Learning graph structure with stationary graph signals via first-order approximation

Estimating the graph structure plays an important role in the emerging field of signal processing on graph (SPoG). A graph acts as a filter when a signal propagates on it with the frequency response being a polynomial of the eigenvalues of the graph Laplacian matrix. In this paper, we endeavor to learn the underlying graph structure (a.k.a. graph connectivity) from stationary graph signal observations. To this end, we first make use of the first-order approximation to linearize the polynomial representation of the frequency response. Then we formulate a convex optimization problem to recover the eigenvalues of the graph Laplacian. We further show the problem can be efficiently solved by applying the alternating-direction method of multipliers (ADMM). Accordingly, the graph structure can be learned from the reconstructed graph Laplacian. Numerical tests demonstrate the proposed approach offers accurate estimates of the connections in a graph.