Abstract
Clustering refers to the process of extracting maximally coherent groups from a set of objects using pairwise, or high-order, similarities. Traditional approaches to this problem are based on the idea of partitioning the input data into a predetermined number of classes, thereby obtaining the clusters as a by-product of the partitioning process. In this chapter, we provide a brief review of our recent work which offers a radically different view of the problem and allows one to work directly on non-(geo)metric data. In contrast to the classical approach, in fact, we attempt to provide a meaningful formalization of the very notion of a cluster in the presence of non-metric (even asymmetric and/or negative) (dis)similarities and show that game theory offers an attractive and unexplored perspective that serves well our purpose. To this end, we formulate the clustering problem in terms of a non-cooperative “clustering game” and show that a natural notion of a cluster turns out to be equivalent to a classical (evolutionary) game-theoretic equilibrium concept. Besides the game-theoretic perspective, we exhibit also characterizations of our cluster notion in terms of optimization theory and graph theory. As for the algorithmic issues, we describe two approaches to find equilibria of a clustering game. The first one is based on the classical replicator dynamics from evolutionary game theory, the second one is a novel class of dynamics inspired by infection and immunization processes which overcome their limitations. Finally, we show applications of the proposed framework to matching problems, where we aim at finding correspondences within a set of elements. In particular, we address the problems of point-pattern matching and surface registration.
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Notes
- 1.
We note that although we restrict ourselves to games where all players share the same set of pure strategies and payoff function, in more general settings each agent can well be associated to its own pure strategy set and payoff function.
- 2.
In the original paper, Motzkin and Straus proved the “only-if” part of this theorem. The converse, however, is a straightforward consequence of their result [40].
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Pelillo, M., Rota Bulò, S., Torsello, A., Albarelli, A., Rodolà, E. (2013). A Game-Theoretic Approach to Pairwise Clustering and Matching. In: Pelillo, M. (eds) Similarity-Based Pattern Analysis and Recognition. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-5628-4_8
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