Integer Programming Relaxations for Integrated Clustering and Outlier Detection

In this paper we present methods for exemplar based clustering with outlier selection based on the facility location formulation. Given a distance function and the number of outliers to be found, the methods automatically determine the number of clusters and outliers. We formulate the problem as an integer program to which we present relaxations that allow for solutions that scale to large data sets. The advantages of combining clustering and outlier selection include: (i) the resulting clusters tend to be compact and semantically coherent (ii) the clusters are more robust against data perturbations and (iii) the outliers are contextualised by the clusters and more interpretable, i.e. it is easier to distinguish between outliers which are the result of data errors from those that may be indicative of a new pattern emergent in the data. We present and contrast three relaxations to the integer program formulation: (i) a linear programming formulation (LP) (ii) an extension of affinity propagation to outlier detection (APOC) and (iii) a Lagrangian duality based formulation (LD). Evaluation on synthetic as well as real data shows the quality and scalability of these different methods.