Geodesic Generative Topographic Mapping

Nonlinear dimensionality reduction (NLDR) methods aim to provide a faithful low-dimensional representation of multivariate data. The manifold learning family of NLDR, methods, in particular, do this by defining low-dimensional manifolds embedded in the observed data space. Generative Topographic Mapping (GTM) is one such manifold learning method for multivariate data clustering and visualization. The non-linearity of the mapping it generates makes it prone to trustworthiness and continuity errors that would reduce the faithfulness of the data representation, especially for datasets of convoluted geometry. In this study, the GTM is modified to prioritize neighbourhood relationships along the generated manifold. This is accomplished through penalizing divergences between the Euclidean distances from the data points to the model prototypes and the corresponding geodesic distances along the manifold. The resulting Geodesic GTM model is shown to improve not only the continuity and trustworthiness of the representation generated by the model, but also its resilience in the presence of noise.