Topological Dynamic Bayesian Networks

The objective of this research is to embed topology within the dynamic Bayesian network (DBN) formalism. This extension of a DBN (that encodes statistical or causal relationships) to a topological DBN (TDBN) allows continuous mappings (e.g., topological homeomorphisms), topological relations (e.g., homotopy equivalences) and invariance properties (e.g., surface genus, compactness) to be exploited. The mission of TDBN is not limited only to classify objects but to reveal how these objects are topologically related as well. Because TDBN formalism uses geometric constructors that project a discrete space onto a continuous space, it is well suited to identify objects that undergo smooth deformation. Experimental results in face identification across ages represent conclusive evidence that the fusion of statistics and topology embodied by the TDBN concept holds promise. The TDBN formalism outperformed the DBN approach in facial identification across ages.