Optimal Embedding of Heterogeneous Graph Data with Edge Crossing Constraints

We propose a novel approach to visualization of heterogeneous data characterized by both a relationship graph structure and intrinsic features. Each data point is a node in a graph with a given structure. Each data point is also associated with a set of features that have a corresponding distance or similarity measure. A successful visualization accurately captures the desired proximity structure as measured by some embedding objective while simultaneously optimizing an aesthetic criterion, no edge crossings. The edge-crossing constraint is expressed as a nonlinear constraint which has an intuitive geometric interpretation closely related to support vector machine classification. The approach can be generalized to remove intersections of general convex polygons including node-edge and node-node intersections. We demonstrate the approach on multi-dimensional scaling or equivalently Kamada-Kawai force-directed graph layout, by modifying the stress majorization algorithm to include penalized edge crossings. The resulting Expectation-Maximization-like algorithm can be readily adapted to other supervised and unsupervised optimization-based embedding or dimensionality reduction methods. The method is demonstrated on a problem in tuberculosis molecular epidemiology – creating spoligoforests for visualizing genetic relatedness between strains of the Mycobacterium tuberculosis complex characterized by a phylogenetic forest, and multiple biomarkers with a corresponding non-metric genetic distance.