Blind identification of graph filters with sparse inputs

Network processes are often conceptualized as signals defined on the vertices of a graph. To untangle the latent structure of such signals, one can view them as outputs of linear graph filters modeling underlying network dynamics. This paper deals with the problem of blind graph filter identification, which finds applications in social and brain networks, to name a few. Given a graph signal y modeled as the output of a graph filter, the goal is to recover the vector of filter coefficients h and the input signal x which is assumed to be sparse. While the filtered graph signal is a bilinear function of x and h, y is also a linear combination of the entries of the rank-one matrix xh^T. As with blind deconvolution of time (or spatial) domain signals, it is shown that the blind graph filter identification problem can be tackled via rank minimization subject to linear constraints. Graph-dependent conditions under which the solution set of the optimization problem includes only rank-one matrices are derived. Numerical tests with synthetic and real-world networks corroborate the effectiveness of the blind identification approach.