Building nonlinear data models with self-organizing maps

We study the extraction of nonlinear data models in high dimensional spaces with modified self-organizing maps. Our algorithm maps lower dimensional lattice into a high dimensional space without topology violations by tuning the neighborhood widths locally. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. The performance of the algorithm is demonstrated for one- and two-dimensional principal manifolds and for sparse data sets

Nonlinear principal component analysis

We study the extraction of nonlinear data models in high-dimensional spaces with modified self-organizing maps. We present a general algorithm which maps low-dimensional lattices into high-dimensional data manifolds without violation of topology. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. Moreover we present a second algorithm for the extraction of generalized principal curves comprising disconnected and branching manifolds. The performance of the algorithm is demonstrated for both one- and two-dimensional principal manifolds and also for the case of sparse data sets. As an application we reveal cluster structures in a set of real world data from the domain of ecotoxicology

Building Nonlinear Data Models with Self-Organizing Maps

We study the extraction of nonlinear data models in high dimensional spaces with modified self-organizing maps. Our algorithm maps lower dimensional lattice into a high dimensional space without topology violations by tuning the neighborhood widths locally. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. The performance of the algorithm is demonstrated for one- and two-dimensional principal manifolds and for sparse data sets

Building nonlinear data models with self-organizing maps

We study the extraction of nonlinear data models in high dimensional spaces with modified self-organizing maps. Our algorithm maps lower dimensional lattice into a high dimensional space without topology violations by tuning the neighborhood widths locally. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. The performance of the algorithm is demonstrated for one- and two-dimensional principal manifolds and for sparse data sets

An Algorithm for Generalized Principal Curves with Adaptive Topology in Complex Data Sets

Generalized principal curves are capable of representing complex data structures as they may have branching points or may consist of disconnected parts. For their construction using an unsupervised learning algorithm the templates need to be structurally adaptive. The present algorithm meets this goal by a combination of a competitive Hebbian learning scheme and a self-organizing map algorithm. Whereas the Hebbian scheme captures the main topological features of the data, in the map the neighborhood widths are automatically adjusted in order to suppress the noisy dimensions. It is noteworthy that the procedure which is natural in prestructured Kohonen nets could be carried over to a neural gas algorithm which does not use an initial connectivity. The principal curve is then given by an averaging procedure over the critical uctuations of the map exploiting noise-induced phase transitions in the neural gas

Nonlinear principal component analysis

We study the extraction of nonlinear data models in high-dimensional spaces with modified self-organizing maps. We present a general algorithm which maps low-dimensional lattices into high-dimensional data manifolds without violation of topology. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. Moreover we present a second algorithm for the extraction of generalized principal curves comprising disconnected and branching manifolds. The performance of the algorithm is demonstrated for both one- and two-dimensional principal manifolds and also for the case of sparse data sets. As an application we reveal cluster structures in a set of real world data from the domain of ecotoxicology

Nonlinear principal component analysis

We study the extraction of nonlinear data models in high-dimensional spaces with modified self-organizing maps. We present a general algorithm which maps low-dimensional lattices into high-dimensional data manifolds without violation of topology. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. Moreover we present a second algorithm for the extraction of generalized principal curves comprising disconnected and branching manifolds. The performance of the algorithm is demonstrated for both one- and two-dimensional principal manifolds and also for the case of sparse data sets. As an application we reveal cluster structures in a set of real world data from the domain of ecotoxicology

An Algorithm for Generalized Principal Curves with Adaptive Topology in Complex Data Sets

Generalized principal curves are capable of representing complex data structures as they may have branching points or may consist of disconnected parts. For their construction using an unsupervised learning algorithm the templates need to be structurally adaptive. The present algorithm meets this goal by a combination of a competitive Hebbian learning scheme and a self-organizing map algorithm. Whereas the Hebbian scheme captures the main topological features of the data, in the map the neighborhood widths are automatically adjusted in order to suppress the noisy dimensions. It is noteworthy that the procedure which is natural in prestructured Kohonen nets could be carried over to a neural gas algorithm which does not use an initial connectivity. The principal curve is then given by an averaging procedure over the critical uctuations of the map exploiting noise-induced phase transitions in the neural gas

Nonlinear Principal Component Analysis

We study the extraction of nonlinear data models in high-dimensional spaces with modified self-organizing maps. We present a general algorithm which maps low-dimensional lattices into high-dimensional data manifolds without violation of topology. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. Moreover we present a second algorithm for the extraction of generalized principal curves comprising disconnected and branching manifolds. The performance of the algorithm is demonstrated for both one- and two-dimensional principal manifolds and also for the case of sparse data sets. As an application we reveal cluster structures in a set of real world data from the domain of ecotoxicology