Graph learning from band-limited data by graph Fourier transform analysis

•We reveal the intrinsic relation between the unknown frequency-domain representation of general band-limited graph signals and the graph Fourier transform basis.•A new analytical expression is derived to determine the graph Fourier transform basis of observed band-limited signals.•Given the graph Fourier transform basis and the statistical knowledge of the graph Laplacian, the estimation of the eigenvalues of the graph Laplacian is convex and efficiently solved using the alternating direction method of multipliers. A graph provides an effective means to represent the statistical dependence or similarity among signals observed at different vertices. A critical challenge is to excavate graphs underlying observed signals, because of non-convex problem structure and associated high computational requirements. This paper presents a new graph learning technique that is able to efficiently infer the graph structure underlying observed graph signals. The key idea is that we reveal the intrinsic relation between the frequency-domain representation of general band-limited graph signals, and the graph Fourier transform (GFT) basis. Accordingly, we derive a new closed-form analytic expression for the GFT basis, which depends deterministically on the observed signals (as opposed to being solved numerically and approximately in the literature). Given the GFT basis, the estimation of the graph Laplacian, more explicitly, its eigenvalues, is convex and efficiently solved using the alternating direction method of multipliers (ADMM). Simulations based on synthetic data and experiments based on public weather and brain signal datasets show that the new technique outperforms the state of the art in accuracy and efficiency.