GTM: The Generative Topographic Mapping

GTM: The Generative Topographic Mapping

April 16, 1997 | Christopher M. Bishop, Markus Svensén, Christopher K. I. Williams
The paper introduces the Generative Topographic Mapping (GTM), a non-linear latent variable model that represents data in a high-dimensional space using a smaller number of latent variables. Unlike the Self-Organizing Map (SOM), GTM overcomes limitations such as the lack of a cost function, theoretical basis for parameter selection, and general convergence proofs. GTM uses a constrained mixture of Gaussians, optimized using the EM algorithm, to map data from a high-dimensional space to a lower-dimensional manifold. The model is defined by specifying points in the latent space and basis functions, with parameters determined through maximum likelihood. GTM is particularly useful for data visualization, where Bayes' theorem is used to invert the mapping. The paper also discusses the relationship between GTM and SOM, highlighting key differences such as the explicit probability density and well-defined objective function in GTM. Experimental results on toy problems and real-world data from multi-phase oil pipelines demonstrate the effectiveness of GTM.The paper introduces the Generative Topographic Mapping (GTM), a non-linear latent variable model that represents data in a high-dimensional space using a smaller number of latent variables. Unlike the Self-Organizing Map (SOM), GTM overcomes limitations such as the lack of a cost function, theoretical basis for parameter selection, and general convergence proofs. GTM uses a constrained mixture of Gaussians, optimized using the EM algorithm, to map data from a high-dimensional space to a lower-dimensional manifold. The model is defined by specifying points in the latent space and basis functions, with parameters determined through maximum likelihood. GTM is particularly useful for data visualization, where Bayes' theorem is used to invert the mapping. The paper also discusses the relationship between GTM and SOM, highlighting key differences such as the explicit probability density and well-defined objective function in GTM. Experimental results on toy problems and real-world data from multi-phase oil pipelines demonstrate the effectiveness of GTM.
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