TY - GEN

T1 - Random walks on hypergraphs with edge-dependent vertex weights

AU - Chitra, Uthsav

AU - Raphael, Benjamin J.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Hypergraphs are used in machine learning 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 7e(v) for each incident hyperedge e that describes 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 advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.

AB - Hypergraphs are used in machine learning 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 7e(v) for each incident hyperedge e that describes 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 advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.

UR - http://www.scopus.com/inward/record.url?scp=85077989196&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85077989196&partnerID=8YFLogxK

M3 - Conference contribution

T3 - 36th International Conference on Machine Learning, ICML 2019

SP - 2002

EP - 2011

BT - 36th International Conference on Machine Learning, ICML 2019

PB - International Machine Learning Society (IMLS)

T2 - 36th International Conference on Machine Learning, ICML 2019

Y2 - 9 June 2019 through 15 June 2019

ER -