SLoG-Net: Algorithm Unrolling for Source Localization on Graphs

We present a novel model-based deep learning solution for the inverse problem of localizing sources of network diffusion. Starting from first graph signal processing (GSP) principles, we show that the problem reduces to joint (blind) estimation of the forward diffusion filter and a sparse input signal that encodes the source locations. Despite the bilinear nature of the observations in said blind deconvolution task, by requiring invertibility of the diffusion filter we are able to formulate a convex optimization problem and solve it using the alternating-direction method of multipliers (ADMM). We then unroll and truncate the novel ADMM iterations to arrive at a parameterized neural network architecture for Source Localization on Graphs (SLoG-Net), that we train in an end-to-end fashion using labeled data. This supervised learning approach offers several advantages such as interpretability, parameter efficiency, and controllable complexity during inference. Our reproducible numerical experiments corroborate that SLoG-Net exhibits performance on par with the iterative ADMM baseline, but with markedly faster inference times and without needing to manually tune step-size or penalty parameters. Overall, our approach combines the best of both worlds by incorporating the inductive biases of a GSP model-based solution within a data-driven, trainable deep learning architecture for blind deconvolution of graph signals.