NG with local PCA models. We demonstrate the behavior of matrix NG and SOM in several benchmark examples and in an application to image compression.