Compositional Generative Mapping for Tree-Structured Data—Part II: Topographic Projection Model

We introduce GTM-SD (Generative Topographic Mapping for Structured Data), which is the first compositional generative model for topographic mapping of tree-structured data. GTM-SD exploits a scalable bottom-up hidden-tree Markov model that was introduced in Part I of this paper to achieve a recursive topographic mapping of hierarchical information. The proposed model allows efficient exploitation of contextual information from shared substructures by a recursive upward propagation on the tree structure which distributes substructure information across the topographic map. Compared to its noncompositional generative counterpart, GTM-SD is shown to allow the topographic mapping of the full sample tree, which includes a projection onto the lattice of all the distinct subtrees rooted in each of its nodes. Experimental results show that the continuous projection space generated by the smooth topographic mapping of GTM-SD yields a finer grained discrimination of the sample structures with respect to the state-of-the-art recursive neural network approach.