Continuously indexed Potts models on unoriented graphs

This paper introduces an extension to undirected graphical models of the classical continuous time Markov chains. This model can be used to solve a transductive or unsupervised multi-class classification problem at each point of a network defined as a set of nodes connected by segments of different lengths. The classification is performed not only at the nodes, but at every point of the edge connecting two nodes. This is achieved by constructing a Potts process indexed by the continuum of points forming the edges of the graph. We propose a homogeneous parameterization which satisfies Kolmogorov consistency, and show that classical inference and learning algorithms can be applied. We then apply our model to a problem from geomatics, namely that of labelling city blocks automatically with a simple typology of classes (e.g. collective housing) from simple properties of the shape and sizes of buildings of the blocks. Our experiments shows that our model outperform standard MRFs and a discriminative model like logistic regression.