Hypergraphs are used in many machine learning methods to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex weights: hypergraphs where every vertex v has a weight $\gamma_e(v)$ for each incident hyperedge e, describing the contribution of v to the hyperedge e. We derive a random walk-based hypergraph Laplacian, and bound the mixing time of random walks on such hypergraphs. Moreover, we give conditions under which random walks on such hypergraphs are equivalent to random walks on graphs. As a corollary, we show that current machine learning methods that rely on Laplacians derived from random walks on hypergraphs with edge-independent vertex weights do not utilize higher-order relationships in the data. Finally, we demonstrate the effectiveness of hypergraphs with edge-dependent vertex weights on ranking applications using both synthetic and real-world datasets.
Uthsav Chitra (Princeton University)
Benjamin Raphael (Princeton University)
Related Events (a corresponding poster, oral, or spotlight)
2019 Poster: Random Walks on Hypergraphs with Edge-Dependent Vertex Weights »
Thu Jun 13th 06:30 -- 09:00 PM Room Pacific Ballroom