Further nonlinear dimensions are then added, produced by combining the original dimensions. The enlarged latent space is then projected back into the 1D data space. The probability of a given projection is, as before, given by the product of the likelihood of the data under the Gaussian noise model with the prior on the deformation parameter. Unlike conventional spring-based deformation modelling, this has the advantage of being analytically optimizable. The disadvantage is that it is a 'data-mining' approach, i.e. the shape of the deformation prior is unlikely to be meaningful as an explanation of the possible deformations, as it is based on a very high, artificial- and arbitrarily constructed nonlinear latent space. For this reason the prior is learned from data rather than created by a human expert, as is possible for spring-based models. Comparison with Kohonen's self-organizing maps While nodes in the self-organizing map (SOM) can wander around at will, GTM nodes are constrained by the allowable transformations and their probabilities. If the deformations are well-behaved the topology of the latent space is preserved. The SOM was created as a biological model of neurons and is a heuristic algorithm. By contrast, the GTM has nothing to do with neuroscience or cognition and is a probabilistically principled model. Thus, it has a number of advantages over SOM, namely: it explicitly formulates a density model over the data. it uses a cost function that quantifies how well the map is trained. it uses a sound optimization procedure (EM algorithm). GTM was introduced by Bishop, Svensen and Williams in their Technical Report in 1997 (Technical Report NCRG/96/015, Aston University, UK) published later in Neural Computation. It was also described in the PhD thesis of Markus Svensen (Aston, 1998). Applications See also Self-organizing map (SOM) Artificial Neural Network Connectionism Data mining Machine learning Nonlinear dimensionality reduction Neural network software Pattern recognition External links Bishop, Svensen and Williams Generative Topographic Mapping paper Generative topographic mapping developed at the Neural Computing Research Group os Aston University (UK). ( Matlab toolbox ) Artificial neural networks