A Game-Theoretic Approach to Pairwise Clustering and Matching

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

On the Informativeness of Asymmetric Dissimilarities (abstract)

Nearest-neighbor (NN) classification has been widely used in many research areas, as it is a very intuitive technique. As long as we can defined a similarity or distance between two objects, we can apply NN, therefore making it suitable even for non-vectorial data such as graphs. An alternative to NN is the dissimilarity space [2], where distances are used as features, i.e. an object is represented as a vector of its distances to prototypes or landmarks. This representation can be used with any classifier, and has been shown to be potentially more effective than NN classification on the same dissimilarities. Defining distance measures on complex objects is not a trivial task. Due to human judgments, suboptimal matching procedures or simply by construction, distance measures on non-vectorial data may often be asymmetric. A common solution for NN approaches is to symmetrize the measure by averaging the two distances [2]. However, in the dissimilarity space, symmetric measures are not required. We explore whether asymmetry is an artifact that needs to be removed, or an important source of information. This abstract highlights one example of informative asymmetric measures, covered in [1].Intelligent SystemsElectrical Engineering, Mathematics and Computer Scienc