Graph Filtering Over Expanding Graphs

Abstract

Data processing tasks over graphs couple the data residing over the nodes with the topology through graph signal processing tools. Graph filters are one such prominent tool, having been used in applications such as denoising, interpolation, and classification. However, they are mainly used on fixed graphs although many networks grow in practice, with nodes continually attaching to the topology. Re-training the filter every time a new node attaches is computationally demanding; hence an online learning solution that adapts to the evolving graph is needed. We propose an online update of the filter, based on the principles of online machine learning. To update the filter, we perform online gradient descent, which has a provable regret bound with respect to the filter computed offline. We show the performance of our method for signal interpolation at the incoming nodes. Numerical results on synthetic and graph-based recommender systems show that the proposed approach compares well to the offline baseline filter while outperforming competitive approaches. These findings lay the foundation for efficient filtering over expanding graphs.

Online Edge Flow Prediction Over Expanding Simplicial Complexes

“…To account for the underlying topological dynamics, we propose an online framework to perform edge flow prediction over an expanding simplicial complex. In detail, we first use the simplicial convolutional filter proposed in [10,21] as the flow predictor on the new edge; then we consider an online gradient descent algorithm to update the filter parameters [22,23], generalizing the methods developed for the node signal space [24,25,26]. Online gradient descent, as a simple update rule, has sub-linear regret bounds [22].…”