Source localization on graphs via ℓ1 recovery and spectral graph theory

We cast the problem of source localization on graphs as the simultaneous problem of sparse recovery and diffusion kernel learning. An ℓ1 regularization term enforces the sparsity constraint while we recover the sources of diffusion from a single snapshot of the diffusion process. The diffusion kernel is estimated by assuming the process to be as generic as the standard heat diffusion. We show with synthetic data that we can concomitantly learn the diffusion kernel and the sources, given an estimated initialization. We validate our model with cholera mortality and atmospheric tracer diffusion data, showing also that the accuracy of the solution depends on the construction of the graph from the data points.