Topological Volterra Filters

To deal with high-dimensional data, graph filters have shown their power in both graph signal processing and data science. However, graph filters process signals exploiting only pairwise interactions between the nodes, and they are not able to exploit more complicated topological structures. Graph Volterra models, on the other hand, are also able to exploit relations between triplets, quadruplets and so on. However, they have only been exploited for topology identification and are only based on one-hop relations. In this paper, we first review graph filters and graph Volterra models and then merge the two concepts resulting in so-called topological Volterra filters (TVFs). TVFs process signals over multiple hops of higher-level topological structures. First-level TVFs are basically similar to traditional graph filters, yet higher-level TVFs provide a more general processing framework. We apply TVFs to inverse filtering and recommender systems.