Blind Identification of Graph Filters

Network processes are often represented 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 joint identification of a graph filter and its input signal, thus broadening the scope of classical blind deconvolution of temporal and spatial signals to the less-structured graph domain. Given a graph signal $\mathbf{y}$ modeled as the output of a graph filter, the goal is to recover the vector of filter coefficients $\mathbf{h}$, and the input signal $\mathbf{x}$ which is assumed to be sparse. While $\mathbf{y}$ is a bilinear function of $\mathbf{x}$ and $\mathbf{h}$, the filtered graph signal is also a linear combination of the entries of the lifted rank-one, row-sparse matrix $\mathbf{x} \mathbf{h}^T$. The blind graph-filter identification problem can thus be tackled via rank and sparsity minimization subject to linear constraints, an inverse problem amenable to convex relaxations offering provable recovery guarantees under simplifying assumptions. Numerical tests using both synthetic and real-world networks illustrate the merits of the proposed algorithms, as well as the benefits of leveraging multiple signals to aid the blind identification task.