Blind Deconvolution on Graphs: Exact and Stable Recovery

We study a blind deconvolution problem on graphs, which arises in the context of localizing a few sources that diffuse over networks. While the observations are bilinear functions of the unknown graph filter coefficients and sparse input signals, a mild requirement on invertibility of the diffusion filter enables an efficient convex relaxation leading to a linear programming formulation that can be tackled with off-the-shelf solvers. Under the Bernoulli-Gaussian model for the inputs, we derive sufficient exact recovery conditions in the noise-free setting. A stable recovery result is then established, ensuring the estimation error remains manageable even when the observations are corrupted by a small amount of noise. Numerical tests with synthetic and real-world network data illustrate the merits of the proposed algorithm, its robustness to noise as well as the benefits of leveraging multiple signals to aid the (blind) localization of sources of diffusion. At a fundamental level, the results presented here broaden the scope of classical blind deconvolution of (spatio-)temporal signals to irregular graph domains.