Characterization and inference of weighted graph topologies from observations of diffused signals

In the field of signal processing on graphs, a key tool to process signals is the graph Fourier transform. Transforming signals into the spectral domain is accomplished using the basis of eigenvectors of the graph Laplacian matrix. Such a matrix is dependent on the topology of the graph on which the signals are defined. Therefore it is of paramount importance to have a graph available. We consider the problem of inferring the graph topology from observations of signals. In this article we model the observations as being measured after a few steps of diffusing signals that are initially mutually independent and have independent entries. We propose a way to characterize and recover the possible graphs used for this diffusion. We show that the space of feasible matrices on which mutually independent signals can have evolved, based on the observations, is a convex polytope, and we propose two strategies to choose a point in this space as the topology estimate. These strategies enforce the selection of graphs that are, respectively, simple or sparse. Both methods can be formulated as linear programs and hence solved efficiently. Experiments suggest that in the ideal case when we are able to retrieve the exact covariance matrix of the signal, we are able to recover the graph on which the signals were diffused, with no error. We also apply our method to a synthetic case in which a limited number of signals are obtained by diffusion on a known graph, and show that we are able to recover a graph that is very close to the ground truth one, with an error decreasing as the number of observations increases. Finally, we illustrate our method on audio signals, and show that we are able to recover a graph that is globally one-dimensional.