Online Filtering over Expanding Graphs

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.

Graph filtering over expanding graphs

Our capacity to learn representations from data is related to our ability to design filters that can leverage their coupling with the underlying domain. Graph filters are one such tool for network data and have been used in a myriad of applications. But graph filters work only with a fixed number of nodes despite the expanding nature of practical networks. Learning filters in this setting is challenging not only because of the increased dimensions but also because the connectivity is known only up to an attachment model. We propose a filter learning scheme for data over expanding graphs by relying only on such a model. By characterizing the filter stochastically, we develop an empirical risk minimization framework inspired by multi-kernel learning to balance the information inflow and outflow at the incoming nodes. We particularize the approach for denoising and semi-supervised learning (SSL) over expanding graphs and show near-optimal performance compared with baselines relying on the exact topology. For SSL, the proposed scheme uses the incoming node information to improve the task on the existing ones. These findings lay the foundation for learning representations over expanding graphs by relying only on the stochastic connectivity model.

Learning Expanding Graphs for Signal Interpolation

Performing signal processing over graphs requires knowledge of the underlying fixed topology. However, graphs often grow in size with new nodes appearing over time, whose connectivity is typically unknown; hence, making more challenging the downstream tasks in applications like cold start recommendation. We address such a challenge for signal interpolation at the incoming nodes blind to the topological connectivity of the specific node. Specifically, we propose a stochastic attachment model for incoming nodes parameterized by the attachment probabilities and edge weights. We estimate these parameters in a data-driven fashion by relying only on the attachment behaviour of earlier incoming nodes with the goal of interpolating the signal value. We study the non-convexity of the problem at hand, derive conditions when it can be marginally convexified, and propose an alternating projected descent approach between estimating the attachment probabilities and the edge weights. Numerical experiments with synthetic and real data dealing in cold start collaborative filtering corroborate our findings.