Learning Nonlinear Principal Manifolds by Self-Organising Maps

Abstract

Summary. This chapter provides an overview on the self-organised map (SOM) in the context of manifold mapping. It first reviews the background of the SOM and issues on its cost function and topology measures. Then its variant, the visualisation induced SOM (ViSOM) proposed for preserving local metric on the map, is introduced and reviewed for data visualisation. The relationships among the SOM, ViSOM, multidimensional scaling, and principal curves are analysed and discussed. Both the SOM and ViSOM produce a scaling and dimension-reduction mapping or manifold of the input space. The SOM is shown to be a qualitative scaling method, while the ViSOM is a metric scaling and approximates a discrete principal curve/surface. Examples and applications of extracting data manifolds using SOM-based techniques are presented.Key words: Self-organising maps, principal curve and surface, data visualisation, topographic mapping IntroductionFor many years, artificial neural networks have been studied and used to construct information processing systems based on or inspired by natural biological neural structures. They not only provide solutions with improved performance when compared with traditional problem-solving methods, but also give a deeper understanding of human cognitive abilities. Among the various existing neural network architectures and learning algorithms, Kohonen's self-organising map (SOM) [35] is one of most popular neural network models. Developed for an associative memory model, it is an unsupervised learning algorithm with simple structures and computational forms, and is motivated by the retina-cortex mapping. Self-organisation in general is a fundamental pattern recognition process, in which intrinsic inter-and intra-pattern relationships within the data set are learnt without the presence of a potentially biased or subjective external influence. The SOM can provide topologically

A Chilean seismic regionalization through a Kohonen neural network

“…That is, the application of a Kohonen neural network allowed us to determine the natural number of classes or seismic regions without resorting to the Principal Components Analysis [26], to Fuzzy Inference Systems or to seismological subjective criteria.…”

III Pre-Specific Modeling: Computational Machines in a Coexistence with Concrete Universals and Data Streams

“…For example, one can compare the SOM with other clustering methods or with space transformation and feature extraction methods such as the Principal Component Analysis (PCA). 59 It is possible to explain and compare the SOM with vector quantization methods. 60 Further, it is possible to explain the SOM as a nonlinear function approximation method and to see it as a type of neural network methods and radial basis function.…”

Comparison of Self Organizing Maps Clustering with Supervised Classification for Air Pollution Data Sets

“…Patterns of air quality have been searched for Mexico City by Neme and Hernandez [10], whereas Karatzas and Voukantsis have done the same for the city of Thessaloniki [7]. Skön et al, have analyzed indoor air quality using SOM [12]. Li and Chou have investigated air pollution spatial variation with SOM [9].…”