From Points to Nodes: Inverse Graph Embedding through a Lagrangian Formulation
In this paper, we introduce a novel concept: Inverse Embedding. We formulate inverse embedding in the following terms: given a set of multi-dimensional points coming directly or indirectly from a given spectral embedding, find the mininal complexity graph (following a MDL criterion) which satisfies the embedding constraints. This means that when the inferred graph is embedded it must provide the same distribution of squared distances between the original multi-dimensional vectors. We pose the problem in terms of a Lagrangian and find that a fraction of the multipliers (the smaller ones) resulting from the deterministic annealing process provide the positions of the edges of the unknown graph. We proof the convergence of the algorithm through an analysis of the dynamics of the deterministic annealing process and test the method with some significant sample graphs.