Visualization of Tree-Structured Data Through Generative Topographic Mapping

摘要

In this paper, we present a probabilistic generative approach for constructing topographic maps of tree-structured data. Our model defines a low-dimensional manifold of local noise models, namely, (hidden) Markov tree models, induced by a smooth mapping from low-dimensional latent space. We contrast our approach with that of topographic map formation using recursive neural-based techniques, namely, the self-organizing map for structured data (SOMSD) (Hagenbuchner , 2003). The probabilistic nature of our model brings a number of benefits: 1) naturally defined cost function that drives the model optimization; 2) principled model comparison and testing for overfitting; 3) a potential for transparent interpretation of the map by inspecting the underlying local noise models; 4) natural accommodation of alternative local noise models implicitly expressing different notions of structured data similarity. Furthermore, in contrast with the recursive neural-based approaches, the smooth nature of the mapping from the latent space to the local model space allows for calculation of magnification factors—a useful tool for the detection of data clusters. We demonstrate our approach on three data sets: a toy data set, an artificially generated data set, and on a data set of images represented as quadtrees.