Abstract
We introduce hypernode graphs as weighted binary relations between sets of nodes: a hypernode is a set of nodes, a hyperedge is a pair of hypernodes, and each node in a hypernode of a hyperedge is given a non negative weight that represents the node contribution to the relation. Hypernode graphs model binary relations between sets of individuals while allowing to reason at the level of individuals. We present a spectral theory for hypernode graphs that allows us to introduce an unnormalized Laplacian and a smoothness semi-norm. In this framework, we are able to extend spectral graph learning algorithms to the case of hypernode graphs. We show that hypernode graphs are a proper extension of graphs from the expressive power point of view and from the spectral analysis point of view. Therefore hypernode graphs allow to model higher order relations whereas it is not true for hypergraphs as shown in [1]. In order to prove the potential of the model, we represent multiple players games with hypernode graphs and introduce a novel method to infer skill ratings from game outcomes. We show that spectral learning algorithms over hypernode graphs obtain competitive results with skill ratings specialized algorithms such as Elo duelling and TrueSkill.
This work was supported by the French National Research Agency (ANR). Project Lampada ANR-09-EMER-007.
Chapter PDF
References
Agarwal, S., Branson, K., Belongie, S.: Higher Order Learning with Graphs. In: Proc. of ICML, pp. 17–24 (2006)
Berge, C.: Graphs and hypergraphs. North-Holl Math. Libr. North-Holland, Amsterdam (1973)
Cai, J., Strube, M.: End-to-end coreference resolution via hypergraph partitioning. In: Proc. of COLING, pp. 143–151 (2010)
Easley, D., Kleinberg, J.: Networks, crowds, and markets: Reasoning about a highly connected world. Cambridge University Press (2010)
Elo, A.E.: The Rating of Chess Players, Past and Present. Arco Publishing (1978)
Gallo, G., Longo, G., Pallottino, S., Nguyen, S.: Directed hypergraphs and applications. Discrete Applied Mathematics 42(2-3), 177–201 (1993)
Hamilton, S.: PythonSkills: Implementation of the TrueSkill, Glicko and Elo Ranking Algorithms (2012), https://pypi.python.org/pypi/skills
Hein, M., Setzer, S., Jost, L., Rangapuram, S.S.: The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited. In: Proc. of NIPS, pp. 2427–2435 (2013)
Herbrich, R., Minka, T., Graepel, T.: TrueSkillTM: A Bayesian Skill Rating System. In: Proc. of NIPS, pp. 569–576 (2006)
Herbster, M., Pontil, M.: Prediction on a Graph with a Perceptron. In: Proc. of NIPS, pp. 577–584 (2006)
Klamt, S., Haus, U.-U., Theis, F.: Hypergraphs and Cellular Networks. PLoS Computational Biology 5(5) (2009)
Lasek, J., Szlávik, Z., Bhulai, S.: The predictive power of ranking systems in association football. International Journal of Applied Pattern Recognition 1(1), 27–46 (2013)
Lee, H.: Python implementation of Elo: A rating system for chess tournaments (2013), https://pypi.python.org/pypi/elo/0.1.dev
Lee, H.: Python implementation of TrueSkill: The video game rating system (2013), http://trueskill.org/
Penrose, R.: A generalized inverse for matrices. In: Proc. of Cambridge Philos. Soc., vol. 51, pp. 406–413. Cambridge University Press (1955)
Von Luxburg, U.: A tutorial on spectral clustering. Statistics and computing 17(4), 395–416 (2007)
Zhang, S., Sullivan, G.D., Baker, K.D.: The automatic construction of a view-independent relational model for 3-D object recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(6), 531–544 (1993)
Zhou, D., Huang, J., Schölkopf, B.: Learning from labeled and unlabeled data on a directed graph. In: Proc. of ICML, pp. 1036–1043 (2005)
Zhou, D., Huang, J., Schölkopf, B.: Learning with hypergraphs: Clustering, classification, and embedding. In: Proc. of NIPS, pp. 1601–1608 (2007)
Zhu, X., Ghahramani, Z., Lafferty, J., et al.: Semi-supervised learning using gaussian fields and harmonic functions. In: Proc. of ICML, vol. 3, pp. 912–919 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ricatte, T., Gilleron, R., Tommasi, M. (2014). Hypernode Graphs for Spectral Learning on Binary Relations over Sets. In: Calders, T., Esposito, F., Hüllermeier, E., Meo, R. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2014. Lecture Notes in Computer Science(), vol 8725. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44851-9_42
Download citation
DOI: https://doi.org/10.1007/978-3-662-44851-9_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44850-2
Online ISBN: 978-3-662-44851-9
eBook Packages: Computer ScienceComputer Science (R0)