  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.
